The science of finance/Printable version

The science of finance

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Introduction

Too often, even a hard worker does not earn enough to support his or her family, has not really been trained, except by himself and some friends or colleagues, and has neither good tools nor safe working conditions .

Good wages, training, good equipment and safety impose costs that can not always be covered by previous revenues. So we have to advance funds. In other words, we have to finance our project. If we advance the funds ourselves, it is self-financing, if others bring their funds, it is external financing. The science of finance studies how human beings finance their projects.

The financing of a project can be conceived in real terms: what are the raw materials, the supplies, the labor force and the capital goods that must be advanced to launch the project, until its revenues are sufficient to cover its costs? It can also be conceived in monetary terms: what are the sums of money needed to buy these raw materials, these supplies and this labor force, and to rent or buy these capital goods?

Introduction to economics

Economics is the science of the production and consumption of goods and services.

Goods and services

Services (labor) are consumed when they are produced.

The goods produced are consumed after a certain period, short (fresh products) or more or less long (durable goods, including stocks of non-perishable commodities).

Some durable goods are almost eternal (quality housing, jewelry, works of art ...). Others are consumed by use during their lifetime. Even near-eternal goods generally require work to be maintained.

Final consumption is the consumption of goods and services which directly improve the quality of life (in principle, because they can also deteriorate it): food, clothing, housing, health, education, transport, sport and entertainment, long distance communication ...

Intermediate consumption is the consumption of goods and services that serve in the production chain of final goods and services.

Certain goods such as means of transport, computers and smartphones can be used both as intermediate goods and as final goods.

Quality of life

The quality of life does not only depend on final consumption: having a good job and benefiting from good working conditions, feeling safe in the present, for our future, that of our children, our country and all humanity, respect and be respected, love and be loved, know how to meditate and relax, be at peace with oneself and with others, do not despair, breathe good air, benefit from a good climate and a welcoming nature ...

Real wealth and market wealth

Real wealth (capital) at any given time is the sum of all durable goods.

We can also include in the real wealth the intelligence, the competence and the health of human beings (human capital) and the natural wealth (seas, oceans, rivers, lakes and rivers, landscapes, natural fauna and flora ...).

Market wealth is the market value of real wealth. It is assessed with market prices. When goods are not sold, their market value is assessed from the market prices of equivalent goods. Since human beings are not sold as slaves, their market value cannot be assessed, except by very questionable indirect means (discounted lifetime income or risk price).

Market wealth depends on long-term expectations. Durable goods have a market value because it is anticipated that they will be used, and that they can be sold. But lifestyles, and expectations of future lifestyles, can vary. Such variations are difficult to predict. If, for example, humans give up tourism by air, all the infrastructure and equipment intended to produce and consume the planes, including the planes themselves, automatically lose their value. If fiancés lose the habit of offering diamonds, the market value of diamond stocks will be greatly diminished.

Expectations are very fluctuating. They vary with the occurrence of unforeseen events (disasters ...) and are often irrational (animal spirits) because no one can predict with certainty what the future holds for us. This is why the market value of shares can change suddenly. Billions of dollars can disappear in a day without a note being burned, simply because humans have changed their minds.

Investment and long-term vision

Investment is the change in real wealth over a period. A negative investment is a decrease in real wealth (non-renewal of durable goods consumed or deteriorated).

To invest intelligently, we need to have a long-term vision. What goods and services will human beings want to enjoy in the coming decades? Will they have the means to buy them? What intermediate goods must be produced to facilitate the production of final goods and services?

Training is an investment in human capital. Intelligent vocational training also requires a long-term vision.

The market economy is generally short-sighted. The ideal would be for all citizens to agree on the priorities to be given to the economy and for an intelligent state to correct the failings of the market economy.

Work and dignity

Work is the main source of wealth. When we consume goods and services, we give work to all those who participated in their production. Being deprived of work is generally a great suffering, at the same time economic, social and psychological. Giving work is giving dignity, provided it is a good job, in good conditions (hours, equipment, security, respect ...) and honestly paid.

Evaluation of projects

We want to finance the most profitable projects. We must therefore be able to evaluate them beforehand. This chapter introduces the most basic principles of economic and financial evaluation, as well as some important examples. The subject of project evaluation is further developed in a supplementary chapter devoted to risk assessment.

What makes economic value?

Use value ​​

Anything of value can be called a good. Services are ephemeral goods, consumed as soon as they are produced. The other goods are in general material goods which one can preserve more or less long. There are also intangible goods, such as intellectual property rights.

The value of a good depends on its use and therefore its usefulness. Final goods are directly useful goods, because they give pleasure or comfort, or because they reduce pain or discomfort. Intermediate goods prove their utility by serving the production of other goods.

Land and labor are the first two sources of use values. Land refers to all natural resources, mainly the biosphere, that is, the thin film on the surface of the Earth that houses life. Labor creates value as soon as it is useful. It is a service so a good. Land and labor are the primary resources, capital goods are second resources, because they are produced from primary resources.

The use value is variable. If one is thirsty, the use value of a glass of water is very large, but it becomes almost zero when one is quenched.

Value creation by trade and money

The same good can be used in various ways, by different agents, with different values. This explains the possibility of gains in the exchange. If the agent 1 owns the good A which has for him a value of use inferior to that of the good B, possessed by the agent 2, and if 2 estimates inversely that A is worth more than B, then 1 and 2 have both interest in exchanging A versus B. This is a win-win exchange. When trade thus increases the use values ​​of traded goods, it creates value.

When trade is reduced to barter, it is hindered by the problem of double-coincidence of needs. 1 must have a good desired by 2 and vice versa. Money makes it possible to solve this problem, because in a monetary economy, all the exchangers are willing to give up their goods for money. This solution to the problem of double-coincidence of needs is sometimes cited as the explanation of the origin of money. Human beings would have invented money to develop trade, which was previously reduced to barter. But it is a myth. Trade is not the only cause of the appearance of money. We must also think of the loots of war, taxes, debts ... It remains true that money is tremendously useful for trade. It has become like an indispensable oil for the functioning of the economy. If we tried to do without it, no one could work and consume as he or she does today.

Since trade is value-creating and money increases trade opportunities, money creates value.

Exchange value

The exchange value of a good is the value of another good against which it can be exchanged. When a good is sold, ie traded for money, its exchange value is simply its selling price. But what is the value of money? It is still an exchange value. Money is valuable because it can be exchanged for other goods, because it allows us to buy them. It is even the definition of money: money is a good that is accepted in all transactions, not for its value in use but for its exchange value, because it makes it possible to buy all goods available for sale.

Of course, money is put to use in wars, crimes and all forms of violence and domination, and it incites selfishness and greed. Economic value is not moral value. Even in the absence of violence or egoism, use values ​​are morally questionable, since they depend only on the satisfaction of agents. And even if there is a gain in the exchange, the distribution of gains can be unbalanced and unfair.

This book of economics will discuss value especially in the economic sense. The economic value is either its use value, and it depends on the satisfaction of the agents, or its exchange value, and it depends on the commercial possibilities.

The economic value is a priority in the order of the means, because we need a minimum of satisfaction if only to survive, but perhaps not in the order of the ends, because it is reduced to the satisfaction of all agents (which is still a lot).

The labor theory of value

This book has adopted an orthodox approach to the theory of economic value. The basic economic values ​​are the use values. Exchange values ​​come from the fact that agents have an interest in exchanging. Relative prices of goods are observed in the markets. This theory of value allows a fair price to be defined only as the average price observed in a market, provided that it is sufficiently developed, ie that there are many buyers and sellers. It is a fair price only in the economic sense, not necessarily in a moral sense, since market prices, for example wages, do not always go in the direction of social justice.

A competing approach, initiated by David Ricardo and taken over by Karl Marx ("The Capital", Chapter 1), postulates that the real economic value of a good must be estimated not by its market price but by the amount of labor necessary for its production, the amount of work that has been as incorporated into it. If a good requires 10 hours of work to be produced, it should be worth 10 times more than a good that requires only one hour. But this theory comes up against a difficulty, because the hours of work are not all equivalent, they are not all equally useful. Marx recognizes this difficulty but evacuates it by stating that there is an average hourly unit of work, to which all other hours of work can be compared. But this comes down to solving the problem of economic value (what explains the differences in value between goods?) by postulating that it is already solved, since the relative values ​​of the hours of work must be known in advance to calculate the value of the incorporated amount of work.

Of course, work can create value, but saying it is not enough to explain the differences in the value of economic goods.

Work and value are not necessarily tied up. Many natural goods, which are of great value, are sometimes obtained without work, or with very little work compared to their value. Conversely, work can be ineffective, or produce useless goods, or do more harm than good. To say that a good is valuable because we had to work to produce it is to put things backwards. We work to produce goods because they are valuable. If they have no value, there is no sense in working to produce them.

Marx and the Marxists support the labor theory of value to denounce economic injustice: owners can earn a lot of money without working to the detriment of workers. The surplus value is defined by Marx as the difference between the true value of the produced goods and the paid wages. It is therefore a kind of theft. But we can not define a true economic value except by market prices. Nevertheless renouncing the Marxist theory of surplus value is not a way of denying economic injustice. Most businesses need financing hence owners who lend or entrust their money and in return receive interest or profit. Finance gives owners the means to systematically collect some of the wealth produced by labor. We do not need the Marxist theory of surplus value to denounce this injustice.

Profit and the rate of profit

Costs, revenues and profit

A project mobilizes resources, goods, to produce other goods.

Costs are workloads, raw materials and supplies consumed, capital goods used, taxes and any financial charges such as debt repayment. The revenues are the goods (including services) produced and sold, to which can be added loans, donations, grants, and any financial income. A loan can be considered a revenue that precedes a cost, repayment of the loan.

When goods can be bought and sold in a market, they can be priced and valued. From the point of view of an accountant and a mathematician, a project is a series of costs and revenues, evaluated by their prices, staggered over time. When dates and amounts of revenues and costs are known in advance, the project is certain, otherwise it is uncertain.

The valuation method based on monetary costs and revenues is more general than it seems. Of course, many projects have non-monetary costs and gains, but they can sometimes be estimated by a monetary equivalent, especially if these projects participate in the economic activity.

In the following it will be assumed that all costs and revenues are estimated by a monetary equivalent. We can then evaluate a project from its profit or loss.

Profit can be calculated simply as the difference between the sum of all revenues and the sum of all costs. If this difference is negative, it is a loss. This definition of profit ignores the opportunity cost of cash advances and cash costs. To take this into account requires a definition of profit that discounts the values. It is presented later in this chapter. As long as the projects are short-term, there is usually a negligible error in calculating profit simply as the difference between revenue and costs, without discounting them.

For a company one usually wants to know the profit during a given period. The company's assets at the beginning of the period are counted as an initial cost at the beginning of the period, and as a final revenue at the end of the period, because it is reinvested for the following period. The valuation of a company's assets is therefore very important to evaluate its annual profit.

Cash advances and the rate of profit

Projects often require funding, because the costs precede the revenues. Cash advances are the costs that can not be paid by previous revenues. When we commit to a project, we commit ourselves to advance such funds.

There are projects that can yield a profit without requiring cash advances, because the revenues precede the costs and are sufficient to cover them. For example, if one sells cash with a commercial gain that one bought on credit, one can make a good profit without advancing any money. When one can realize such a project, it is obviously a boon. In general, if we can borrow money, we do not have to advance it.

When all funds are advanced at an initial date, and discounting values is neglected, the rate of profit can be calculated simply as the ratio of profit to cash advances. If, for example, a project has a single initial cost of 100 and a single final revenue after one year of 110, the profit is 10, the cash advance is 100 and the profit rate is 10/100 = 0.1 = 10%

Leverage

We can benefit from leverage if a project has a higher rate of profit than the rate at which money can be borrowed. Leverage increases the rate of profit to infinity by borrowing all or part of the funds needed for the project. If we can borrow all the funds, there is no money to advance and the rate of profit is infinite. If we only borrow a portion of the funds, we increase the rate of profit, because we gain on the difference between the rate of profit of the project and the rate at which we borrow.

An example: if we invest 100 in a company with a profit rate of 20% a year, we make a profit of 20 after one year. If we borrowed 50 at the rate of 10%, we have to pay back 55 after one year, the profit is only 15, but we have advanced only 50. The profit rate is therefore 15/50 = 30%. By borrowing, the rate of profit has been increased by leverage from 20 to 30%.

Leverage, when one can benefit from it, looks like a magnificent windfall, since it allows to increase the rate of profit as much as we want. If the project is not risky, there is no reason to deprive oneself of such a windfall. But projects are usually risky. If the realized rate of profit is lower than the rate at which one has borrowed, one must support a loss, which is all the more important that one borrowed more. Leverage increases the risk of a project and can lead to bankruptcy. This is why companies are generally required to have sufficient capital, not be solely financed by loans. These funds are like a sort of cushion, which allows the company to bear possible losses (Admati & Hellwig 2013). If a company is abusing leverage, having low capital compared to what it borrows, it runs the risk of bankruptcy and puts lenders at risk of default. Leverage is therefore a way to increase the expected rate of profit while increasing risks, and by offloading some of these risks on lenders.

It is desirable, if only for reasons of social justice, so that even the less fortunate can undertake, that some projects be financed solely by borrowing, without requiring initial capital, so that they benefit from infinite leverage. But in this case the lenders must know that they take on the project risks.

Banks are the primary beneficiaries of leverage, because they can borrow at a very low rate, possibly zero, when bank accounts are unpaid.

Value creation by composition of projects

The composition of projects is a two-way expression. This involves both the design of a new project (mobilizing resources to produce goods) and the assembly of several projects, initially separated, into a more complex project that combines them. The two senses are closely related and can not always be clearly distinguished, because by assembling several projects into one, one may have to modify and reassign some of their resources, which ultimately amounts to designing a new project. .

A profitable project is value-creating, since revenue is worth more than costs. Profit is the part of the created value retained by the owner, but the project can also create value for everyone involved.

Exchange gains can create value by assembling two projects (the production of the two goods traded) and exchanging their products.

When a project increases the value of other projects, it has positive externalities. If, on the contrary, it reduces the value of other projects, it has negative externalities.

When two or more projects have positive externalities for each other, value is created by assembling them, because each will bring more in the presence of others than without them. In this case, the value of the composite project is greater than the sum of the values ​​of the component projects taken separately.

If a company has the opportunity to acquire or merge with a competing company, it gives itself a project that composes the projects of the two companies initially separated. If there are increasing returns, the profits of the merged companies are higher than those of the separate companies. It is a form of value creation by project composition. Investment banks, advising their clients on possible mergers or acquisitions, are looking for such opportunities for value creation. Even if there are no increasing returns, the merger can be profitable for the companies, but at the expense of the consumers, because it reduces the pressure of competition on the decrease of profits. The new company can therefore hope to make higher profits by exploiting a monopoly position.

A project can have a value greater than the sum of the values ​​of the projects of which it is constituted by a risk reduction effect, because a composite project can be much less risky than the projects of which it is constituted.

If two separate projects are very risky, a project that contains both of them can still be risk free. Such a reduction in the risk increases the value of the composite project, because for the same average profit, a project is worth more if it is less risky. For example, suppose a project 1 makes a profit after one year, only in case the uncertain event A is realized, and another project 2, brings the same profit, after one year , only in case event A is not realized. Projects 1 and 2, done separately, are risky. On the other hand, to achieve them together is not risky, because we are sure to obtain the same profit whatever the realization of event A. By composing projects, we can sometimes cancel, or at least reduce, the risk to which we are exposed. As risk reduces the value of a project, canceling or reducing the risk increases its value.

A risky bet can always be offset by another risky bet so that the two bets taken together are a risk free venture. If for example I bet on heads, I only need to bet also to bet on tails so that the risk of the first bet is canceled by the risk of the second. This is why the financial markets paradoxically resemble both a casino and an insurance company. By negotiating bets, we can increase our risk but we can also reduce it, because the risks of different bets can offset each other.

Betting on both heads and tails to make profit is a method often used. When a player takes a bet, the owner of a casino always agrees to take the opposite bet. He bets every night on both red and black, even and odd, and so on, and he makes regularly a profit since the chances of winning are slightly in his favor. Banks do the same thing. They can regularly make safe profits, because they can offset all their risks, always with a slight chance of winning in their favor. Bank customers take risks, but not necessarily the bank itself, which can behave like a casino owner.

Project diversification is a way of offsetting risks with other risks. By making many bets, or betting on all possible outcomes, one can reduce or cancel the risks. That's why prudent investors are advised to diversify their investments, not to put all their eggs in one basket. Business diversification is also important for a company that wants to reduce risk. It is sometimes said that companies should not diversify their activities in order to reduce their risk because shareholders can already do so by diversifying their portfolios. But it is ignoring the additional costs imposed by bankruptcies. Bankruptcy is more expensive than the only commercial loss related to a decline in activity. Bankruptcies are expensive for owners, their creditors, employees and all economic agents because they increase the counterparty risk to which everyone is exposed. It is therefore in the general interest that companies reduce their risks as much as they can, not just shareholders, and therefore diversify their activities if they can.

Why can investment banks charge for services at levels that seem exorbitant? If the work of the bankers was similar to that of a dating agency between lenders and borrowers, it should not be paid much more than a marriage agency. But the expertise in corporate wealth management is part of the competence of investment banks. They manage their own wealth and sell their expertise to their clients. When it comes to large companies, the opportunities for value creation by project composition offer sometimes very high profit opportunities. There can be very, very big money to earn. This is why corporate wealth management services can sometimes be sold at a very high price.

Value creation by project composition encourages agents to associate. All communities, associations, companies, partnerships, cooperatives, unions ... can create value by composing with the projects and resources of their agents.

The economic value of the common order

The invisible hand of the market and the interest to cooperate

The metaphor of the invisible hand of the market comes from the fact that sometimes agents act in the general interest while they are driven only by their particular interests, as if an invisible hand guides them towards a common goal that they are not looking for. If one understands it correctly, the existence of this invisible hand is not stupendous, but it is above all often doubtful. The invisible hand is neither absurd nor miraculous in a market economy, because in order to maximize their incomes, agents are encouraged to produce salable goods and thus useful goods to others. To seek their individual interest they must make themselves useful to others. But to believe that they can always thus reach the general interest ignores the importance of externalities. When they commit themselves to their projects, the agents are not encouraged to take into account their externalities, positive or negative. By pursuing exclusively their selfish interests they can harm others, or not take advantage of the opportunities offered by cooperation.

When their projects have negative externalities, agents are encouraged by the competition in the market to ignore them. Reducing or eliminating negative externalities has a cost and decreases profits. An unscrupulous agent can therefore sell at lower prices than competitors who refuse to harm others, and thus eliminate them from the market. To remain competitive, all competitors are encouraged to give up their scruples. When negative externalities are ignored, market rules do not select the best but only the least altruistic or the most dishonest (Akerlof and Shiller 2015).

The prisoner's dilemma is a simple theoretical example that shows why the invisible hand does not always work. Imagine two prisoners to whom an interrogator offers separately the following deal. If both denounce each other, they will take 5 years. If none denounce, they will remain 1 year in prison, before being released. If one denounces the other without having been denounced by him, he goes out immediately and the other takes 10 years. Each prisoner can then hold the following reasoning: if the other denounces me, I have interest to denounce him, since I would take 5 years instead of 10, and if the other does not denounce me, I also have interest to denounce him, since I would go out immediately instead of staying 1 year. The two prisoners are thus incited to denounce and they obtain together a result inferior (from their point of view) to that which they would have obtained if they had decided to cooperate. The selfish search for their interest deprives them of value creation by cooperation.

Suppose two companies A and B can both be involved in projects 1 or 2, and that if they both chose project 1, they will compete, whereas if they both choose project 2 they will be complementary. Specifically suppose that they will each earn 30 if they both commit to Project 1, and 40 if they both commit to Project 2, but if only one commits to Project 1, he will win 50 and the other in project 2 will only win 10. A's project 1 has a negative externality on B's project 1, because it lowers B's gain from 50 to 30. On the other hand, A project 2 has a positive externality on B project 2, since it raises the gain from 10 to 40. If they do not cooperate, A and B are incited to choose the least profitable project, as in the prisoner's dilemma. But if they cooperate, they can create value by composing their projects.

The interdependence of all projects

A profitable project not only benefits its owners, but also all those who participate: customers, because they gain buying opportunities, suppliers, because they gain sales opportunities, employees, because they earn a salary, possibly a bank, or other lenders, because they receive interest, and the state, which levies taxes. The project therefore has positive externalities for all these participants.

In a market economy, all agents generally have an interest in others being prosperous, because to be prosperous we have to sell, so we need buyers who are prosperous enough to buy. A profitable project is so for everyone because it contributes to overall prosperity. Without such shared prosperity, it is much harder to be prosperous oneself. A profitable project therefore has positive externalities on most other projects.

In some cases, projects may be competing, or another form of antagonism, and the profit of one is a loss to others. Such projects have negative externalities on each other.

Agents are not encouraged to consider externalities when pursuing their particular interests, which leads them to ignore or neglect the benefits and losses of their interdependence. If they want to benefit fully from it they have an interest in associating, one way or another.

Financial markets and risk

Financial contracts can be used to reduce risks, but they can also increase them.

Financial contracts can reduce risks by either transferring them to a counterparty that insures us against risk, or by diversification, because a project made up of risky and properly diversified projects can be less risky, and even sometimes almost risk-free, than the projects of which it is composed.

An example of a risk transfer contract: Suppose a project has an initial cost of 40 and yields 100 three times out of five and 0 otherwise. The average profit is 20, but it is risky. We risk losing 40 two times out of five. Such a risk may deter us from realizing the project, although its average profit makes it attractive. If a counterparty agrees to sell a contract that guarantees that if the project does not pay, he will pay the 100 that the project did not pay and if the price of this contract is 50, we can engage in a project without risk which costs 40 + 50 = 90 and which will yield 100 without risk. The counterparty has an interest in offering us such a contract, because he sells 50 a contract which will cost 40 on average. Both parties may be interested, one to turn a risky project into a risk-free project, the other to benefit from a positive average profit. One party transfers its risk to the other party. One is the insured, the other is the insurer.

Of course, when we are looking for a counterparty that protects us against the risk, we have to be reasonably sure that he will be able to fulfill his commitments. To be insured by an insurer who risks going bankrupt is not to be insured at all. A counterparty on which a risk is transferred is good insurance either because he has sufficient wealth to withstand potential losses, or because he knows how to reduce or eliminate risk through diversification.

Financial markets can increase economic risks because they can entice agents to take risks when they are not able to bear them. If other agents believe they are insured when they are not, because their insurer may go bankrupt, there may be a contagion of bankruptcies. Even projects that we believe to be safe because we think we are insured are really very risky. Everyone takes the risk of going bankrupt simultaneously.

Common order, prosperity and freedom

Economic agents collectively have an interest in associating and adopting laws to prohibit or limit negative externalities, to favor positive externalities, to stabilize and secure the economy, and in general, so that everyone benefits as much as possible of the economic conditions, or at least not suffer too much from them. It is not only a matter of social justice, it is also more prosaically a way to increase profits, because a common order, if it is adapted to reality, benefits everyone, because it makes us able to create value by unifying our projects.

Integral laissez-faire, which is sometimes promoted by some theoreticians of utopia, can only lead to anarchy and economic catastrophe. Economies need a strong and well-managed State to thrive. Without such a State one loses most of the economic benefits of organization and cooperation.

In a true democracy, a strong State is not necessarily against freedoms. If everyone participates in the design, evaluation, and decision-making of the common order, which then is not left to the arbitrariness of a leader or a minority, we can hope that this collective order promotes the freedom of all.

Options

We can distinguish two kinds of action: those we are obliged to do and those for which we can decide before acting. The theory of options studies the seconds, the actions that one is free to do or not.

The decisions we can make depend on our means of action. To acquire options is to become capable, to acquire means of action. To exercise the option is simply to act.

American and European options

There are many ways to empower oneself and therefore several kinds of options:

• If the means of action can only be used at a fixed date, fixed in advance, this is a European option. The option can not be exercised before or after its maturity date.
• If the means of action can be used every day, but disappears as soon as it is used, it is an American option. In the financial markets, they generally have a maturity: the option is retained as long as it is not exercised only until a certain date, after which it disappears, whether it has been exercised or not. But we can also think of American options without maturity, which disappear only if they are exercised.
• If the means of action can be kept and used every day, and if it is not consumed by its use, it is an unlimited succession of European options: one for each day, or each period, of use.

The ubiquity of options

Options are ubiquitous in economics, as in all human activities, to the extent that we are free to act.

• When designing a feasible project, we acquire the option to realize it. This is an American option with no deadline, since we can postpone the completion of the project.

• A sustainable consumer good is an option to consume. One acquires the option to consume by acquiring the consumable good. We exercise the option when we consume the good. It is an American option whose deadline is the consumption deadline.

• A piece of equipment is an option on its use. If it is not used by its use, it is an unlimited succession of European options, an option for each day, or period, of use. But if it is used by its use, it is a package of American options. Each time we use it, we consume part of its potential use, which amounts to exercising an American option.

• A professional skill is an option on its exercise. It's a succession of European options for every day, or every period, of work.

• A natural wealth is an option on its use. If it is renewable, as a land that is not degraded by its use, it is an unlimited succession of European options, one for each day, or each period, of use. If it is consumed by its use, like a natural oil reserve, it is a package of American options.

• Money is an option to buy. Before we buy, we usually need to have the money available. This money gives you the means to buy or not to buy. It is an American option with no maturity, because we keep the option to buy until we exercise it.

• A bond is an option on the repayment of a debt. It can be conceived as a European option for the repayment day, but the reality is usually more complicated, since repayments can be spread out over time, and the default can be followed by partial repayment.

• Options traded on the financial markets are generally options on the purchase or sale of financial products. They are explained below.

• There may be option on options. These are ways of acquiring other means.

A durable and salable good is always accompanied by an option to sell it. This option is acquired by acquiring the good and is exercised by selling it. This is an American option whose maturity depends on the duration of the good.

When an option 1 is salable, buying it is equivalent to acquiring two options, option 1 itself plus option 2 for selling option 1. But we can not evaluate separately the value of the option 2 and add it to option 1 as if it were a free gift, because options 1 and 2 can not be exercised simultaneously. If option 1 is exercised, it can no longer be sold and option 2 can not be exercised. Conversely, if option 2 is exercised, we lose the right to exercise option 1. We have the choice between option 1 and option 2, two possible exercises, but it is one or the other, never both together.

Options can be combined in many ways, more or less complicated, because the exercise, or non-exercise, of one or more options may be a condition for exercising other options.

All decisions we make are always ways of exercising options, because before we decide we are free to decide. The general theory of options is therefore simply the theory of decision-making. Since the economy as a whole is the result of all the economic decisions of all agents, economic science can be conceived as the theory of economic options, ie options whose exercise has an economic value, a use value or an exchange value.

A project can always be seen as a succession of options and liabilities, which are often interdependent.

The exercise value of an option

An option is exercised only if the agent believes it has value. But this value is not known in advance, the day when one acquires the option. The exercise day, it is sometimes known exactly, if for example the exercise leads to a monetary revenue, otherwise it must be estimated by the agent who must decide. This known or estimated value is the exercise value of the option.

Acquiring options amounts to increasing one's freedom, since by acquiring means of action one becomes more free to act. But this freedom does not make economic behavior unpredictable. On the contrary, economic behavior is very often predictable, provided that some assumptions are satisfied. If an agent feels that his action has value, he acts, if not, he does not act. When an agent has an option, he acts if and only if he believes that the exercise of the option has a positive value. If one can predict the value he estimates, one can predict his action. That is s why options whose exercise is to receive an income immediately are predictable. If the income is not zero, the agent always chooses to exercise the option, to receive the income.

A European option looks like a lottery ticket. On the exercise day of the option, a draw determines its exercise value. If it is positive, the option is exercised and its owner receives its exercise value. An American option is similar, except that the draw takes place every day and you can refuse a present gain in the hope of a higher future gain.

If the exercise value is negative, an option is not exercised. An option therefore never exposes to losses, only to gains. So it is an asset, a right to future payments, never a liability. But it is risky because future payments are random, and they can be zero. Options can only increase the value of a wealth. The more options we acquire, the more we get rich, as long as the options have value.

The exercise value of a consumption option may not be known in advance. If for example we bought a bottle of Champagne for a romantic dinner, and if finally the dinner is canceled, the exercise value of the Champagne consumption option this night is significantly reduced.

The exercise value of a project completion option is the value that is assigned to the project on the day it is decided to make it.

The exercise value of the option on the use of a capital good is the service it renders that day.

When we sell a good, we exercise the option of selling it. The exercise value of the option is the selling price.

A stock that pays dividends is accompanied by a succession of European options for all dividend payment days. The exercise values ​​of these options are the dividends.

A stock that does not pay dividends is like any salable commodity with the option of selling it. It is this option, this possibility of selling it, that makes the value of the stock (in addition to other rights attached to the property).

Money, as an option to buy, has a variable exercise value, which depends on buying opportunities and inflation, or deflation.

The economic value of imagination

We can acquire options by buying them, but also by our work, or because we were lucky, or because we were given them. We can even acquire them by creating them by the imagination. Just imagine a feasible and profitable project to acquire the option to realize it. As an option on a profitable project has value, we have increased our wealth by the mere use of the imagination.

Since projects are options until we commit to them, creating value by project composition is also a value creation by composition of options. The value of a sum of options may be greater than the sum of the values ​​of the options taken separately. By combining options with the imagination, we can acquire new options on more profitable projects and thus increase our wealth.

The illiquidity premium

When a portfolio is made up of liquid assets, the funds advanced to build it are not blocked. They can be recovered, at least in part, by selling the portfolio back to its current value. On the other hand, if the assets are illiquid, the funds are locked in, we can not use them to invest in other projects. If very profitable opportunities arise, we can not take advantage of them. The liquidity of a portfolio is therefore an option on future opportunities. If a portfolio is illiquid, we give up this option.

A project can be said to be liquid if we can disengage at any time by reselling our participation to its present value. Acquiring a portfolio of liquid assets is a liquid project. But the projects in which one engages are not in general liquid. There are disengagement costs that can be very high. Even for a liquid portfolio, there are generally disengagement costs, because there are transaction fees, but they are low. But for some illiquid projects, we risk losing all the expected profits if we disengage before the end. A project is illiquid when the disengagement costs are dissuasive. Cash advances for an illiquid project are locked-in funds. When we lock in funds, we give up an option on future opportunities. Since this option is valuable, an illiquid project must be more profitable than a liquid project with the same funds to compensate for the loss of this option. The option lost on future opportunities is an illiquidity premium, the loss of value due to illiquidity.

If there were not stock markets, the ownership of a company would be a very illiquid asset, because it is generally difficult to find a buyer. Equity markets mean that shares of traded companies are highly liquid. They have therefore eliminated the loss of value due to the illiquidity of corporate ownership.

The call and put options

A call option is an option to purchase at a pre-agreed price a good traded in a liquid market. A put option is an option to sell such a good at a pre-agreed price. The good is called the underlying asset. Call and put options are financial derivatives of the underlying asset. The price agreed in advance is the strike price of the option. For a call option, if on the exercise day the market price of the underlying asset is higher than the strike price, the exercise value of the option is positive and equal to their difference, otherwise it is zero and the option is not exercised. For a put option, it is the opposite. It has a positive exercise value when the market price is lower than the strike price. The call and put options are therefore like gambles whose earnings depend on the price variation of an underlying asset (Hull 2011).

Call and put options are bought and sold on the financial markets for many underlying assets. Option buyers are similar to lottery ticket buyers. Option sellers agree to pay any future gain against an upfront payment, the option price.

The net present value of a project

The opportunity cost of cash advances and the cash costs

If cash advances are required, their opportunity cost is assessed based on the interest they would have earned if they had been invested without risk. This opportunity cost is all the higher as the project is long and interest rates for risk-free investments are high.

A profit calculated as the difference between all revenues and all costs does not represent the value of a project that requires cash advances because their opportunity cost is ignored. If the profit is less than the opportunity cost of cash advances, it is not worthwhile to engage in the project.

If the project treasury is in cash, such as a cash box filled with notes and coins, or if it is placed in an unpaid bank account, the project profit is correctly calculated by the difference between revenues and costs. But calculating this way, it underestimates the potential profit because it ignores that the project could have yielded more if the cash had been better managed, if efforts had been made to eliminate cash costs. These are the costs of the money that sleeps. When a cash box stays full for a long time, it would be interesting to place its contents so that it earns interest, all the time the money is not used. The cash costs are precisely those interests that we did not collect when we could have received them if we had placed the money left in the cash box. Cash costs can be reduced if the contents of the cash box can be placed to earn a risk free interest. In principle it is possible to eliminate the cash costs, if one arranges so that the cash box remains almost always empty, if one always places the cash surpluses so that they yield an optimal interest without risk. Since cash costs can be eliminated in principle, a project has a potential profit that is independent of these costs.

The discount rate and the net present value

The net present value of a project is calculated from its potential profit, independent of cash costs, from which the opportunity cost of cash advances has been deducted.

To calculate the net present value, it is necessary to define the discount rate. It is estimated from the interest rates of risk-free and liquid investments.

Because of the cash costs, adding the values of cash flows ​​over time by making a simple sum of payments is not appropriate, because the value of the money received today is not equal to the value of the money received at a later date. The existence of risk-free investments that earn interest means that the money paid at a later date has a lower value than the money paid today, because the money placed today is equivalent to higher future payments. Conversely past payments have a higher present value, because it is sufficient to invest them so that they yield an interest. The interest rate of a risk-free investment can therefore be interpreted as a discount rate for past and future payments. It enables us to calculate the present or current value of a series of past and future payments (Merton & Bodie 2000):

$V_{AN}=\sum _{i}{\frac {V_{i}}{(1+r)^{t_{i}}}}$

$r$ is the annual discount rate. $t_{i}$ is the date (in years) of the payment $V_{i}$. The $V_{i}$ are all payments, positive if they are revenues, negative if they are costs, associated with the project. $V_{AN}$ is the net present value of the project.

(Specifically, the discount rate $r$ may vary depending on the maturity date, because interest rates vary depending on their term. In the following, this complication will generally be neglected.)

When the $t_{i}$ are positive, these are future payments. Since $r$ is generally positive, future payments are all the more devalued as they are distant in time. The higher the discount rate, the more the future payments are devalued relative to the value of the present payments. When the $t_{i}$ are negative, they are past payments.

A project that pays a profit can be considered as an asset. If it is risk free, its net present value is the price that should be paid to acquire such an asset.

A project that makes losses can be considered a liability. Its net present value is negative. If the project is certain, ie if the losses are known in advance, its net present value measures the amount of the liability on the day it is valued.

When you have to value a wealth (the difference between assets, everything you own, and liabilities, all you need) you have to include the net present value of all the projects in which you are engaged. If, for example, a company is risk-free, and if its profit rate is higher than the risk-free rate, the company's assets are not properly valued if it is estimated at the value of the initial capital initially paid, because the net present value of the difference between the expected profit and the profit at the risk-free rate must be taken into account. In general, engaging in a profitable project is a creation of value and an increase of wealth. But it is difficult to assess when projects are risky. Conversely, engaging in a project at a loss is a destruction of value and a loss of wealth.

The annual discount rate has been defined in the usual way: an investment of $1$ today is worth $1+r$ in one year, if $r$ is the annual discount rate. But we can also measure this rate in another way, which is often better for mathematical models. The discount rate if the period was six months would not be $r/2$ but ${\sqrt {1+r}}-1$. If on the other hand we define a discount rate $R$ such that $e^{R}=1+r$ then $R/2$ is the six-month rate, because $e^{R/2}e^{R/2}=e^{R}$. $R$ is the logarithmic discount rate. When $r$ is small, it is very little different from $R$.

The profit of a project and the surplus profit

Profit is the difference between revenue and costs. But since the payments can be on different dates, they must be updated.

Profit is what is earned after advancing funds, if the project requires capital, or after not having advanced nothing, if the capital required is zero. The closing day of the project, we recover the capital plus the profit. If the profit is negative, we have lost part of the advanced funds.

In order to ignore cash costs, it is assumed that the project bank account is remunerated at the risk free investment rate. So everything happens as if any cash surplus was invested at the risk-free rate until it is spent.

To calculate the profit of a project, the sum of the discounted values, at the end of the project, of all revenues and costs covered by previous revenues is first calculated.

Advanced funds are not discounted in the same way as costs covered by previous revenues. The sum of the discounted values, at the project start date, of all cash advances is an estimate of the project's start-up cost. It is the capital that one must be ready to invest to get started in the project.

The profit of the project is thus calculated as the sum of the discounted values, at the closing date, of all revenues minus the costs covered by previous revenues, minus the sum of the discounted values, on the launching day, of all cash advances. .

If all the project treasury is in cash or an unpaid bank account, it is easier to calculate the profit, because it is not necessary to discount the revenues and costs covered by previous revenues. We simply calculate the sum of the revenues and subtract all those costs. But it is still necessary to discount the advanced funds, if they are staggered in time, to correctly evaluate the committed capital and thus the profit. If we do not discount the cash advances, it is like assuming that they were all deposited into an unpaid account, or in cash, on the day the project was launched. If the cash flow of the project is not remunerated and all the advanced funds are paid on the launch day, the realized profit is correctly calculated without discounting the revenue, the advanced funds or the other costs. But we do not evaluate the cash costs. It ignores that the same project could have paid more if the cash flow had been better managed.

The rate of profit is the rate of interest which would have allowed the capital invested to yield the profit if it had been invested at this rate. For projects without cash advances, so without capital, they just have to yield a profit for their rate of profit be considered infinite. If the profit is negative, the rate of profit, when it can be calculated, is then negative. It can be interpreted as the rate of depreciation of capital that would have caused the same loss.

A simple example: a project consisting of a single initial cost of 100 ($V_{0}=-100$) and a single final revenue obtained after one year <math> V_1 = 110 < /math> without risk. Suppose the discount rate is 5% = 0.05. The present value at the closing date of the revenue is 110. The discounted value on the launching day of the advanced cash is 100. The profit is therefore 10 and the profit rate is 10%. The net present value of the project on the day of its launch is -100 + 110 / 1.05 = 4.76. If an investor bought the project at its net present value on the day of its launch, it would have to pay that day 4.76 + 100 = 104.76 and it would receive 110 after one year. The difference, 5.24 is the interest of the same investment paid at the risk-free rate of 5%.

If the profit is positive, the project provides a gain but that is not enough to make it interesting, because if the profit of the project is lower than the profit that one would obtain by investing the capital at the risk-free rate, one would not want to start it, it is better to invest one's money at the risk-free rate. The term "project at a loss" is ambiguous. On one side a loss is a negative profit and a project at a loss is a negative profit project. But on the other hand, if the rate of profit of a project is positive but lower than the discount rate, it is a project that we do not have interest to launch and which must be regarded as a project at a loss, even if its profit is positive. From this point of view, a project is at a loss when its net present value is negative. Even if the profit is positive, the net present value may be negative.

When a project is capitalless, its net present value is the present value, the day the project is launched, of the expected profit, or the loss that will have to be incurred if it is negative. When a project requires capital, its net present value is the present value, the day of the project's launch, of the difference between the expected profit and the profit that would be obtained by placing its capital at the risk-free rate. This difference is the surplus profit. The net present value of a project is the present value of its surplus profit. A risk-free project is interesting when its surplus profit is positive. This leads to stating the net present value rule:

One must go into a risk-free project, where all the costs and all the gains have been evaluated by a monetary equivalent, if and only if its net present value is positive.

But this rule ignores the possibilities of creating value by composition of projects. The profit of a composite project can be greater than the sum of the profits of the separate projects, because the projects can have positive externalities.

Evaluation of assets, liabilities and portfolios

An asset is a right on future payments.

The same asset can enter into various projects. If the resale of the asset is part of the project, then the selling price is an expected future revenue.

Some assets, such as ownership of a company for which one is fully responsible (no limited liability), are rights to future payments with the obligation to pay any losses. They may be called ambivalent assets because they can turn into debt. In general, financial assets are not ambivalent, they are always rights on payments, they do not turn into obligations to pay debts.

Future payments associated with an asset may be considered as revenue from a project without costs. The current value of the project is then the current value of the asset.

An asset can also be associated with the project to buy it on an initial day to receive future payments. The purchase price of the asset is then counted as the sole initial cost of the project. The net present value of this project is positive if and only if the present value of the asset is greater than or equal to its purchase price. This leads to a particular case of the net present value rule:

An asset must be purchased if and only if its present value is greater than or equal to its purchase price.

But this rule does not take into account value creation by asset composition. By combining assets with each other, so by building a portfolio, one can obtain a portfolio discounted value greater than the sum of the discounted values ​​of the assets of which it is constituted. The discounted value of an isolated asset therefore does not always make it possible to estimate its contribution to the increase in value of a portfolio in which it is incorporated.

If the asset is intended to be resold, the project of acquiring it for resale is usually risky, if one does not know the day and the price of the sale. To calculate the average net present value of the project, it is then necessary to know the probabilities of the possible sales, for all the possible prices and all the possible days, hence it is necessary to know the strategy of resale of the asset. If we do not know this strategy we can not calculate the net present value of the project. Even if we know how to calculate it and if it is positive, it is not enough to justify the project, because the risks of a low profit and especially a loss are dissuasive.

An asset is liquid when there is a market where agents are willing to buy or sell it every day. A liquid asset can therefore be resold easily. An asset is illiquid if it is difficult to find a buyer for an acceptable price.

The projects in which we are engaged are often illiquid because we can not resell them or dispose of them easily.

A liability is an obligation of future payments. If the dates and amounts of payments are known in advance, the liability is certain, otherwise it is uncertain.

Future payments associated with a liability can be considered as the costs of a non-revenue project. Its present value is the price one has to ask to commit to such a liability if one does not want to expose oneself to a loss.

A loan is a project that imposes a liability on us, the obligation to repay the loan. The loan amount is the only initial income from the project. The net present value rule seems to show that one should never go into debt. The project is generally certain, because the payment obligations are known in advance, and its net present value is generally negative, because the interest rate is higher than the risk-free rate. This does not prove that one should never go into debt, but only that the cost of indebtedness must be offset by the surplus profit of the projects with which it is associated.

A liability is generally illiquid because one can not get rid of one's obligations by buying them back at a market price. But it is sometimes liquid. A company can buy back its bonds if they are traded on a market.

Ambivalent assets and commitments are neither really assets nor liabilities. They are risky commitments that expose us to both a chance of gain and a risk of loss, such as when we bet on a gain while accepting a possible loss. They can always be thought of as a combination of a risky asset and a risky liability. Forward and futures, presented below, are examples of ambivalent commitments.

A permanent portfolio is a set of assets, liabilities and ambivalent commitments that are held indefinitely to collect income while meeting payment obligations. A dynamic portfolio is a set of assets, liabilities and ambivalent commitments that are managed on a daily basis, by selling and buying assets, making or releasing liabilities, or ambivalent commitments, .

If a portfolio is permanent, if all its assets and liabilities are risk free and if they are independent, in the sense that their income does not depend on the presence of other assets or liabilities in the portfolio, then the net present value of the portfolio is the sum of the present values ​​of all its assets and liabilities (the present value of a liability is negative).

If the assets or liabilities are risky, or if they are not independent, or if the portfolio is managed dynamically, the calculation of the average net present value of the portfolio is obviously more complicated.

To evaluate a portfolio, we need to know how it is managed. A manager who knows how to make commercial gains will on average make higher profits than a less good manager, the average net present value of the portfolio is increased by the same amount.

All risky assets can always be valued as options because they are expectations of future earnings. If one ignores the liabilities and the ambivalent commitments, a wealth is always a wealth of safe assets and options. But since everything is risky, there are no safe assets, there are only options left.

Forwards and futures

A forward contract is a contract on the forward sale of an economic good. The sale price is negotiated on the day of the conclusion of the contract. If the day of the sale, the price on the market is higher than the negotiated price, the seller loses the difference, if it is lower, it is the buyer who loses it. Committing to the sale or purchase on a forward contract is therefore similar to a bet where the gains and losses of the players are decided by the price changes in the market.

A forward contract makes it possible to create value by composition of projects. Each party can increase the value of it project by adding the bet associated with the forward contract. An example: a supplier of commodities is exposed to the risk of falling price on the market. Symmetrically, a buyer of commodities is exposed to the risk of rising price. If they enter into a forward contract together, they both cancel the risk (Hull 2011). With a risky gamble (the forward contract) everyone turns their risky initial project (the sale or purchase of commodities) into a risk-free project. This is not entirely true, because there is still the counter-party risk. It is the risk that the other will not fulfill its commitments.

Forward contracts are over-the-counter transactions. Futures are forward contracts for future prices traded daily in a market, where many agents commit to buy or sell. The market price balances these supplies and demands. The existence of a market means that futures can be used in a very different way from forward contracts, because they can be cancelled before term. We cancel a future by engaging in another futures in the opposite direction. If we had committed to the sale, we commit to the purchase. If we had committed to the purchase, we commit to the sale. The price difference between the sale and the purchase is a profit or a loss. It is paid by a clearing house or it must be paid to it.

To evaluate risks, we must conceive of futures as part of a lasting game of chance, because in each period we can record a loss or gain. But we are always free to leave the game, that is to say to cancel the future. The option to leave the game is a real option. Its exercise value is equal to the difference between the present gain on the futures and the present value of the anticipated gain, or between the present loss (negatively counted) and the present value of the anticipated loss. If it is positive, it is better to exercise the option and to settle the future.

Following chapter >>>

Inequalities in front of finance

"If you give a fish to a poor man, he will eat one day; but if you teach him to fish, he will eat every day. "

Chinese proverb

And we can add: if a fisherman does not have a boat, you can make him pay every day the boat that you lend him.

The private property based financial system is unfair because it favors owners over less fortunate workers. It allows the richest to earn a lot of money without working and thus aggravates social inequalities (Stiglitz 2012, Piketty 2013). It can sometimes partially correct some economic inequalities by offering the less fortunate the means to undertake, but even so it remains deeply unequal.

The problem of the property of the means of production

When workers do not own their means of production (equipment, buildings, vehicles, machines, tools ...), they must yield the profits to the owners of the company, the shareholders. When the amounts involved are not too large, it is a problem that can in principle be solved by finance. If workers can borrow at a low interest rate, they can become owners of their means of production. If they have good projects and if they do not have the financial means to achieve them, the job of bankers and financial intermediaries is to solve their problem. If all good projects could always be properly funded, the less fortunate could undertake as much as the rich (Shiller 2012). Of course this condition is not often realized. Human beings are not all equal in the face of borrowing, and the interest rates offered to them, when they are, are often too high to encourage business ventures.

The solution to the problem of ownership of the means of production can not be that workers always own their means of working. Some companies require very expensive equipment for a relatively small number of workers. All of them should be very wealthy to own their company, so all low-income workers should be prohibited from participating. This would surely not be considered a solution to the problem.

Shareholding allows in principle to democratize the freedom of entrepreneurship. Anyone can try to sell shares on a business plan and finance it as long as he or she finds investors. Startups are usually funded this way. Crowdfunding exploits the same principle. This allows an entrepreneur with no personal fortune to finance her projects without going into debt. This does not solve the problem of the separation between labor and ownership of the means of production, since the shares are shares in the ownership of a company, but the designer, responsible for a project, can ask to be partially paid in shares. In this way, she can retain part of the property of her business without having advanced funds at the outset.

Why are there interest rates?

When the investment is risk-free, collecting interest means collecting income without working and without worrying, it is like having a goose that lays the golden eggs, and the richer we are the bigger the eggs.

When the debts result from taxes or fines, the rate of interest is fixed by the power which imposes this obligation. Other debts generally result from loans. In theory, and except in the case of a forced loan, loan agreements are freely concluded. Both parties are volunteers. There is an interest rate because some people want to lend their money with interest and others want to borrow it. Borrowers often agree to pay high rates, far in excess of the rates at which they could safely place money if they had any. A loan agreement therefore always has a negative net present value for them. It seems that they engage in a project at a loss and that they are wrong to borrow. However, we often have excellent reasons to apply for a loan. We must distinguish investment loans, consumer loans and loans to repay previous debts.

• An investment loan finances a project to make a profit. If the rate of profit is higher than the interest rate of the loan, the borrower will be able to cash the difference between the profit and the paid interest. Even if the rate of profit is slightly lower than the borrowing rate, it may be interesting to borrow part of the cash advances if it enables us to realize projects with a profit rate higher than the risk-free rate, and if we can not do them without borrowing. As soon as the agents have profitable projects, they have an interest in borrowing to realize them.

A home loan can always be considered an investment loan, even if you buy the property to live in it and not to rent it. In this case the housing service provided can be considered as profit, as if one had paid a rent to oneself.

• A consumer loan finances the purchase of a consumer good. Many reasons, not just unconsciousness or lack of foresight, may make a present purchase better than a delayed purchase due to lack of available money.
• When you can not repay your debts, getting a new loan can be a big help, because the costs of insolvency are generally quite high. But if we do not have prospects of future income sufficient to repay, we are of course only sinking into debt.

It would be wrong to believe that the existence of interest rates is exclusively linked to indebtedness. Loan interest is just one form of ownership income, and as soon as there is income from ownership, we can calculate an interest rate. A rent is with respect to a housing identical, from an accounting point of view, to an interest with respect to a loan. The rent rate (the annual rent discounted at the end of the year, divided by the price of housing) is the interest rate of the real estate property. The same calculation can be made for any rental of durable goods, provided that the charges are correctly counted. The profits of a project are the income from the ownership of the committed funds. The rate of profit is the interest rate of the business property.

These three forms of property income, interest on loans, rent on rented property and business profits compete. The supply of loans with interest depends on the other forms of property income, because one can choose between lending money, or buying real estate to rent, or buying shares to receive profits. The demand for interest-bearing loans also depends on the other forms of property income, because one can borrow to buy real estate, and because profit expectations encourage businesses to borrow. The existence of other forms of property income means that interest rates on loans can not be arbitrarily reduced to zero. If, for example, a central bank agreed to lend to everyone at zero rate (technically it could do it, because it can create the money lent) it could seriously disrupt the economy, because everyone would be encouraged to borrow to invest in real estate or in companies, and because companies would be encouraged to invest too much. Interest rates work as a brake on the desire for debt. If this brake were removed, an economy founded on private property would be exposed to a high risk of high inflation and other imbalances.

There are interest rates simply because there is an interest in being an owner. Even if everyone owned their home and their equipment goods and no one was in debt, we could still calculate interest rates - but it would be more difficult. The existence of interest rates is not even unique to human economies. Animals have a supply of energy that is vital to their daily activities. At the beginning of the period, this reserve is like a cash advance. At the end of the period, after the animal has researched its food and performed all its other activities, all the energy consumptions can be counted as expenses, and all the food eaten as recipes. If there is a surplus, it can be counted as a profit, because the animal has increased its reserves. We can therefore calculate a rate of profit, which is the interest rate on reserve property. That the animal is owner of its means of production (its body, including the reserves) does not prevent that there is a rate of interest.

As soon as we have a project, we can evaluate its profit or loss. If the project requires cash advances, a rate of profit can be calculated. As soon as there are profitable projects that require cash advances, then there are interest rates.

If an equipment good is required by a project, it must be counted as a cash advance. If it is still there the end of the project, we count it among the final revenues, otherwise it was consumed by the project. Whether or not workers own their means of production does not change the fact that profit rates can be calculated for projects that make use of these means.

Real estate income exists even when the property is not rented. A house helps because we are better in it at night than on the street. The housing service, whether paid or not, is an income from the property. Even if everyone owned their homes, virtual rents could be assessed, because homeowners can choose between different ways to find lodgings and various ways to invest their funds in a variety of other profitable projects. The profit rates of the other projects are thus representative of the value that the owners give to the services provided by their housing.

If interest rates measure the interest of projects, so the interest in living, why do we say that they must be very low, in order to promote economic development? The interest rates of the loans are not representative of the interest of all our projects, they play the role of a limit between the projects that can be realized and the others. If the rate of profit of a project is higher than the interest rate at which one can borrow, it is better to borrow, otherwise one is dissuaded from it. The higher the interest rates, the more difficult it is to complete projects, even if they are profitable. The lower the interest rates, the more it becomes possible to realize many projects.

All investments are always risky, because there is always the risk of an economic catastrophe so serious that it would affect all investments, even those usually considered risk free. Even the debts of the US federal state are risky if we take into account the risk of inflation (there is no counterparty risk because the Federal Reserve always has the possibility of creating dollars to repay. As debts are denominated in dollars, it is certain that they can be so repaid). But as long as the economies are a little prosperous or not too badly hit by a recession, it is usually quite easy to find low risk investments because many states or large corporations can be considered very reliable borrowers and because some real estate investments are low risk.

The existence of some property income can be considered a serious injustice, because it is a way of making money without working and because it aggravates social inequalities. It is possible to design an economic system that removes some income from property (interest on loans, shareholder profits, and homeowners who rent real estate) while retaining other economic freedoms, i.e. other property rights, freedom to undertake, work, sell, buy and consume all goods and services produced. A very powerful state could finance the entire economy and all the real estate available for rent. But one may fear that such a public power is a remedy worse than evil, because it is likely to be a dictatorship. In the sequel such a possibility is ignored. We think about an economy such as ours, where property income is respected. The resulting aggravation of inequalities can sometimes be partially compensated and corrected other than by completely renouncing capitalism.

Inequality in front of risks

Agents are not equal in front of the risk of loss. A poor agent who goes into debt for a risky project and then suffers heavy losses will probably pay dearly for the rest of his life. When we can not pay our debts, we add additional costs (fines, court fees and lawyers, loss of reputation ...) to the initial costs that can not already be paid. These costs of insolvency are very dissuasive, even if we are no longer going to jail because of unpaid debts. As a result, poor people are discouraged from going into debt to engage in risky projects. If on the other hand we have a substantial fortune, we are more solid to bear any losses. If the project is ultimately losing, we lose a part of our fortune, but in general we keep a large part, even after paying all our debts.

The projects that earn the most on average are most often the most risky projects. Those with fortune and a bold mind are in the best position to take advantage of these opportunities. The less fortunate are much less well placed, and they are generally encouraged to abstain, if only to avoid the risk of being drowned in debt.

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How can we create money from scratch?

It seems that to advance funds it is necessary to dispose of it in advance. This would be true if all the money was in gold, if there were no bank notes, or deposits on bank accounts. But the banking system creates money out of nothing, just ink and paper, or computerized accounting. It allows to advance funds that did not exist before, which are created at the very moment they are loaned. It looks like counterfeit money, but it's very different, because money is created in exchange for promises of repayment.

One of the oldest ways to create money from scratch was seigniorage. The lord of the kingdom demanded that he be put in deposit the existing coins, he then made them coined at a different, higher value, and he returned the deposits to their face value, by cashing the difference, which allowed him to repay his debts or increase his treasure.

Bank notes and exchanges of debt recognitions

Bank notes first appeared as deposit securities. We deposited our gold in the bank and we received in return a deposit security that could be used instead of gold for payments. Simply writing a deposit security is not enough to create money. The amount of money in circulation does not increase. The gold that was in circulation is immobilized in deposit and is replaced by a security that is the only one to circulate. But these deposit securities played the role of bearer repayment promises and the banks could put into circulation more promises of repayment than they had gold in their coffers, because the promises of repayment were not all required at the same time. They could be used as gold and they were easy to circulate. During normal periods, they remained in circulation and their repayment was not required. But if there was a loss of confidence in the bank's ability to repay its notes, all the holders could come forward at the same time to demand gold, and the bank could not meet its obligations.

The privilege of issuing notes could be very profitable and not very risky, because the banks lent with interest created money which did not cost them anything. If the borrowers were reliable, a bank feared little for not being able to repay the notes it had issued. But it still faced a risk of illiquidity, meaning that it could be asked to repay its notes before the loans it had granted were repaid.

Another way to create money from scratch is to print counterfeit notes. If their quality is sufficient so that they are not distinguished from others, the amount of money in circulation has been increased.

Imagine two friends who are mutually signing IOUs of equal value. An IOU can work as currency, provided the debtor's reputation is good. If no one is aware of the deception, and the two friends have a good reputation each, each can pay with the other's debt recognition. They thus circulate their papers as money. So they created money from scratch - almost, only a little ink and paper.

If the debts are repaid, the IOUs are destroyed and disappear from circulation. Generally, money can be created with IOUs and destroyed when debts are repaid.

Individuals generally can not circulate IOUs as currency. On the other hand, bankers always can, because their IOUs are considered very reliable. When a bank issued a loan with notes that it had created, it exchanged a debt acknowledgment, the notes, against another, that of his client.

Central banks

Banks that had acquired the right to print banknotes have gradually lost this privilege, except for central banks. Originally a central bank was the bank of the prince, or the ruler. Rich individuals who lent their gold to the sovereigns in exchange for privileges together formed banks to strengthen their position (which was sometimes tricky: a way for the princes to free themselves of a debt was simply to cut the head of the creditor). Thus they gradually acquired the monopoly of issuing banknotes.

Bank notes have gradually ceased to be convertible into gold, or silver metal. Such convertibility is dangerous and costly for banks, because they have to keep gold reserves that may not be enough if there are too many holders of notes who ask to be reimbursed. Even central banks, which held large gold reserves, were constrained in their monetary creation by the gold convertibility requirement. In fact during a crisis, they did not hesitate to suspend this convertibility as soon as they feared for their reserves.

With the abandonment of gold convertibility, central banks have ceased to be constrained by the need to keep gold reserves, they can create all the money they decide. Gold as a currency has become useless. Now economies no longer need gold for currency trading.

Imagine someone finding a huge treasure, gigantic gold reserves. He has no interest in selling such wealth too quickly, because if he floods everyone with his gold, he raises prices and it diminishes the value of his treasure. Central banks have the same problem. They have an unlimited supply of money, a sort of huge treasure, but if they sell it too quickly, they drive up prices.

How does the money created by the central banks flow into the economy? In principle, they can buy anything they want, including gold stocks, and even all wealth available for sale. But of course grabbing all the riches is not their role.

A central bank puts money into circulation by a technique that looks like an exchange of IOUs, because in general it lends the money it creates, either directly, by lending to banks, it is then a bank of banks, or indirectly, by purchasing IOUs on the financial markets. When the central bank grants a loan, or when it buys a bond, it puts money into circulation. When the loan or bond is repaid, the money is withdrawn from circulation. But it is not really an exchange of IOUs, because a central bank owes nothing. Its notes are not rights to real wealth but only rights to be exchanged for real wealth at a price that is a priori indeterminate. Since gold convertibility has been completely abandoned, central banks do not have to pay off debts any more because they no longer have debt. It all happened as if they had refused to repay their debts, since originally the notes were IOUs.

A central bank can also give the money it creates, either to the state, which allows it to reduce taxes or repay its debts, either directly to the citizens, giving them all an equal amount, for example, as an egalitarian Christmas gift. In this case, it is sometimes referred to as helicopter money, as if the central bank was distributing its notes by helicopter. The term helicopter money is obviously a joke, but not the reality. Creating money and giving it to all citizens can be a very effective way of fighting deflation (see next chapter).

The economies of today are generally debt economies. The amounts of all debts are important in relation to the wealth produced. This allows central banks, and other banks, to easily circulate the money they create, lending it to those who want to borrow it. If the agents refused to go into debt, if they paid off all their debts, would the money disappear from the economy? All the money today in circulation would be destroyed by the repayment of debts. To replace the destroyed money, the central bank would be led to give the money it can create.

Bank accounts

Losing the privilege of bank notes emission has not deprived banks of the money-creation privilege, because bank accounts are a form of money. A creditor bank account is an acknowledgment of the bank's debt to its client. To grant a credit a banker does not need to have the money beforehand, he just has to click on his computer and the credit is automatically displayed on the customer's account. That's all. When a banker gives credit, it is simply an exchange of IOUs. The client signs an acknowledgment of the debt vis-à-vis the bank. In exchange, the bank increases the customer's current account.

One would think that banks do not create money because they would only lend the money they are loaned, but this is not true. A bank does not have the right to lend more than its depositors have entrusted to it, but that does not prevent it from creating money. When it grants a credit, it does not debit the account of its depositors. Since the money supply includes all current accounts, it increases each time a bank grants a credit: the current account of the borrower is increased without any other current account being debited.

Of course, if the bankers were completely free to grant credit, we would live in complete monetary anarchy, because accomplice bankers could lend each other money and create as much as they wish, to the point of buy all the wealth of humanity! What prevents them from becoming crooks? Only the rule that forbids them to grant more credits than the money that was given to them by the depositors, because if they spent the money they created, the deposits would not be enough to cover the granted credits. This rule does not prevent them from creating money, but it prevents them from creating as much as they could wish.

Only a central bank can create money without limit. When it lends money, it is limited only by its own rules of operation. It can create billions with one click. It just has to decide it. Of course it does not have the right to abuse this privilege, otherwise it would sink the economy.

Deposits make credits and credits make deposits

Banks have the right to grant new credits as soon as they receive new deposits. So deposits make the credits.

When a bank gives credit, the money created is usually left in the same bank, or in another if the credit has been spent. Only a portion of the credit granted is in the form of notes in circulation. Most of it is usually somewhere in bank accounts. So the credits make the deposits.

Each credit makes a new deposit that can make a new credit, and so on. Is it then to be feared that banks can create money without limit and drown the economy in a monetary deluge?

Since some of the money created by banks is kept in the form of notes and is not redeposited, the sequence of credits and deposits can not lead to unlimited creation, because each new deposit is only a fraction of the credit that created it. In addition, central banks generally impose mandatory reserves on banks. For each deposit, they must keep a fraction in reserve and therefore do not use it to grant a new credit. As a result of these two effects, the money created by the banks can only be a multiple of money put into circulation by the central bank. This monetary multiplier depends on the reserves of the banks, mandatory or surplus, and the mass of cash in circulation.

If we removed cash money and if there were no reserve requirements, would the money supply increase to infinity? Presumably no, because banks lend only if they find reliable borrowers, and it is not always easy to find. Even if they were not obliged to keep reserves, they would still retain them because of the lack of borrowers to lend them to.

Should banks be prohibited from creating money?

From the point of view of a depositor, his money in the bank is kept, since he can withdraw it at any moment, but from the point of view of the bank, everything happens as if it were lent, since it can lend almost as much money as the one deposited. Monetary creation by banks is a way of turning sleeping money into circulating money. By depositing our money in a bank, we keep it without letting it sleep, since it makes more money circulating into the economy.

From an accounting point of view, the money deposited in the bank is lent to borrowers. A depositor may rebel against this practice. If the bank goes bankrupt, it could lose its deposit. We can therefore think of 100% reserve deposits (Fisher), that is to say that the bank would promise not to lend them, it would refrain from using it to create bank money. Today's banks could offer such a 100% reserve bank money service, but depositors would have to pay more bank fees for the management of their accounts, because the banks could not pay themselves by lending the money entrusted to them. Such an increase in fees would be a deterrent. In addition, depositors would have nothing to gain, because demand deposits are already guaranteed. Even if his bank goes bankrupt, the depositor does not lose his money. Why then want a 100% reserve account?

Are the interests of a bank loan a theft?

We may find that there is something dishonest in collecting interest by lending money that was not had beforehand. Should we accuse the banks of organized fraud and ask them to repay all interest improperly received?

Legally banks, except central banks, do not create money, they only lend the money that is lent to them by depositors. We count, however, a monetary creation by summing all deposit accounts. Where is the truth ? Is the money created by the banks, or only lent? This ambiguity about money creation by banks comes from the ambiguity of demand deposits: is it money kept by banks, or money they put back into circulation in the economy by lending? Both answers are legitimate. From the depositor's point of view, a bank account works like a cash register, and the money he deposits there is kept, since he can withdraw it whenever he wants. But from the point of view of the bank, the money deposited is put back into circulation as soon as it is lent.

Banks do a useful and important job of evaluating borrowers' projects. They must be paid for it. In addition, they are necessary for the economy, if only to manage our means of payment. That they are paid is therefore legitimate. Only if the interest they ask for is too high, or for other forms of dishonest loans, can one really speak of theft.

If banks were not allowed to charge interest on their loans, they would not be able to lend because they could no longer cover their costs. This would deprive the economy of a major source of funding and prevent bankers from doing their job, while it is useful and important for everyone.

Deposit banks do not own the funds they lend. The interest they receive is therefore not income from property, it is only income from work.

Individuals could complain about unfair competition from banks. To collect interest, they must first earn the money they lend. Banks compete with them by lending money that they have not earned. The wealthy owners are thus competing with workers, the bankers, who lend money they have not even acquired.

Should we return to the convertibility of bank notes into gold?

Since the convertibility of their notes to gold has been abandoned, central banks can create unlimited money. This allows them, in particular, to intervene in the market for lendable funds to lower interest rates, if this seems desirable to them. They only have to lend the money they create at a rate as low as they decide. Other lenders must then lower their own rates if they want to remain competitive. The market for loans is therefore not an ordinary market where individuals meet on an equal footing. Central banks are clearly favored. They lead the dance. This is why they are regularly accused of dangerously disturbing the economy. Their decisions do not always benefit everyone.

The return to gold convertibility of their notes would be a way of limiting the power of central banks. If we are opposed to state intervention in the economy, this may seem desirable, but then we ignore that such convertibility can not be a lasting solution. Gold reserves can not generally grow at the same rate as the rest of the economy. But to meet the needs, a central bank must increase the money supply in a growing economy. Gold reserves increase little while the mass of notes in circulation increases more. It can not last forever. Sooner or later agents will understand that the gold reserves are not enough and they will ask to convert their notes. From this point of view it is surprising that gold convertibility has been abandoned so late. It could only be a temporary system, which prompted agents to accept paper money and bank money, but which could not last.

Why money creation is not necessarily bad

When central banks and other banks create money, they do not do it in the manner of a fraudulent crook who would squander an ill-gotten fortune, they do it in the manner of a serious banker who assesses the quality of the borrower and who asks for guarantees. Money is created only in exchange for reliable repayment promises, not for playing poker. This helps to meet the financing needs of the economy without being limited by the amount of gold available. Money can be created to meet the real investment needs, and the ability of the economy to engage in profitable projects. A good project is a sufficient guarantee to legitimize the creation of the money that finances it.

It must be remembered that monetary creation by banks is not final. It is enough that they decide to lend less than the loans that are repaid to them to destroy some of the money they have created. The money lent can always be eventually destroyed if necessary. Only helicopter money, given by the central bank, can not be taken out of circulation.

The higher the interest rates, the more owners take a significant share of the domestic income, to the detriment of non-owners. With money creation, central banks generally have the power to impose low interest rates permanently. If inflation is already under control, a slight rise in interest rates is usually sufficient to contain possible inflationary trends. Even in times of full employment, rates can remain quite low without excessive inflation. With money creation, central banks therefore have redistribution power. As in the legend of Robin Hood, they can take to the rich, the owners, to give to the poor, non-owner workers. They do it daily by charging interest rates as low as possible.

Financing the economy

« Give them the power to collect the grain during those good years and to store it in your cities. It can be stored until it is needed during the seven years when there won’t be enough grain in Egypt. This will keep the country from being destroyed because of the lack of food. » (Genesis, 41,35-36)

In the absence of monetary creation by banks, the financing of the private economy would depend exclusively on the self-financing capacities and the goodwill of agents who are fortunate enough to entrust or lend their money to entrepreneurs. Such an economy would therefore face the risk of a shortage of investment, because owners can choose to keep their money instead of lending it. Monetary creation by banks makes it possible to cancel this risk of shortage of investors. Money lent by banks is like a loan without lenders, because banks lend money they have created for the occasion, not money that existed beforehand. Monetary creation can finance the economy even if there is a shortage of investors, it is enough that the banks take over. But an economy is then confronted with the risk of excess investment. If all we have to do is create money to finance all our projects, we risk funding too many projects at the same time. To benefit from sustainable prosperity, economies must navigate between these two pitfalls, too little investment, lack of private investors or of sufficient money creation, and too much investment, because of the exaggerated optimism of private investors or an excess of money creation.

Macroeconomic fluctuations

All economic activity is the realization of all economic projects of all agents. Every day, projects are carried out, continued, renewed or abandoned, and new projects are launched.

At one point, the ability of an economy to simultaneously carry out projects is limited, because raw materials and supplies, stored or soon available, labor and capital goods are limited. If all agents simultaneously engage in projects that exceed the capacity of the economy, they will not all be able to carry them out at the same time. Some will be forced to delay or abandon their projects, either because of the shortage or because of rising prices. When the projects implemented exceed the capacity of the economy, it is said to be overheating, or overcapacity. The labor shortage is driving up wages, the scarcity of raw materials, supplies and capital goods is driving up their prices, all of which puts upward pressure on the prices of consumer goods.

If the projects in which an economy is committed are below its capacity, it is said to be under capacity. Available means are not used or they could be better used by being assigned to better projects. Underemployment of the available labor force is the main sign of an under-capacity economy.

More precisely overheating and under-capacity can be distributed in a differentiated way within the economy. Some sectors may overheat, with scarcity or price increases while other sectors are under capacity.

Whether an economy is overheating or under-capacity depends on the agents' decisions. If the atmosphere is pessimistic, they will be reluctant to engage in projects, and conversely if they are optimistic. That's why it has been said since Keynes that "animal spirits", that is mood swings, determine macroeconomic trends. The alternation of periods of high and low activity resembles manic depression (bipolarity). Episodes of depression, remission and euphoria follow each other.

But macroeconomic fluctuations do not depend only on animal spirits. They may have causes beyond the will of the agents. If, for example, the prices of raw materials imported or exported in large quantities vary significantly, this is sufficient to considerably affect the activity of an economy. In general, any unforeseen changes in the conditions of their activity may encourage agents to give up their projects or, on the contrary, to form new ones.

Overcapacity and under-capacity are not symmetrical. When an economy is overheated, there is necessarily a tendency for prices to rise unless prices are capped. On the other hand, under-capacity does not necessarily lead to lower prices, because many prices are rigid downward, especially wages, and because expectations of price increases are sufficient to raise them, even if the economy is in recession.

A policy or effect is procyclical when it tends to destabilize the economy, to amplify overheating and inflation during an upward phase, and to aggravate the recession, unemployment and perhaps deflation in a down phase. A policy or effect is counter-cyclical if it tends to stabilize the economy, to cool down overheating and to reduce the inflation that accompanies it, or to counter recession, unemployment and deflation.

Price stability

Inflation and purchasing power

When the currency was convertible into gold, the price of gold, $ 35 per ounce, was a tautological price. Prices for all other goods were valued in dollars, but a dollar was equivalent, by definition, to 1/35 of an ounce of gold. To say that the price of gold was $ 35 an ounce was simply to say that an ounce of gold is an ounce of gold. And the price of gold was considered constant. But this constancy was an illusion. Gold had a value that could vary over time. How to measure this value?

The value of gold could vary because the same amount of gold could be used to buy varying amounts of other goods. If all the prices of the other goods increased, it would automatically decrease the value of the gold, and conversely if all the prices decreased. To measure changes in the value of gold, it was necessary to measure changes in all prices in the economy. The same is true in an economy where money is not convertible into gold. To measure changes in the value of money, we must measure the variations of all prices.

The value of money is defined by its purchasing power (Fisher 1911). It is calculated by reasoning on a defined basket of economic goods. The variations in the price of this basket then represent the variations in purchasing power. A consumer price index is thus calculated on a basket that contains all the goods and services consumed during the year. The inflation rate is then defined as the annual rate of change of this index.

Changing lifestyles mean that the rate of inflation over a long period (decades) is impossible to calculate accurately, nor can it be interpreted without ambiguity. Each year new goods and services appear and old ones disappear or are modified. This poses a problem for the calculation of inflation, because the basket of goods and services varies over time. The statisticians solve this difficulty by reasoning on equivalent goods, but the result of the calculation depends on their judgment on these equivalences between the old and the new. As from one year to another, the consumption basket varies little, the uncertainty on the resulting annual rate of inflation is low. But if we compare very different baskets, because consumption varies a lot in one or more decades, the uncertainty is much higher.

A very high rate of inflation, called hyperinflation, is obviously harmful to the economy. If employees have to be paid every day and spend their salary immediately, because purchasing power drops sharply from one day to the next, the economy can not function normally. But a high inflation rate, 20% annually, for example, is not so obviously harmful. The monthly rate is $1.20^{1/12}-1\approx 0.015=1.5\%$ is low enough for employees to spend their monthly salaries normally. What are the costs of such moderately high inflation? If inflation could be exactly and certainly anticipated, the costs could be relatively low. Agents would have to regularly change their prices (the waltz of labels, menus ...) and take into account inflation in all their economic calculations. But inflation can not be exactly and certainly anticipated. This makes economic activity more risky. Lenders are hurt by an unanticipated rise in inflation, and borrowers by a decline. It is this increase in risk that is probably the highest cost that high but moderate inflation charges. By targeting a very low rate of inflation, central banks make economic activity much less risky (and eliminate the cost of the waltz of labels).

The danger of deflation

Deflation occurs when the inflation rate is negative. It must be considered as an economic disaster, because it encourages the abandonment of our projects and because it drives debtors into more debt and pushes them into bankruptcy. When deflation sets in, everything happens as if sleeping money brought in an income, because its value increases over time, and all the agents are encouraged to delay their purchases, because they anticipate a fall in prices. Debts increase in value because future repayments are fixed while their value increases.

That wages, and other prices, are rigid downward can be seen as an economic benefit, because it reduces the risk of deflation (but it does not cancel it).

Price stability could be defined by an inflation rate of zero, but that is not desirable because the economy is constantly on the brink of a precipice. Deflation tends to feed itself. It is pushing the recession and thus weakening demand, pushing prices down, and therefore further deflation. Even a weak initial deflation can have catastrophic effects. Since the inflation rate remains variable, even if inflation is properly controlled, it is better for it to remain above zero in order to reduce the risk of deflation. An average rate of 2% for example (this is the inflation target usually adopted by central banks) gives the monetary authorities leeway to react in the event of deflationary tendencies. If the target was 0%, deflation could set in before the monetary authorities have time to react (Bernanke, Laubach, Mishkin, Posen, 1999).

The underemployment of the domestic wealth

Wealth is the difference between assets, all owned goods, including options, and liabilities, all debts. Ambivalent commitments are counted as assets or liabilities depending on whether their estimated value is positive or negative.

Domestic wealth is the sum of the wealth of all resident agents in an economy. Since all agents' liabilities are assets of other agents, one can ignore all domestic debts when one counts the domestic wealth. One can also ignore the options sold on the markets, because they are assets for the buyer offset by liabilities for the seller. Assets in bank accounts are bank debts to their customers and can be ignored. The bank notes in circulation are in the assets of their holders and the liabilities of the central bank. This is purely accounting logic, because the central bank owes nothing to anyone when it puts its notes into circulation, since they are no longer convertible into gold. But this makes it possible to count the domestic wealth correctly. If the central bank puts new notes into circulation, it does not directly create wealth. Banknotes are just paper, not real wealth. Domestic wealth is thus made up of all the durable goods preserved in an economy and all the options of all the agents (except those sold on the markets).

The value of durable goods depends on the projects to which they are assigned. When a good remains unused, it is assigned to a default project, which pays nothing, and its value is no more than the option to use it again.

Domestic wealth can therefore be considered as the portfolio of all the projects in which agents are committed and all their options. This portfolio is shared between all agents. Its management results from the management by agents of their own portfolios of projects.

The projects of an economy can require more or less labor. There is unemployment if there are too few projects or if they do not require enough labor. The problem of unemployment is therefore a problem of managing the portfolio of all the projects of an economy.

The central bank can indirectly increase or decrease the domestic wealth, when it creates money or when it refuses to create it, because it can help the agents to realize profitable projects, or on the contrary prevent them.

Monetary power

The intervention of the banking system on the loanable funds market

The existence of banks and a central bank, and the resulting monetary creation, make the market for loanable funds much more than a market for private lenders and private borrowers.

If a private lender decides to stop lending money, he has to keep it at home in the form of cash, because if he deposits it at the bank, it is like lending it, and it increases the capacity of the bank to grant loans. By depositing one's funds in the bank, there is little reduction in the loanable funds available, because the bank must keep only a small fraction of required reserves. The supply of loanable funds can therefore only decrease if the agents keep more of their money in cash, or if the banks refuse to lend, and keep excess reserves, or if the central bank withdraws money from circulation.

Illiquidity, insolvency and loan of last resort

If there is a bank run, that is, many depositors want to recover their money at the same time, a bank can be illiquid while being solvent. To say that it is illiquid means here that it does not have enough cash to fulfill its obligations. It must therefore declare bankruptcy. But it is solvent because its assets, and in particular all the loans it has made, are greater than its liabilities. It would be enough that one lends it the necessary cash so that it escapes the bankruptcy and that it pays later all its creditors. In order to protect the banking system against this risk of illiquidity, central banks are now accepting the role of lender of last resort (Bagehot 1873, Bernanke 2015). If a bank is solvent, and no one wants to lend to it, the central bank is committed to providing the necessary funds at a rate slightly above the market rate. Of course, the central bank must judge that the bank is solvent, and it requires guarantees.

Countercyclical monetary policy and the control of inflation

A central bank always has the means to cool an overheated economy and thus to fight against inflation, because it is in a dominant position in the market for loanable funds. If it refuses to lend the money it can create, it raises interest rates and discourages agents from engaging in some of their projects. It is said that it removes the bowl of punch while the party has just begun - overheating offers generous opportunities for speculators.

Central banks do not want to abuse rate hikes, because high rates are bad for economic development. In times of prosperity, they seek to keep rates low, and only raise them to fight against inflation. If all goes well, the economy benefits from both low and controlled inflation and low interest rates. From this point of view, we can rejoice that the central banks have acquired the power that is theirs today.

If the economy suffers from underemployment, the power of a central bank is much more limited. It is encouraged to lower interest rates, if only to counter deflationary trends, to keep inflation at a level sufficient to move away from the risk of deflation. But there is a floor, it can not, or hardly, lower rates below zero, and even rates too close to zero can be considered dangerous, because they hurt the agents that need to place their money without risk and in a liquid way, like insurance companies for example, and because they incite to take thoughtless risks.

When an economy is under-capacity, monetary policy may not be enough to revive activity. Central banks can incentivize agents to borrow, lowering rates, but they can not force them to engage in projects. In times of pessimism, even very low rates are not enough to get involved, because one always have to pay back the principal, and if the project is at risk of being lost, one is afraid of staying in debt.

If the recession is so severe that there is a danger of deflation, and if the central bank has already lowered interest rates to the lowest, it still has the power to give the money it creates to revive the activity and fight against deflation.

Permanent inflation and monetary policy with clubs

When inflation has settled, it is very difficult to reduce, because agents anticipate future inflation, and because their expectations are self-fulfilling. If the agents think that the prices will increase, they increase their prices, not to be damaged, and by increasing their prices, they cause the inflation that they anticipated.

Central banks still have the means to fight against rising prices. It is enough to cause a rise in interest rates. But this has a cost. If a permanent inflation has settled, it is necessary to convince the agents that they must change their expectations. To make promises to them is not enough, they need proofs that inflation can decrease. By raising rates to a sufficient level, a central bank can cause an economic recession. The increase in unemployment and the economic recession are leading agents to change their expectations, and after a few months or years of suffering, expectations having changed, the central bank can lower interest rates without restarting inflation. This monetary policy, which consists of raising interest rates to trigger a recession and thereby reducing inflation expectations and hence future inflation, is sometimes called the Volcker method, after the name of the director of the US Federal Reserve, which applied it, in the early 1980s. It could be called the monetary policy with clubs: give blows of the club until people no longer want to buy. When they are stunned enough to stop buying, the upward pressure on prices is eliminated and inflation is under control.

Fiscal power

Public investment

The state is not just an economic agent. To consider it solely as a for-profit enterprise would obviously lead to ignoring its most important functions. But it is still a very powerful economic agent.

Complementarity between the public authorities and the market economy is one of the keys to economic development. Citizens, as owners, workers and consumers, need a powerful state that protects their interests and empowers them to engage in profitable projects. In return, the state needs a prosperous economy to raise taxes.

As soon as they promote economic development, public spending can be considered an investment. If it leads to an increase in activity, it brings at the same time additional tax receipts. The initial expenses are costs and the additional taxes revenues. If the revenues exceed the costs, the investment is profitable.

The crowding out effect

In periods of full employment or overheating, an increase in public expenditure can not generally lead to an increase in economic activity because it is already at its peak. If the state spends more, it will use means that would otherwise have been used in private projects. Private investment is thus squeezed out by public investment. This crowding out effect must not prevent the state from spending when its objectives have priority, because the general interest goes before private profits, but it obviously affects the profitability of public investments.

The crowding out effect is sometimes stated in financial terms: the extra public spending is financed by indebtedness, and the money lent to the state is squeezed out of private projects that could otherwise have been financed. But when one thinks in this way, one ignores that loanable funds are not reduced to private funds, that the central bank and other banks create money by lending it.

Countercyclical fiscal policy

In times of underemployment, there is no crowding out effect (except perhaps in some sectors that have remained at full employment) and the profitability of public investment is increased accordingly.

During a recession, individuals are not generally encouraged to invest. They all have an interest in everyone investing more to get out of the crisis, but none is encouraged to do so. If an agent engages in expensive investments, it will increase the activity, and its suppliers will benefit, but not him, because an agent alone can not restart the activity of all agents, and if the expected recovery does not occur, he will have to bear losses. On the other hand, the state has an interest in investing, because it will automatically benefit from any additional activity with additional tax revenue, and if it is sufficiently powerful it can cause the expected recovery.

A recession is very expensive for the state, because tax revenues are greatly reduced. But it is precisely at these moments when it earns the least, that the state must spend the most and go into debt. In times of full employment, the state has much less interest in investing, and it is better that they use their additional tax revenues to repay their debts. Paradoxically the state has to spend less when it earns more and it has to spend more when it earns less.

Imagine a country with years of famine and years of plenty. In times of plenty, owners are encouraged to make large stocks. In times of famine, the state goes into debt to buy and gradually distribute stocks. When abundance comes back, the state pays off its debts and lets the owners rebuild their stocks. It is a modern version of the story of Joseph and Pharaoh. Countercyclical fiscal policy is similar.

Is fiscal austerity necessary to fight inflation?

In times of full employment or overheating, any increase in public spending, in the absence of additional taxes, and therefore through indebtedness, puts upward pressure on prices. To control inflation, the central bank must tighten the credit lines and thus raise interest rates. In such circumstances, fiscal laxity increases the costs of indebtedness and the difficulty of controlling inflation.

If, in addition, permanent inflation has taken hold, the increase in public spending is not likely to reduce inflation expectations, rather the opposite. Fiscal austerity announced and realized is then a way to change expectations and to support a monetary policy of reducing inflation.

On the other hand, in times of underemployment, and if inflation is already under control, the increase in public spending should not have an inflationary effect. In such circumstances, the problem is not inflation, but rather deflation, and any increase in investment is welcome, provided that it is for good projects.

Financing the economy through monetary creation

Financial work is essentially about finding and evaluating projects. If we have found a good project, we have to finance it. When agents were obsessed with gold or other precious metals, when they believed that nothing could have value if it was not convertible into gold, financiers were constrained in their willingness to finance. It was not enough to find good projects, it was also necessary to find gold, or to have some in its trunk. Now this bridle is cut. The financiers can work normally, they only have to evaluate projects, they no longer have to worry about gold reserves. The most important thing is to find good projects. Once they are found, it only remains to create the money that finances them. The only real financial problem is the design and evaluation of projects.

To benefit from prosperity, agents must design and implement good projects. They often have an interest in associating in various ways because they can create value by composing their projects. The only thing that really matters is the choice of these projects, not the funds available, because the money that finances them can always be created. Money is never a problem unless there are too many projects going on simultaneously. This is why agents need to set up institutions to regulate financial resources, in order to avoid overheating and inflation, or recession, underemployment and deflation.

When the economy is not in full employment, or if it suffers shortages, or inflation, or deflation, it must be seen as an error in the composition of all its projects. Domestic wealth is collectively poorly managed. Agents have an interest in agreeing and organizing to better benefit from the wealth they share.

Because it is the largest investor, the state has additional means for global investment to be responsive to the needs and capabilities of the economy. The state must always be ready to invest in profitable projects, in order to revive the economy when private projects do not seize the available opportunities enough, but it is also necessary that it gives up its ardours and that it puts some of its projects on standby, when the economy is at full employment, so as not to hinder the development of the private economy.

Are low rates responsible for financial crises?

Low rates promote economic development because they encourage investment in the real economy. But they also increase financial profits. With money borrowed at low rates one can increase financial profits through leverage. Easy money encourages speculation and makes possible to earn very high profits as long as there is no crisis. But when the crisis occurs leverage goes in the opposite direction. It increases losses instead of increasing profits.

Borrowing at low rates and lending at higher rates is the main source of income for banks. When they invest in the real economy by lending money directly to non-financial businesses they contribute to economic development. But if they invest in the financial markets by speculating on the rise in stock prices, they feed financial bubbles. This shows the need to discipline banks and financial firms (Admati & Hellwig 2013). Low rates should be used for the development of the real economy not for financial speculation.

Probabilities and evaluation of risks

Probabilities in economics

To give an objective meaning to a probability, it is necessary and sufficient that it be measurable. For that it is necessary and sufficient that an experiment be reproducible. By reproducing the same experiment many times, the probabilities of its results are measured from their frequencies of appearance. But in economics experiments are never reproducible. Each economic event is unique. All agents are always different, and the conditions in which they are placed are never exactly the same. Therefore, there can be no objective meaning for economic probabilities. But probabilistic models are of fundamental importance for economics. Where is the error? Are probabilistic models in economics always false because the probabilities in economics have no objective significance?

The uncertainty of games of chance can be considered purely probabilistic, provided they are not faked, because they are generally reproducible experiments. Economic uncertainty is not strictly probabilistic, but economic decision-making about risk is very similar to gamblers' decision-making. When we face a risk, we reason as if we made a bet with reality. When we attribute probabilities to expected economic events, they can never be true and accurate, but they can still be estimates that make better decisions. An insurance company, for example, must estimate probabilities of death. It does so from the statistics of recorded deaths. If it did not do so, it would not be able to anticipate its payment obligations and it would run the risk of bankruptcy.

Probabilistic models in economics can never be strictly true, but they may be sufficiently similar to reality to merit comparison. All that is needed is a relevant analogy to justify a model. And models can be useful even when they are very wrong, because the gap between the model and reality can be very significant.

The measurement of risk

Risk is commonly referred to as a measurable magnitude: the risk is zero in a situation of certainty and increases when uncertainty increases. The ratio of the standard deviation to the average profit of a project is often a good measure of its risk because it increases with uncertainty and because it makes it possible to compare projects of different average profits. But a single magnitude can never suffice to account for the diversity of risks. Standard deviation measures the spread of a probability distribution, but the shape of this distribution is also important for assessing risk. It determines the distribution of probabilities between very high and moderate gains and moderate and very high losses. The standard deviation measures spread, but it does not say anything about the diversity of forms. It is therefore useful especially for comparing distributions of the same form. If for example two distributions are normal (gaussian), the riskiest is clearly the one with the greatest standard deviation. But if distributions have different forms, comparing their standard deviations is not necessarily a good way to compare their risks. In addition, the standard deviation is not the only quantity that measures the spread of a distribution. The mean of the absolute values ​​of the deviations from the mean is sometimes a more natural estimate of the spread than the square root of the mean of the squares of the deviations from the mean, which places great importance on the large differences to the detriment of the small ones.

When we measure the spread of a distribution, we make an average both on the chances of winning, when the return is above its average, and on the chances of loss, when it is lower. The risks of loss are particularly important when one questions the viability of a company or project. We are particularly attentive to the left side of the probability distribution, while the spread of the right side, the chances of winning, does not inform us.

Value at risk, is an indicator of a risk of loss widely used to study the viability of companies.

The p value at risk of a random variable $X$ is by definition the number $V_{X}^{R}(p)$ such that $Pr(X\leq -V_{X}^{R}(p))=p$. The positive values of $X$ are profits, the negative ones are losses.

$Pr(X\leq x)=F_{X}(x)$ is the probability that $X\leq x$. $F_{X}$ is the cumulative distribution function of the random variable $X$. $F_{X}(x)=\int _{-\infty }^{x}f_{X}(y)dy$ where $f_{X}$ is the probability density of $X$.

$V_{X}^{R}(p)=-F_{X}^{-1}(p)$

The value at risk makes it possible to estimate the loss that may be incurred. We know that with a probability $1-p$, the loss will not be greater than $V_{X}^{R}(p)$. $p$ is usually a small number, $0.05=5\%$ or less. Losses that have a probability lower than $p$ can be considered extraordinary losses. The value at risk is used to estimate the maximum amount of ordinary losses, but it does not say anything about extraordinary losses. If losses have a probability lower than $p$, they have no influence on $V_{X}^{R}(p)$. Two companies, one of which is exposed to gigantic but rare losses, and not the other, may have the same value at risk.

Risk indicators such as standard deviation, or the mean of absolute values ​​of deviations from the mean, or the value at risk, can never provide complete information on the risks to face. It is necessary to know the probability density of the expected profits and losses, or in an equivalent way their cumulative distribution function, to be completely informed on the risk.

The price of risk

In general, it is believed that risk decreases the value of a project. The risks of low profit and especially loss are generally dissuasive, even if the chances of a high profit make that the average profit is substantial. Of two projects that have the same average profit, the riskiest is usually the least valuable. This decrease in value is a risk premium. In order for a risky project to be attractive, it must generally yield more than the risk-free interest rate, it must have a sufficient average profit to offset risk taking. The risk premium can be calculated from the surplus profit rate required to offset the risk, ie the difference between the required rate of profit and the risk-free interest rate.

Risk aversion is not universal. There are many exceptions. The risk may not diminish the value of a project but instead be sought for itself. In most games of chance, for example, one always loses on average, the average profits are negative, but even very small chances of a large gain are enough to convince the gamblers to participate. The value they attribute to the sought risk offsets their average loss.

Since economic agents are not equal in the face of risks, they evaluate them very differently. This promotes risk trading. The same risk that is very dissuasive for one agent may be very little for another. The second then has an interest in selling the first one a guarantee against risk.

Discounting uncertain projects

A risk-adjusted discount rate is sometimes defined by adding to the risk-free discount rate the surplus profit rate which measures the risk premium. But to calculate the net present value of an uncertain project, this definition makes sense only if there is only one initial cost and one final revenue. If there are uncertain losses, it is obviously foolish to depreciate them with a higher discount rate than the risk-free rate. And when uncertain payments are staggered over time there is usually no reason to discount them with the same risk-adjusted discount rate because they may have very different uncertainties.

The discount rate is the risk-free interest rate. It is estimated from the investment rates considered safe but this does not prevent it from being applied to the evaluation of uncertain projects. It is an exchange rate between the money paid today and the money paid later, a time exchange rate. That a project is certain and risk-free or very uncertain and very risky should not change anything in the way of discounting the payments. To evaluate risky projects, all revenues and costs must be discounted at the risk-free investment rate, because this is the best way to add payments staggered over time. Whether or not these payments were predicted with certainty or not does not change the case. The discount rate depends on the economic reality at a given moment, not the projects to which it is applied. It is the same for all projects facing the same reality, whether they are risky or not.

If a project is risky, and if the probabilities of revenue and costs are known, we can calculate with the true discount rate, not a risk-adjusted rate, an average discounted value of revenues, cash advances and other costs, and therefore an average profit, an average profit rate and an average net present value, but this is not the net present value of the project, because it does not take into account the risk premium.

The net present value of an uncertain project is its average net present value minus the risk premium. The risk premium is the present value of the average surplus profit required to offset the risk. It is the present value of the difference between the average profit required and the profit that would be obtained if one placed one's capital at the risk-free interest rate. Since the average net present value is the present value of the average surplus profit, the net present value of an uncertain project is the present value of the difference between the average surplus profit and the surplus profit required to offset the risk, hence the present value of the difference between the average profit and the profit required to offset the risk.

Statistical independence, anticorrelation and risk reduction

There are two techniques to reduce risk. One makes use of the statistical independence of economic events, the other their anticorrelations.

• When many projects are statistically independent, their sum is much less risky than each individual project. A simple example suffices to be convinced: if $n$ statistically independent projects have the same expected average profit $\pi$ with the same standard deviation $\sigma$, $\sigma /\pi$ measures the risk of an individual project. Their sum has a mean expected profit $n\pi$ with a standard deviation $\sigma {\sqrt {n}}$, because the variance of a sum of independent variables is the sum of their variances. The risk of the sum of all projects is therefore measured by ${\frac {\sigma {\sqrt {n}}}{n\pi }}={\frac {\sigma }{\pi {\sqrt {n}}}}$ which approaches zero if $n$ is very large. In general, projects do not have the same expected average profits or the same risks, but if they are statistically independent, the risk of their sum is generally much lower than the risk of each of them taken in isolation, and approaches zero when the number of projects is very large.

• Two projects are anticorrelated, or negatively correlated, when the success of one is correlated with the failure of the other. The covariance of their profits is negative. For example, the price rise of a stock is correlated with the decline in the value of a put option on that stock. Positive and negative correlations between the values ​​of assets and corresponding options result in risk-free portfolios that form the basis of option pricing methodology of Merton, Black and Scholes presented later. Positive correlations can be used by betting down, which can be done by selling short, buying put options or selling calls. Selling short is selling shares that have been borrowed by committing to buy them later to return them. Risk hedging strategies generally involve betting at the same time on the increase of certain assets and the decline of other assets. But assets are often positively correlated, their values ​​often increase or decrease at the same time. The rise of some is therefore often anticorrelated with the decline of others. Hedging strategies therefore reduce the risks associated with changes in the value of assets. One tries to win on both counts, both when the market is rising and when it is falling.

The casino economy

A project is usually affected by many uncertain events that can increase or decrease its profit. Two types of events must be distinguished, those that affect only the current profit of the project, unforeseen costs or revenues that momentarily affect the profit of a project but have no influence on its future profits, and those that permanently affect the project's ability to make a profit. The effects of events of the first type must be added to obtain their cumulative effect, while those of the second type must be multiplied.

Suppose that a project is affected only by events of the second type: many random events $i$ can influence its value $V$ during a time interval $dt$. Each event $i$ if it occurs multiplies $V$ by a factor $R_{i}>1$ if it promotes the success of the project and $<1$ if it is unfavorable. We assume that the events $i$ have probabilities $p_{i}=\rho _{i}dt$ and that they are independent. The variation of $\ln V$ is then a sum of random variations $\ln R_{i}$. According to the central limit theorem, if these variations are very numerous, small compared to their sum and independent, the distribution of probabilities of this sum is a normal law. Let $\mu dt$ be its average and $\sigma ^{2}dt$ its variance. If the project environment is constant, $\mu$ and $\sigma$ do not depend on $t$. On a time $T$, the variation of $\ln V$ then follows a normal distribution of mean $\mu T$ and variance $\sigma ^{2}T$.

Let $V_{0}$ be the initial value of the project. Since $\ln V$ follows a normal distribution of mean $\ln V_{0}+\mu T$ and of variance $\sigma ^{2}T$, $V$ follows a log-normal law whose probability density is:

$f(V)={\frac {1}{V\sigma {\sqrt {t}}{\sqrt {2\pi }}}}e^{-(\ln {\frac {V}{V_{0}}}-\mu T)^{2}/2T\sigma ^{2}}$

The average value of $V$ is $V_{0}e^{(\mu +\sigma ^{2}/2)T}$ and its standard deviation is $V_{0}e^{(\mu +\sigma ^{2}/2)T}{\sqrt {e^{\sigma ^{2}T}-1}}$. If we measure the risk by the ratio of the standard deviation on the mean, we obtain ${\sqrt {e^{\sigma ^{2}T}-1}}$ which increases very quickly with $T$.

Suppose now that a project is affected only by events of the first type. Many random events $i$ can influence the current profit $d\Pi$ during a time interval $dt$. Each event $i$ if it occurs increases $d\Pi$ by an amount $d\Pi _{i}>0$ if it is a receipt and $<0$ if it is a cost. We assume that the events $i$ have probabilities $p_{i}=\ rho_{i}dt$ and that they are independent. The current profit $d\Pi$ during $dt$ is then a sum of random variations $d\Pi _{i}$. According to the central limit theorem, if these variations are very numerous, small compared to their sum and independent, the distribution of probabilities of this sum $d\Pi =\sum _{i}d\Pi _{i}$ is a normal law. Let $\mu dt$ be its mean and $\sigma ^{2}dt$ its variance. If the project environment is constant, $\mu$ and $\sigma$ do not depend on $t$. On a duration $T$, the profit $\Pi$ follows a normal distribution with mean $\mu T$ and variance $\sigma ^{2}T$. The ratio of the standard deviation on the average is ${\frac {\sigma }{\mu {\sqrt {T}}}}$ and approaches zero when $T$ is large.

Over long periods of time, the risk caused by events of the second type tends to dominate that of events of the first type. To evaluate the risks we can therefore often ignore events of the first type, which leads us to retain a log-normal distribution. But this rule is not universal. If the risks associated with events of the first type are important, they should not be ignored, especially if the time considered is short.

All the assumptions that justify the normal distributions of $\Pi$ or log-normal of $V$ are a priori very doubtful. The $d\Pi _{i}$ and the $\ln R_{i}$ are not necessarily small compared to their sum because a single event can have a great influence on the success or failure of a project. Nor are they necessarily independent, because failure can lead to more failures or more success, or failure can precede success by resilience, or success can lead to failure, because greatness sometimes precedes the fall. Moreover, there is generally no reason to assume that the probabilities $p_{i}=\rho _{i}dt$ are constant over time, because economic events depend on circumstances that vary. In fact, there is usually no reason that these probabilities can be properly defined because economic events are never exactly reproducible. The real economy can be very different from the casino economy. This is just a mathematical model that can help us understand reality but can also mislead us.

For a log-normal distribution, the ratio ${\sqrt {e^{\sigma ^{2}T}-1}}$ of the standard deviation on the mean is not necessarily a good indicator of the risk , especially when the distribution is very flattened, for values ​​of $\sigma ^{2}T>1$. In general, one prefers $\sigma {\sqrt {T}}$, which is also an indicator of the dispersion of expected values ​​around the mean. $\sigma {\sqrt {T}}$ is the standard deviation of the logarithmic yield. It is called volatility. If time is measured in years, $\sigma$ is the annual volatility.

In this section we have reasoned on the value of a project but we could have kept the same reasoning on the prices on the stock market. In this case the log-normal distribution is a fairly good approximation, but it slightly underestimates the probabilities of large deviations from the mean (Luenberger 1997).

To reason more easily, we will make several simplifying hypotheses in the following:

• Stock price changes are always represented by log-normal distributions whose parameters $\mu$ and $\sigma$ parameters do not vary.

• We reason on stocks that do not pay dividends. The profits are therefore only the capital gains. This amounts to assuming that the paid dividends are systematically reinvested.

• Inflation is ignored. This means that we reason on the real values ​​and not on the nominal ones.

• Transaction costs and taxes are ignored.

• We assume that there is a single risk-free interest rate. We therefore ignore its term structure. Above all, we ignore that even the "risk free" interest rate is risky because even if there is no risk of default by the borrower, there is always a risk of inflation.

These simplifying hypotheses make it possible to concentrate the reasoning on some of the most important points, but if we want to apply the conclusions to the real world, we must of course be aware of the complications that the theory ignores.

Arithmetic, logarithmic and average yields

Arithmetic yield, or simply yield, is the rate of profit of an asset. If $V_{i}$ is its initial value and $V_{f}$ its final value, the profit is $V_{f}-V_{i}$ and the profit rate is $r={\frac {V_{f}-V_{i}}{V_{i}}}$ therefore $V_{f}=V_{i}(1+r)$

The logarithmic yield is $R=\ln({\frac {V_{f}}{V_{i}}})=\ln(1+r)$ therefore $V_{f}=V_{i}e^{R}$

When $r\ll 1$, $R\approx r$

The logarithmic yield is more convenient than the arithmetic yield when one has to compose yields for successive periods:

$r_{12}=(1+r_{1})(1+r_{2})-1=r_{1}+r_{2}+r_{1}r_{2}$

while

$R_{12}=\ln(e^{R_{1}}e^{R_{2}})=R_{1}+R_{2}$

Arithmetic yield is more convenient than logarithmic yield when calculating a portfolio yield:

$r_{pf}={\frac {\sum _{n=1}^{N}V_{ni}r_{n}}{\sum _{n=1}^{N}V_{ni}}}=\sum _{n=1}^{N}\omega _{n}r_{n}$

where the $V_{ni}$ are the initial values ​​of the $N$ assets that make up the portfolio and $r_{n}$ their respective yields. $\omega _{n}$ is the weight of the asset $n$ in the portfolio.

$R_{pf}=\ln(1+r_{pf})\neq \sum _{n=1}^{N}\omega _{n}R_{n}$

When $R$ is a random variable, the average of logarithmic yield is not the logarithmic yield calculated from the average profit, that is why the logarithmic average yield $R_{m}$ is defined not by the average of logarithmic yield but by the logarithm of the ratio between the average final value and the initial value:

$R_{m}=\ln({\frac {\langle V_{f}\rangle }{V_{i}}})\neq \langle R\rangle$

If ${\frac {V_{f}}{V_{i}}}$ has a log-normal distribution with parameters $\mu$ and $\sigma$, the logarithmic average yield is $\mu +\sigma ^{2}/2$ because $e^{\mu +\sigma ^{2}/2}=\langle {\frac {V_{f}}{V_{i}}}\rangle ={\frac {\langle V_{f}\rangle }{V_{i}}}$ but the average of logarithmic yield is $\mu$, because $\ln({\frac {V_{f}}{V_{i}}})$ has a normal distribution with mean $\mu$. The standard deviation of the logarithmic yield is $\sigma$. It measures the dispersion of the logarithmic yield around its mean $\mu$ not around the logarithmic average yield $\mu +\sigma ^{2}/2$ but the difference is often quite small. For a risky asset $\mu =0.1$ and $\sigma =0.15$ are typical values ​​for the annual logarithmic yield, and $\mu +\sigma ^{2}/2\approx 0.11$

The evolution of the expected value of an asset

It is assumed that the changes in the price of an asset are random and are described by the lognormal distribution presented above.

If $V_{0}$ is the present value of the asset, its future value $V_{T}$ on date $T$ has a log-normal probability density :

$f(V)={\frac {1}{V\sigma {\sqrt {T}}{\sqrt {2\pi }}}}e^{-(\ln V-\ln V_{0}-\mu T)^{2}/2T\sigma ^{2}}$

The average, or expected value, of future prices is

$\langle V_{T}\rangle =e^{\ln V_{0}+\mu T+\sigma ^{2}T/2}=V_{0}e^{(\mu +\sigma ^{2}/2)T}$

The Capital Asset Pricing Model

The Capital Asset Pricing Model (CAPM) is a simplified model that shows how idealized financial markets could evaluate risk premiums. The main lesson of this model is that a risk premium does not depend on the standard deviation of the value of an asset but on its covariance with the general economic conditions. This measures a non-eliminable risk while the standard deviation includes a risk eliminable by diversification.

If all economic projects were statistically independent, we could always eliminate their risks through diversification. A mutual fund could be an almost risk-free portfolio from a very large number of risky assets. Since the risk is almost eliminated, its cost would be negligible and the risk premiums would be almost zero. All risky projects would then be attractive as soon as their yield is at least the risk-free interest rate. But economic projects are not usually independent. On the contrary, they are generally dependent on the same economic conditions, because all the agents are sensitive to general prosperity or its absence, so they are generally correlated with each other. It is not always possible to eliminate risk by diversification. Risky projects whose risk can not be eliminated must have a higher average yield. The Capital Asset Pricing Model measures non-eliminable risks and the risk premiums they generate. It is based on some simplifying assumptions:

• Agents always measure the performance of a portfolio based on its average yield, and the risk based on the standard deviation of that yield. They are all equally informed about average asset yields, their standard deviations and their covariances.

• They can lend and borrow at the risk free rate freely and without cost.

• They can always build a diversified portfolio. All weighted sums of assets are potential portfolios.

The half-line of optimal portfolios

Let $r_{m}$ be a yield (arithmetic, not logarithmic) higher than the risk-free interest rate $r_{f}$. Let $\Omega (r_{m})$ be the set of all risky portfolios that have the same average yield. Let $\sigma _{min}(r_{m})$ be the standard deviation of the yield of an optimal portfolio of $\Omega (r_{m})$: for any portfolio in $\Omega (r_{m})$ the standard deviation of its yield is greater than or equal to $\sigma _{min}(r_{m})$. Then the half-line of optimal portfolios in the half-plane $\sigma r$ is given by the equation:

$r=r_{f}+(r_{m}-r_{f}){\frac {\sigma }{\sigma _{min}(r_{m})}}$

All $(\sigma ,r)$ of possible portfolios are below this half-line. All points on this half-line are optimal $(\sigma ,r)$ of portfolios. They are the least risky for a given expected yield, and the most profitable for a given risk.

Proof: Let us first show that all the points of the half-line are possible portfolios. Consider a portfolio consisting of $\alpha$ at the risk-free rate and $(1-\alpha )$ of the optimal portfolio at the rate $r_{m}$. Its average yield is $\alpha r_{f}+(1-\alpha )r_{m}$, the standard deviation of this yield is $(1-\alpha )\sigma _{min}(r_{m})$, therefore it is on the half-line of the optimal portfolios. If $0\leq \alpha \leq 1$, $r$ is between $r_{f}$ and $r_{m}$. If $\alpha <0$, $r>r_{m}$. In this case, one borrows funds at the risk-free rate $r_{f}$ to invest them at the risky rate $r_{m}$, and increases the average yield by leverage.

Let $(\sigma ,r)$ be a point above the half-line of optimal portfolios and assume that it represents a possible portfolio. By the same argument as above, the half-line that starts from $(0,r_{f})$ and goes through $(\sigma ,r)$ would also represent possible portfolios. In particular $(\sigma {\frac {r_{m}-r_{f}}{r-r_{f}}},r_{m})$ would represent a possible portfolio. Since $(\sigma ,r)$ is above the half-line of optimal portfolios:

${\frac {r-r_{f}}{\sigma }}>{\frac {r_{m}-r_{f}}{\sigma _{min}(r_{m})}}$

hence

$\sigma _{min}(r_{m})>\sigma {\frac {r_{m}-r_{f}}{r-r_{f}}}$

But it is contrary to the definition of $\sigma _{min}(r_{m})$ which is the smallest of the standard deviations of the possible portfolios of the same yield $r_{m}$. So $(\sigma ,r)$ can not represent a possible portfolio if it is above the half-line of optimal portfolios.

It can be concluded that:

${\frac {\sigma _{min}(r_{m})}{\sigma _{min}(r_{M})}}={\frac {r_{m}-r_{f}}{r_{M}-r_{f}}}$

for all $r_{m}$ and $r_{M}$ greater than $r_{f}$, since $(\sigma _{min}(r_{m}),r_{m})$ and $(\sigma _{min}(r_{M}),r_{M})$ are both on the half-line of optimal portfolios.

The hypothesis of optimal markets

The market portfolio includes in principle all the assets, risky or not, that can be traded in an economy. It includes all the wealth that can make a profit, securities or real estate, as soon as they have a market price. Its average yield depends on the prosperity of the economy. Its standard deviation represents the general risk that all agents face, because it represents the risk of a general lack of prosperity. Because the market portfolio is the most diversified, it eliminates all the risks that can be eliminated through diversification. Very diversified portfolios, such as the SP500, are also eliminating risks that can be eliminated through diversification.

The hypothesis of optimal markets postulates that the market portfolio is on the half-line of optimal portfolios.

This hypothesis is obviously wrong, since agents are not always rational when they invest their money. But as a first approximation we can assume that very diversified portfolios like the SP500 are not very different from an optimal portfolio. We then place the half-line of optimal portfolios by estimating the average yield of the SP500, or another well-diversified and well-chosen portfolio, and its standard deviation.

The covariance with the general economic conditions

If we assume that the market portfolio is optimal and if $r_{M}$ is its average return, $\sigma _{min}(r_{M})$ is the standard deviation $\sigma _{M}$ of its yield.

The $\beta _{i}$ of an asset $i$ is by definition:

$\beta _{i}={\frac {Cov(x_{i},x_{M})}{\sigma _{M}^{2}}}$

where $x_{i}$ and $x_{M}$ are the random yields of the asset $i$ and the market portfolio respectively.

$\beta _{i}$ is proportional to the covariance with the yield of the market portfolio, so with general economic conditions.

We will show that

$\beta _{i}={\frac {r_{i}-r_{f}}{r_{M}-r_{f}}}$

where $r_{i}$ is the average yield of the asset $i$.

Consider a portfolio that contains a value $\alpha$ of the asset $i$ and $(1-\alpha )$ of an optimal portfolio with the same average return $r_{i}$. The yield of this portfolio is $r_{i}$. Let $v(\alpha )$ be the variance of its yield. Since $v(\alpha )$ is minimal at $\alpha =0$, ${\frac {dv}{d\alpha }}(0)=0$ .

Let $x_{i}$ be the random yield of the asset $i$ and $x_{opt}$ the random yield of the optimal portfolio with the same average return.

$r_{i}=E(x_{i})=E(x_{opt})$

${\frac {dv}{d\alpha }}={\frac {d}{d\alpha }}[\alpha ^{2}\sigma _{i}^{2}+2\alpha (1-\alpha )cov(x_{i},x_{opt})+(1-\alpha )^{2}\sigma _{opt}^{2}]$

$=2\alpha \sigma _{i}^{2}+2(1-2\alpha )cov(x_{i},x_{opt})+(2\alpha -2)\sigma _{opt}^{2}$

${\frac {dv}{d\alpha }}(0)=2Cov(x_{i},x_{opt})-2\sigma _{opt}^{2}=0$

hence

$Cov(x_{i},x_{opt})=\sigma _{opt}^{2}=\sigma _{min}(r_{i})^{2}$

The portfolio with yield $x_{opt}$ can consist of ${\frac {r_{i}-r_{f}}{r_{M}-r_{f}}}=\beta _{i}'$ of the market portfolio and ${\frac {r_{M}-r_{i}}{r_{M}-r_{f}}}=(1-\beta _{i}')$ of a risk-free asset. So

$Cov(x_{i},x_{opt})=\beta _{i}'Cov(x_{i},x_{M})$

Now

$Cov(x_{i},x_{opt})=\sigma _{min}(r_{i})^{2}=\beta _{i}'^{2}\sigma _{M}^{2}$

Therefore

$\beta _{i}'={\frac {Cov(x_{i},x_{M})}{\sigma _{M}^{2}}}=\beta _{i}$

$\beta$ measures the risk which cannot be eliminated by diversification

We define the random variable $\epsilon _{i}$ by

$x_{i}=r_{f}+\beta _{i}(x_{M}-r_{f})+\epsilon _{i}$

Then

$E(\epsilon _{i})=E(x_{i})-\beta _{i}E(x_{M}-r_{f})-r_{f}=0$

$Cov(\epsilon _{i},x_{M})=Cov(x_{i}-\beta _{i}x_{M}-(1-\beta _{i})r_{f},x_{M})=Cov(x_{i},x_{M})-\beta _{i}Cov(x_{M},x_{M})=Cov(x_{i},x_{M})-\beta _{i}\sigma _{M}^{2}=0$

$\sigma _{i}^{2}=\beta _{i}^{2}\sigma _{M}^{2}+\sigma _{\epsilon _{i}}^{2}$

The variance of $x_{i}$ is the sum of two terms. $\beta _{i}^{2}\sigma _{M}^{2}$ represents a risk that can not be eliminated by diversification. $\sigma _{\epsilon _{i}}^{2}$ represents the risk that can be eliminated by diversification, because it is the variance of a random variable that is not correlated with the market portfolio.

The risk premium depends only on the covariance with the general economic conditions

The risk premium of an asset $i$ is the present value of the surplus profit required to offset its risk. It is obtained from the required surplus profit rate $r_{i}-r_{f}$. Since $r_{i}-r_{f}=\beta _{i}(r_{M}-r_{f})$, the risk premium of an asset $i$ depends only on $\beta _{i}$ therefore only on the covariance with the general economic conditions.

If $\beta _{i}=0$, the risk premium of the asset $i$ is zero. It is possible that this asset is also very risky, with a very high $\sigma _{i}$, but this risk can be eliminated by diversification with other assets whose $\beta$ is zero. That is why the risk premium is zero.

When $\beta$ is negative

If $\beta _{i}$ is negative, $r_{i}=r_{f}+\beta _{i}(r_{M}-r_{f})<r_{f}$. Why agree to hold the risky asset $i$ when it exposes us on average to profits lower than the risk free profit or to losses?

The $\beta$ of a portfolio is the weighted average of the $\beta _{j}$ of the assets $j$ of which it is composed, because the covariance of a sum of random variables with the same variable is the sum of their covariances with that variable. An asset $i$ whose $\beta _{i}$ is negative decreases the $\beta$ of a portfolio in which it is included, so the risk non-eliminable by diversification. The decrease of the yield is offset by the decrease of the risk.

A put option is an example of an asset with negative $\beta$ if the $\beta$ of the underlying asset is positive, because a put option is negatively correlated with the underlying asset. The $\beta$ can even be negative enough for the average yield to be negative. Such an option is more expensive to buy than it yields on average. Why then buy put options? Because they reduce the risk of a portfolio that would be very risky without them.

The three kinds of risk

• If $\beta _{i}>0$, the risk of the asset $i$ is positively correlated with the market. It is a risk that can not be eliminated without reducing profits.
• If $\beta _{i}=0$, the risk of the asset $i$ is decorrelated from the market and can be eliminated by diversification without reducing profits.
• If $\beta _{i}<0$, the risk of the asset $i$ is negatively correlated with the market. Anticorrelations of $i$ with the market reduce the risk of a portfolio whose $\beta$ is positive while reducing its yield.

Optimizing a portfolio

The CAPM can be used even if the optimal markets hypothesis is abandoned because it provides a method for optimizing a portfolio:

Having initially chosen a sensible portfolio, the SP500 for example, we begin by estimating its average yield $r_{I}$ and its variance $\sigma _{I}^{2}$. For each asset $i$ that it does not contain, its average yield $r_{i}$ and its covariance $cov_{i}$ with the initial portfolio are then estimated. We then calculate the $\beta _{i}$ of this asset relative to the initial portfolio:

$\beta _{i}={\frac {cov_{i}}{\sigma _{I}^{2}}}$

If ${\frac {r_{i}-r_{f}}{r_{I}-r_{f}}}>\beta _{i}$ then it is valuable to incorporate the asset $i$ in the initial portfolio. If the new assets so chosen are well diversified, this method gives a better portfolio than the initial one. It can have both higher yield and lower risk. By iterating the process, one can hope to find an optimal portfolio and beat the market.

The CAPM was introduced by Jack Treynor (1961, 1962), William F. Sharpe (1964), John Lintner (1965) and Jan Mossin (1966) independently, building on the earlier work of Harry Markowitz on diversification and modern portfolio theory. (Wikipedia)

Expected values of yields are usually not measurable

To measure the expected value $\mu$ of a random variable $X$ with standard deviation $\sigma$, we must have at least $100({\frac {\sigma }{\mu }})^{2}$ independent measurements.

Proof: The average $\mu _{n}$ measured with $n$ independent measurements is itself a random variable:

$\mu _{n}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}$ where the $X_{i}$ are independent random variables with the same law as $X$.

The expected value of $\mu _{n}$ is obviously $\mu$ and its standard deviation is

$\sigma _{n}={\frac {\sigma }{\sqrt {n}}}$

For the measurement to be meaningful, $\sigma _{n}$ must be small in front of $\mu _{n}$. With ${\frac {\sigma _{n}}{\mu _{n}}}<{\frac {1}{10}}$, we can hope to have a correct estimate of $\mu$ but this remains a very imprecise measurement, because there is a non-negligible probability, approximately 0.04, that $\mu _{n}$ deviates from $\mu$ by more than 20%.

${\frac {\sigma _{n}}{\mu }}<{\frac {1}{10}}$ requires $n>100({\frac {\sigma }{\mu }})^{2}$

The annual yields of stocks are generally between 6% and 30% and sometimes lower. 12% is a typical value. The standard deviations of these yields are generally between 10% and 60%. 15% is a typical value. So $\sigma /\mu \approx 1$ or higher. It would take more than a century of annual yield measurements to get a rough estimate of its expected value. But we do not have a century, only a few years, because there is no reason for the expected value of a yield to remain constant for more than a few years. If we measure the monthly return, we increase the number of measurements by a factor of 12 but $(\sigma /\mu )^{2}$ also increases by a factor of 12. So the conclusion is not changed. In general, expected values of yields are not measurable (Luenberger 1998).

The measured average yields are not estimates of the theoretical average yields, unless the investments are low risk, because then $\sigma /\mu$ is small in front of 1. This implies that a probabilistic model of a variation of yield can not be directly confronted with reality, since one can not measure its most important parameter, the expected value of the yield. This is not surprising since, in general, theoretical probabilities have no objective significance in economics, as economic events are not reproducible.

Since probabilistic models of yield variations can not be directly confronted with reality, one is tempted to conclude that they have no scientific value, but it is an exaggerated conclusion, because these models can still lead to observable predictions. They are sometimes very useful to explain our observations even if we can not measure all their parameters.

The CAPM assumes that agents make their investment choices knowing the expected values of yields. Since these are usually not measurable, agents can not know them. In fact, these expected values of yields are purely theoretical and have no real existence. But it should not be concluded that the CAPM is completely unrealistic. Agents estimate expected yields and make decisions based on their estimates. If the agents are properly informed, the expected yields may be not too bad estimates of the yields finally achieved.

Risk reduction by temporal diversification

Consider a very risky investment that allows one to multiply one's capital by a factor $k>2$ over a short period of time with a probability $0.5$ or to lose everything. Its average efficiency is $k/2-1$. But if we invest all our capital for $n$ successive periods, we have a probability $1-(1/2)^{n}$ of being ruined. If $n$ is large, the ruin is almost certain and we will not have benefited from the average yield. Is it not possible to take advantage of this average yield without taking risks?

Instead of risking all our capital, we can decide to manage it dynamically and to bet each period only a fraction $\alpha$ on this very risky investment. At each period, the average yield is only $\alpha (k/2-1)$. We have a chance on two to multiply our capital by $1-\alpha$ and a chance on two to multiply it by $1+(k-1)\alpha$. Let $V_{0}=1$ the initial value of the capital and $V_{n}$ its value after n periods. The logarithmic efficiency after n periods $\ln(V_{n}/V_{0})=\ln(V_{n})$ is a sum of independent random variables with mean $[\ln(1-\alpha )+\ln(1+(k-1)\alpha )]/2$ and variance $[\ln(1-\alpha )-\ln(1+(k-1)\alpha )]^{2}/4$. When n is large, the distribution of $\ln V_{n}$ tends to a normal distribution with mean $n[\ln(1-\alpha )+\ln(1+(k-1)\alpha )]/2$ and variance $n[\ln(1-\alpha )-\ln(1+(k-1)\alpha )]^{2}/4$.

The ratio ${\frac {|\ln(1-\alpha )-\ln(1+(k-1)\alpha )|}{{\sqrt {n}}[\ln(1-\alpha )+\ln(1+(k-1)\alpha )]}}$ of the standard deviation on the mean of logarithmic yield tends to zero when n is very large. If the average of the logarithmic yield per period is positive, $[\ln(1-\alpha )+\ln(1+(k-1)\alpha )]/2>0$ then we can do a risk free profit when n is very big. If $\alpha <(k-2)/(k-1)$ this dynamic management strategy delivers a certain profit as soon as the number of periods is large enough.

This risk reduction strategy is based on time diversification. It assumes that profits over a given period are independent of previous periods. This reduces risk by adding risky but statistically independent profits.

The formulas of Black and Scholes

The formulas of Black and Scholes make it possible to calculate the present value of a European call or put option from the present value $S_{t}$ of the underlying asset, its volatility $\sigma$, the time $T$ to maturity, the option's strike price $K$, and the risk-free interest rate $r$.

The easiest way to demonstrate them is to reason on a risk-neutral economy. It means that risk premiums are always zero. This hypothesis is obviously completely false but very surprisingly it leads to correct formulas. The Black and Scholes equation presented below does not require the hypothesis of risk neutrality and justifies the formulas of the same name. More generally, the assumption of risk neutrality often makes it possible to correctly evaluate derivative products which are nevertheless very risky, because these products are valued on the basis of the present prices of their underlying assets. These market prices take into account the required risk premiums. If risk premiums change, because agents are more afraid of risk for example, market prices also change, but their relation with the prices of derivatives does not change (Hull 2011). This is why risk premiums can often be ignored when valuing derivative prices from the underlying prices. The binomial tree model presented later allows to justify this more clearly.

Risk neutrality and value of an asset

It is assumed that agents are risk neutral. The present value of an asset is then equal to the average of the present values of its anticipated values:

$S_{0}=\langle e^{-rT}S_{T}\rangle$

where $r$ is the risk-free interest rate. We deduce for a stock whose price variations are determined by a log-normal distribution with parameters $\mu$ and $\sigma$ :

$\langle S_{T}\rangle =e^{rT}S_{0}=e^{(\mu +\sigma ^{2}/2)T}S_{0}$

Hence

$r=\mu +\sigma ^{2}/2$

The present value of a European call option

If $S_{T}$ is the anticipated value of the underlying asset, $max(S_{T}-K,0)$ is the anticipated value $C_{T}$ of a call option whose strike price is $K$. Risk neutrality requires that its present value $C_{0}$ is the average of the present values of its anticipated values:

$C_{0}={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{K}^{+\infty }{\frac {S-K}{S}}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS$

$={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}(\int _{K}^{+\infty }e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS-\int _{K}^{+\infty }{\frac {K}{S}}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS)$

$=C_{+}-C_{-}$

With $x=\ln {\frac {S}{S_{0}}}$, $S=S_{0}e^{x}$, $dS=S_{0}e^{x}dx$

$C_{+}={\frac {S_{0}e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{\ln {\frac {K}{S_{0}}}}^{+\infty }e^{x-(x-\mu T)^{2}/2T\sigma ^{2}}dx$

Now $x-(x-\mu T)^{2}/2T\sigma ^{2}=-(x-(\mu +\sigma ^{2})T)^{2}/2T\sigma ^{2}+\mu T+\sigma ^{2}/2$

Hence

$C_{+}=S_{0}e^{(-r+\mu +\sigma ^{2}/2)T}{\frac {1}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{\ln {\frac {K}{S_{0}}}}^{+\infty }e^{-(x-(\mu +\sigma ^{2})T)^{2}/2T\sigma ^{2}}dx=S_{0}{\frac {1}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{\ln {\frac {K}{S_{0}}}}^{+\infty }e^{-(x-(\mu +\sigma ^{2})T)^{2}/2T\sigma ^{2}}dx$

since $r=\mu +\sigma ^{2}/2$

With $y=(x-(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}$, $dx=\sigma {\sqrt {T}}dy$

$C_{+}=S_{0}{\frac {1}{\sqrt {2\pi }}}\int _{(\ln {\frac {K}{S_{0}}}-(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}}^{+\infty }e^{-y^{2}/2}dy$

Let $N(.)$ be the cumulative distribution function of the standard normal law:

$N(x)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-y^{2}/2}dy$

Since ${\frac {1}{\sqrt {2\pi }}}\int _{x}^{+\infty }e^{-y^{2}/2}dy={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{-x}e^{-y^{2}/2}dy=N(-x)$

$C_{+}=S_{0}N[(\ln {\frac {S_{0}}{K}}+(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}]=S_{0}N[(\ln {\frac {S_{0}}{K}}+(r+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

$C_{-}={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{K}^{+\infty }{\frac {K}{S}}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS={\frac {Ke^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{\ln {\frac {K}{S_{0}}}}^{+\infty }e^{-(x-\mu T)^{2}/2T\sigma ^{2}}dx$

With $z=(x-\mu T)/\sigma {\sqrt {T}}$, $dx=\sigma {\sqrt {T}}dz$

$C_{-}=Ke^{-rT}{\frac {1}{\sqrt {2\pi }}}\int _{(\ln {\frac {K}{S_{0}}}-\mu T)/\sigma {\sqrt {T}}}^{+\infty }e^{-z^{2}/2}dz$

$=Ke^{-rT}N[(\ln {\frac {S_{0}}{K}}+\mu T)/\sigma {\sqrt {T}}]=Ke^{-rT}N[(\ln {\frac {S_{0}}{K}}+(r-\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

Finally

$C_{0}=S_{0}N[(\ln {\frac {S_{0}}{K}}+(r+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]-Ke^{-rT}N[(\ln {\frac {S_{0}}{K}}+(r-\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

This is the Black and Scholes formula for European call options.

The present value of a European put option

If $S_{T}$ is the anticipated value of the underlying asset, $max(K-S_{T},0)$ is the anticipated value $P_{T}$ of a put option whose exercise price is $K$. Risk neutrality requires that its present value $P_{0}$ is the average of the present values of its anticipated values:

$P_{0}={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{0}^{K}{\frac {K-S}{S}}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS=P_{+}-P_{-}$

$P_{+}={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{0}^{K}{\frac {K}{S}}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS={\frac {Ke^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{-\infty }^{\ln {\frac {K}{S_{0}}}}e^{-(x-\mu T)^{2}/2T\sigma ^{2}}dx$

$=Ke^{-rT}{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{(\ln {\frac {K}{S_{0}}}-\mu T)/\sigma {\sqrt {T}}}e^{-z^{2}/2}dz$

$=Ke^{-rT}N[(\ln {\frac {K}{S_{0}}}-\mu T)/\sigma {\sqrt {T}}]=Ke^{-rT}N[(\ln {\frac {K}{S_{0}}}-(r-\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

$P_{-}={\frac {e^{-rT}}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{0}^{K}e^{-(\ln {\frac {S}{S_{0}}}-\mu T)^{2}/2T\sigma ^{2}}dS=S_{0}e^{(-r+\mu +\sigma ^{2}/2)T}{\frac {1}{\sigma {\sqrt {T}}{\sqrt {2\pi }}}}\int _{-\infty }^{\ln {\frac {K}{S_{0}}}}e^{-(x-(\mu +\sigma ^{2})T)^{2}/2T\sigma ^{2}}dx$

$=S_{0}{\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{(\ln {\frac {K}{S_{0}}}-(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}}e^{-y^{2}/2}dy$

$=S_{0}N[(\ln {\frac {K}{S_{0}}}-(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}]=S_{0}N[(\ln {\frac {K}{S_{0}}}-(r+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

Finally

$P_{0}=Ke^{-rT}N[(\ln {\frac {K}{S_{0}}}-(r-\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]-S_{0}N[(\ln {\frac {K}{S_{0}}}-(r+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

This is the Black and Scholes formula for European put options.

The Black and Scholes equation

In principle, risk hedging strategies can be used to build risk-free portfolios with options and shares or other assets. The technique is simply to compensate for increases and decreases. Any increase above the risk-free rate for one or more items in the portfolio is offset by a decrease in other items, or a lower increase than the risk-free rate. Such a portfolio is therefore remunerated at the risk-free rate. Increases and decreases compensate mechanically. But risk-free portfolios are the same in an economy that pays risk premiums as in an economy that does not pay for them, and the Black and Scholes formulas determine their composition. Different formulas would necessarily lead to portfolios without risk of different compositions. Since risk-free portfolios are the same whether or not the economy is risk neutral, the validity of the Black and Scholes formulas does not depend on the existence of risk premiums.

More precisely, we will show from the existence of risk-free portfolios that the function that determines the value of an option must satisfy a partial differential equation, the Black and Scholes equation. The formulas of the same name are the only solutions in this equation that satisfy the boundary conditions.

Let $V(t,S)$ be the value of an option based on the current price $S$ of the underlying asset, at time $t$. $V(t,S)$ can be known empirically if it is a market price. A priori it could depend on risk premiums, because buying and selling options are risky transactions. Black and Scholes reasoning about a risk-free portfolio shows that $V(t,S)$ is in fact independent of risk premiums.

A risk-free portfolio is created at time $t$ by holding an option and $-{\frac {\partial V}{\partial S}}$ units of the underlying. If ${\frac {\partial V}{\partial S}}>0$ the underlying asset must be sold short.

As before, we assume that the random variations of the underlying asset are described by a log-normal distribution.

The $dS$ variation of $S$ during a time interval $dt$ is the sum of two terms:

${\frac {dS}{S}}=mdt+\sigma \epsilon {\sqrt {dt}}$

where $\epsilon$ is a standard normal random variable.

To compute $dV$, the formula $dV=({\frac {\partial V}{\partial t}}+{\frac {\partial V}{\partial S}}{\frac {dS}{dt}})dt$ is not appropriate, because ${\frac {dS}{dt}}$ diverges when $t$ tends to zero.

The following reasoning on orders of magnitude is not rigorous but it leads to an exact formula, which can be proved with Ito's lemma.

$dV={\frac {\partial V}{\partial t}}dt+{\frac {\partial V}{\partial S}}dS+{\frac {1}{2}}{\frac {\partial ^{2}V}{\partial S^{2}}}dS^{2}$

${\frac {dS^{2}}{S^{2}}}=m^{2}dt^{2}+m\sigma \epsilon dt^{3/2}+\sigma ^{2}\epsilon ^{2}dt$

The first two terms are negligible compared to the third when $dt$ tends to zero.

Assuming that we can replace $\epsilon ^{2}$ with its average value $1$ we obtain:

$dV=({\frac {\partial V}{\partial t}}+mS{\frac {\partial V}{\partial S}}+S^{2}{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}V}{\partial S^{2}}})dt+S{\frac {\partial V}{\partial S}}dW$

The change in value of the risk-free portfolio is

$d\Pi =dV-{\frac {\partial V}{\partial S}}dS=({\frac {\partial V}{\partial t}}+S^{2}{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}V}{\partial S^{2}}})dt$

The random term $dW$ has disappeared. This confirms that the portfolio is risk free.

$\Pi$ must vary at the risk-free rate $r$:

$d\Pi =r\Pi dt=r(V-{\frac {\partial V}{\partial S}}S)dt$

We obtain

${\frac {\partial V}{\partial t}}+S^{2}{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}V}{\partial S^{2}}}=r(V-{\frac {\partial V}{\partial S}}S)$

This is the Black and Scholes equation:

${\frac {\partial V}{\partial t}}+S^{2}{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}V}{\partial S^{2}}}+rS{\frac {\partial V}{\partial S}}-rV=0$

One can verify that the formulas of Black and Scholes are solutions of the equation of the same name. This equation does not require the hypothesis of a risk-neutral environment. The formulas of Black and Scholes are therefore true in an environment that is not indifferent to risk. The binomial tree method provides a simpler explanation of this result.

Binomial trees and risk neutrality

The following examples are very simple and very unrealistic, but they are sufficient to understand why the relation between option and stock prices does not depend on risk premiums:

It is assumed that an action worth 100 today can be 110 or 90 after one period with the probabilities p and 1-p respectively. One wonders about the present price P of a put option whose exercise price is 110. It is assumed for simplicity that the risk-free interest rate is zero. The present price of a portfolio that contains an action and an option is 100 + P. Its future value is 110 in all cases. It is therefore risk free. We deduce that P = 10. We do not need to know the probability of a rise to know the price of the option. We do not need to know the risks taken by investors and their risk premiums.

More generally, a single period binomial tree is defined by the following parameters: p, u, d and r. p is the probability that the price of the stock is multiplied by u (up) after one period, 1-p the probability that it is multiplied by d (down). r is the risk-free interest rate over one period.

It is assumed that the present price of the action is 100, that u> 1 and d <1. To evaluate the present price P of a put option whose exercise price is K, we reason on a portfolio that contains $\Delta$ shares and one option. We assume that $100d\leq K\leq 100u$. The current price of the portfolio is $100\Delta +P$. Its future value is either $100u\Delta$ or $100d\Delta +K-100d$. This portfolio is risk free when

$100u\Delta =100d\Delta +K-100d$

that is to say

$\Delta ={\frac {K-100d}{100(u-d)}}$

Since it is safe, one must have

$(100\Delta +P)(1+r)=100u\Delta$

so

$P={\frac {100\Delta (u-1-r)}{1+r}}={\frac {(K-100d)(u-1-r)}{(u-d)(1+r)}}$

To find out the present price C of a call option whose exercise price is K, we think about a portfolio consisting of having sold $\Delta$ shares short and having bought an option. Selling short is selling shares that have been borrowed by committing to buy them later to return them. The current price of the portfolio is $-100\Delta +C$. Its future value is either $-100u\Delta +100u-K$ or $-100d\Delta$. This portfolio is risk free when

$-100u\Delta +100u-K=-100d\Delta$

that is to say

$\Delta ={\frac {K-100u}{100(d-u)}}$

Since it is safe, one must have

$(-100\Delta +C)(1+r)=-100d\Delta$

so

$C={\frac {100\Delta (1+r-d)}{1+r}}={\frac {(K-100u)(1+r-d)}{(d-u)(1+r)}}$

As with the first example, we do not need to know the probability p of a rise to know the price of the options. So we do not need to know the risk premiums to evaluate the put and call options.

It is of course perfectly unrealistic to assume that a share can only have two values ​​after one period. But it is enough to reason on multiperiod binomial trees to make the model very realistic, as soon as the number of periods is large enough. With two periods, three possible future values ​​are obtained: 100uu, 100ud, and 100dd. With N periods, we obtain N + 1 possible future values. If N is very large, the variations of the share price follow a log-normal law. We can prove the formulas of Black and Scholes by reasoning on binomial trees. The independence of the price of options with respect to risk premiums established with binomial trees is therefore much more realistic than what the simplicity of the model suggests at the outset.

The article by Cox, Ross and Rubinstein, Option pricing, a simplified approach (1979) is the pioneering article that showed the value of this model.

The average yield of a put option may be negative

A put option has a negative $\beta$ if the $\beta$ of the underlying stock is positive. We show on an example that the $\beta$ can be negative enough for the average yield to be negative:

Consider a stock whose average yield (arithmetic annual) is 10% and volatility 20%:

$\mu =0.08$

$\sigma =0.2$

$e^{\mu +\sigma ^{2}/2}-1\approx 0.1$

We reason on a a six-month put option whose strike price is equal to the current price of the share plus its average yield.

Let $V_{0}=100$ be the present price of the action. The strike price of the option is

$K=V_{0}e^{(\mu +\sigma ^{2}/2)T}=100e^{(0.08+0.2^{2}/2)0.5}\approx 105.13$

The present price of the option is obtained with the formula of Black and Scholes:

$P_{0}=Ke^{-rT}N[(\ln {\frac {K}{S_{0}}}-(r-\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]-S_{0}N[(\ln {\frac {K}{S_{0}}}-(r+\sigma ^{2}/2)T)/\sigma {\sqrt {T}}]$

With $r=0.02$, $P_{0}\approx 8.02$,

The option will yield $\langle P_{f}\rangle$ on average on the day of exercise:

$\langle P_{f}\rangle =KN[(\ln {\frac {K}{S_{0}}}-\mu T)/\sigma {\sqrt {T}}]-S_{0}e^{rT}N[(\ln {\frac {K}{S_{0}}}-(\mu +\sigma ^{2})T)/\sigma {\sqrt {T}}]$

$\approx 7.87$

The average annual logarithmic yield of the option is therefore

$2\ln {\frac {7.87}{8.02}}\approx -0.038=-3.8\%$

Mathematical supplements

The normal law

Probability density

• A random variable $X$ follows a standard normal law when its probability density is:

$f_{X}(x)={\frac {1}{\sqrt {2\pi }}}e^{-x^{2}/2}$

${\frac {1}{\sqrt {2\pi }}}$ is a normalization factor for the sum of probabilities to be equal to one:

$\int _{-\infty }^{+\infty }f_{X}(x)dx={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }e^{-x^{2}/2}dx=1$

It is proved by the following reasoning:

$\int _{-\infty }^{+\infty }\int _{-\infty }^{+\infty }e^{-(x^{2}+y^{2})}dxdy=\int _{-\infty }^{+\infty }e^{-x^{2}}dx\int _{-\infty }^{+\infty }e^{-y^{2}}dy=(\int _{-\infty }^{+\infty }e^{-x^{2}}dx)^{2}$

$=\int _{0}^{2\pi }\int _{0}^{+\infty }e^{-r^{2}}rd\phi dr=2\pi [-{\frac {1}{2}}e^{-r^{2}}]_{0}^{+\infty }=\pi$

hence $\int _{-\infty }^{+\infty }e^{-x^{2}}dx={\sqrt {\pi }}$

With $x=y/{\sqrt {2}}$, $dx=dy/{\sqrt {2}}$, we get $\int _{-\infty }^{+\infty }e^{-x^{2}}dx={\frac {1}{\sqrt {2}}}\int _{-\infty }^{+\infty }e^{-y^{2}/2}dy$

hence

$\int _{-\infty }^{+\infty }e^{-y^{2}/2}dy={\sqrt {2\pi }}$

• A random variable $X$ follows a centered normal law when its probability density is:

$f_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-x^{2}/2\sigma ^{2}}$

With $y=x/\sigma$, $dx=\sigma dy$, we check that $\int _{-\infty }^{+\infty }f_{X}(x)dx={\frac {\sigma }{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{-y^{2}/2}dy=1$

• A random variable $X$ follows a normal law when its probability density is:

$f_{X}(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-(x-\mu )^{2}/2\sigma ^{2}}$

With $y=x-\mu$, $dy=dx$, we check that $\int _{-\infty }^{+\infty }f_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{-y^{2}/2\sigma ^{2}}dy=1$

The red curve is the standard normal distribution.

Expected value

A random variable $X$ is centered when its mean, or expected value $E(X)$, is zero. It is easy to verify that centered normal laws have a zero mean since their probability densities are even. $E(X)=\int _{-\infty }^{+\infty }xf_{X}(x)dx$ is the integral of an odd function and is therefore zero.

With $y=x-\mu$, we obtain for the expected of the normal distribution:

$E(X)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }xe^{-(x-\mu )^{2}/2\sigma ^{2}}dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }(y+\mu )e^{-y^{2}/2\sigma ^{2}}dy=\mu$

Variance and standard deviation

• We obtain the variance of a standard normal random variable $X$ with an integration by parts:

$Var(X)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }x^{2}e^{-x^{2}/2}dx={\frac {1}{\sqrt {2\pi }}}([-xe^{-x^{2}/2}]_{-\infty }^{+\infty }-\int _{-\infty }^{+\infty }-e^{-x^{2}/2}dx)=1$

• With $y=x/\sigma$, $dx=\sigma dy$, we then obtain the variance of the centered normal law:

$Var(X)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }x^{2}e^{-x^{2}/2\sigma ^{2}}dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }\sigma ^{2}y^{2}e^{-y^{2}/2}\sigma dy=\sigma ^{2}$

• The variance of the normal distribution is identical:

$Var(X)={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }(x-\mu )^{2}e^{-(x-\mu )^{2}/2\sigma ^{2}}dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }y^{2}e^{-y^{2}/2\sigma ^{2}}dy=\sigma ^{2}$

$\sigma$ is therefore the standard deviation ${\sqrt {Var(X)}}$ of $X$.

For the normal distribution, $\sigma$ is also a good estimate of the mean of the absolute values of the deviations from the mean:

${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{+\infty }|x|e^{-x^{2}/2}dx=2{\frac {1}{\sqrt {2\pi }}}\int _{0}^{+\infty }xe^{-x^{2}/2}dx$

$={\sqrt {\frac {2}{\pi }}}[-e^{-x^{2}/2}]_{0}^{+\infty }={\sqrt {\frac {2}{\pi }}}$

hence

${\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }|x-\mu |e^{-(x-\mu )^{2}/2\sigma ^{2}}dx=\sigma {\sqrt {\frac {2}{\pi }}}\approx 0.8\sigma$

The lognormal law

Probability density

Let $X$ be a random variable such that $\ln X$ follows a normal law with mean $\mu$ and standard deviation $\sigma$.

The cumulative distribution function of $X$ is

$F_{X}(x)=Pr(X\leq x)=Pr(\ln X\leq \ln x)=F_{\ln X}(\ln x)$

The probability density of $X$ is therefore

$f_{X}(x)={\frac {d}{dx}}F_{X}(x)={\frac {1}{x}}f_{\ln X}(\ln x)={\frac {1}{x\sigma {\sqrt {2\pi }}}}e^{-{\frac {(\ln x-\mu )^{2}}{2\sigma ^{2}}}}$

Some log-normal density functions with identical parameter $\mu$ but differing parameters $\sigma$

Expected value

$\langle X\rangle ={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{0}^{+\infty }xf_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{0}^{+\infty }e^{-(\ln x-\mu )^{2}/2\sigma ^{2}}dx$

With $y=\ln x$, $x=e^{y}$, $dy={\frac {dx}{x}}$, $dx=e^{y}dy$

$\langle X\rangle ={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{y-(y-\mu )^{2}/2\sigma ^{2}}dy$

Now $y-(y-\mu )^{2}/2\sigma ^{2}=-(y-(\mu +\sigma ^{2}))^{2}/2\sigma ^{2}+\mu +\sigma ^{2}/2$

Hence

$\langle X\rangle =e^{\mu +\sigma ^{2}/2}{\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{-(y-(\mu +\sigma ^{2}))^{2}/2\sigma ^{2}}dx$

$=e^{\mu +\sigma ^{2}/2}$

Variance and standard deviation

$\langle X^{2}\rangle ={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{0}^{+\infty }x^{2}f_{X}(x)dx={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{0}^{+\infty }xe^{-(\ln x-\mu )^{2}/2\sigma ^{2}}dx$

With $y=\ln x$, $x=e^{y}$, $dx=e^{y}dy$

$\langle X^{2}\rangle ={\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{2y-(y-\mu )^{2}/2\sigma ^{2}}dy$

Now $2y-(y-\mu )^{2}/2\sigma ^{2}=-(y-(\mu +2\sigma ^{2}))^{2}/2\sigma ^{2}+2(\mu +\sigma ^{2})$

Hence

$\langle X^{2}\rangle =e^{-2(\mu +\sigma ^{2})}{\frac {1}{\sigma {\sqrt {2\pi }}}}\int _{-\infty }^{+\infty }e^{-(y-(\mu +2\sigma ^{2}))^{2}/2\sigma ^{2}}dx$

$=e^{2(\mu +\sigma ^{2})}$

$Var(X)=\langle X^{2}\rangle -\langle X\rangle ^{2}=e^{2(\mu +\sigma ^{2})}-(e^{\mu +\sigma ^{2}/2})^{2}$

$=(e^{2\mu +\sigma ^{2}})(e^{\sigma ^{2}}-1)$

The standard deviation of $X$ is therefore

${\sqrt {Var(X)}}=e^{\mu +\sigma ^{2}/2}{\sqrt {e^{\sigma ^{2}}-1}}$

Bibliography

Admati, Anat, Hellwig, Martin, The bankers' new clothes : what's wrong with banking and what to do about it (2013)

Akerlof, George A., Shiller, Robert J., Phishing for phools : the economics of manipulation and deception (2015)

Bagehot, Walter, Lombard Street (1873)

Bernanke, Ben S., Mémoires de crise (2015)

Bernanke, Ben S., Laubach, Thomas, Mishkin, Frederic S. Posen, Adam Simon, Inflation targeting : lessons from the international experience (1999)

Böhm von Bawerk, Eugen, The positive theory of capital (1889)

Fisher, Irving N., The nature of capital and income (1906), The purchasing power of money (1911)

Hull, John, Options, futures et autres actifs dérivés (2011)

Kindleberger, Charles, Manias, panics and crashes: a history of financial crises (2000)

Krugman, Paul, Sortez-nous de cette crise, maintenant (2012)

Krugman, Paul, Obstfeld, Maurice, Économie internationale (2008)

Krugman, Paul, Wells, Robin, Macroéconomie (2012)

Marx, Karl, Le capital (1867)

Merton, Robert C., Bodie, Zvi, Finance (2000)

Rodrik, Dani, The globalization paradox, why global markets, states and democracy can't coexist (2011)

Shiller, Robert, Finance and the good society (2012)

Stiglitz, Joseph E., Principes d'économie moderne (1999), La grande désillusion (2002), Quand le capitalisme perd la tête (2003), Un autre monde : contre le fanatisme du marché (2006), Le triomphe de la cupidité (2010), Le prix de l'inégalité (2012), L'Euro, comment la monnaie unique menace l'avenir de l'Europe (2016)

Summary

We want to finance the most profitable projects. We must therefore be able to evaluate them beforehand. Evaluation of projects presents the most basic principles of economic and financial evaluation. It explains the dual origin of economic value, use value and exchange value, the creation of value by composition of projects, options, because it is perhaps the most important concept for economic evaluation, because our wealth is essentially a wealth of options, the net present value of certain and uncertain projects, and the economic value of the common order. The subject of project evaluation is further developed in a supplementary chapter (not yet written) devoted to risk assessment.

Inequalities in front of finance explains the origin of interest rates and other property income, which make the financial system very unequal, since it privileges the owners to the detriment of the less fortunate. Finance sometimes makes it possible to partially correct certain economic inequalities by offering means of entrepreneurship, but even so it remains profoundly inegalitarian.

It seems that to advance funds it is necessary to dispose of it in advance. But it is to count without the banking system capable of advancing funds that are created at the very moment they are loaned. It looks like counterfeit money, but it's very different, because money is created in exchange for promises of repayment. How can we create money from scratch? shows how the banking system works and why money creation is not necessarily an evil. Money can be created to meet the real investment needs and the ability of the economy to engage in profitable projects. A good project is a sufficient guarantee to legitimize the creation of the money that finances it.

To benefit from sustainable prosperity, economies must navigate between two pitfalls, too little investment, lack of private investors or of sufficient money creation, and too much investment, because of the exaggerated optimism of private investors or an excess of money creation. Financing the economy presents the basic knowledge that shows how to proceed, and why it is difficult. It emphasizes the importance of countercyclical monetary and fiscal policies. Paradoxically the state has to spend less when it earns more and it has to spend more when it earns less.

This book is translated from the French language Wikibook La science de la finance.