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

Template:Colored definition Template:Colored remark 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. Template:Colored definition Template:Colored remark An example to illustrate how we can use net present value to determine profitability of investment is in the following. Template:Anchor Template:Colored exercise

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. Template:Colored definition Template:Colored remark Let's illustrate the split of initial investment mentioned in the remark in the following exercise.

Template:Anchor Consider the following cash flows for an investment.

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

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.

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

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

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 '$\le T$' 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.

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.

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.

$◻$

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.

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.

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: Template:Anchor Template:Colored definition Template:Colored remark Template:Anchor Template:Colored exercise

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 $C_s((1+j_1{)}^{t-s}-1)$. Therefore, the total amount of reduction of loan and interest is

$\underset{\text{loan}}{\underbrace{C_s}}+\underset{\text{interest}}{\underbrace{C_s((1+j_1{)}^{t-s}-1)}}=\underset{\text{total}}{\underbrace{C_s(1+j_1{)}^{t-s}}}.$

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

$\underset{\text{balance at }{t}^{*}}{\underbrace{A(t^{*})}}\stackrel{\text{ depositing }A(t^{*})}{\overbrace{(1+j_2{)}^{T-t^{*}}}}+\underset{\text{net cash flows after discounted payback period}}{\underbrace{\sum _{t>t^{*}}{C}_t(1+j_2{)}^{T-t}}}.$

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.

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

$\underset{\text{principal}}{\underbrace{1}}+\underset{\text{accumulated value of interest}}{\underbrace{i{s}_{\bar{n}|j}}}.$

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

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

In particular, it equals $s_{\bar{n}|i}$ 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

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 $a+b\le 1$ (because we are considering one measurement period).

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

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

Then, by definitions,

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

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

$I=\underset{{}_1{i}_0}{\underbrace{i}}A+\sum _t{C}_t{(}_{1-t}{i}_t).$

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 Template:Colored em 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 Template:Colored em simple interest throughout every period from time $t$ to $1$, (each period corresponds to one net cash flow.) then

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

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

$I=iA+\sum _t{C}_t(1-t)i⟺i=\frac{I}{A+\sum _t{C}_t(1-t)}$.

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

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

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

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 $0\mathrm{\%}$ because $I=\underset{B}{\underbrace{100}}-\underset{A}{\underbrace{100}}-\underset{C}{\underbrace{0}}=0$.

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

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

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 $(1-50\mathrm{\%})(1+100\mathrm{\%})-1=0\mathrm{\%}$ which measures the 'performance' of the fund. In two of the situations, the dollar-weighted rate of return differ from $0\mathrm{\%}$ 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.

Template:Colored definition Template:Colored remark It can be observed that the time-weighted rate of return for each of three previous situation is

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

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 Template:Colored exercise

Stocks

We will discuss two types of stock: Template:Colored em (or preference share) and Template:Colored em (or ordinary share) in this section.

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

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. $$P=D{a}_{\bar{\mathrm{\infty }}|i}=\frac{D}{i}.$$

$◻$

Template:Colored em stock is also a type of ownership security. However, it does not earn a Template:Colored em dividend rate. Instead, common stock dividends are paid only Template:Colored em 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 Template:Colored em price (which is generally not the price in reality) of common stock based on the Template:Colored em, i.e. the price should represent the present value of future dividends, similar to that of preferred stock.

Proof. The theoretical price is $$Dv+\underset{Dv[v(1+k)]}{\underbrace{D(1+k)v^2}}+\cdots =\frac{Dv}{1-v(1+k)}=\frac{Dv(1+i)}{1+i-v(1+i)(1+k)}=\frac{D}{i-k}.$$

$◻$

Illustration of the dividends paid:

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

Term structure of interest rate

The effective interest rates vary according to the term of investment, which is shown by Template:Colored em. Template:Colored definition Template:Colored remark There are some theories explaining the change of effective interest rate when the term changes in length, which are explained in the [[../Determinants of Interest Rates]] chapter.

Spot rates

When we are computing yield rate of arbitrary Template:Colored em of fixed interest securities at arbitrary given date, the interest rates vary according to the Template:Colored em 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 Template:Colored em of (notional) Template:Colored em, if we assume that there is no Template:Colored em (i.e. risk-free trading profit) (this is called Template:Colored em).

After assuming there is no Template:Colored em, 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 Template:Colored em), 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 Template:Colored em is quite related to the zero-coupon bond. Template:Colored definition Template:Colored remark Template:Colored example

Forward rates

Template:Colored definition 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 $k=C{v}_{i_t}^t$) 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, $$k(1+i_t{)}^t(1+f_{t\to t+r}{)}^r=k(1+i_{t+r}{)}^{t+r}\Rightarrow (1+i_t{)}^t(1+f_{t\to t+r}{)}^r=(1+i_{t+r}{)}^{t+r}.$$ Template:Colored example

Par yields

Template:Colored definition Template:Colored remark Illustration of Template:Colored em:

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

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

Template:Colored em measures the Template:Colored em or Template:Colored em of a financial transaction.

There is another way to measure the average term to maturity, namely the Template:Colored em. The index is computed as the Template:Colored em 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 $$\bar{t}=\frac{\sum _{}^{}t{s}_t}{\sum _{}^{}{s}_t}.$$ However, it does Template:Colored em consider the effect of interest rate, and Template:Colored em considers the effect, and thus is generally a better index. Template:Colored definition Illustration:

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

Template:Colored remark Template:Colored example To measure the sensitivity of a series of cash flows to movements in the interest rates is the Template:Colored em. Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark Template:Colored definition Template:Colored remark The usage of Template:Colored em is as follows: consider a small change in interest from $i$ to $i+ϵ$. By Taylor series expansion, $$P(i+ϵ)=P(i)+ϵP^{\prime }(i)+\frac{{ϵ}^2}{2}{P}^{″}(i+\delta )$$ in which $0<|\delta |<|ϵ|$. Thus, using both Template:Colored em and Template:Colored em gives an Template:Colored em to the change in $P(i)$ by a small change in interest rates. This will be quite useful in [[../Immunization/]] chapter. Template:Colored example

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

We can use Macaulay duration or modified duration to Template:Colored em the change in present value from a Template:Colored em change in interest rate. Template:Colored proposition

Proof.

• First, $P(i)\approx P(i_0){\left(\frac{1+i_0}{1+i}\right)}^{{\mathrm{D}}_{\text{mac}}(i_0)}\iff P(i)(1+i{)}^{{\mathrm{D}}_{\text{mac}}(i_0)}\approx P(i_0)(1+i_0{)}^{{\mathrm{D}}_{\text{mac}}(i_0)}$.
• Define function $Q_t(i)=P(i)(1+i{)}^t$.
• Then, since for each $i$,

$$\begin{array}{cccc}& & {Q_t}^{\prime }(i)& =0\\ & \iff & P(i)t(1+i{)}^{t-1}+(1+i{)}^t{P}^{\prime }(i)& =0\\ & \iff & t& =\frac{-P^{\prime }(i)(1+i{)}^t}{P(i)(1+i{)}^{t-1}}\\ & & & =\underset{={\mathrm{D}}_{\text{mod}}(i)}{\underbrace{\frac{-P^{\prime }(i)}{P(i)}}}(1+i)\\ & & & ={\mathrm{D}}_{\text{mac}}(i),\end{array}$$

• using the first-order Taylor approximation (because the change in interest rate is small) to $Q_t(i)$ about $i_0$ with $t={\mathrm{D}}_{\text{mac}}(i_0)$, we have

$$\begin{array}{cccc}& & Q_t(i)& \approx Q_t(i_0)+(i-i_0)\underset{=0\text{ when }t={\mathrm{D}}_{\text{mac}}(i_0)}{\underbrace{{Q_t}^{\prime }(i_0)}}\\ & & & =Q_t(i_0)\\ & \iff & P(i)(1+i{)}^{D_{\text{mac}}(i_0)}& \approx P(i_0)(1+i_0{)}^{D_{\text{mac}}(i_0)},\end{array}$$

• and the result follows.

$◻$

Proof.

• Using first -order Taylor approximation (because the change in interest rate is small) to $P(i)$ about $i_0$,

$$P(i)\approx P(i_0)+(i-i_0)P^{\prime }(i_0)=P(i_0)-P(i_0)(i-i_0)\cdot \underset{{\mathrm{D}}_{\text{mod}}(i_0)}{\underbrace{\frac{-P^{\prime }(i_0)}{P(i_0)}}}=P(i_0)(1-(i-i_0){\mathrm{D}}_{\text{mod}}(i_0)),$$ as desired.

$◻$