When I ask my students this question, most prefer to receive their dollar imme-diately. The reasons they give vary widely, but usually involve avoiding future risks such as possible inflation, forgetting about the promised dollar, or even facing a small risk of death. Some mention their personal preference for spending the dollar now, regardless of the risk or reward from waiting. Others mention the fact that if they took the dollar now and saved it for a year, they would collect interest in the meantime, and would therefore have more than one dollar a year from now.
All of these reasons for valuing a current dollar more highly than one received in the future have a certain degree of validity, particularly when considering an individual decision to spend or save.3However, only two or three factors seem to apply to public policy decisions. These are the rate of interest, the rate of inflation, and possible elements of risk. The rate of interest represents an oppor-tunity cost of waiting for a given amount of income because, as stated above, waiting for future income involves giving up the chance to earn interest on that income between now and the time of the payoff. Secondly, inflation erodes the purchasing power of each dollar, so any future payoff involving a fixed number of dollars will be less valuable in the presence of inflation. The concept of risk, which will be introduced in Chapter 9, also has implications for the relative value of present and future income, although there is disagreement about the role of risk when benefits are spread across a large number of individuals.
Other versions of the earlier question about saving suggest the challenge facing a policy analyst or an individual when weighing this choice. Would I rather have
$1 now or $1.20 (or $1.50, or $1.90) in two years? Would I rather have
$1,000 now, or $1,200 (or more) in two years? Would I rather have $1,000 now or $2,000 in 20 years? By varying the amounts of time and money involved in the question one can get a feeling for one’s own relative value of present versus future income. This relative value might vary for different lengths of time.
Your Turn 7.1:If you had a choice of $100 now or $X in two years, how high would $X have to be before you would be willing to delay receiving the money? Discuss with classmates or roommates. Answer the same question for different lengths of time, such as five years and 20 years.
Decisions involving investment in long-lived assets raise similar questions. A new science building at your university could bring in additional students, more qualified faculty and research grants. Are these future sources of income worth the investment? The value of present versus future net benefits is one of several important dimensions in such a policy question. The same issue applies to high-way or mass transit construction, new sports stadiums, and training and education programs for the poor. These types of issues will be discussed in more detail in later chapters.
Investment versus saving
The economic definitions of the terms investment and saving are somewhat different from their common meanings. In economics, investment is the purchase of long-lasting capital goods such as factories, shopping malls, machines, etc.
This definition applies to private or public investments, and is equally true for human investment (education and training) or capital investment. When business, government, non-profit institutions or households decide to invest, they are usually
spending money in the present or near future in order to attain net benefits which occur primarily in the more distant future.
Capital investment is one form of action that one can take with one’s wealth.
Another is saving. To the economist, saving means any action which stores wealth for future use. Saving includes any common form of personal financial investment such as the purchase of stocks, bonds, mutual funds, cash-value life insurance, or simple savings account deposits. This is a far broader definition of saving than the public would normally expect. Saving offers benefits in the form of interest, dividends, or financial capital gains. The income from saving repre-sents an important opportunity cost of capital investment. Let’s consider the benefits of saving, and then the corresponding opportunity costs of investment.
For simplicity, the following discussion will concentrate exclusively on interest payments.
If you save: compound interest
Saving involves delaying the spending of one’s assets in order to collect interest.
For example, if the annual interest rate on savings is 3 percent, each dollar a person saves will produce 3 cents in interest after one year. Moreover, if a person allows the savings to remain in place, in future years she will begin to earn interest not only on the original deposit (the principal), but also on the interest accumulated during earlier time periods. This process of collecting interest on earlier interest as well as the principal is called compound interest, and can be computed for annual interest payments using the following formulas:
Future wealth next year: W1= W0(1 + r) (7.1)
Future wealth in t years: Wt= W0(1 + r)t (7.2) where W0= wealth today, W1= wealth after one year, Wt= wealth after t years, and r is an interest rate in decimal form.
Example: Don deposits $1,000 in a money market fund and leaves it there, along with any interest earned on the $1,000, for three years. If the annual interest rate on a money market fund is 5 percent, or 0.05, how much will Don have at the end of this time? To start, let’s consider this process one year at a time. I will work out this example, and let you work out a similar example below.
How much interest will Don earn during his first year?
Don will earn $1,000×0.05, or $50 in interest.
How much will Don’s savings account be worth after one year?
At the end of the year, his account will be worth $1,000 + 50, or $1,050. Notice that
$1,050 equals $1,000×(1.05), where 0.05 equals the rate of interest.
Your Turn 7.2: Assume that Deirdre has $1,000 and access to an account that pays a guaranteed return of 10 percent (or 0.10) per year.
(A) How much interest will Deirdre earn during her first year? How much will her account be worth after the first year?
(B) What will her fund be worth at the end of two years? Why did her fund grow by a greater amount during the second year than the first?
(C) How much will Deirdre’s savings account be worth after three years?
(D) Why is the difference between Deirdre’s and Don’s payoffs larger in the third year than in the first?
How much will Don’s account be worth after two years?
According to the compound interest formula, after two years his account will be worth $1,000×(1.05)2, or $1,000 ×1.1025, or $1,102.50.
During the second year Don earns $52.50 in interest. This amount is made up of $50 of interest earned on the original principal plus $2.50 in additional interest earned on the $50 he earned during the first year. Earning interest in year 2 on the interest collected in year 1 is the basis for compound interest.
How much will Don’s account be worth after three years?
According to the formula for compound interest, after three years his account will be worth $1,000 ×(1.05)3, or $1,000 ×1.1576, or $1,157.63.
In year 3 his account grew by slightly more than $55, demonstrating that the total value of the account will increase at an increasing rate due to compound interest.
To summarize this example, see Table 7.1.
Table 7.1 Don’s compound interest
Year 0 (now) After 1 year After 2 years After 3 years
$1,000 $1,000(1 + 0.05) $1,000(1.05)2 $1,000(1.05)3
= 1,000 $1,050 $1,102.50 $1,157.63
In words, after the first year Don will have the original deposit (the principal) plus interest income which equals the interest rate times the principal. During the second year he will collect interest on the principal plus the first year’s interest payment, which will make the second year’s interest payment somewhat larger than the first, and so on.
Now we can consider the implications of the previous examples more fully.
If you completed Your Turn 7.2 correctly, you will notice that the value of the interest earned by Deirdre is twice as high as Don’s in the first year due to the higher interest rate, but is more than twice as high as Don’s in the second and third years. The annual payments resulting from the higher interest rate grow
more rapidly in future years for two related reasons: (1) Naturally, the higher interest rate will pay more money on any given amount of savings, as in the first year. (2) Because of the higher interest payments in early years, one earns inter-est on an increasingly greater amount of accumulated wealth as time goes by. In fact, wealth that is saved at a higher interest rate tends to move away from wealth saved at a lower interest rate quite explosively as the number of years grows.
Table 7.2 demonstrates these two properties of compound interest.
In Table 7.2, two trends seem noteworthy. First, compound interest at any inter-est rate is most significant when allowed to accumulate for many years, because the amount of accumulated interest upon which compound interest can be earned is much higher. Secondly, higher interest rates bring increasingly substantial benefits in the more distant future. Notice that $1 saved for 50 years at 5 percent interest will be worth $11.47, while the same amount saved at 15 percent interest will be worth $1,083.62, or almost 100 times as much. At the 15 percent rate,
$1,000 invested now will make you a millionaire ($1,083,620) in 50 years. It is no wonder that Albert Einstein once referred to compound interest as one of the most powerful forces in the world!
Your Turn 7.3:If your calculator can handle it, find the future value of
$1,000 saved for 50 years at a 5 percent interest rate. Also calculate the future value of $1,000 saved for 100 years at 5 percent and $1,000 saved for 50 years at 10 percent.
If you invest: foregone interest and present value
Recall that to economists saving is the storing of assets for later use, often in an account which pays interest, and investment is the purchase of long-lasting capital goods. While savings generally earns compound interest, spending on capital goods does not. Similarly, no interest is earned on one’s investment in college, a business’s purchase of a factory or store, or the govern-ment’s spending on roads, environmental protection, or job-training programs.
Table 7.2 Examples of the future value of $1
5 percent interest 15 percent interest
Year Formula Future value Year Formula Future value
0 $1(1 + 0.05)0 $1.00 0 $1(1 + 0.15)0 $1.00
Therefore, compound interest is an opportunity cost of each of these long-term investments. In order to account for the opportunity cost of foregone interest, the future benefits and costs of investment spending must be discounted to their present value in order to account for the cost of foregone compound interest.
The concept of present value can be displayed in the following example. If a person has a chance to collect $120 in two years, how much would that amount be worth in money available today? In this case we must reduce, or discount, this future amount of income by removing the compound interest which one could earn between now and two years from now. If the future value (with compound interest) in year t equals the present value times (1 + r)t, then the present value (PV) will equal the future value (FV) divided by (1 + r)t. In symbols,
FV = PV•(1 + r) t and PV = FV/(1 + r)t (7.3) One can get from a compound interest equation to a present value equation by dividing both sides of equation (7.2) by (1 + r)t.
An example should prove very helpful at this point.
Example: Tina gets her annual allowance on a delayed basis. Her insufferably cheap parents have given her the choice of $120 in two years or $100 today. If she accepts the $100, it will be invested in a fund paying 10 percent interest. Which choice should she make? The $100 is already in the present, so it does not need to be discounted. Therefore, finding the present value of the $120 and comparing the answer to $100 is one way to analyze this problem.
Present value of $120 acquired in two years
In 2 years Now Now Present value
(7.4)
Concept: The present value of a dollar received in some future year t is the amount which, if put in a savings account and allowed to collect compound interest, would equal $1 in year t.
This present value problem says that $120 in two years is equivalent to about
$99 today because if Tina deposited $99 at 10 percent interest, it would grow to
$120 in two years. Therefore $99 is the present value of the $120 of future income discounted at the 10 percent interest rate. By comparing the present value of the
$120 in two years with the $100 available now, we can see that Tina would be slightly better off choosing the $100 now. In this case, one could also calculate the future value of the $100 and compare it to the $120 available in two years.
Because $100 ×(1.10)2= $121, taking the $100 is still the best option.
Conceptually, present value involves asking a modified version of the question asked earlier in the chapter: How much would I accept today in order to give up a dollar I could receive at some point in the future? For example, if the interest rate is 5 percent, how much would a person accept today instead of
$1 two years from now? The concept of foregone interest says that the person would accept less than $1 today because that person could take today’s income, put it in an interest-bearing account, and have additional funds in two years.
More specifically, receiving $1 after two years involves foregoing two years of compound interest at 5 percent per year. Therefore the present value (PV) of
$1 received two years from now would be
(7.5) In other words, a person would slightly prefer 91 cents today to a dollar paid two years from now given a 5 percent interest rate. Similarly, $1 to be paid in four years would be equivalent to 82.6 cents [$1/(1.05)4] today because 82.6 cents invested at 5 percent would equal $1 after four years. Similarly, if the interest rate is 10 percent, $1 expected in two years would have a present value of 82.6 cents [$1/(1 + 0.10)2= $0.826], and $1 expected in four years would have a present value of 68.3 cents. The present value of future net benefits is obviously quite sensitive to the interest rate used to discount the future sums to their present values.
This relationship between interest rates and present values is demonstrated further in Table 7.3. Table 7.3 is closely related to Table 7.2. Table 7.2 shows us
PV=$ • =
. $ .
1 1
1 052 0 907
Table 7.3 Examples of the present value of $1
5 percent interest 15 percent interest
Year Formula Present value Year Formula Present value
0 $1/(1 + 0.05)0 $1.000 0 $1/(1 + 0.15)0 $1.000
the possible future values of one current dollar, while Table 7.3 shows us the pres-ent values of $1 we might receive in various future years. Both logically and mathematically these tables have an inverse relationship. For example, receiving
$1 in three years without the opportunity to earn compound interest at a 5 percent rate is equivalent to saving an initial deposit of $0.846 for three years. That is why
$1 in the third year discounted at 5 percent has a present value of $0.846. As in Table 7.2, present values based on different interest rates in Table 7.3 diverge significantly in the distant future.
Your Turn 7.4:One of America’s most common financial fantasies is winning the lottery. However, the way most States pay their lottery winnings makes the present value of the payoff less than its face value. For example, a person who wins $10 million in the Pennsylvania Lottery receives $250,000 immediately, $250,000 per year for 19 years, and then a $5 million payment at the end of 20 years. Assume that you are a winner. If the interest rate is 5 percent, find the present value of the $250,000 you will receive one year from now, two years from now, and three years from now. (Hint: Be sure that your answer is less than $250,000 in each case.) If your calculator can handle it, find the present value of the $5,000,000 you would receive in the 20th year.
By the way, if the discount rate is 5 percent, the present value of winning
$10 million in the lottery paid in this manner over 20 years is $5,053,750, still a tidy sum.
The present value formula
Any public or private decision with net benefits extending into the future can be analyzed using the present value concept. The model for present value analysis follows a simple two-step process; (1) find the present value for each year’s net benefits, and (2) add the present values of different years in order to determine the total present value of the decision. The formula for the total present value of an investment is
(7.6)
where Σequals the sum of all years 1 to N, t equals the number of years elapsed since the beginning of the project, N is the final year of the project’s net benefits, Bt= the benefits in year t, Ct= costs in year t, B0 and C0equal the immediate benefits and costs, and r equals the discount rate. Notice that in the present value
PV B C
formula benefits in the present (year 0) do not have to be discounted, since net benefits received immediately have no time to collect foregone interest. According to the general formula, current benefits in year zero could be divided by (1+r)0, but since anything raised to the zero power equals one, discounting current benefits amounts to dividing by one. Therefore, not discounting immediate net benefits is consistent with the present value formula as well as with common sense.
Applying the formula involves calculating the net benefits for each year, discounting each year’s net benefit separately to present value and then adding the annual present values to determine the total net benefits. The order of these steps is very important. Do not add the benefits and/or costs across different years until the net benefits for all years are discounted to present value. Some examples will help to understand this process.
If the present value is negative, the policy should not be adopted. There is no reason to calculate the foregone compound interest separately. The opportunity cost of foregone compound interest is calculated through discounting.
If interest rates change, so will the opportunity cost of investment. An invest-ment decision which would be rejected at high rates of interest may be accepted if the interest rate falls.
Example: In February 2003 Senator Barbara Boxer of California introduced a bill requiring that missile defense technology be installed on all commercial airliners.4 Supporters of the bill cited several reported missile attacks on airliners over the past 25 years. Most attacks were against propeller planes in the Third World, but a few attempts have also been made against larger commercial jets. Technology for such a defense system is frequently used on military aircraft and has apparently been installed on some El Al planes (the Israeli airline). The primary benefits of this program are possible savings of lives and property. The defense equipment may also increase travel demand by decreasing fear among passengers. Opponents of the law cited its cost, the lack of proof of a domestic threat of this type, and some uncertainty about the best alternative technology for the project.
In order to calculate a simplified present value, assume that the full installation
In order to calculate a simplified present value, assume that the full installation