In our computations of future amounts, we have assumed that interest is paid (compounded) or payments are made annually. Therefore, in using the tables, we used annual periods and an annual interest rate. Investment payments or interest payments may be made on a more
Our original formula . . .
restated to find the amount of the periodic payments
Exhibit B–4
FUTURE VALUE OF AN ORDINARY ANNUITY Table FA–2
Future Amount of $1 Paid Periodically for n Periods Number
of
Periods Interest Rate
(n) 1% 11⁄2% 5% 6% 8% 10% 12% 15% 20%
1 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
2 2.010 2.015 2.050 2.060 2.080 2.100 2.120 2.150 2.200
3 3.030 3.045 3.153 3.184 3.246 3.310 3.374 3.473 3.640
4 4.060 4.091 4.310 4.375 4.506 4.641 4.779 4.993 5.368
5 5.101 5.152 5.526 5.637 5.867 6.105 6.353 6.742 7.442
6 6.152 6.230 6.802 6.975 7.336 7.716 8.115 8.754 9.930
7 7.214 7.323 8.142 8.394 8.923 9.487 10.089 11.067 12.916
8 8.286 8.433 9.549 9.897 10.637 11.436 12.300 13.727 16.499
9 9.369 9.559 11.027 11.491 12.488 13.579 14.776 16.786 20.799 10 10.462 10.703 12.578 13.181 14.487 15.937 17.549 20.304 25.959 20 22.019 23.124 33.066 36.786 45.762 57.275 72.052 102.444 186.688 24 26.974 28.634 44.502 50.816 66.765 88.497 118.155 184.168 392.484 36 43.077 47.276 95.836 119.121 187.102 299.127 484.463 1014.346 3539.009
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frequent basis, such as monthly, quarterly, or semiannually. Tables FA–1 and FA–2 may be used with any of these payment periods, but the rate of interest must represent the interest rate for that period.
As an example, assume that 24 monthly payments are to be made to an investment fund that pays a 12 percent annual interest rate. To determine the future amount of this investment, we would multiply the amount of the monthly payments by the factor from Table FA–2 for 24 periods, using a monthly interest rate of 1 percent—the 12 percent annual rate divided by 12 months.
Present Values
As indicated previously, the present value is today’s value of funds to be received in the future. While present value has many applications in business and accounting, it is most easily explained in the context of evaluating investment opportunities. In this context, the present value is the amount that a knowledgeable investor would pay today for the right to receive an expected future amount of cash. The present value is always less than the future amount, because the investor will expect to earn a return on the investment. The amount by which the future cash receipt exceeds its present value represents the investor’s profit.
The amount of the profit on a particular investment depends on two factors: (1) the rate of return (called the discount rate ) required by the investor and (2) the length of time until the future amount will be received. The process of determining the present value of a future cash receipt is called discounting the future amount.
To illustrate the computation of present value, assume that an investment is expected to result in a $1,000 cash receipt at the end of one year and that an investor requires a 10 percent return on this investment. We know from our discussion of present and future val- ues that the difference between a present value and a future amount is the return (interest) on the investment. In our example, the future amount would be equal to 110 percent of the original investment, because the investor expects 100 percent of the investment back plus a 10 percent return on the investment. Thus, the investor would be willing to pay $909 ($1,000 ⫼ 1.10) for this investment. This computation may be verified as follows (amounts rounded to the nearest dollar):
Compute the present values of future cash flows.
L e a r n i n g O b j e c t i v e C
o e a
LO5
Exhibit B–5 FUTURE AMOUNT OF A SERIES OF INVESTMENTS
Year 1 Year 2 Year 3 Year 4 Year 5
Sinking fund payment 1
$1,638,000
Sinking fund payment 2
$1,638,000
Sinking fund payment 3
$1,638,000
Sinking fund payment 4
$1,638,000
Sinking fund payment 5
$1,638,000 Future amount
$10,000,000
Year 1
Sinking fun ar 2
Sinking fun ar 3
Sinking fun ar 4
Sinking fun ar 5
Sinking fun
Amount to be invested (present value) . . . $ 909 Required return on investment ($909 ⫻ 10%) . . . 91 Amount to be received in one year (future value) . . . $1,000
Confirming Pages
Present Values B-7
As illustrated in Exhibit B–6 , if the $1,000 is to be received two years in the future, the investor would pay only $826 for the investment today [($1,000 ⫼ 1.10) ⫼ 1.10]. This com- putation may be verified as follows (amounts rounded to the nearest dollar):
Amount to be invested (present value) . . . $ 826
Required return on investment in first year ($826 ⫻ 10%) . . . 83
Amount invested after one year. . . $ 909
Required return on investment in second year ($909 ⫻ 10%) . . . 91 Amount to be received in two years (future value) . . . $1,000
The amount that our investor would pay today, $826, is the present value of $1,000 to be received two years from now, discounted at an annual rate of 10 percent. The $174 difference between the $826 present value and the $1,000 future amount is the return (interest revenue) to be earned by the investor over the two-year period.
USING PRESENT VALUE TABLES
Although we can compute the present value of future amounts by a series of divisions, tables are available that simplify the calculations. We can use a table of present values to find the present value of $1 at a specified discount rate and then multiply that value by the future amount as illustrated in the following formula:
Present Value ⴝ Future Amount ⴛ Factor (from Table PV–1)
Referring to Table PV–1 in Exhibit B–7 , we find a factor of .826 at the intersection of two periods and 10 percent interest. If we multiply this factor by the expected future cash receipt of
$1,000, we get a present value of $826 ($1,000 ⫻ .826), the same amount computed previously.
WHAT IS THE APPROPRIATE DISCOUNT RATE?
As explained earlier, the discount rate may be viewed as the investor’s required rate of return.
All investments involve some degree of risk that actual future cash flows may turn out to be less than expected. Investors will require a rate of return that justifies taking this risk. In today’s market conditions, investors require annual returns of between 2 percent and 6 percent on low-risk investments, such as government bonds and certificates of deposit. For relatively high-risk investments, such as the introduction of a new product line, investors may expect to earn an annual return of perhaps 15 percent or more. When a higher discount rate is used, the present value of the investment will be lower. In other words, as the risk of an investment increases, its value to investors decreases.
Formula for finding present value Year 1 Year 2
Ye
Present value
$826
Future amount
$1,000
$ ,
Exhibit B–6
PRESENT VALUE OF
$1,000 TO BE RECEIVED IN A SINGLE SUM IN TWO YEARS
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