1.4 Time Value of Money and Other Cash Flow Patterns

There are a variety of other common cash flow patterns for which we can perform time value of money calculations. In reality, we can evaluate any stream of cash flows by using FV = PV × (1 + i)ⁿ or PV = FV ÷ (1 + i)ⁿ for each cash flow. In some instances, however, this technique is not practical and can be circumvented by clever application of the concepts we already know. Here we will discuss three additional cash flow patterns: perpetuities, annuities due, and uneven cash flows.

Perpetuity

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:

${\text{PV}}_{\text{annuity}}=\text{PMT}\times \left\{\frac{1-[1÷(1+i){]}^n}{i}\right\}$

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

${\text{PV}}_{\text{perpetuity}}=\frac{\text{PMT}}{i}$

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

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

Though it appears suspiciously easy, the solution to this problem is simply the $10,000 annual payment divided by the 8% interest rate:

${\text{PV}}_{\text{perpetuity}}=\frac{\text{PMT}}{i}=\$\text{10,000}÷0.08=\$\text{125,000}$

With an 8% required rate of return, this infinite cash stream is actually worth $125,000 to us today.

Annuity Due

Earlier we examined ordinary annuities, or annuities in which the payments occur at the end of the period, that is, after a one-period delay. An annuity due, by contrast, is an annuity whose payments occur at the beginning of the period. Thus, an ordinary annuity that starts today, at time 0, will not make its first payment until time 1, while an annuity due starting at the same time will make its first payment at time 0. Consider Figure 1.6. We see a three-payment, $1,000 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 $1,000 with a discount rate of 8% starting today) but change the timing of the payments to the beginning of the year. This means that the first payment will come at time 0.

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

Method 1

Use the ordinary annuity method to find the PV of all payments other than the first payment.

In our example, we are looking for the present value at time 0. The first payment of $1000 comes at time 0. Therefore, the present value of the first payment is known (the value today of $1,000 to be received today is obviously $1,000). All we need to do is find the present value of the other payments using the ordinary annuity method and then simply add in the first payment. Using your financial calculator (after setting it up), the keystrokes will be as follows:

Method 2

Set calculator to “Begin Mode” and solve for value of all payments.

Your financial calculator is set up to handle annuity due problems. Hence, you can change your calculator to “Begin Mode” and solve for present value of the annuity due as follows:

At first glance, you may prefer method 2. However, you need to be very careful with this approach. Most annuity questions involve ordinary annuity calculations. If you set your calculator to Begin Mode and then forget to change it back to End Mode, you will probably be very disappointed in your score on the next exam.

Future Value of an Annuity Due

Solving for the future value of an annuity due is easiest when using the Begin Mode on your calculator. We can calculate the future value of this same annuity similar to how we calculated the present value by doing the following:

Without looking, can you figure out whether the annuity due or the ordinary annuity is more valuable? If you’re having trouble, look at the picture of the cash flows in Figure 1.8 and ask yourself if you’d rather have the annuity due shown or an ordinary annuity (hint: the first payment for the ordinary annuity would come at time 1).

Recall that in previous examples we calculated the future and present values of this same three-year, $1,000 payment annuity as an ordinary annuity. With end-of-year payments, we calculated a future value at time 3 of $3,246.40 and a present value at time 0 of $2,577.10. Looking back to our annuity-due calculations, we calculated the future value to be $3,506.11 and the present value to be $2,783.26. In both cases, the value of the annuity due is greater than the value of the ordinary annuity. Hopefully, you were able to reason this to be the case. With an annuity due, we receive our payments earlier than we would with an ordinary annuity. As we learned at the beginning of this topic, because of inflation, risk, and opportunity, money today is worth more than money tomorrow. Hence, earlier payments make an annuity due more valuable both today and in the future than an ordinary annuity of equal length and payments.

Uneven Cash Flows

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 Table 1.1.

It’s important 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 1.10 is a good example—if all else fails, simply use the brute force approach of discounting/compounding each cash flow individually.

Deferred Annuity

Figure 1.10 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 can the process of saving for the education of your children. There are also many applications in financial markets.

Exercise: Present Value of a Deferred Annuity

Cash flows from the investment illustrated above are expected to be $40 per year at the end of years 4, 5, 6, 7, and 8 (see Figure 1.10). If you require a 20% rate of return, what is the present value of these cash flows?

We can find the present value of a deferred annuity in a number of ways. For instance, just as we did with uneven cash flows, we can discount each individual cash flow back to time 0 separately using the formula PV = FV ÷ (1 + i)ⁿ. Alternatively, we can find the present value of the annuity at its beginning (which, if we consider it to be an ordinary annuity, will be one period before the first payment) then discount the resultant single sum back to time 0. For this deferred annuity, we would calculate the present value of the annuity at time 3, one year before the first payment is made.

Are we done? Remember, we want the present value at time 0. What we have calculated here is the single sum value of the annuity at time 3. Hence, we need to finish the problem by discounting this single sum ($119.62) back to time 0:

What if we had used a different approach to the valuation of the deferred annuity in Figure 1.10? No matter how you discount the cash flows, the present value of this deferred annuity will always be $69.22 as long as we properly account for the timing of the cash flows. You might want to take a few minutes to experiment with other approaches to calculating the present value of this deferred annuity.