An expression for can be derived in an analogous manner as an equation of value at the end of the nlh period. The accumulated value of a payment of 1 made at the end of the first period is (1 + i)" . The accumulated value of a payment of 1 made at the end of the second period is (1 + 0"". This process is continued until the accumulated value of a payment of 1 made at the end of the nth period is just 1. The total accumulated value 5 must equal the sum of the accumulated values of each payment, i.e.

= 1 + (1 + 0 + • • • + (1 + i)" + (1 + if. (3.3)

Again, a more compact expression can be derived by summing the geometric progression

+ (1 + i)"- + (1 + 0""

1 + (1 + 0 + (1 + 0" - 1 (1 +0-1 (1 + 0" -1

(3.4)

Values of and s-;] at several rates of interest and for values of n from

1 to 50 appear in the interest tables in Appendix I. On occasion, the interest rate is written to the lower right of the symbol, e.g. - q-j and Sj gg. Since this tends to clutter the symbols, we will do this only if there could be any ambiguity concerning the interest rate to be used in evaluating the function. It is possible to give a verbal interpretation to formula (3.2) written as

1= 1 +

Consider the investment of 1 for n periods. Each period the investment of 1 will yield interest of i paid at the end of the period. The present value of these interest payments is ia. At the end of n periods the original investment of 1, whose present value is v", is returned. Thus, both sides of the equation represent the present value of an investment of 1 at the date of investment. A similar verbal inte retaUon for formula (3.4) is left as an exercise. There is a simple relationship between and

5 = {1 + " . (3.5)

This relationship is obvious from a comparison of either formulas (3.1) and (3.3) or from formulas (3.2) and (3.4). It is also obvious from the time diagram, since 5 is the value of the same payments as , only the value is taken n periods later.

Another relationship between and is

This relationship can be derived as follows:

(3.6)

+ i =

(1 + 0" - 1 / + i{\ + 0" - /

(1 + 0" - 1 i

1 -v« 1

Values of 1/ appear in the interest tables in Appendix I. Formula (3.6) will be quite significant in another context in Chapter 6. A verbal interpretation of the formula will be given at that time.

Example 3.1 Find the present value of an annuity which pays $500 at the end of each half-year for 20 years if the rate of interest is 9% convertible semiannually.

The answer is

500 5 = 500(18.4016) = $9200.80.

Example 3.2 If a person invests $1000 at 8% per annum convertible quarterly, how much can be withdrawn at the end of every quarter to use up (he fund exactly at the end of 10 years?

Let R be the amount of each withdrawal. The etjuation of value at the date of investment is

««401.02 = 1000-

Thus, we have

1000 401.02 = 1000

i 27.3555

= $36.56

Example 3.3 Compare the total amount of interest that would be paid on a $1000 loan over a 10-year period, if the effectiie rate of interest is 9% per annum, under the following three repayment meftods:

(1) The entire loan plus accumulated interestis paid in one lump-sum at the end of 10 years.

(2) Interest is paid each year as accrued and principal is repaid at the end of 10 years.

(3) The loan is repaid by level payments over the 10-year period.

1. The accumulated value of the loan at the end of 10 years is

1000(1.09)° = $2367.36 Thus, the total amount of interest paid is equal to

$2367.36 - 1000.00 = $1367.36.

2. Each year the loan earns interest of 1000(.09) = $90, so that the total amount of interest paid is equal to

10-90 = $900.00.

3. Let the level payment be . An equation of value for R at the inception of the loan

= 1000

which gives

lOOC

1000

= $155.82.

6.417658

Thus, the total amount of interest paid is equal to

10(155.82) - 1000 = $558.20.

The reader should justify by general reasoning the relative answers under the three repayment methods. The later the payments are made on a loan, the higher the total amount of interest will be. Conversely, the sooner the payments are made, the smaller the amount of interest. Although the total amount of payments under the three methods are different, the present value of payments is equal to $1000, the amount of the loan, for all three.

3.3 ANNUITY-DUE

In Section 3.2 the annuity-immediate was defined as an annuity in which payments are made at the end of the period. In this section, we will consider the annuity-due in which payments are made at the beginning of the period instead. The use of the terms "annuity-immediate" and "annuity-due" is unfortunate, since these terms are not descriptive of the properties of these annuities.

1 1

1 1

n-2 /1-1

Figure 3.2 Time diagram for an annuity-due

Figure 3.2 is a time diagram for an n-period annuity-due. Arrow 1 appears at the time the first payment is made. The present value of the annuity at this point in time is denoted by c- Arrow 2 appears n periods after arrow 1, one period after the last payment is made. The accumulated value of the aimuity at this point in time is denoted by s-

We can write an expression for analogous to formula (3.1)

a = 1 4- V -f v- -b • • Again summing the geometric progression

1 -v" 1 -V 1 -v"

iv 1 -v"

,1-1

(3.7)

(3.8)

which is analogous to formula (3.2).

Similarly for i-,, we have the following formulas analogous to formulas (3.3) and (3.4)

= (1 + 0 + (1 + 0 + • • • + (1 + 0"" + (1 + 0"

= (l+OL±i (1-1-0-1

(1 + 0"-l

(l + 0"-l

(3.9)

(3.10)

It is instructive to compare formulas (3.2) and (3.8). The numerators are identical; however, the denominator of (3.2) is / and the denominator of (3.8) is d. Under the annuity-immediate, payments are made at the end of the period and i is a measure of interest payable at the end of the period. Under the annuity-due, payments are made at the beginning of the period and d is a measure of interest payable at the begiiming of the period. A comparison of formulas (3.4) and (3.10) gives similar results.

The above property, relating the time aimuity payments are made to the measure of interest in the denominator, is helpful in remembering annuity formulas. Moreover, this property can be generalized to the more complex annuities which are discussed in Chapter 4.

It is immediately obvious that

a formula analogous to formula (3.5). It can also be shown that

(3.11)

(3.12)

a formula analogous to formula (3.6). The derivation of formula (3.12) is left as an exercise.

It is possible to relate the annuity-immediate and the annuity-due. One type of relationship is

= {\ + 1) (3.13)

=*Hl(l+0. (3.14)

Formula (3.13) can immediately be derived by comparing formulas (3.1) and (3.7), or formulas (3.2) and (3.8). Since each payment under is made one period earlier than each payment under , the total present value must be larger by one periods interest. Formula (3.14) can be derived similarly.

There is another type of relationship between the annuity-immediate and the armuity-due

dj = l+o- (3.15)

This formula can be derived from Figure 3.2. The n payments made under can be split into the first payment and the remaining n - 1 payments. The present value of the first payment is 1, and the present value of the remaining ra - 1 payments is . The sum must give the total present value Similarly, we can obtain

= s-

- 1 .

(3.16)

This formula can also be derived from Figure 3.2. Temporarily, assume that an imaginary payment of 1 is made at the end of the nib. period. Then, the total accumulated value of the -I- 1 payment is s-j. However, we must remove the accumulated value of the imaginary payment, which is just 1. The difference gives the accumulated value i.

Most compound interest tables, including those in Appendix I, do not include values for annuities-due. Thus, formulas such as (3.13) or (3.15) and (3.14) or (3.16) must be used in finding numerical values for annuities-due.

Considerable confiision has often been created by treating the annuity-immediate and the annuity-due as if they were greatly different. Actually they refer to exactly the same series of payments evaluated at different points in fime. Figure 3.3 clarifies this point.

««1

" 1

Figure 3.3 Time diagram comparing an annuity-immediate with an annuity-due

Example 3.4 An investor wishes to accumulate $1000 in a fund at the end of 12 years. To accomplish this the investor plans to make deposits at the end of each year, the final payment to be made one year prior to the end of the investment period. How large should each deposit be if the fund earns 7% effective?

Since we are interested in the accumulated value one year after the last payment, the equation of value is

Rs\ = 1000

where R is the annual deposit. Solving for R we have

1000

1.07i

1000

(1.07)(15.7836)

= $59.21.

3.4 ANNUITY VALUES ON ANY DATE

Thus far we have considered evaluating annuities only at the beginning of the term (either one period before, or on the date of, the first payment), or at the end of the term (either on the date of, or one period after, the last payment). However, it is often necessary to evaluate annuities on other dates. We will discuss the following three cases: (1) present values more than one period before the first payment date, (2) accumulated values more than one period after the last payment date, and (3) current values between the first and last payment dates. We will assume that the evaluation date is an integral number of periods from each payment date.

The value of an annuity on any date could be found by accumulating or discounting each separate payment and summing the results. However, this method would become inefficient if a large number of payments are involved.