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where

i = | the rate of interest |

t = | the number of periods in the future that an amount of money is to be paid or received |

FV_{t} = | the future amount to be paid or received |

PV_{t} = | the present value of the future amount, given that the interest rate is equal to i. |

Here we generalize this formula and examine some variations that are likely to prove useful to you in everyday living.

**MULTIPLE FUTURE AMOUNTS**

The formula shown above illustrates how to calculate the present value of a single future amount. What method should be used when there are multiple future amounts? The general rule is quite simple:

The present value of a series of future amounts is the sum of the present values of each of the elements in the series.

To see how this rule operates, let's consider a simple example. Assume that on your eighteenth birthday your Uncle Jamal offers you a choice between two presents. You may have either (a) $1,500 in cash immediately, *or* (b) $1000 on your twenty-first birthday and an additional $1000 on your twenty-fifth birthday. Which do you prefer? Clearly, the answer will depend on the rate of interest. This is because the interest rate will determine the present value of each of the two future amounts, and thus will determine the attractiveness of waiting for these future amounts rather than accepting the $1,500 immediately.

Consider option (b), which begins with the receipt of a $1,000 payment on your twenty-first birthday. This first payment would be three years from today, so t = 3. The future amount is $1,000, so FV_{3} = $1,000. Suppose that the interest rate is 10 percent, so that i = .10. We can now compute the present value of $1,000 payment due on your twenty-first birthday as being

Thus, the present value of the $1,000 due three years from now is $751.31.

The present value of the payment due on your twenty-fifth birthday can be calculated in a similar manner, recognizing that in this case, t = 7. Thus,

so that the present value of the $1,000 due in seven years is $513.08.

Now, as the general rule above notes, the present value of the option that contains the two future payments is simply the sum of the present values of the components:

Clearly, given a choice between $1,500 immediately, versus a series of future amounts that have a present value of only $1,264.39, you will choose the option of receiving $1,500 now, if the proper rate of interest is 10 percent.

In general, of course, the preferability of receiving the present amount rather than the series of future amounts will depend on the precise value of the rate of interest. Indeed, as you will see at the end of the chapter when you extend the analysis, if the interest rate were 5 percent rather than 10 percent, the series of future payments would have a present value in excess of $1,500 and would thus be the preferable option. Nevertheless, the general point is that whenever dealing with a series of future amounts, all that is required to determine the present value of the series is to calculate the present value of each element, and then add those present values.

**ANNUITIES**

An **annuity** consists of a series of future amounts that are the same in each period. The annuity gets its name from the fact that the length of the period is most commonly one year--the payments come annually. The process of computing the present value of an annuity parallels the process of computing the present value of any series of future amounts, although the structure of the annuity permits one simplification.

Suppose your Aunt Samantha also offers you a choice of presents on your eighteenth birthday. One possibility is $1,500 immediately, but the other option is to receive $603.38 at the end of each of the next three years (i.e., on your nineteenth, twentieth, and twenty-first birthdays). Which option should you choose? If the interest rate is 10 percent, the present value of the $603.38 per year for three years is given by

But note that each of the future values is the same, namely $603.38. Thus, using A = $603.38 as the designator for the common value of each of the future amounts, we have FV1 = FV2 = FV3 = A. We can factor out the common factor A, leaving us with

PV = | A • { [1 / (1+.10)^{1}] + [1 / (1+.10)^{2}] + [1 / (1+.10)^{3}] } |

= | A • { 0.909 + 0.826 + 0.751} |

= | $603.38 • { 2.486} |

= | $1,500. |

Because the annuity has a present value ($1,500) equal to the present value ($1,500) of the immediate payment, you should be **indifferent** between the two options.

**PERPETUITIES**

One special type of annuity is called a *perpetuity* because it involves a *perpetual* annual series of payments. Thus, a **perpetuity** is an annuity that lasts *forever*, i.e., the equal annual-valued amounts never stop. Following the analysis that we used for the annuity, if the annual amount involved is A, and the interest rate is i, then the present value of the perpetuity (P) is given by

[1 / (1+ i)

where the series of periods at the end (...) indicates that more and more terms continue to be added on forever. This appears to be quite messy, but in fact it turns out to have an elegant solution. The series of terms in braces is called an *infinite geometric series*, which can be shown to be equal to (1 / i). Thus,

[1 / (1+ i)

so that the present value of the perpetuity is given by

To illustrate with a numerical example, suppose Uncle Lamar offers you a choice between $1,000 immediately or $100 a year forever. If the interest rate is 10 percent, the $100 perpetuity is worth

so the two options have the same present values. Note also that if the interest rate were 12.5 percent, the $100 perpetuity would be worth $100 / 0.125 = $800, while if the interest rate were 5 percent, the perpetuity would have a value of $100 / 0.05 = $2,000.

**THE POWER OF COMPOUNDING**

As you can see, the value of any perpetuity is extremely sensitive to changes in the interest rate. For example, if the interest rate is 10 percent, the present value of a *single* payment of $1,100 due one year from now is

Similarly, the present value of a $100 perpetuity is given by

If the interest rate falls to 5 percent, the present value of the single future payment rises to

while the value of the perpetuity rises to

Thus, the same drop in the interest rate causes the present value of the single future payment to rise by $47.62, while it causes the present value of the perpetuity to *double*.

In a similar vein, if the interest rate were to jump to 20 percent, the present value of the single future payment would decline to

while the present value of the perpetuity would plunge to

The reason that changes in the interest rate have such a powerful impact on the value of a perpetuity can be seen by examining the initial equation we used to determine the value of the perpetuity:

[1 / (1+ i)

As is evident, the interest rate enters the denominator of the term in brackets time after time. Moreover, the interest rate ends up being multiplied by itself time after time. [Recall, for example, that the term (1 + i)^{5} = (1 + i) • (1 + i) • (1 + i) • (1 + i) • (1 + i).] Thus, when the interest rate rises, its depressing effect on the value of the perpetuity *compounds* itself over and over, while when the interest rate falls, its elevating effect is compounded many times over. The result is that the present value of a perpetuity is extremely sensitive to changes in the interest rate.

**HOW THE PERPETUITY WORKS**

It may seem odd that an *infinite* series of future payments can have a finite present value. And it may seem stranger still that Uncle Lamar can offer you payments that last forever, when his own lifetime surely will not last so long. As it turns out, a simple example illustrates the fact that neither of these phenomena is as odd as it seems at first glance.

Suppose Uncle Lamar takes $1,000 down to his bank, which is presently paying 10 percent on deposits. He offers the bank the following deal: He agrees to deposit the $1,000 and *never withdraw it*, if the bank will pay 10 percent interest on it for as long as the money remains deposited. Then Uncle Lamar instructs the bank to permit you or anyone designated by you to withdraw the accrued interest, *but not the original $1,000*. Thus, Uncle Lamar gives up (and the bank receives) forever the $1,000 he deposits. In return, you (or whoever you designate) can go to the bank at the end of one year and withdraw $100 [= ($1,000) • (0.10)]. Similarly, you can return at the end of the second year and withdraw $100 [= ($1,000) • (0.10)]. Indeed, you (or whoever you designate, such as your children, grandchildren, and so forth) can return to the bank at the end of each year *forever* and withdraw $100 each time.

Thus, by forever giving up a present amount equal to $1,000, if the interest rate is 10 percent, one can receive in return an annual amount of $100 forever. For this reason, we conclude that if the interest rate is 10 percent, $100 per year forever is equivalent to $1,000 immediately: The present value of a perpetuity that pays $100 per year is $1,000, if the interest rate is 10 percent.

**GROWTH AND THE RULE OF 72**

If you were to deposit $1,000 in the bank and leave it as well as the interest on it to accumulate, the value of your bank account would clearly grow over time. This sort of growth over time is fundamental to understanding many important economic forces. Whether we are thinking about population, real income, the price level, or the value of your bank account, we must understand the principles of growth if we are to understand how and why the levels of these variables got where they are today, and what is likely to happen to them in the future. Here we illustrate a simple rule that yields great insight into the process of growth over time.

**THE RULE OF 72**

To find the approximate time it takes for any amount to double, multiply its annual growth rate by 100, and then divide that number into 72. This is called the **Rule of 72**, and it can be expressed algebraically as

where

T = | time in years for doubling to take place, and |

R = | annual percentage growth, multiplied by 100 (so that, for example, 0.10 = 10 percent is treated as being equal to 10). |

**An Example** Let's consider a concrete example. For many years, the aggregate production of goods and services (sometimes called *output*) in the United States has been growing at an average rate of approximately 3 percent per year. Because the annual growth rate is 0.03, we multiply it by 100 to obtain

Following the rule of 72, we have

This means that the total output of goods and services in the United States has *doubled* approximately every 24 years.

In a similar vein, suppose you deposit $1,000 in a bank account that pays 8 percent interest. In this case, R = (.08) • (100) = 8, so that

Thus, if you make no withdrawals, and if the interest rate remains at 8 percent, the amount of money in your account will double in nine years: You will have $2,000 after that period of time, i.e., nine years.

**Turning the Rule Around** The Rule of 72 can also be "turned around." If we know the doubling period of some amount, we can calculate the approximate annual growth rate of that amount. Thus, we can write

where R and T are defined as before. For example, suppose you know that real income (income corrected for change in average prices) in the mythical nation of Ruritania has doubled roughly every 15 years (that is, T = 15). From this information, we can compute the rate at which per capita income in Ruritania has been growing:

Thus, real income has been growing at a rate of 4.8 percent per year.

**MULTIPLE DOUBLINGS AND THE RULE OF 72**

So far, we have only been concerned about periods of time in which the doubling in value takes place *once*. We can also use the Rule of 72 to deal with *multiple* doublings. To see this, we return to our example of output growth in the United States. As we saw, output has been growing at 3 percent per year, so that the level of output has been doubling every 24 years (T = 72/3). Clearly, if we have *one* doubling in 24 years, at the end of 48 years we will have *two* doublings. Similarly, at the end of 72 years, we will have *three* doublings. So, if we were to start with $1 growing at the rate of 3 percent per year, we would have $2 at the end of 24 years, $4 (= $2 • 2) at the end of 48 years, and $8 (= $4 • 2) at the end of 72 years.

Now, with a bit of algebra, we can be systematic about this process. In particular, if we start with an amount A, which is growing at the rate R, then at the end of N years we will have the amount A_{n}, where

where

T = | doubling period at the growth rate R, and |

2^{(N/T)} = | 2 raised to the power (N/T), where (N/T) is just the number of doublings that occur in N years. |

Here's how to make it work:

- Use the rule of 72 to calculate the doubling period, T.
- Divide the overall time period, N, by the doubling period, T, to obtain the number of doublings that occur in N years.
- Multiply the original amount, A, by "2 raised to the power (N/T)," where N/T is the number you calculated in step 2.

Returning to the outcome-growth experience of the United States, we can put this to practical use. Following the steps outlined above, we have:

- At 3 percent growth per year, the Rule of 72 tells us that output doubles every 24 years. That is, T = 72/3 = 24.
- Suppose we want to find out what has happened to output over the past 48 years, so N = 48. Then we have (N/T) = 48/24 = 2. That is, output has doubled
*two*times over the course of the last 48 years. - Multiply the level of output 48 years ago (A = original amount) by the factor "2 raised to the power (N/T)," which is 22 = (2 • 2) = 4.

Thus, the level of output (the production of goods and services) in the United States is *four* times higher than it was 48 years ago.

If all of this seems a bit abstract, let's end with an example that deals more closely with your future. If you haven’t already, at some point, you are likely to get married and start a family. Suppose that, when your first child is born, you put $10,000 into a savings account that pays 8 percent interest. How much money will be available to pay for the child's college education when he or she reaches age eighteen? To determine the answer, we simply follow the steps outlined above.

- Because the growth rate is 8 percent, the doubling period is T = 9, which is simply 72/8.
- We are interested in what happens over the next 18 years, so N = 18, and of course (N/T) = 18/9 = 2. That is, the value of the money in the account will double twice over the next 18 years.
- In this case the original amount is A = $10,000, so we solve for
A

_{n}= A • 2^{(N/T)}= ($10,000) • (2^{2}) = ($10,000) • 4 = $40,000.

Thus, if you invest $10,000 now at 8 percent interest, then in 18 years you will have $40,000. At current prices, that would just about pay for four years at most state universities. But because the dollar cost of a college education is likely to double over the next 18 years,* the $10,000 you set aside now will only pay for two years of his or her public college education.

* What are we assuming about the rate at which that cost will grow over the next 18 years?