Financial Math FM/General Cash Flows and Portfolios

Learning objectives

The Candidate will understand key concepts concerning yield curves, rates of return, and measures of duration and convexity, and how to perform related calculations.

Learning outcomes

The Candidate will be able to:

• Define and recognize the definitions of the following terms: yield rate/rate of return, dollar-weighted rate of return, time-weighted rate of return, current value, duration (Macaulay and modified), convexity (Macaulay and modified), portfolio, spot rate, forward rate, yield curve, stock price, stock dividend.
• Calculate:

• The dollar-weighted and time-weighted rate of return.
• The duration and convexity of a set of cash flows.
• Either Macaulay or modified duration given the other.
• The approximate change in present value due to a change in interest rate,

• The approximate change in present value due to a change in interest rate,
• Using 1st-order approximation based on Macaulay duration.

• The price of a stock using the dividend discount model.
• The present value of a set of cash flows, using a yield curve developed from forward and spot rates.

Discounted cash flow analysis

Definition. (Net cash flow) Net cash flow at time $t$ equals cash inflow at time $t$ minus cash outflow at time $t$.

Remark. We use $C_{t}$ to denote net cash flow at time $t$.

Then, we can express a series of cash flows as follows:

Net cash flow	$C_{0}$	$C_{1}$	$C_{2}$	$\cdots$	$C_{n}$
Time	$0$	$1$	$2$	$\cdots$	$n$

After defining net cash flows, we can also define net present value.

Definition. (Net present value) The net present value (NPV), denoted by $P(i)$, is sum of present value of every net cash flow, i.e. $P(i)=\sum _{t}{C_{t}v^{t}}.$

Remark.

• Net present value is a function of $i$.
• An investment is profitable if and only if its net present value is positive, i.e. the series of cash flows in the investment is equivalent to one cash inflow at time 0.
• The present value of cash flow can be interpreted as discounted cash flow: cash flow at time $t$ is discounted by the factor $v^{t}$.

An example to illustrate how we can use net present value to determine profitability of investment is in the following.

Exercise.

Ivan gives you an offer to invest in his project. His project requires an investment from you of $10,000 now and the project certainly pays $3,000 at the end of 2nd year, $4,000 at the end of 3rd year and $5,000 at the end of 10th year to you. At an annual effective interest rate of 10%, should you take his offer?

	yes
	no

Apart from determining whether the investment is profitable, we sometimes also want to know how profitable the investment is. A natural way to measure how profitable the investment is using its rate of return which can be measured by internal rate of return.

Definition. (Internal rate of return) The internal rate of return (IRR) of an investment is the effective interest rate per measurement period at which the net present value of the investment equals zero.

Remark.

• It is also called yield rate.
• Equivalently, the internal rate of return is the interest rate at which the initial investment, i.e. cash flow at time zero, equals the net present value of the future net cash flows. Therefore, we may split the initial investment into several portions, and each of them will accumulate to the amount of each of the future cash flow correspondingly at the internal rate of return, at the corresponding time of the cash flow. (See the exercise below.)
• Internal rate of return is also the interest rate at which the total present value of cost (i.e. cash outflow(s)) equals total present value of benefit (i.e. cash inflow(s)).
• It is internal in the sense that the calculation excludes external factors, e.g. financial risk, inflation, etc.
• In general, it is very difficult to calculate internal rate of return directly by hand. Therefore, financial calculator, e.g. BA II Plus, is usually used to calculate it. We may also use linear interpolation in the interest table to approximate it.
• It may not exist and may not be unique. There can be no internal rate of return, or multiple internal rates of return, although it is uncommon in practice.

Let's illustrate the split of initial investment mentioned in the remark in the following exercise.

Consider the following cash flows for an investment.

Time	Net cash Flow
0	-500
1	100
2	0
3	300
4	0
5	500

Exercise.

Refer to the above cash flows. It is given that the internal rate of return of this investment is 17.05% (correct to two decimal places). Split the initial investment (with amount 500) into three portions that will accumulate to the amount of each of the future cash flow correspondingly at the internal rate of return (i.e. 17.05% in this case, use exactly this percentage for calculation), at the corresponding time of the cash flow. Correct your answers to two decimal places.

The portion with amount will accumulate to 100 at time 1.

The portion with amount will accumulate to 300 at time 3.

The portion with amount will accumulate to 500 at time 5.

at the interest rate of 17.05%, i.e. internal rate of return.

BA II Plus calculator usage

The TI BA II plus has a cash flow function to deal with uneven cash flow streams. Here is an example of cash flows.

Time	Net cash Flow
0	-500
1	100
2	0
3	300
4	0
5	500

kk

You can use the following command to calculate the net present value at effective interest rate of 10% per measurement period.

2ND CE|C CF 500 +|- ENTER ↓ 100 ENTER ↓ ↓ 0 ENTER ↓ ↓ 300 ENTER ↓ ↓ 0 ENTER ↓ ↓ 500 ENTER NPV 10 ENTER ↓ CPT

You should get NPV=126.76. The following are some explanations on the command.

1. 2ND CE|C will clear any previous work saved in the calculator.
2. CF Enters into cash flow mode.
3. Next, you will see CF0= for which you need to enter the first cash flow of -500 which is made at time zero. You should enter negative value for cash outflow. This is the convention used in BAII Plus calculator.
4. After entering -500 and then pressing ↓, you will see C01, for which you need to enter the second cash flow of 100 which is made at time one.
5. After pressing ↓ again, you will see F1, for which you need to enter the frequency of the cash flow, i.e. how many times this cash flow is repeated. The default value is 1, which is correct in this case. We will later see an exercise in which the frequency is higher than 1.
6. Then, we enter the remaining cash flows. We need to enter all zero cash flows to ensure that the later nonzero cash flow is entered at a correct time.
7. After entering the last cash flow of 500, press NPV instead of ↓. Then you will see I, and need to enter the effective interest rate (only the number in the percentage expression). You should enter 10 in this case.
8. After pressing ↓ and CPT, you should see "NPV=126.76" which is the net present value.

To calculate internal rate of return, press IRR and then CPT after entering the last cash flow. You should get IRR=17.05, i.e. the internal rate of return is 17.05%. The calculator will only display the number before the % sign of percentage expression. (This result is used in exercise about spliting initial investment.)

Here is another example of cash flows.

Time	Net cash Flow
0	-500
1	200
2	200
3	200
4	200
5	200

Press 2ND CE|C CF 500 +|- ENTER ↓ 200 ENTER ↓ 5 ENTER NPV 10 ENTER ↓ CPT to calculate net present value at effective interest rate of 10% per measurement period. To calculate internal rate of return, press IRR CPT after 5 ENTER instead. You should get NPV=258.16 for net present value, and IRR=28.65 for internal rate of return.

After pressing 200 ENTER ↓, we will see F01 for which we need to enter the frequency of this cash flow. Because this cash flow repeats five times, we enter 5.

In the later commands, 2ND CE|C will be omitted, and it is assumed that you always press this sequence of commands before entering the cash flows.

Let's practice the calculator usage in the following exercise.

Exercise.

Continue from exercise about determining profitability using net present value, calculate the internal rate of return of the project. Correct your answer to two decimal places.

%

Idealized practical situation

In this subsection, we assume that we can borrow and lend money at a fixed interest rate freely. Practically, although we can borrow and lend money, we may not be allowed to borrow and lend money freely, and we typically cannot borrow and lend money at a fixed interest rate for a long time, because interest rate will change, and interest rate for borrowing and lending are usually different (and the former is usually higher than latter).

Under this assumption we can calculate 'net accumulated value', which is analogous to net present value. This term is nonstandard and rarely used. We can connect the net cash flows from one project with another project in which interest is payable (when net cash flow is positive, we can put the money in, just like lending money) or credited (when net cash flow is negative, we need to take the money out, just like borrowing money) at a fixed rate $i$. Let's also assume we can put money in and take money out freely, in arbitrary amount (and of course they are subject to the interest payable or credited). When the another project ends (say at time $T$), the accumulated value will be

$\sum _{t\leq T}C_{t}(1+i)^{T-t}.$

in which $C_{t}$ is the net cash flow from the first project at time $t$. If the first project ends before the another project ends, then we can remove the '$\leq T$' because $t$ is always smaller than or equal to $T$, i.e. the accumulated value will be

$\sum _{t}C_{t}(1+i)^{T-t}.$

If the another project continues indefinitely, this value is undefined (because it tends to infinity). However, for a project that continues indefinitely, in which there are net cash flows, its net present value may be defined, just like perpetuity.

Exercise.

Project A requires an initial investment of 1 million, and brings you cash inflows of 200000 at the end of 2nd year, 500000 at the end of 3rd year, and 800000 at the end of 7th year. However, it requires a maintenance fee of 300000 at the end of 5th year. The semiannual effective interest rate is 5%.

1 Calculate the net present value of project A.

	-242471.42
	-234803.25
	-53188.09
	125876.53
	416927.61

2 Suppose we connect the cash flows associated with project A with project B. In project B, interest is payable or credited at a fixed annual rate of 10%, and project B continues indefinitely. Calculate the accumulated value of the net cash flows connected with project B at the end of 5th year.

	It is undefined.
	-1039310
	-390505.65
	571200
	1171200

In this subsection, let's also assume that the internal rate of return exists and $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and negative at the interest rate that is strictly greater than the internal rate of return in this subsection. Actually, this is usually the case in practice, unless there are multiple internal rates of return. Then, we have the following proposition.

Proposition. (Necessary and sufficient condition for a project to be profitable) Assume that the internal rate of return exists and $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and negative at the interest rate that is strictly greater than the internal rate of return. A project is profitable if and only if the effective interest rate per measurement period is strictly smaller than internal rate of return.

Proof. It follows from the assumption that $P(i)$ is positive at the interest rate that is strictly smaller than the internal rate of return, and the fact that a project is profitable if and only if $P(i)$ is positive.

$\Box$

Remark.

• An equivalent condition is that the internal rate of return is strictly greater than the interest rate at which the investor may lend

or borrow money.

• Indeed, this idea is also mentioned in the explanation of answer in the exercise about calculating internal rate of return.

Comparison of two investment projects

Illustration of the comparison between project A and B. When $i$ is at the orange region but not the red region, $P_{A}(i)<P_{B}(i)$ . When $i$ is at the red region but not the orange region, $P_{B}(i)>P_{A}(i)$ . When $i$ is at the intersection point of red and orange region, $P_{B}(i)=P_{A}(i)$ .

Sometimes, we want to compare two investment projects to decide which one has a higher profitability, and then we invest in the project with higher profitability. Naturally, we may think that the project with higher internal rate of return should always have a higher profitability. However, this is not always the case.

To decide the profitability of each project, we should compare the profit at time $T$, which is the date at which the later of the two projects ends, of each project. Equivalently, this is the net present value calculated at the rate of interest $i_{1}$ at which the investor may lend or borrow money. If the net present value of project A, $P_{A}(i_{1})$, is strictly greater than that of project B, $P_{B}(i_{1})$, project A is more profitable than project B.

Because the internal rate of return of project B ($i_{B}$) is strictly higher than project A ($i_{A}$) may not imply $P_{B}(i_{1})>P_{A}(i_{1})$ (whether the inequality holds depends on value of $i_{1}$), project with strictly higher internal rate of return does not necessarily have a strictly higher profitability.

Exercise.

Project A requires initial investment of 1000, and gives investor double of the amount of initial investment at the end of first year, and project B requires initial investment of 2000, and gives investor double of the amount of initial investment at the end of second year. We can only invest in at most one of the projects.

1 Calculate internal rate of return of project A and B. Correct your answer to two decimal places if the answer is not exact.

Internal rate of return of project A is %.

Internal rate of return of project B is %.

2 Assume the annual interest rate is 10% at which we can lend or borrow money freely. Based on their profitability, should we invest in project A or B?

	Project A
	Project B
	Neither of them.

3 Someone claims that project A should have a higher profitability because it can give double of the amount of initial investment in a shorter time, and then the investor can use the time difference to lend money to earn extra interest. Do you agree with the claim?

	yes
	no

4 Someone claims that we should invest neither of project A and B if the annual interest rate is strictly higher than both of internal rate of return of project A and B. Do you agree with the claim?

	yes
	no

Different interest rates for lending and borrowing

In the subsection of idealized practical situation, we assume that the investor may borrow or lend money at the same rate of interest. However, in practice, the investor may need to pay a higher interest rate on borrowing than the rate earned on money invested, e.g. interest rate earned in deposit account. (When we deposit money, the money in the account may be used for lending.)

In these circumstances, the concepts of net present value and yield are not meaningful anymore in general. We must calculate the accumulation of net cash flow from first principles. Let's illustrate how the calculation looks like in the following exercise.

Exercise.

A project requires initial investment of 30000, and another investment of 20000 at the end of first year. It gives investor 10000 at the end of first year, and 60000 at the end of third year. We can borrow money only at effective interest rate of 10%, earn interest for money deposited in our deposit account at the effective interest rate of 5%, and we can repay the loan and deposit money at any time. We must borrow money for every payment needed. Calculate the maximum possible accumulated value of the net cash flows at the end of tenth year.

	12700
	15531.28
	17870.18
	20045.90

Discounted payback period

In practice, the net cash flow usually changes sign only once, and this change is from negative to positive. In these circumstances, the balance in the investor's account will change from negative to positive at a unique time $t^{*}$. If the balance will always be negative, the project is always not profitable, and thus no such time exists. If such time $t^{*}$ exists, it is the time at the end of discounted payback period. We define it more formally as follows:

Definition. (Discounted payback period) Discounted payback period is the smallest positive integer $t$ in the unit of time such that

$A(t)=\sum _{s\leq t}C_{s}(1+i)^{t-s}\geq 0$

in which $i$ is the effective rate of interest at which the net cash flows accumulate.

Remark. Assume that a project ends at time $T$. If $A(T)<0$, the project has no discounted payback period and is not profitable. This is because the balance in the investor's account will only change from negative to positive at a unique time. If the balance is still negative at the end of the project, it shows that the balance is negative throughout the project, i.e. the net present value is negative.

Exercise.

A project requires an initial investment of 50000, and pays 6000 to each of its investor at the end of each year starting from the end of year 3. Calculate the discounted payback period if the effective annual interest rate is 10%.

years

In particular, we may need to calculate discounted payback period for project in which we borrow money to invest, when given the effective interest rate of borrowing money, say $j_{1}$, which may not be the same as the effective interest rate of depositing money, say $j_{2}$ as suggested in the subsection of different interest rates for lending and borrowing. However, $j_{2}$is not involved in and does not affect our calculation.

Naturally, we may think that we can just replace every $i$ by $j_{1}$ in the definition of discounted payback period and use it to calculate the discounted payback period. However, there is a problem. Although we can borrow money for cash outflow, so it makes sense to accumulate that amount of money ,which is negative for net cash flow, at $j_{1}$, we cannot 'borrow money for cash inflow'.

Instead, we can only use the cash inflow to repay loan, i.e. reduce the amount we borrow. In view of this, we need to have an assumption, namely assuming that repayment can be made at arbitrary time. (The longer the time the payment is borrowed, the higher the amount of interest accumulated. Therefore, to minimize the interest payment and thus minimize the time needed for the accumulated value of the net cash flow to be nonnegative, we should repay the loan as soon as possible, when we receive some cash inflow.)

Then, we can also accumulate the cash inflow, or positive net cash flow, at $j_{1}$, because they can be treated as the reduction of accumulated value of loan at effective interest rate of $j_{1}$ (the accumulated value includes both the amount of loan that is not repaid, and the interest accumulated). To be more precise, for each positive net cash flow $C_{s}$, it is used to repay the loan at time $s$ and then the loan repaid will stop accumulating interest after time $s$. Therefore, it reduces the loan by $C_{s}$ at time $s$, and also reduces the future interest that will be kept accumulating if the loan is not repaid until the time $t$, to zero, i.e. reduces by $C_{s}\left((1+j_{1})^{t-s}-1\right)$. Therefore, the total amount of reduction of loan and interest is

$\underbrace {C_{s}} _{\text{loan}}+\underbrace {C_{s}\left((1+j_{1})^{t-s}-1\right)} _{\text{interest}}=\underbrace {C_{s}(1+j_{1})^{t-s}} _{\text{total}}.$

Then, at a certain cash inflow, its amount is sufficient for the loan to be completely repaid (the interest is also paid in the progress of repaying the loan using cash inflow), and then at that time point, the accumulated value of sum of each net cash flow will be greater than or equal to zero. Therefore, that cash flow is the last repayment of loan, and the time at which that cash flow is made is time $t$ (so the future interest equals zero, because there is no time for accumulating).

As a result, we can replace every $i$ with $j_{1}$ in the definition of discounted payback period, with the assumption that repayment can be made at arbitrary time.

If the project is profitable, the accumulated profit when the project ends at time $T$ is

$\underbrace {A(t^{*})} _{{\text{balance at }}t^{*}}\overbrace {(1+j_{2})^{T-t^{*}}} ^{{\text{ depositing }}A(t^{*})}+\underbrace {\sum _{t>t^{*}}C_{t}(1+j_{2})^{T-t}} _{\text{net cash flows after discounted payback period}}.$

in which $t^{*}$ is the time at the end of discounted payback period.

If the project is not profitable, then the accumulated profit is negative (i.e. we have loss) or zero (then we are indifferent in investing in the project), and we cannot use the above formula.

Exercise.

Continue from previous exercise. Assume that repayment can be made at arbitrary time. The effective annual interest rate for borrowing is 10% and that for depositing money is 6%.

1 Calculate the discounted payback period.

months

2 Suppose the project ends at the end of year 20. Calculate the accumulated profit at the end of the project. Correct your answer to two decimal places. Type n/a if the accumulated profit is negative.

3 Suppose the project ends at the end of year 21. Calculate the accumulated profit at the end of the project. Correct your answer to two decimal places. Type n/a if the accumulated profit is negative.

Reinvestment rates

In the section of idealized practical situation, we examine what happens if we connect the cash inflow to another project. The main idea in this subsection is similar to that case, and can be interpreted as the generalized version of this case, because we discuss some more ways of reinvestment in this subsection, not limited to connect to another project.

Suppose we invest 1 in a project, and the project pays interest at the effective interest rate of $i$ at the end of each period for $n$ periods, and the interest received is reinvested at effective interest rate of $j$. Then, accumulated value at the end of $n$ periods is

$\underbrace {1} _{\text{principal}}+\underbrace {is_{{\overline {n}}|j}} _{\text{accumulated value of interest}}.$

In particular, it equals $(1+i)^{n}$ if $i=j$, which is the same as the case in compound interest because this is equivalent to the definition of compound interest: interest earned is automatically reinvested back to the project at the same rate $i$.

                            -|
    -----------------------> |
    |   -------------------> | rate j
    |   |              ----> |
    |   |              |    -|
    |   |              |   
1   i   i              i   i
↓   ↑   ↑    ...       ↑   ↑
|---|---|--------------|---|--- 
0   1   2    ...      n-1  n

Suppose we invest 1 at the end of each period for $n$ periods at the effective interest rate of $i$, and the interest is reinvested at the effective interest rate of $j$. Then, the accumulated value at the end of $n$ periods is

$n+i(Is)_{{\overline {n-1}}|j}=n+i\left({\frac {{\ddot {s}}_{{\overline {n-1}}|j}-(n-1)}{j}}\right)=n+i\left({\frac {s_{{\overline {n}}|}-n}{j}}\right).$

In particular, it equals $s_{{\overline {n}}|i}$ if $i=j$.

    1   2   3   ...    n-1  n   total investment

    1   1   1   ...     1   1
    ↓   ↓   ↓   ...     ↓   ↓   
|---|---|---|-----------|---|--- 
0   1   2   3   ...    n-1  n
        ↓   ↓           ↓   ↓
        i  2i  ...  (n-2)i (n-1)i

Exercise.

Suppose we invest $k$ at the end of each quarter for $n$ years at the semiannual effective interest rate of $10\%$, and the interest is reinvested at the annual effective interest rate of $10\%$. Choose the correct expression for the accumulated value at the end of $n$ years.

	$ks_{{\overline {n}}|0.1}$
	$kn+(1.1^{1/2}-1)k(Is)_{{\overline {n-1}}|1.1^{1/4}-1}$
	$kn+(1.1^{1/4}-1)k(Is)_{{\overline {n-1}}|1.1^{1/2}-1}$
	$4kn+4(1.1^{1/2}-1)k(Is)_{{\overline {4n-1}}|1.1^{1/4}-1}$
	$4kn+4(1.1^{1/4}-1)k(Is)_{{\overline {4n-1}}|1.1^{1/2}-1}$

Interest measurement of a fund

To calculate the yield rate earned by an investment fund, we can use its earned effective interest rate. Recall that the definition of effective interest rate assumes that principal remains constant throughout the period, all interest earned throughout the period is paid at the end of the period. These assumption are usually not satisfied in practice, because there are usually irregular principal deposits and withdrawals (they are net cash flows), and irregular interest earning (possibly at different effective rate for each interval) throughout the period. (There may also be some time periods at which no interest is earned, i.e. the effective rate is zero.) To illustrate this, consider the following figure.

↓ ↑ ↑↑   ↓   ↓  ...  ↓↑   ↑    irregular principal deposits and withdrawals
|---|---|---|---------|---|---
0   1   2   3   ...  n-1  n
 |--------| |----|    |---|      irregular interest earning
 rate i_1  rate i_2 ... rate i_k

There are two ways to calculate a reasonable effective interest rate for such complicated situation, namely dollar-weighted rate of return (or interest) and time-weighted rate of return (or interest). These two ways can simplify the calculations involved.

Dollar-weighted rate of return

The aim is finding the effective interest rate $i$ earned by a fund over one measurement period. For simplicity, let us use the following notations:

• $A$ is the amount in fund at the start of the period
• $B$ is the amount in fund at the end of the period
• $I$ is the amount of interest earned during the period
• $C_{t}$ is the net amount of principal contributed at time $t\in [0,1]$. This value can be positive, negative or zero. (This is net cash flow)
• $C$ is the total net amount of principal contributed during the period, i.e. $\sum _{t}C_{t}$.
• $_{a}i_{b}$ is the amount of interest earned by 1 invested at time $b$ over the following period of length $a$, i.e. to time $a+b$, in which $a,b$ are positive real numbers such that $a+b\leq 1$ (because we are considering one measurement period).

     rate _{a}i_b 
     |---------|

-----|---------|-----
     b        a+b

Then, by definitions,

$\underbrace {B} _{\text{end}}=\underbrace {A} _{\text{start}}+\underbrace {C} _{\text{contribution}}+\underbrace {I} _{\text{interest}},$

and if we assume that all the interest earned, $I$, is received at the end of the period to be consistent to the definition of effective interest rate, the exact equation of $I$ is

$I=\underbrace {i} _{_{1}i_{0}}A+\sum _{t}{C_{t}}(_{1-t}i_{t}).$

To solve for $i$ in the above equation, the term for which we need to calculate in a more complicated way is $_{1-t}i_{t}$. Without any assumption, it is very hard or even impossible to calculate it directly. Therefore, we need to have an assumption to simplify the calculation and approximate its value.

If we assume compound interest throughout every period from time $t$ to $1$ (each period corresponds to one net cash flow.),

$_{1-t}i_{t}=(1+i)^{1-t}-1$

because the length of time involved is $1-t$. If we put this into the exact equation of $I$, the equation can be solved by iteration using computer or financial calculator. This is not the main focus in this subsection. Instead, the following is the main focus.

If we want to simplify our calculation, we can assume simple interest throughout every period from time $t$ to $1$, (each period corresponds to one net cash flow.) then

$_{1-t}i_{t}=\underbrace {(1-t)} _{\text{time length}}i.$

Putting it into the exact equation of $I$, we can solve for $i$ and

$I=iA+\sum _{t}C_{t}(1-t)i\iff i={\frac {I}{A+\sum _{t}C_{t}(1-t)}}$.

This is referred as the dollar-weighted rate of return (or interest). Let's define it formally as follows:

Definition. (Dollar-weighted rate of return) Dollar-weighted rate of return is the simple interest rate approximation to the yield rate (or the effective interest rate), and is given by

$i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-t)}}.$

Remark.

• This approximation is close to the exact effective interest rate if $C_{t}$'s are small compared to $A$.

• This is because when $C_{t}$'s are small, then $\sum _{t}^{}C_{t}(1-t)\approx 0$, and thus $i^{DW}\approx {\frac {I}{A}}$.
• On the other hand, the effective interest rate $i={\frac {I}{A}}$ by definition. So, $i^{DW}\approx i$.

• It is dollar-weighted in the sense that the terms involved are mostly related to dollar ($A,C_{t},I$)

Because the calculation of the numerator term can be tedious, we may further assume that every net principal contribution occurs at time $t=0.5$, then we have

$i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-0.5)}}={\frac {I}{A+0.5C}}={\frac {I}{A+0.5(B-A-I)}}={\frac {2I}{A+B-I}}$

   A                       B = A+C+I
---|-----------|-----------|---
   0          0.5          1
               +C

Apart from the advantage of simpler calculation, another advantage is that we can calculate $i$ using $A,B,I$ only, and we do not need to know the values of $C_{t}$'s.

Exercise.

Consider a fund with amount of 20000 at the beginning of year 1.

During year 1, interest of 1000 is earned for the fund, and there are net cash flows of 2000, 3000, 4000

at the time $t=0.2,0.3,0.4$ respectively for the fund.

1 Calculate dollar-weighted rate of return assuming that every net principal contribution occurs at time $t=0.5$.

	3.33%
	3.83%
	4%
	4.08%
	5%

2 Calculate dollar-weighted rate of return without assuming that every net principal contribution occurs at time $t=0.5$.

	3.33%
	3.83%
	4%
	4.08%
	5%

3  

The dollar-weighted rate of return assuming that every net principal contribution occurs at time $t=$ equals

the dollar-weighted rate of return without assuming when the net principal contribution occurs. (Correct your answer to two decimal places.)

Exercise. Prove that the dollar-weighted rate of return assuming that every net principal contribution occurs at time $t=k\in [0,1]$ is

${\frac {I}{kA+(1-k)B-(1-k)I}}.$

Proof

The dollar-weighted rate of return with such assumption is

$i^{DW}={\frac {I}{A+\sum _{t}C_{t}(1-k)}}={\frac {I}{A+(1-k)C}}={\frac {I}{A+(1-k)(B-A-I)}}={\frac {I}{A+(1-k)B-(1-k)A+(1-k)I}}={\frac {I}{kA+(1-k)B+(1-k)I}}.$$\Box$

Time-weighted rate of return

For dollar-weighted rate of return, it is sensitive to the amount of money invested during different subperiods. To see this, consider the following situations for a fund.

Situation 1:

        +50            contribution
   100   50    200     balance
----|-----|-----|----
    0    0.5    1
     \   / \   /
      \ /   \ /
     -50%  +100%      effective interest rate

Dollar-weighted rate of return for the entire period is

${\frac {\overbrace {200} ^{B}-\overbrace {50} ^{C}-\overbrace {100} ^{A}}{\underbrace {100} _{A}+\underbrace {50} _{C_{0.5}}(1-\underbrace {0.5} _{t})}}=40\%.$

Situation 2:

   100   50    100      balance
----|-----|-----|----
    0    0.5    1
     \   / \   /
      \ /   \ /
     -50%  +100%      effective interest rate

Dollar-weighted rate of return for the entire period is $0\%$ because $I=\underbrace {100} _{B}-\underbrace {100} _{A}-\underbrace {0} _{C}=0$.

Situation 3:

        -25            contribution
   100   50    50      balance
----|-----|-----|----
    0    0.5    1
     \   / \   /
      \ /   \ /
     -50%  +100%      effective interest rate

Dollar-weighted rate of return for the entire period is

${\frac {\overbrace {50} ^{B}-(\overbrace {-25} ^{C})-\overbrace {100} ^{A}}{\underbrace {100} _{A}\underbrace {-25} _{C_{0.5}}(1-\underbrace {0.5} _{t})}}\approx -28.57\%.$

The dollar weighted rate of return for each of the situation is very different. However, the effective interest rate of the fund at each subperiod is the same in every situation, and the effective interest rate for the entire period is $(1-50\%)(1+100\%)-1=0\%$ which measures the 'performance' of the fund. In two of the situations, the dollar-weighted rate of return differ from $0\%$ a lot. Therefore, this is inaccurate to measure the performance of the fund. A more accurate way is using the time-weighted rate of return which is not affected by the amount of contribution and balance.

Definition. (Time-weighted rate of return) Consider the following diagram.

            C_{t_1} C_{t_2}   C_{t_3}  ... C_{t_{m-1}}  Net contribution to the fund
----|----------|-----|---------|---------------|-----|
  t_0=0       t_1   t_2       t_3    ...     t_{m-1} t_m=1
  B'_0       B'_1   B'_2      B'_3    ...    B'_{m-1} B'_m    Fund value 
     \       / \   /  \       /                 \   /
      \    /    \ /    \    /                    \ /
       \ /      j_2     \  /          ...        j_m       Yield rate
       j_1               j_3

in which

• $C_{t_{n}}$ is the net contribution to the fund at time $t_{n}\in [0,1]$
• $C_{t_{0}}=C_{t_{m}}=0$ because there is no net contribution at time $t_{0}$ and $t_{m}$.
• $B'_{n}$ is the fund value just before the $n$th contribution, and $n\in [1,m-1]$ because there are only $m-1$ net contributions. For $B'_{0},B'_{m}$, they are defined as follows to make the following formula of time-weighted rate of return valid for each $k\in 1,2,\ldots ,m$
• $B'_{0}$ is the fund value at the beginning of the period.
• $B'_{m}$ is the fund value at the end of the period.
• $j_{n}$ is the time-weighted rate of return over the $n$th subinterval.

Assume that there are $m-1$ net contributions to the fund in total, and they do not occur exactly at the beginning or the end of the period. Then there are $m$ subinterval corresponding to the $m-1$ net contributions, as shown in the above diagram. The time-weighted rate of return over the $k$th subinterval (i.e. from time $t_{k-1}$ to time $t_{k}$), is defined by

$1+j_{k}={\frac {\overbrace {B'_{k}} ^{{\text{Balance just before }}k{\text{th contribution}}}}{\underbrace {B'_{k-1}+C_{t_{k-1}}} _{{\text{Balance at time }}t_{k-1}}}},$

and the yield rate over the whole period ($i^{TW}$) by time-weighted rate of return is defined by

$1+i^{TW}=(1+j_{1})(1+j_{2})\cdots (1+j_{m}).$

Remark.

• This rate is not consistent with the assumption of compound interest, because the amount of principal changes throughout the period.
• It is time-weighted in the sense that the yield rate by time-weighted rate of return is affected by the time points at which the net contributions are made.
• Time-weighted rate of return over the $k$th subinterval is the effective interest rate over the $k$th subinterval, because the balance at time $t_{k-1}$ accumulates at the effective interest rate to the balance just before the $k$th net contribution, and there are no net contributions during the accumulation process.
• The yield rate over the whole period is the effective interest rate over the whole period.

It can be observed that the time-weighted rate of return for each of three previous situation is

$(1-50\%)(1+100\%)-1=0\%$

which is not affected by the balance and contribution. Therefore, it is more accurate than dollar-weighted rate of return in this situation. Although it is more accurate, it requires more information than dollar-weighted rate of return, because we need the information about contributions at different time points and the corresponding balance at each time point to calculate the overall yield rate. On the other hand, only information about the balance at the beginning and the end, and the contributions are needed for dollar-weighted rate of return for dollar-weighted rate of return Therefore, it is sometimes impossible to calculate the overall yield rate using time-weighted rate of return, but possible when we use the dollar-weighted rate of return

Exercise.

Consider a fund with amount 10000 at the beginning of a year, 15000 at the end of first quarter of the year, 17000 at July 31 of the year, and 22000 at the end of the year. At the end the first quarter, an extra 3000 is deposited in the fund, and at July 31 of the year, 1000 is withdrawn from the fund.

1 Calculate the yield rate over the year using dollar-weighted rate of return over the year. Type n/a if it is impossible. Correct your answer to two decimal places.

%

2 Calculate the yield rate over the year using time-weighted rate of return. Type n/a if it is impossible. Correct your answer to two decimal places.

%

Stocks

We will discuss two types of stock: preferred stock (or preference share) and common stock (or ordinary share) in this section.

In general, preferred stock is a fixed-income security which provides fixed dividends periodically. However, it differs from bond, which also gives fixed income, since preferred stock is an ownership security instead of a debt security (bond is a debt security), and also preferred stock ranks behind bonds and other debt instructments in terms of the degree of security. This is because all payments on indebtedness must be made before preferred stock receives a dividend.

Proposition. (Price of preferred stock without maturity date) The price of preferred stock without maturity date, with amount of each periodic dividend equal $D$ is

$$P={\frac {D}{i}}.$$

Proof. To set a 'fair' price for the stock that reflects its value, the price is set to be the present value of all periodic future dividends, which last forever, i.e.

$$P=Da_{{\overline {\infty }}|i}={\frac {D}{i}}.$$

$\Box$

Common stock is also a type of ownership security. However, it does not earn a fixed dividend rate. Instead, common stock dividends are paid only after interest payments on all bonds and other debt and dividends on preferred stock are paid. Thus, common stock has an even lower degree of security than preferred stock. Also, the common stock price is usually quite volatile.

We can compute the theoretical price (which is generally not the price in reality) of common stock based on the dividend discount model, i.e. the price should represent the present value of future dividends, similar to that of preferred stock.

Proposition. (Price of common stock with dividends varying in geometric sequence) Let $D$ be the common stock dividend at the end of current period. Suppose the dividends are expected to change in a geometric sequence with common ratio $1+k$ indefinitely, in which $-1<k<i$. Then, the theoretical price is

$${\frac {D}{i-k}}.$$

Proof. The theoretical price is

$$Dv+\underbrace {D(1+k)v^{2}} _{Dv[v(1+k)]}+\cdots ={\frac {Dv}{1-v(1+k)}}={\frac {Dv(1+i)}{1+i-v(1+i)(1+k)}}={\frac {D}{i-k}}.$$

$\Box$

Illustration of the dividends paid:

       D  D(1+k)  ...
---|---|---|------------
   0   1   2      ...

Example. Suppose a common stock pay annual dividends with amount $10$ at the end of each year forever. The common stock dividend is expected to increase by $5\%$ per year.

An investor purchases this common stock at the price of $200$. Compute the yield rate $i$.

Solution:

$$200=10v+10(1.05)v^{2}+\cdots \Rightarrow 200={\frac {10v}{1-1.05v}}\Rightarrow 200-210v=10v\Rightarrow v\approx 0.90909\Rightarrow i=0.1$$

Exercise.

1 Suppose there is now more pessimistic view on the common stock, and thus the common stock dividend is now expected to increase by $3\%$ per year only. Then, the yield rate

	increases.
	decreases.
	remains unchanged.
	has uncertain change.

2 Suppose the common stock now pays annual dividends with amount $5$ at the end of each year forever instead. The dividend is expected to increase by $x\%$ such that the yield rate remains unchanged. Then, $x$ is

	$\geq 5$ and $<6$.
	$\geq 6$ and $<7$.
	$\geq 7$ and $<8$.
	$\geq 8$ and $<9$.
	$\geq 9$ and $<10$.

Optional topic: short sales

Some investors in securities use short sales (or short transactions) when they think that the price of a security is likely to decline. When there is a short sale, sale occurs first and the purchase occurs later (ideally at a price that is lower than selling price).

Because of this terminology, we say the normal transaction (purchase occurs first, then sale occurs later), as long transactions.

Typically, with a short sale, the investor borrows (not purchases) the security from a 2nd party, and sells it in the market to a 3rd party. Later, the investor buys back the security from the market (ideally at a lower price than selling price), to return the security to the 2nd party. The process of buying back the security is called 'covering the short'.

Practically, when the sale is made, the short seller need to make a deposit of a percentage of the price, and the percentage is called margin and cannot be recovered by the short seller, until the short position is covered (i.e. 'covering the short' is done). Short seller will be credited interest on the margin deposit.

Also, if the security pays dividends, then the short seller need to pay these to the purchaser of the security, without receiving them from the seller of security, e.g. company. That is, the short seller plays the role of the company selling the security temporarily, and need to pay the dividend to the 3rd party.

Example. An investor sold a stock short for $10000$ and bought back for $9500$ at the end of 1st year. The stock paid $700$ in dividend during the year. Given that the margin requirement is $30\%$, and the interest rate on the margin is $10\%$. Compute the yield rate ofthe investor (which is ${\frac {\text{net profit}}{\text{margin deposit}}}$, since the margin deposit can be viewed as the amount of investment, and the yield rate is the rate of return on the investment).

Solution:

• gain on short sale is $10000-9500=500$
• interest on margin is $0.1(10000(0.3))=300$
• dividend on stock is $700$

So, the net profit is

$$500+300-700=100.$$

The yield rate is

$${\frac {100}{10000(0.3)}}\approx 0.033333.$$

Exercise.

1 Suppose the stock paid $D$ in dividend instead during the year, such that the yield rate is zero. Compute $D$.

	$500$.
	$600$.
	$700$.
	$800$.
	$900$.

2 Suppose the investor had a wrong prediction on the price of the stock, and the stock price actually increased to $10500$ at the end of 1st month (or at the end of $1/12$th of 1st year) after the investor sold the stock short. So, the investor bought back the stock for $10500$ to avoid further losses from the short sale. Holding other conditions constant, the yield rate of the investor is

	$\geq -0.4$ and $<-0.3$.
	$\geq -0.3$ and $<-0.2$.
	$\geq -0.2$ and $<-0.1$.
	$\geq -0.1$ and $<0$.
	$\geq 0$ and $<0.1$.

Term structure of interest rate

The effective interest rates vary according to the term of investment, which is shown by yield curve.

Definition. (Yield curve) Yield curve is a curve showing the relationship between the effective interest rate and term of investment.

Remark.

• Typically, the effective interest rate increases when the term is longer.
• A yield curve is normal or upward-sloping if the interest rate increases with term.
• A yield curve is inverted or downward-sloping if the interest rate decreases with term.
• A yield curve is bowed if the interest rate increases and then decreases, or decreases and then increases with term</ref>.
• A yield curve is flat if the interest rate is the same for each term.

There are some theories explaining the change of effective interest rate when the term changes in length, which are explained in the Financial Math FM/Determinants of Interest Rates chapter.

Spot rates

When we are computing yield rate of arbitrary group of fixed interest securities at arbitrary given date, the interest rates vary according to the term of the investment, as shown in the previous subsection. So, we need to take this variation into consideration.

If there are factors other than term that vary, e.g. frequency of coupon payments, then it makes the comparison between different groups of the securities complicated. So, to avoid complications, we compare short-term and long-term interest rates with reference to zero-coupon bonds, by considering each security as a combination of (notional) zero-coupon bond(s), if we assume that there is no arbitrage (i.e. risk-free trading profit) (this is called no-arbitrage assumption).

After assuming there is no arbitrage, it is impossible for the fixed-interest security and the combination of zero-coupon bonds that replicates the security to have two different prices (this is called the law of one price), otherwise, investor may be able to gain a risk-free trading profit using the price difference. We will not discuss the strategy to obtain arbitrage in this book. However, such arbitrage opportunity is rare in modern financial market, and also, even if it exists, it will be quickly eliminated after being spotted by some investors, and exploited by them.

Actually, the spot rate of interest is quite related to the zero-coupon bond.

Definition. (Spot rate of interest) The n-year spot rate of interest is the annual yield rate of zero-coupon bonds of term $n$, denoted by $i_{n}$.

Remark.

• since interest rate varies according to the term of investment, $i_{n}$ has different values for different value of $n$ generally.
• the term structure of interest rates means the spot rate vector $(i_{1},i_{2},\ldots ,i_{n})$ (the numbers inside are spot rates)
• $n$ is often positive integer, with year as unit, but in some rare cases, it may be positive decimal number, with different unit

Example. (Considering a fixed-interest bond as a combination of zero-coupon bonds) Consider a 10-year bond that pays coupons with amount $20$ at the end of each year and has the redemption value of $500$ at the end of 10th year. Given that the $k$-year spot rate is $0.01k$, compute $P$.

Then, we can consider this bond as a combination of zero coupon bonds as follows:

• for the ten annual coupons, we replicate them by ten zero coupon bonds, with redemption value $20$ at the end of 1st, 2nd, ... , 10th year
• for the redemption payment, we replicate it by a zero coupon bond, with redemption value $500$ at the end of 10th year

Thus, the bond price is

$$P=20(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{10}}^{10})+500v_{i_{10}}^{10}=20(1.01^{-1}+1.02^{-2}+\cdots +1.1^{-10})+500(1.1)^{-10}\approx 644.759{\text{ (by direct computation)}}$$

Forward rates

Definition. (Forward rates) The forward rate $f_{t\to t+r}$ represents the annual interest rate payable on a risk-free investment made at time $t$ and maturing at time $t+r$.

The forward rates can be computed from the spot rates, vice versa. This is because an investment of $k$ in a $t$-year zero-coupon bond followed by an investment of the redemption value (which is $k(1+i_{t})^{t}$, since $k=Cv_{i_{t}}^{t}$) from this bond in an $r$-year zero-coupon bond is worth the same as an investment of $k$ in a $(t+r)$-year zero-coupon bond. That is,

$$k(1+i_{t})^{t}(1+f_{t\to t+r})^{r}=k(1+i_{t+r})^{t+r}\Rightarrow (1+i_{t})^{t}(1+f_{t\to t+r})^{r}=(1+i_{t+r})^{t+r}.$$

Example. Consider a 3-year bond, with coupon of amount $10$ paid at the end of year 1 and year 3, and its redemption value is $50$. Given that the 1-year spot rate is $5\%$ and 1-year deferred 2-year forward rate (i.e. $f_{1\to 3}$) is $7\%$, compute $P$. Solution: The 3-year spot rate is $\left((1.05)(1.07)^{2}\right)^{1/3}-1\approx 0.063291360$.

$$P=10(1.05)^{-1}+10(1.063291360)^{-3}+50(1.063291360)^{-3}\approx 59.43459.$$

Par yields

Definition. (Par yield) The $n$-year par yield, denoted by $i_{p_{n}}$ is the coupon per unit nominal (i.e. $Fr/F=r$) that would be payable on a bond with a term of $n$ years, that would give the bond a current price under current term structure of $1$ per unit nominal (i.e. $A=P/F$), assuming the bond is redeemed at par (i.e. $C=1$). That is,

$$i_{p_{n}}\left(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{n}}^{n}\right)+v_{i_{n}}^{n}=1.$$

(by the formula $P/F=Fr/F\left(v_{i_{1}}+v_{i_{2}}^{2}+\cdots +v_{i_{n}}^{n}\right)+Cv_{i_{n}}^{n}$)

Remark.

• the par yields given an alternative measure of the relationship between the yield and the term of investments
• the difference between $n$-year par yield and $n$-year spot rate is called the $n$-year coupon bias

Illustration of par yield:

   1  ipn ipn                 ipn 1
   ↓   ↑   ↑                   ↑↗ 
---|---|---|-------------------|---
   0   1   2                   n

Example. Given that the annual term structure of interest rate is $(0.01,0.02,0.03,\ldots ,)$, compute the 3-year par yield.

Solution: Since

$${\begin{aligned}&&i_{p_{3}}(v_{i_{1}}+v_{i_{2}}^{2}+v_{i_{3}}^{3})+v_{i_{3}}^{3}&=1\\&\Rightarrow &i_{p_{3}}(1.01^{-1}+1.02^{-2}+1.03^{-3})&=1\\&\Rightarrow &i_{p_{3}}\approx 0.3488685.\end{aligned}}$$

Exercise.

1 Compute the 3-year spot rate.

	$0.01$
	$0.02$
	$0.03$
	$1.02$
	$1.03$

2 Compute the 3-year coupon bias.

	$0.049$.
	$0.149$.
	$0.319$.
	$0.349$.
	$0.489$.

3 Compute the 2-year deferred 1-year forward rate.

	$0.00980$.
	$0.00985$.
	$0.01980$.
	$0.00489$.
	$0.00494$.

Macaulay duration, modified duration, modified convexity, and Macaulay convexity

Macaulay duration measures the average length or term of a financial transaction.

There is another way to measure the average term to maturity, namely the method of equated time. The index is computed as the weighted average of different payments in which the weights are the amount paid. E.g., if the payment is $s_{k}$ at time $t=k$, then the average term to maturity is

$${\overline {t}}={\frac {\sum _{}^{}ts_{t}}{\sum _{}^{}s_{t}}}.$$

However, it does not consider the effect of interest rate, and Macaulay duration considers the effect, and thus is generally a better index.

Definition. (Macaulay duration) The Macaulay duration $\operatorname {D} _{\text{mac}}(i)$ is

$${\frac {t_{1}v_{i}^{t_{1}}C_{t_{1}}+\cdots +t_{n}v_{i}^{t_{n}}C_{t_{n}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}$$

in which cash flow $C_{t_{k}}$ is made at time $t_{k}$.

Illustration:

      Ct1 Ct2        Ctk     Ctn
---|---|---|----------|-------|---    
   0  t_1 t_2   ...  t_k ... t_n

Remark.

• The Macaulay duration is the mean term of cash flows weighted by their present values (instead of simply amount, which is used in method of equated time)
• The Macaulay duration is a function of $i$, i.e. it changes as $i$ changes
• Alternative names of Macaulay duration include discounted mean term.
• Unless otherwise specified, "duration" means Macaulay duration.

Example. A 20-year bond with par value $10$ has a $3\%$ semiannual coupon rate. Suppose its redemption value is $20$, and the semiannual yield rate is $i=5\%$. Compute its Macaulay duration.

Solution: Its Macaulay duration is

$${\frac {10(0.03)(Ia)_{{\overline {40}}|0.05}+40(20)(1.05)^{-40}}{10(0.03)a_{{\overline {40}}|0.05}+20(1.05)^{-40}}}\approx 23.4893.$$

Exercise.

1 Compute its average term to maturity using method of equated time.

	$20.5$
	$23.2857$
	$32.6183$
	$32.6875$
	$32.7647$

2 Compute the duration if $i=0\%$.

	$20.5$
	$23.2857$
	$32.6183$
	$32.6875$
	$32.7647$

To measure the sensitivity of a series of cash flows to movements in the interest rates is the volatility.

Definition. (Modified duration) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. Let $P(i)$ be the present value of the cash flows at rate $i$, i.e.

$$P(i)=C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}.$$

The modified duration is

$$\operatorname {D} _{\text{mod}}(i)={\color {darkgreen}-}{\frac {P'(i)}{P(i)}}={\frac {C_{t_{1}}t_{1}v_{i}^{t_{2}}+\cdots +C_{t_{n}}t_{n}v_{i}^{t_{n+1}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}={\frac {\operatorname {D} _{\text{mac}}(i)}{1+i}}$$

Remark.

• It measures the rate of change of $P(i)$ with $i$ and is independent of the size of the present value (because of the denominator)
• It is analogous to force of interest, and the negative sign is added to ensure that the volatility is positive, since $P'(i)<0$, considering that ${\frac {d}{di}}v^{t}<0$ for each $t$
• There is close relationship between Macaulay duration and modified duration, and hence the name modified duration
• Alternative name includes volatility.

Definition. (Modified convexity) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. Let $P(i)$ be the present value of the cash flows at rate $i$, i.e.

$$P(i)=C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}.$$

The modified convexity is

$$\operatorname {C} _{\text{mod}}(i)={\frac {P''(i)}{P(i)}}={\frac {C_{t_{1}}(t_{1})(t_{1}+1)v_{i}^{t_{3}}+\cdots +C_{t_{n}}(t_{n})(t_{n}+1)v_{i}^{t_{n+2}}}{C_{t_{1}}v_{i}^{t_{1}}+\cdots +C_{t_{n}}v_{i}^{t_{n}}}}.$$

Remark.

• Unless otherwise specified, "convexity" refers to modified convexity.
• In mathematics, whether a function is "convex" is related to its second derivative, and hence the name "convexity".

• The convexity is modified when compared to the Macaulay convexity.

Definition. (Macaulay convexity) Consider a series of cash flows $C_{t_{1}},C_{t_{2}},\ldots ,C_{t_{n}}$. The Macaulay convexity is

$$\operatorname {C} _{\text{mac}}(i)={\frac {t_{1}^{2}C_{t_{1}}v^{t_{1}}+\dotsb +t_{n}^{2}C_{t_{n}}v^{t_{n}}}{C_{t_{1}}v^{t_{1}}+\dotsb +C_{t_{n}}v^{t_{n}}}}$$

Remark.

• Since the expression for this convexity is similar to the one for Macaulay duration (each time in the numerator is squared only) and it is also similar to the one for (modified) convexity, it is named Macaulay convexity.

The usage of convexity is as follows: consider a small change in interest from $i$ to $i+\epsilon$. By Taylor series expansion,

$$P(i+\epsilon )=P(i)+\epsilon P'(i)+{\frac {\epsilon ^{2}}{2}}P''(i+\delta )$$

in which $0<|\delta |<|\epsilon |$. Thus, using both volatility and convexity gives an approximation to the change in $P(i)$ by a small change in interest rates. This will be quite useful in Immunization chapter.

Example. Suppose there are cash inflows of $100$, $200$ and $300$ at the end of year 2,3 and 4 respectively, and that the annual interest rate is $5\%$. Its duration is

$$\operatorname {D} _{\text{mac}}(i)={\frac {100(2)v_{0.05}^{2}+200(3)v_{0.05}^{3}+300(4)v_{0.05}^{4}}{100v_{0.05}^{2}+200v_{0.05}^{3}+300v_{0.05}^{3}}}={\frac {200(1.05)^{-2}+600(1.05)^{-3}+1200(1.05)^{-4}}{100(1.05)^{-2}+200(1.05)^{-3}+300(1.05)^{-4}}}\approx 3.30593.$$

Its modified duration is

$$\operatorname {D} _{\text{mod}}(i)={\frac {d(i)}{1.05}}\approx 3.1485.$$

Its convexity is

$$\operatorname {C} _{\text{mod}}(i)={\frac {100(2)(3)v_{0.05}^{4}+200(3)(4)v_{0.05}^{5}+300(4)(5)v_{0.05}^{6}}{100v_{0.05}^{2}+200v_{0.05}^{3}+300v_{0.05}^{4}}}\approx 13.4267$$

Exercise. Let $P(i)$ be the net present value of the cash flows in this example.

1 Compute $P(0.05)$.

	$-535.80$
	$-510.28$
	$510.28$
	$535.80$
	$1686.95$

2 Compute $P'(0.05)$.

	$-1686.95$
	$-1606.62$
	$3.15$
	$1606.62$
	$1686.95$

3 Compute $P''(0.05)$.

	$-7194.03$
	$-6851.38$
	$13.43$
	$6851.38$
	$7194.03$

Approximation for change in present value due to a small change in interest rate

We can use Macaulay duration or modified duration to approximate the change in present value from a small change in interest rate.

Proposition. (First-order Macaulay approximation of change in present value) Let $P(i)$ be the present value under interest rate $i$. Then, when interest rate changes slightly from $i_{0}$ to $i$, the new present value is

$$P(i)\approx P(i_{0})\left({\frac {1+i_{0}}{1+i}}\right)^{\operatorname {D} _{\text{mac}}(i_{0})}$$

given that the change in interest rate is small.

Proof.

• First, $P(i)\approx P(i_{0})\left({\frac {1+i_{0}}{1+i}}\right)^{\operatorname {D} _{\text{mac}}(i_{0})}\Leftrightarrow P(i)(1+i)^{\operatorname {D} _{\text{mac}}(i_{0})}\approx P(i_{0})(1+i_{0})^{\operatorname {D} _{\text{mac}}(i_{0})}$.
• Define function $Q_{t}(i)=P(i)(1+i)^{t}$.
• Then, since for each $i$,

$${\begin{aligned}&&Q_{t}'(i)&=0\\&\Leftrightarrow &P(i)t(1+i)^{t-1}+(1+i)^{t}P'(i)&=0\\&\Leftrightarrow &t&={\frac {-P'(i)(1+i)^{t}}{P(i)(1+i)^{t-1}}}\\&&&=\underbrace {\frac {-P'(i)}{P(i)}} _{=\operatorname {D} _{\text{mod}}(i)}(1+i)\\&&&=\operatorname {D} _{\text{mac}}(i),\end{aligned}}$$

• using the first-order Taylor approximation (because the change in interest rate is small) to $Q_{t}(i)$ about $i_{0}$ with $t=\operatorname {D} _{\text{mac}}(i_{0})$, we have

$${\begin{aligned}&&Q_{t}(i)&\approx Q_{t}(i_{0})+(i-i_{0})\underbrace {Q_{t}'(i_{0})} _{=0{\text{ when }}t=\operatorname {D} _{\text{mac}}(i_{0})}\\&&&=Q_{t}(i_{0})\\&\Leftrightarrow &P(i)(1+i)^{D_{\text{mac}}(i_{0})}&\approx P(i_{0})(1+i_{0})^{D_{\text{mac}}(i_{0})},\end{aligned}}$$

• and the result follows.

$\Box$

Proposition. (First-order modified approximation of change in present value) Let $P(i)$ be the present value under interest rate $i$. Then, when interest rate changes from $i_{0}$ to $i$, the new present value is

$$P(i)\approx P(i_{0}){\big (}1-(i-i_{0})\operatorname {D} _{\text{mod}}(i_{0}){\big )}$$

given that the change in interest rate is small.

Proof.

• Using first -order Taylor approximation (because the change in interest rate is small) to $P(i)$ about $i_{0}$,

$$P(i)\approx P(i_{0})+(i-i_{0})P'(i_{0})=P(i_{0}){\color {darkgreen}-}{\color {blue}P(i_{0})}(i-i_{0})\cdot \underbrace {\frac {{\color {darkgreen}-}P'(i_{0})}{\color {blue}P(i_{0})}} _{\operatorname {D} _{\text{mod}}(i_{0})}=P(i_{0}){\big (}1-(i-i_{0})\operatorname {D} _{\text{mod}}(i_{0}){\big )},$$

as desired.

$\Box$