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Pension Finance By David Blake Copyright © 2006 David Blake
Appendix A: Financial Arithmetic This appendix covers some of the most important aspects of financial arithmetic that are be required in the book. It covers such issues as simple interest, compound interest, future values, present values, internal rates of return and time-weighted rates of return. A.1
FUTURE VALUES: SINGLE PAYMENTS
The future value of a sum of money (the principal) invested at a given annual rate of interest will depend on: whether interest is paid only on the principal (this is known as simple interest), or, in addition, on the interest that accrues (this is known as compound interest); (in the case of compound interest) the frequency with which interest is paid (e.g. annually, semi-annually, quarterly, monthly, daily, continuously). A.1.1 Simple interest With simple interest, the future value is determined by: F
P(1
rT)
(A.1)
where: P F r T
principal amount future value (or end-of-period value or terminal value) rate of interest (annual) number of years.
For example, if P
£1000, r
F
0 1 (10%) and T
1000 [1 (0 1)(2)] 1200 00
411
2 years, then:
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A.1.2 Compound interest: annual compounding With compound interest, the future value (also known as the compound value) is determined by: F
P(1
r )T
(A.2)
In this case, interest is earned on the interest that accrues. This can be seen by looking at the case when T 2 years. After one year, the principal will have increased to P(1 r ), the same as in the simple interest case. But in the second year, interest is earned on the accrued interest, so that at the end of the second year, the principal will have increased to: F
P(1 P(1
r )(1 r )2
r)
Using the information in the last example, the compound value after two years is: F
1000(1 0 1)2 1210 00
which is £10 greater than with simple interest.
A.1.3 Compound interest: more frequent compounding Sometimes compounding takes place more frequently than once a year. For example, if compounding takes place semi-annually, then an interest payment (equal to half the annual interest payment) will be made at the end of six months, and that interest payment will itself earn interest for the next six months. After two years, four interest payments will have been made. In general, if compounding takes place m times per year, then, at the end of T years, mT interest payments will have been made and the future value of the principal will be: F
P 1
r m
mT
(A.3)
For example, with semi-annual compounding, the future value of £1000
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at the end of two years when the interest rate is 10% is: P 1
F
r 2
2T
(2)(2)
01 2
1000 1 1215 51
We can examine the effect of the frequency of compounding by examining the annualised interest-rate factors, 1
Interest-rate factor Assuming that r
r m
m
0.1 (10%):
Compounding frequency Annual Semi-annual Quarterly Monthly Daily
Interest-rate factor (1 r ) 1 100000 r 2 1 102500 1 2 4 r 1 1 103813 4 12 r 1 1 104713 12 365 r 1 1 105156 365
Clearly, the more frequent the compounding, the greater the interest-rate factor. The limit to this process occurs when interest is compounded continuously. This limit is derived as follows. Equation (A.3) can be rewritten: F
m r rT
1
r m
P
1
1 m r
P
1
1 n
P
m r
rT
n rT
(A.4)
where n m r . As m, and hence n, approach infinity (and compounding becomes continuous), the expression in square brackets in (A.4) tends
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to the value known as e: e
lim
1 n
1
n
n
2 71828
Substituting this into (A.4) gives: F
Per T
(A.5)
in the case of continuous compounding. In (A.5), er T is known as the exponential function of r T . It provides the continuously-compounded interest-rate factor. If r 0 1 (10%) and T 1 year, then: er
(2 71828)0 1
1 105171
This is the limit to the process of more frequent compounding. To illustrate continuous compounding, the future value of £1000 at the end of two years when the interest rate is 10% is: F
1000e(0 1)(2) 1221 40
A.1.4 Flat and effective rates of interest The flat rate of interest is the interest rate that is quoted on a deposit, loan, etc. But the effective rate of interest (sometimes called the annual percentage rate or APR) will be greater than the flat rate if compounding takes place more than once a year. The effective rate re is the compounded interest rate: r m 1 1 (A.6) re m For example, if the flat rate of interest is 10% and compounding takes place 12 times per year, then the effective interest rate is: re
A.2
0 1 12 1 12 0 1047 (i e 10 47%) 1
PRESENT VALUES: SINGLE PAYMENTS
If an amount F is to be received in T years’ time, the present value of that amount is the sum of money P which, if invested today, would generate
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the compound amount F in T years’ time. The process of finding present values is known as discounting and is the exact inverse of the process of finding future values. A.2.1 Present values: annual discounting With annual discounting, a sum of money F to be received in T years’ time has a present value of: F
P
(1 r )T F(1 r )
T
(A.7)
This is found by dividing both sides of (A.2) by (1 r )T . For example, the present value of £1000 to be received in five years’ time when the interest rate is 10% is: P
1000(1 620 92
0 1)
5
This is because £620.92 invested for five years at 10% generates £1000. The rate of interest, r, involved in this calculation is known as the discount rate and the term (1 r ) T is known as the T -year discount factor, DT : DT
(1
r)
T
(A.8)
Hence, (A.7) can be written in the equivalent form: P
F
DT
(A.9)
The five-year discount factor when the discount rate is 10% is: DT
(1 0 1) 0 62092
5
A.2.2 Present values: more frequent discounting If discounting takes place m times per year, then we can use (A.3) to derive the appropriate present value formula: P
F 1
r m
mT
(A.10)
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For example, with semi-annual discounting, the present value of £1000 to be received in five years’ time when r 0 1 (10%) is: P
01 2
1000 1
(2)(5)
613 91 The more frequent the discounting, the lower the present value. In the limiting case of continuous discounting, we can use (A.5) to derive the appropriate present value formula: Fe
P
rT
(A.11)
Using (A.11), the present value of the £1000 to be received in five years’ time is: P
A.3
1000e (0 1)(5) 606 53
FUTURE VALUES: MULTIPLE PAYMENTS
So far we have considered the future value of a single payment. But we can also calculate the future value of a stream of payments. Initially we will assume that the stream of payments is an irregular one. Then we will consider the simpler case of a regular stream of payments. A.3.1 Irregular payments To calculate the future value of an irregular stream of payments, the appropriate future value formula is applied to each individual payment and the resulting individual future values are then summed. The formula for this is: T
dt (1
F
r )T
t
(A.12)
t 1
where dt payment in year t (assuming the payment is made at the end of year). Since interest does not accrue until the end of each year, the first payment will accrue interest for (T 1) years, the second payment for (T 2) years, and so on. To illustrate, the future value of the following stream of annual payments, d1 1000, d2 1100, d3 1200, when the interest rate is
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10% is: F
1000(1 1)3 1 1210 1210 3620 00
1100(1 1)3 1200
2
1200(1 1)3
3
A.3.2 Regular payments A regular stream of payments (for a given number of years) is called an annuity. (If the payments are made at the end of the year, as we are assuming here, the annuity is known as an immediate annuity or an annuity in arrears; if the payments are made at the beginning of the year, the annuity is known as an annuity due or an annuity in advance; if the annuity commences further than a year ahead, it is called a deferred annuity.) With an annuity, the payments dt in (A.12) are identical and can be denoted as d. This allows the formula (A.12) to be simplified, as we shall now see. If dt d for all t, then (A.12) becomes: T
F
d
r )T
(1
t
(A.13)
t 1
If we multiply both sides of (A.13) by (1 from (A.13), we get:
r ) and subtract the result
T
F
(1
r )F
d
T
r )T
(1
t
(1
t 1
d[(1
r )T
t 1
t 1
r)
T
(A.14)
1]
which, on rearranging, yields: F
d
(1
r )T r
1
(A.15)
To illustrate this formula, we can calculate the future value of a threeyear annuity, paying £1000 per year for three years, when the interest rate is 10%: F
(1 1)3 1 01 3310 00 1000
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As another example of the use of (A.15), we can calculate the size of an annuity necessary to accumulate a particular sum of money at a particular future date when the interest rate is known. An example of this would be the establishment of a pension fund of a particular size at some future date; the pension fund could then be used to provide a pension annuity to retired workers. The required size of the annuity is given by rearranging (A.15): F
d
(1
r r )T
(A.16)
1
Suppose that a pension fund of £100 000 is required in 20 years’ time. What should the annual pension contribution be if the rate of interest is 10%? Using (A.16): d
01 (1 1)20
100 000
1
1745 96 A.4
PRESENT VALUES: MULTIPLE PAYMENTS
In a similar way, we can calculate the present value of a stream of future payments. Again, the solution depends on whether the future payments are regular or irregular. A.4.1 Irregular payments To calculate the present value of an irregular stream of payments, the appropriate present value formula is applied to each individual payment and the resulting individual present values are then summed. The formula for this is: T
dt (1
P
r)
t
(A.17)
t 1
where dt payment in year t (payment made at end of year). To illustrate, the present value of the following stream of annual payments, d1 1000, d2 1100, d3 1200, when the interest rate is 10% is: P
1000(1 1) 1 1100(1 1) 2 909 09 909 09 901 58 2719 76
1200(1 1)
3
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A.4.2 Regular payments: annual payments with annual discounting The present value of an annuity is found very simply by finding the present value of (A.15): P
F r )T (1 r )T 1 d r 1 (1 r ) T d r (1
1 r )T
(1
(A.18)
To illustrate this formula, we can calculate the present value of a threeyear annuity, paying 1000 per year for three years when the interest rate is 10%: P
1000
1
(1 1) 01
3
2486 85 A.4.3 Perpetuities An annuity that continues indefinitely is called a perpetuity. The future value of a perpetuity is obviously infinite, but its present value is easy to determine using (A.18), recognising that, as T tends to infinity, the term (1 r ) T tends to zero. This leads to: P
d r
(A.19)
as the present value of a perpetuity. An example of a perpetuity is an irredeemable bond, such as a 2.5% Consols, which pays £1.25 per 100 nominal every six months. On a coupon payment date when there is no accrued interest, the price of the bond when the rate of interest is 10% is: d 2 P r 2 1 25 0 05 25 00
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A.5
RATES OF RETURN
So far we have assumed that the rate of interest or the rate of return is given. But sometimes we do not know the rate of interest or the rate of return on an investment and it has to be calculated. There are several different ways of calculating such rates. A.5.1 Single-period rate of return The simplest way of measuring a rate of return is to do so over a single period. If, for instance, we buy a security today for P0 and later sells it for P1 , then the return on holding that security from t 0 to t 1 is given by: r
P0
P1 P0 P1 P0
1
(A.20)
For example, if an investor buys a security for 100 and sells it one week later for 110, his return over the week is: 110 1 0 1 (i e 10%) 100 The ratio P1 P0 in (A.20) is known as the price relative. In many applications, the rate of return over a single period is calculated as a continuously-compounded rate of return, often known as the log price relative. This is given by: r
r
ln
P1 P0
(A.21)
If, for instance, an investor held a portfolio of shares comprising the FTSE 100 index, then the change in the level of that index over a period could be used to measure his return from holding the portfolio. Also, since the level of the index is continuously changing, it might seem reasonable to measure the return in terms of the log price relative. So, if the index is at 6141 (P0 ) at the close of business on one day and falls to 5833 (P1 ) by the close of business the next day, then, from (A.21), his return on the portfolio for that 24-hour period is: r
ln
5833 6141
0 051
(i e
5 1%)
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A.5.2 Internal rate of return or money-weighted rate of return One of the most common ways of measuring the return on an investment is the internal rate of return or the money-weighted rate of return (sometimes called the yield to maturity). This is simple to calculate with single payments. Since F P(1 r )T , the annual internal rate of return on an investment of P paying F in T years’ time is the solution to: (1
r )T
F P
that is: (F P)1
r
T
1
(A.22)
For example, the annual internal rate of return on an investment costing 1000 today and returning 1500 in three years’ time is: (1500 1000)1 3 1 0 1447 (i e 14 47%)
r
With compounding taking place more frequently than once per year, the annual rate of return is the solution to: r mT 1 F P m that is: r
m[(F P)1
mT
1]
(A.23)
Using the last example but with quarterly compounding, the annual internal rate of return is: r
4[(1500 1000)1 (4 3) 1] 0 1375 (i e 13 75%)
The more frequent the compounding, the lower the internal rate of return. With continuous compounding, the internal rate of return is the solution to: er T
F P
(A.24)
In general, if x b y then b is called the logarithm of y to the base x, i.e. b logx (y). In the case of (A.24), we have: rT
loge (F P)
(A.25)
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i.e. r T is the logarithm of (F P) to the base e. Logarithms to the base e are known as natural logarithms and are denoted by b ln(y). Therefore, (A.25) becomes: 1 ln(F P) T
r
(A.26)
Using the last example, but with continuous compounding, the annual internal rate of return is: 1 ln(1500 1000) 3 0 1352 (i e 13 52%)
r
With multiple payments, an analytical solution for the internal rate of return does not generally exist. In general, the solution, r , to either: T
F
dt (1
r )T
dt (1
r)
t
(A.27)
t 1
or: T
P
t
(A.28)
t 1
has to be found numerically, i.e. by trial and error. This is true even if the payments are regular, as with an annuity. Take, for example, the formula for the present value of an annuity: P
d
1
r)
(1 r
T
(A.29)
Suppose we know that a three-year annuity of 1000 has a present value of £2465.12 and we want to find the internal rate of return. We begin with an estimate of the internal rate of return, say, r 0 1 (10%). At r 0 1 we find that P 2486 85, which is greater than 2465.12. This implies that our estimate of r is too low. We should try a larger value for r , say r 0 11 (11%). At r 0 11, we find that P 2443 71, which is less than 2465.12. The correct value of r must therefore lie between 10 and 11%. Suppose that we try r 0 105 (10.5%). At this value of r we find that P 2465 12 as required, so that the internal rate of return is 10.5%.
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Only in the case of a perpetuity is the internal rate of return easy to calculate. From (A.19), we see that the internal rate of return is simply: r
d P
(A.30)
If the present value of a perpetuity of £5 per year is £41.67, then the internal rate of return is: 5 41 67 0 12 (i e 12%)
r
A.5.3 Time-weighted rate of return or geometric mean rate of return The time-weighted rate of return or the geometric mean rate of return takes into account the value of earlier payments at the time that the next payment in the series arises. If P is the initial value of an investment, F is the final value, dt is the payment received by the investment in year t and Vt is the value of the investment when the payment is received, then the time-weighted rate of return is calculated as follows: (1
V1 P
r )T
V2 V1
V3 d1
V2
F d2
VT
1
dT
1
(A.31) But (V1 P) (1 r1 ), one plus the return on the investment in the first period, [V2 (V1 d1 )] (1 r2 ), one plus the return on the investment in the second period, etc., so that this equation can be rewritten: (1
r )T
(1
r1 )(1
r2 )
(1
rT )
(A.32)
1
(A.33)
or, solving for r : r
[(1
r1 )(1
r2 )
(1
r T )]1
T
From (A.33), it is clear that the time-weighted rate of return is the geometric mean of the individual period returns. To illustrate, consider the case of an investment beginning with £100, attracting £50 at the end of year 1 (when the value of the investment was £110), and at the end of year 2 the value of the investment was £225.
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The time-weighted rate of return is calculated as: r
1 2
225 110 50
110 100
[(1 10)(1 406)]1
2
1 1
1 2
1 [1 5469] 0 2437 (24 37%) The comparable money-weighted rate of return is given by the solution to the following calculation: 225
100(1
r )2
50(1
r)
implying a money-weighted rate of return of 27.07%.