2.4 TVM and Other Cash Flow Patterns

A perpetuity is an infinite stream of equally spaced, equal cash flows. Figure 1-5 provides an example of a perpetuity—notice that the payments are of equal size ($1,000), come at equal intervals, and continue forever. In a sense, a perpetuity is just an annuity with an infinite number of periods (n = ∞).

Finding the present value of a perpetuity by using PV = FV / (1 + i)ⁿ to discount each individual cash flow would be impossible given that there is an infinite number of cash flows. Instead, since a perpetuity is just an infinite annuity, we can start by looking at the equation for the present value of an annuity:

We already stated that a perpetuity is an annuity with an infinite n. Think about what will happen mathematically to this formula as the value of n increases to infinity. As n becomes very large, the (1 / (1 + i))ⁿ term becomes essentially zero, leaving us with:

Thus, while using the annuity equation alone to calculate the present value of a perpetuity appears impossible, the truth is that perpetuities are very easy to value—simply divide the annual payment by the discount rate.

Example: Present Value of a Perpetuity

Earlier we examined ordinary annuities, or annuities in which the payments occur at the end of the period, i.e. after a one-period delay. An annuity due, in contrast, is an annuity whose payments occur at the beginning of the period. Thus an ordinary annuityAn annuity that pays at the end of each period. that starts today, at time 0, will not make its first payment until time 1 while an annuity dueAn annuity that pays at the beginning of each period. starting at the same time will make its first payment at time 0. Consider Figure 1-6. We see a three-payment, $1000 per payment annuity. But, is it an ordinary annuity or an annuity due?

In this case, we could evaluate the cash flows either as an ordinary annuity or as an annuity due. But, there is one important difference. Recall from the example for calculating the present value of an annuity that the first annuity payment came at time 1 because we assumed that the annuity started at time 0 and made end-of-period payments (notice that time 1 is the end of the first period). Hence, when we calculated the present value, we obtained the value at time zero—one period before the first payment! This is NOT the case with annuity-due calculations. If we perform an annuity-due calculation, the resulting present value will be at the time of the first payment. This point is subtle, but it is important for two reasons: 1) the difference in value between annuities and annuities due can be significant, and 2) this is a high-likelihood mess-up area on exams for beginning finance students. See Figure 1-7 for an illustration of the timing of the cash flows. Also, note that the difference in the value of an ordinary annuity and an annuity due is caused by the assumption of when the first payment comes and the resulting algebra—there is no deep economic insight.

So, how do we calculate the value of an annuity due? Let's think about the same three-year annuity that we've worked with previously (three payments of $1000 with a discount rate of 8% starting today), only now we change the timing of the payments to the beginning of the year. This means that the first payment will come at time zero.

Generally speaking, there are two ways to calculate the present value of an annuity due:

Method 1

Method 2

Consider the cash flow stream shown in Figure 1-9. Even though the cash flows all come at even intervals, because they are not of equal size this cannot be considered an annuity. It is also not a perpetuity because of its finite length. This cash flow stream falls in the broad category called uneven cash flows. Unfortunately, there is no simplified method for finding the future or present value of an uneven cash flow stream. When all of the cash flows are different, we have to discount or compound each individual flow separately using the present/future value approach that we used for single sums and then add them together. For example, to find the present value of the cash flow stream shown in Figure 1-9 at a 10% discount rate, we would perform the calculations shown in Figure 1-10..

An important insight is to realize that you now have tools to solve almost any TVM problem imaginable. Your approach may involve more or fewer calculations, but as long as you keep track of timing you can solve any problem. The uneven cash flow problem presented in Figure 1-11 is a good example—if all else fails simply use the "brute force" approach of discounting/compounding each cash flow individually.

shows a common cash flow stream called a deferred annuity. As the name implies, this is a standard annuity whose first payment is deferred to some point in the future. You might encounter cash flows with a deferred annuity pattern in your personal financial planning. The retirement income question can be viewed as a deferred annuity as well as the process of saving for the education of your children. There are also many applications in the financial markets.

Exercise: Present Value of a Deferred Annuity