Present Value of Ordinary Annuity

We introduced the concept of present values and future values in our brief discussion of the Time-Value-of-Money Principle in last Posts. We also defined three different rates of return: expected, required and realized. The expected return is the return you expect to earn if you make the investment. 

Single Cash Flows

The required return is the minimum return you must expect to get to be willing to make the investment. The realized return is the return you actually earned on an investment during a given time period. We showed you that finding the present value or the future value of a single cash flow is a simple calculation. After a brief recap, we'll extend its logic to deal with multiple cash flows.

Finding the Future Value of an Investment

The future value (FV) is the value an investment will grow to after a given time period. Let's say you invest $1000 today. Table 1 Shows the amount of money you'll have accumulated at the end of each of the next six years if the bank is paying 10% interest. After one year.

FV₁ = $1000 +$100 = $1100

In the second year, you'll earn $110 more -10% interest on your accumulated investment (=[0.10]1100), for a total of

                                                                       FV₂ = $1100 + $110=$1210 

The extra $10 of interest earned in the second year is called compound interest. Compound interest is a way of computing interest earned where interest is earned on both the original investment and on the reinvested interest. As you can see in table 1, the interest earned each year grows because of compound interest.

Table 1 also shows how fast your $1000 investment grows if invested funds earn simple interest instead of compound interest. Simple interest is a way of computing interest earned where interest is earned on only the original investment. Note that in year 1 with simple interest, the interest earned is $100, the same as with compound interest. However, after that, the story changes. In year 2 with simple interest, the interest earned is again $100. No interest is earned on the first year's $100 interest. All other years also earn only $100, 10% of the original investment.

Would you rather earn compound interest or simple interest? Obviously, if the interest rates are the same, you'll have more money with compound interest than with simple interest. Because of today's technology, the use of simple interest has largely disappeared.

One way to find a future value is to calculate interest each year, adding it to the previous year's balance, and accumulating the result for the desired number of years. In table 1, we stopped at six years. Suppose you were investing for 20 years. It's repetitive and such a large number of hand calculations can cause errors. Consequently, we use shortcut methods whenever we can. One shortcut method of finding future values is to use the future-value formula:

                                                                   The Future-Value Formula

                                                                    FV_n=PV(1+r)ⁿ = PV(FVF_r,n)------------------------(1)

The amount (1+r)ⁿ above is called the future-value factor. The future-value factor, FVFr,n, is the value $1.00 will grow to if it's invested at r per period for n periods. Figure 1 in this post is a graph of FVF_r,n as a function of n and r. As you can see there, future value is directly related to both time and the discount rate. The larger the discount rate, the larger the future value. For positive discount rates, the more time, the larger the future value.

Table 1

    Future Value of an Investment of $1000

Figure 1

The future-value factor, FVF_r,n as a function of time and various discount rates.

An easier way to make our future-value calculation is to use a financial calculator: Put in PV=1000, n=6, r=10%, and PMT = 0, then compute FV = $1771.56. Note that, for most financial calculators, you enter the discount rate as a whole percent, 10, not as a decimal number, 0.10. Throughout the rest of the book, we'll show you such calculator calculations in a standardized format. The amount the calculator solves for is in bold type. The other amount are inputs.

N = 6 r = 10 PV = 1000  PMT = 0 FV = 1771.56

Present Value of a Future Cash Flow

Now, let's find the present value of an expected future cash flow. The present value (PV) is the amount that if invested today at r per period would provide a given future value at time n. We can compute a PV using the present-value formula:

                                                                  The Present-Value Formula

                                                               PV= FV_n[1/(1+r)ⁿ] = FV_n(FVF_r,n)--------------------------(2)

The present-value formula is simply a rearrangement of the future-value formula. We are solving for PV instead of FV. In the present-value formula, the amount [1/(1+r)ⁿ] is called the present-value factor. The present-value factor, PVF_r,n is the amount that, if invested today at r per period will grow to exactly $1.00 n years from today.
Figure 2 is a graph of PVF_r,n as a function of time and various discount rates. It shows that present value is inversely related to both time and the discount rate. That is, the larger the discount rate, the smaller the present value. For positive discount rates, the more time until you get the cash flow, the smaller the present value will be. Like two kinds on a seesaw, when one goes up the other goes down.

Solving for a Return

If you look back at the basic calculator formula, you can see how the present-value formula is part of it. You can also see that if you know any four of the five input variables, the formula can be solved for the fifth.

Figure 2

The present-value factor, PVF_r,n as a function of time and as a function of time and various discount rates.

For example, to find a PV, we put in FV (the expected future cash flow), n (the time the cash flow will occur), r (the required return), and PMT = 0, However, suppose you already know PV from a market price, but you don't know the discount rate. You can rearrange the formula to solve for the expected return. Solving for r, with PMT = 0, we get
                                                                     r = (FV/PV)1/n - 1

Solving for the Number of Time Periods

We also rearrange the basic calculator formula to solve for n, using natural logarithms. However, it's much easier to let the calculator do the work.