Benefits of Compound Interest

Chapter 7: Net Benefits over Time and Present Value I don t believe in princerple, But oh I du in interest. James Russell Lowell My interest is in the future because I am going to spend the rest of my life there. Charles F. Kettering

The key to individual time preference: Would you rather have a dollar today or a dollar a year from now? Why? 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? Answer the same question for different lengths of time, such as 5 years and 20 years.

If you Save

The Benefit of Saving: Compound Interest A fixed amount of saving in an interest bearing account will collect increasing amounts of interest in future years. (7-1) Future Wealth Next Year: W1= W0(1+r), where W0= wealth today, W1= wealth after one year, Wt= wealth after t years, and r is an interest rate in decimal form. Example: With a savings account with a 5% interest rate, $100 today will equal $100 (1+.05) = $105 after one year. (7-2) Future Wealth in t years: Wt= W0 (1+r)t,

3 years of saving at 5% interest 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. The annual interest rate on a money market fund is 5%, or .05 How much interest will Don earn during his first year? How much will Don s savings account be worth after one year? How much will Don s account be worth after 2 years? How much will Don s account be worth after 3 years?

3 years of saving at 5% interest 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. The annual interest rate on a money market fund is 5%, or .05 How much interest will Don earn during his first year? $1,000 x .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 (1.05) or $1,050. How much will Don s account be worth after 2 years? $1,000 x (1.05)2 , or $1,000 x 1.1025, or $1,102.50. How much will Don s account be worth after 3 years? $1,000 x (1.05)3, or $1,000 x 1.1576, or $1157.63

Summarizing this Example To summarize this example, see the following table. Table 7-1: Don s Compound Interest Year 0(now) after 1 yr. after 2 yrs. after 3 years $1,000 $1,000(1+.05) $1,000(1.05)2 $1,000(1.05)3 = 1,000 $1,050 $1,102.50 $1,157.63

Higher Rates of Return Produce Much Higher Payoffs in the Distant Future Your turn 7-2: Assume that Dierdre has $1,000 and access to an account that pays a guaranteed return of 10 percent (or .10) per year. A. How much interest will Dierdre 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 2 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 Dierdre s and Don s payoffs larger in the 3rdyear than in the first?

Compound Interest over the Long Run Table 7-2: Examples of the Future Value of $1 5 percent interest 15 percent interest FUTURE VALUE $1 $1.05 $1.10 $1.16 $1.63 $3.39 $11.47 FUTURE VALUE $1 $1.15 $1.32 $1.52 $4.05 $32.92 $1,083.62 YEAR FORMULA YEAR FORMULA $1(1+.05)0 $1(1+.05)1 $1(1+.05)2 $1(1+.05)3 $1(1+.05)10 $1(1+.05)25 $1(1+.05)50 $1(1+.15)0 $1(1+.15)1 $1(1+.15)2 $1(1+.15)3 $1(1+.15)10 $1(1+.15)25 $1(1+.15)50 0 1 2 3 10 25 50 0 1 2 3 10 25 50 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.

If You Invest

Investing in Non-Interest paying assets When one decides to invest some money in a store or a college education rather than saving the money, one is giving up the chance to collect interest on that money Waiting for profits or income increases in a future year means you are giving up the chance to earn compound interest between now and that time. Therefore, the value of that future payoff should be corrected for the lost interest one experiences while waiting for the return 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.

A Present Value Example Formula: The present value of $1 acquired in year t = $1/(1 + r)t. Example: Tina s insufferably cheap parents have given her the choice of receiving $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 present value of $120 acquired in 2 years (7-4) $120 = 120 = $99 (1.10)2 1.21 She should take the $100 now, which will lead to slightly more than $120 in 2 years.

Another present value example Your Turn 7-4: A person who wins 10 million dollars in the Pennsylvania Lottery might receive $250,000 immediately, $250,000 per year for 19 years, and then a 5 million dollar payment at the end of 20 years. Assume that you are a winner. If the interest rate is 5%, find the present value of the $250,000 you will receive 1 year from now, 2 years from now, and three years from now. If your calculator can handle it, find the present value of the $5,000,000 you would receive in the 20thyear.

The Present Value of a Multi-Year Program or Investment Many types of programs have a pattern with an initial investment followed by a number of years with (hopefully positive) net benefits from the investment. Examples include infrastructure investments such as mass transit, education and training, and pollution control policies. A program with an initial cost and a stream of net benefits extending into the future will have a series of separate present values for each year, which can be added once each future payoff is converted to its present value.

The Present Value of a Multi-Year Program or Investment The Present Value formula for a stream of benefits lasting N years: The steps for solving this problem are important (1) First find the net benefits for each year, and discount them to their present value by dividing. Only after each year is discounted should you add the years together to get the present value of the entire project or investment.

A Present Value Case Study The Job Corps program (discussed in more detail in Chapter 15) is a residential program that provides training and life skills to disadvantaged youth in the U.S. Assume that the cost of the training (including room and board) is $13,000 and occurs when James enters the program. Because of the program James gains $3,000 in earnings (after taxes), and government receives $500 because James pays more taxes and collects less food assistance from the government. If the training costs take place immediately (in year 0) no discounting will be needed for that year. Net benefits total $3,500 each year for 4 years (years 1-4). Set up the net benefits for each year, then discount future years using a 5% interest rate If we are analyzing a 4 year period using a 5% discount rate, is the program a net benefit to society?

The Present Value of a Multi-Year Program or Investment now yr 1 yr 2 yr 3 yr 4 PV = + + + = -13 + 3.33 + 3.17 + 3.02 + 2.88 = -$0.6 (Numbers above are in thousands) Note that converting each year s payoff to present value has to be done before adding different years together. Also note that after 4 years the payoff is negative. If benefits from the program disappeared after 4 years, the investment is not efficient. Only one more year of net benefits would turn the program positive, however.

A Present Value Example: Your Turn Recall that the cost of the training is $13,000 and society gains $3,500 each year for 4 years. Will the Job Corps program be worthwhile if it is analyzed using a 2% discount rate rather than 5%? Work out the present value to prove your answer.

Present Value with Infinitely Long Benefits For investment decisions involving very long time frames, one can arrive at an easy and fairly accurate estimate of the present value of the net benefits by assuming that the stream of net benefits continues forever. If a project results in a constant net benefit each year, the present value of this stream of benefits can be summarized in a very simple formula: PV = $net benefit , where r is the real discount rate. r Your Turn 7-6: Assume that preserving the Grand Canyon will provide $20 billion in annual net benefits forever. If the relevant discount rate is 5 percent, what is the present value of the benefits of preserving the Grand Canyon?

Alternatives to Present Value

Internal Rate of Return Definition: The Internal Rate of Return (IRR) is an interest rate that when substituted into the present value equation produces a discounted present value of zero. As a formula, the internal rate of return appears the same as present value, but it is the discount rate that one needs to find, not the present value itself. (7-9) PV = (Bt-Ct) /(1+IRR)t= 0. Unfortunately, there is no easy way to find this rate except through trial and error.

An Internal Rate of Return Example Example: Assume that Granny, a 63 year old woman, decides to return to college for a one year masters degree, after which she will work until she retires at 65. She currently works at a job paying $20,000. Her tuition will be $2,000 for one year. After she graduates, she will earn $32,000 until retirement. Her benefits and costs are as follows: age (year) 63(0) 64(1) 65(2) benefits $0 $32,000 $32,000 costs $22,000 If you are doing this in class, some calculate the present value using different interest rates. I recommend 5%, 6%, and 7% as examples. If the present value is negative, your interest rate is too high, if it is positive, your interest rate is too low $20,000 $20,000

Internal Rate of Return Decision Rule for the Internal Rate of Return: If the internal rate of return is greater than a comparable market interest rate, the project should be approved. If the internal rate of return is less than the discount rate, the project should be rejected. Another Example: Assume that Mega-man is considering a two year career in the National Football League. His salary will rise from $200,000 to $320,000 for those two years. However, he must also train for one year (with no current income) to develop his football skills. Through trial and error find the internal rate of return. Hints: (1) Start somewhere between 10 and 15 percent. Anything within a few hundred dollars of zero will be close enough. (2) His lost salary during his training year that should be included as part of the net benefits.

Problems with the Internal Rate of Return While the internal rate of return produces a number and a decision rule that are easy to understand, negative net benefits in the distant future might lead to more than one internal rate of return, making the concept meaningless as a guide to policy decisions. Example: Assume that Cleveland is considering hosting an exposition on the history of the steel industry which involves constructing a model steel plant and a set of exhibits, running the exposition for 3 years, then demolishing the plant and cleaning up pollution from the project. Representative values for this project (in billions of dollars) are presented in Table 7-4.

Problems with the Internal Rate of Return Table 7-4: The Cleveland Example NET BENEFITS PER YEAR (MILLIONS) 0% discount rate 5% discount rate 19% discount rate TOTAL PV YEAR 0 YEAR 1 YEAR 2 YEAR 3 YEAR 4 0 -30 -30 -30 20 19.0 16.8 30 27.2 21.2 20 17.3 11.9 -40 -32.9 -19.9 +.06 0 If 5 percent is a reasonable discount rate for a net present value calculation, this project should be approved. However, there are two possible internal rates of return, zero and nineteen percent. If the IRR equals zero, the project should be rejected. With an internal rate of return of 19 percent, the project should be approved.

Another Alternative to PV: The Discounted Benefit/Cost Ratio For the discounted benefit cost ratio one separately calculates the present value of the benefits and the present value of the costs, then divides the discounted benefits by the discounted costs. This is similar to the benefit/cost ratio introduced in chapter 6, and is shown in equation (7-10) below. n Bt = t + Ct t 1 ( ) r (7-10) Discounted B/Cratio = 0 n = t + t 1 ( ) r 0

Discounted Benefit Cost Ratio Decision Rule and Example The decision rule for a single project using the discounted Benefit/Cost ratio is the same as with the basic benefit/cost ratio. If the ratio is greater than one, the project should be adopted, and if it is zero or less, it should not. (7-10) Discounted B/C ratio = Your Turn 7-8: Given the information in Your Turn 7-7, find Mega-Man s discounted B/C ratio given a discount rate of 7 percent. Year 0 Year 1 Year 2 Benefits $0 $320,000 $320,200 Costs $200,000 $200,000 $200,000

Choosing Among Projects of Different Lengths (More Advanced)

The Issue: Projects of differing lengths might be repeated multiple times If this is true, comparing the present values of projects with net benefits extending over different time periods may not be valid. For example, if Project A lasts 4 years and project B lasts two, a simple comparison of their present values will appear as follows: YEAR 1 -10 -9.52 -5 -4.76 YEAR 2 +10 9.07 +12 10.88 YEAR 3 +5 4.32 YEAR 4 +5 4.11 TOTAL xxx $7.98 xxx $6.12 (1) Project A (1) Discounted* (2) Project B (2) Discounted 0 0 0 0 If each project can be undertaken only once, project A should be chosen

Two methods to analyze the choice if one or more projects can be repeated: If both projects can be repeated, repeating one or both projects until a common time frame is reached will correct for the problem of differing project lengths. For example, repeating project B twice provides present value figures for both projects over a 4 year timespan, which could be assumed to extend further into the future. For a simple example, repeat project B in Table 7-5 twice. Table 7-5 YEAR 1 YEAR 2 (1) Project A -10 +10 (1) Discounted* -9.52 9.07 (3) Project B repeated (3) discounted -4.76 10.88 YEAR 3 +5 4.32 -5 YEAR 4 +5 4.11 +12 TOTAL xxx $7.98 xxx -5 +12 -4.32 9.87 $11.61 In this case, 2 repetitions of project B will provide more net benefits in the same time period, making B the better choice.

Another Method of Choosing Among Projects of Different Lengths The Equivalent Annuity Method (EAM) involves 3 steps: 1. Find the present value of a single repetition of each project 2. Find the present value of receiving $1 in each year of each project. For short projects, use a basic present value formula: ? 1 ?? = (1 + ?)? 1 For longer projects, an annuity formula might be faster:

Another Method of Choosing Among Projects of Different Lengths Project A: 4 year annuity value = $1 /1.05 + $1/(1.05)2+ $1/(1.05)3+ (1.05)4 = .952 + .907 + .864 + .823 = $3.546. (See the text for the Annuity formula version of this calculation) The EAV of project A equals its present value from Table 7-5 divided by the PV of the annuity. In equation form, EAV= $7.98/$3.546 = 2.25. Project B: 2 year annuity = $.952 + .907 = $1.859 Present value of project B = $6.12 (from Table 7-5) EAV for B = $6.12/$1.859 = $3.29 For repeatable projects, choose the higher Equivalent Annuity Value

Conclusion The concept of present value provides the basic tool for any analysis of the efficiency of long term private or public investments. The primary lesson of the chapter is that for various reasons, including but not limited to the cost of foregone interest, a dollar of net benefits received in the future should be valued below a dollar received today. You should now be familiar with the mathematics of compound interest, discounting, and the role of present value in analyzing long term policy decisions.