D EF E RRED A NNUITIE S

When the first payment of an annuity occurs some time after the end of the first period, the annuity is deferred. The time line for an ordinary annuity of $1 per period for four periods deferred for two periods is

End of Period 0 1 2 3 4 5 6

The arrow marked P shows the time of the present value calculation; the arrow marked F shows the future value calculation. The deferral does not affect the future value, which equals the future value of an ordinary annuity for four periods.

Notice that the time line for the present value looks like one for an ordinary annuity for six periods minus an ordinary annuity for two periods:

$1 $1 End of Period 0 1 2 3 4 5 6

Calculate the present value of an annuity of n payments deferred for d periods by subtracting the present value of an annuity for d periods from the present value of an annuity for n þ d periods.

Example 15 Refer to the data in Example 12. Recall that Mr. Mason wants to withdraw $50,000 per year

on his 66th through his 75th birthdays. He wishes to invest a sufficient amount on his 63rd, 64th, and 65th birthdays to provide a fund for the later withdrawals.

The time line is

End of Year

As of his 62nd birthday, the series of $50,000 payments on Mr. Mason’s 66th through 75th birthdays is a deferred annuity. The interest rate is 8 percent per year.

You can find the present value using the factor for the present value of an annuity for 13 periods (10 payments deferred for three periods) of 7.90378 and subtracting the factor for the present value of an annuity for three periods of 2.57710. The net amount is 5.32668 ( Multiplying by the $50,000 payment amount, you find the present value of the deferred annuity on Mr. Mason’s 62nd birthday of $266,334 (

Perp etuitie s

A periodic payment promised forever is a perpetuity. Perpetuities have no meaningful future values. They do have present values: One dollar to be received at the end of every period

Implicit Interest Rates: Finding Internal Rates of Return 503

for the present value of an ordinary annuity of $A per payment as n, the number of payments, approaches infinity:

As n approaches infinity, (1 þ r) approaches zero, so that P A approaches A(1/r). If the first payment of the perpetuity occurs now, the present value is A[1 þ (1/r)].

Example 16 The Canadian government offers to pay $30 every 6 months forever in the form of a perpetual

bond. What is that bond worth if the discount rate is 10 percent compounded semiannually? Ten percent compounded semiannually is equivalent to 5 percent per 6-month period. If the first payment occurs six months from now, the present value is $30/0.05 ¼ $600. If the first payment occurs today, the present value is $30 þ $600 ¼ $630.

Implicit Intere st Rate s : Findin g Intern al R ate s of Return

The preceding examples computed a future value or a present value given the interest rate and stated cash payment. Or they computed the required payments given their known future value or their known present value. In some calculations, we know the present or future value and the periodic payments; we must find the implicit interest rate. Assume, for example, a case in which we know that a cash investment of $10,500 will grow to $13,500 in 3 years. What is the implicit interest rate, or market rate of return, on this investment? The time line for this problem is

$10,500

0 0 ($13,500)

End of Year

The implicit interest rate is r, such that

$13,500

$10,500 ¼

3 ð1 þ rÞ (A.1)

$13,500

ð1 þ rÞ 3 (A.2) In other words, the present value of $13,500 discounted three periods at r percent per period is

$10,500. The sum of the present values, discounted at r percent per period, of all current and future cash inflows and outflows nets to zero. In general, to find such an r requires a trial-and-error

procedure. 1 We refer to that procedure as finding the internal rate of return of a series of cash flows. The internal rate of return of a series of cash flows is the discount rate that equates the sum of the net present value of that series of cash flows to zero. Follow these steps to find the internal rate of return:

1. Make an educated guess, called the ‘‘trial rate,’’ at the internal rate of return. If you have no idea what to guess, try zero (or 10 percent).

2. Calculate the present value of all the cash flows (including the one at the end of year 0).

3. If the present value of the cash flows is zero, stop. The current trial rate is the internal rate of return.

4. If the amount found in step 2 is less than zero, try a larger interest rate as the trial rate and go back to step 2.

5. If the amount found in step 2 is greater than zero, try a smaller interest rate as the new trial rate and go back to step 2.

1 In cases where r appears in only one term, as here, you can find r analytically. Here, r ¼ ($13,500/$10,500) 1/3

Implicit Interest Rates: Finding Internal Rates of Return 505

Example 18 In some contexts, such as mortgages or leases, one knows the amount of a series of future periodic

payments, which are identical in all periods, and the present value of those future payments. For example, a firm may borrow $100,000 and agree to repay the loan, in 20 payments of $11,746 each, at the end of each of the next 20 years. To calculate interest expense each period, you must find the interest rate implicit in the loan.

You have the following information: Present Value

Factor for

of an

¼ Periodic

the Present

Payment

Value of an

Ordinary Annuity

The factor to discount 20 payments of $11,746 to a present value of $100,000 is 8.51354. To find the interest rate implicit in the discounting, scan the 20-payment row of Table 4 to find the factor 8.51354. The interest rate at the head of the column is the implicit interest rate, approximately

10 percent in the example. Example 19

An investment costing $11,400 today provides the following after-tax cash inflows at the ends of each of the next five periods: $5,000, $4,000, $3,000, $2,000, $1,000. What is the internal rate of return on these flows? That is, find r such that

Trial rates r produced the following sequences of estimates of the internal rate of return:

Trial

Right-Hand

Iteration Number

Rate

Side of A.4

The process of estimating goes several steps farther than necessary. To the nearest whole percentage point, the internal rate of return is 13 percent.

To the nearest one-hundredth of a percent, the internal rate of return is 13.03 percent. Further trials find an even more precise answer, r ¼ 13.027 percent. Physical scientists learn early in their training not to use more significant digits in calculations than the accuracy of the measuring devices merits. Accountants, too, should not carry calculations beyond the point of accuracy. Given the likely uncertainty in the estimates of cash flows, an estimate of the internal rate of return accurate to the nearest whole percentage point will serve its intended purpose.

Compound Interest and Annuity Tables

TABLE 1

Future Value of $1

F n n = P( 1 + r) r n = interest rate; n = number of periods until valuation; P = $1

Period = n

Present Value of $1 P =F n ( 1 + r) r = discount rate; n = number of periods until payment; F n = $1

Period = n

50 8 TABLE 3

Future Value of Annuity of $1 in Arrears

( n 1 + r)

r r = interest rate; n = number of payments

No. of Payments = n

47.58041 72.03511 109.6868 Note: To convert from this table to values of an annuity in advance, determine the annuity in arrears above for one more period and subtract 1.00000.

n Periods ⫽ Payments

TABLE 4

Present Value of Annuity of $1 in Arrears

$1 $1 Payments P A =

in Arrears

P A P F r = discount rate; n = number of payments

⌺ 冢冣 Individual Values

Value in

冢冣 from Table 2

Table 4

No. of Payments = n

5.84737 4.67547 3.85926 Note: To convert from this table to values of an annuity in advance, determine the annuity in arrears above for one fewer period and add 1.00000.

D EF E RRED A NNUITIE S