# Present Value - Financial Management

The compound, or future, value calculations answer the question:What will be the future value of X dollars invested today, compounded at some rate of interest, i? The financial decision maker, however, is often faced with another type of problem: Given some future value, FVn, what is its equivalent value today? That is, what is its present value, PV0? The solution requires present value calculations, which are used to determine the dollar amount today, PV0, that is equivalent to some promised future dollar amount, FVn. The equivalence depends upon the rate of interest (return) that can be earned on investments during the time period under consideration.

The relationship between compound value and present value can be shown by rewriting Equation to solve for PV0:

FVn = PV0(1 + i)n

or

where 1/(1 + i)n is the reciprocal of the compound value factor. The process of finding present values is frequently called discounting. Equation is the basic discounting formula.

Growth of a $100 Investment at Various Compound Interest Rates To illustrate the use of Equation , suppose your banker offers to pay you$255.20 in five years if you deposit X dollars today at an annual 5 percent interest rate. Whether the investment would be worthwhile depends on how much money you must deposit, or the present value of the X dollars. FVIF tables, such as Table 5.1 presented earlier, can be used to solve the problem as follows:

Thus, an investment of $200 today would yield a return of$55.20 in five years.

Because determining the reciprocals of the compound value interest factors, 1/(1 + i)n, can be a tedious process, present value interest factors (PVIFs) commonly are used to simplify such computations. Defining each present value interest factor as

PVIFi, n = 1/(1+i)

Table at the end of the book provides a number of present value interest factors. A portion of Table II is reproduced here as Table.

For example, Table can be used to determine the present value of $1,000 received 20 years in the future discounted at 10 percent: PV0=FV20(PVIF0.10, 20) =$1,000(0.149) = $149 Thus,$149 invested today at 10 percent interest compounded annually for 20 years would be worth $1,000 at the end of the period. Conversely, the promise of$1,000 in 20 years is worth $149 today, given a 10 percent interest rate. Present Value Interest Factors (PVIFs) for$1 at Interest Rate i for n Periods

Solving for Interest and Growth Rates

Present value interest factors can also be used to solve for interest rates. For example, suppose you wish to borrow $5,000 today from an associate. The associate is willing to loan you the money if you promise to pay back$6,250 four years from today. The compound interest rate your associate is charging can be determined as follows:

Reading across the 4-year row in Table 4.2, 0.800 is found between the 5 percent (0.823) and 6 percent (0.792) columns. Interpolating between these two values yields

Thus, the effective interest rate on the loan is 5.74 percent per year, compounded annually. Another common present value application is the calculation of the compound rate of growth of an earnings or dividend stream. For example, Krispy Kreme had earnings of $0.15 per share in 2000. Suppose these earnings grew to$0.80 at the end of 2005. Over this 5-year period, what is the compound annual rate of growth in Krispy Kreme’s earnings? The answer to this problem can be obtained by solving for the present value interest factor over the 5-year period as follows:

From Table II, we find this approximate present value interest factor in the 5-year row under the 40 percent interest, or growth rate, column. Hence the compound annual rate of growth in Krispy Kreme’s earnings per share has been approximately 40 percent. A more precise calculation using a calculator yields 39.8 percent.

The discounting process can also be illustrated graphically. shows the effects of time, n, and interest rate, i, on the present value of a $100 investment. As the figure shows, the higher the discount rate, the lower the present value of the$100.