Single Amounts

Future (or Compound) Value. To begin with, consider a person who deposits $100 into a savings account. If the interest rate is 8 percent, compounded annually, how much will the $100 be worth at the end of a year? Setting up the problem, we solve for the future value (which in this case is also referred to as the compound value) of the account at the end of the year (FV₁).

FV₁ = P₀(1 + i)= $100(1+0.08) = $108

Interestingly, this first-year value is the same number that we would get if simple interest were employed. But, this is where the similarity ends.

What if we leave $100 on deposit for two years? The $100 initial deposit will have grown to $108 at the end of the first year at 8 percent compound annual interest. Going to the end of the second year, $108 becomes $116.64, as $8 in interest is earned on the initial $100, and $.64 is earned on the $8 in interest credited to our account at the end of the first year. In other words, interest is earned on previously earned interest, hence the name compound interest. Therefore, the future value at the end of the second year is

FV₂ = FV₁(1 + i) = P₀(1 + i)(1 + i) = P₀(1 + i)² = $108(1.08) = $100(1.08)(1.08) = $100(1.08)² = $116.64

At the end of three years, the account would be worth

FV₃ = FV₂(1 + i) = FV₁(1 + i)(1 + i) = P₀(1 + i)³ = $116.64(1.08) = $108(1.08X1.08) = $100(1.08)³ = $125.97

In general, FVn, the future (compound) value of a deposit at the end of n periods, is

FVn =P₀(1+i)ⁿ

Illustration of compound interest with 100$ initial deposit and 8% annual interest rate

YEAR	BEGINING AMOUNT	INTEREST EARNED DURING A PERIOD(8%OF BEGINING AMOUNT)	ENDING AMOUNT FVn
1	100	8	108
2	108	8.64	116.64
3	116.64	9.33	125.97
4	125.97	10.08	136.05
5	136.05	10.88	146.93
6	146.93	11.76	158.69
7	158.69	12.69	171.38
8	171.38	13.71	185.09
9	185.09	14.81	199.9
10	199.9	15.99	215.89

FVn = Po(FVIFi,n)

where we let FVIFi,n (i.e., the future value interest factor (коэф наращения) at i% for n periods) equal (1 + i)ⁿ. Table 3-2, showing the future values for our example problem at the end of years 1 to 3 (and beyond), illustrates the concept of interest being earned on interest. A calculator makes Eq. (3-4) very simple to use. In addition, tables have been constructed for values of (1 + i)ⁿ—FVIF_i,n- —for wide ranges of i and n. These tables, called (appropriately) Future Value Interest Factor (or Terminal Value Interest Factor) Tables, are designed to be used with Eq. (3-5). Table 3-3 is one example covering various interest rates ranging from 1 to 15 percent. The Interest Rate (i) headings and Period (n) designations on the table are similar to map coordinates. They help us locate the appropriate interest factor. For example, the future value interest factor at 8 percent for nine years (FVIF8% 9) is located at the intersection of the 8% column with the 9-period row and equals 1.999. This 1.999 figure means that $1 invested at 8 percent compound interest for nine years will return roughly $2—consisting of initial principal plus accumulated interest.

Future value interest factor of 1$ at i% at the end of n periods

(FVIFi,n) =(1 +i)n
PERIOD (n)	INTEREST RATE (i)
	1%	3%	5%	8%	10%	15%
1	1.010	1.030	1.050	1.080	1.100	1.150
2	1.020	1.061	1.102	1.166	1.210	1.322
3	1.030	1.093	1.158	1.260	1.331	1.521
4	1.041	1.126	1.216	1.360	1.464	1.749
5	1.051	1.159	1.276	1.469	1.611	2.011
6	1.062	1.194	1.340	1.587	1.772	2.313
7	1.072	1.230	1.407	1.714	1.949	2.660
8	1.083	1.267	1.477	1.851	2.144	3.059
9	1.094	1.305	1.551	1.999	2.358	3.518
10	1.105	1.344	1.629	2.159	2.594	4.046
25	1.282	1.094	3.386	6.848	10.835	32.919
50	1.645	4.384	11.167	46.902	117.391	1,083.657

If we take the FVIFs for $1 in the 8% column and multiply them by $100, we get figures (aside from some rounding) that correspond to our calculations for $100 in the final column of Table 3-2. Notice, too, that in rows corresponding to two or more years, the proportional increase in future value becomes greater as the interest rate rises. A picture may help make this point a little clearer. Therefore, in Figure 3-1 we graph the growth in future value for a $100 initial deposit with interest rates of 5,10, and 15 percent. As can be seen from the graph, the greater the interest rate, the steeper the growth curve by which future value increases. Also, the greater the number of years during which compound interest can be earned, obviously the greater the future value.

DISTINCTIONS AMONG VALUATION CONCEPTS

The term value can mean different things to different people. Therefore, we need to be precise in how we both use and interpret this term. Let's look briefly at the differences that exist among some of the major concepts of value