# The science of finance/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.

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.

#### Combinations of interdependent options

When ${\displaystyle N}$ options are independent, there are ${\displaystyle 2^{N}}$ ways to exercise the combination of options. A map ${\displaystyle f}$ from ${\displaystyle \{1...N\}}$ to ${\displaystyle \{0,1\}}$ defines a possible choice. ${\displaystyle f(i)}$ for an integer ${\displaystyle i}$ such that ${\displaystyle 1\leq i\leq N}$ is equal to ${\displaystyle 1}$ if the option ${\displaystyle i}$ is exercised and ${\displaystyle 0}$ otherwise.

But the options are not always independent. They can be mutually exclusive or linked in many ways. An American option for example can be conceived as a series of European options, one for each exercise day, which are mutually exclusive.

Let ${\displaystyle \Omega }$ be the space of all maps from ${\displaystyle \{1...N\}}$ to ${\displaystyle \{0,1\}}$. A non-empty part ${\displaystyle P}$ of ${\displaystyle \Omega }$ is a set of possible choices for a combination of ${\displaystyle N}$ options. For options to really be options, ${\displaystyle P}$ must meet the following condition:

For any integer ${\displaystyle i}$ such as ${\displaystyle 1\leq i\leq N}$, there are elements ${\displaystyle f}$ and ${\displaystyle g}$ of ${\displaystyle P}$ such that ${\displaystyle f(i)=0}$ and ${\displaystyle g(i)=1}$.

If this condition is not satisfied for an ${\displaystyle i}$, then the exercise or non-exercise of the option ${\displaystyle i}$ is imposed in advance, and this one is not an option.

• ${\displaystyle N=2}$

${\displaystyle P=\Omega =\{00,01,10,11\}}$ means that both options are independent.

${\displaystyle P=\{00,01,10\}}$ means that the two options are mutually exclusive.

${\displaystyle P=\{00,10,11\}}$ means that the exercise of the first option is a necessary condition for the exercise of the second.

${\displaystyle P=\{00,01,11\}}$ means that the exercise of the second option is a necessary condition for the exercise of the first.

${\displaystyle P=\{00,11\}}$ means that both options must be exercised together. Either one does not exercise any, or one exercises both, but not one without the other.

${\displaystyle P=\{01,10,11\}}$ means that at least one of the two options must be exercised.

${\displaystyle P=\{01,10\}}$ means that one of the two options must be exercised, but not both.

• ${\displaystyle N>2}$

We can think of some special cases:

All options are independent.

All options are mutually exclusive but we do not have to exercise any. We can therefore choose at most one option to exercise among the possible ${\displaystyle N}$. Such a combination of European options for every day of exercise is an American option.

All the options are mutually exclusive but we have to exercise one. It is therefore necessary to choose an option to exercise among the ${\displaystyle N}$ possible.

The exercise of an option is always a necessary condition for the exercise of the next. A project that we can stop when we want, but we can not start it again once it is stopped, no second chance, can be represented by a series of such options.

All options must be exercised together. It's all or nothing. In this case the combination of options is equivalent to a single option.

### 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):

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

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

(Specifically, the discount rate ${\displaystyle r}$ 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 ${\displaystyle t_{i}}$ are positive, these are future payments. Since ${\displaystyle r}$ 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 ${\displaystyle t_{i}}$ 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 ${\displaystyle 1}$ today is worth ${\displaystyle 1+r}$ in one year, if ${\displaystyle r}$ 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 ${\displaystyle r/2}$ but ${\displaystyle {\sqrt {1+r}}-1}$. If on the other hand we define a discount rate ${\displaystyle R}$ such that ${\displaystyle e^{R}=1+r}$ then ${\displaystyle R/2}$ is the six-month rate, because ${\displaystyle e^{R/2}e^{R/2}=e^{R}}$. ${\displaystyle R}$ is the logarithmic discount rate. When ${\displaystyle r}$ is small, it is very little different from ${\displaystyle R}$.

#### 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 (${\displaystyle V_{0}=-100}$) and a single final revenue obtained after one year [itex] 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.