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More free books @ www.BingEbook.com Chapter 12: Figuring Out Interest and Return on Investment Be careful: “compounding” in these longer-term contexts (which run 5,10, 20, or more years) takes on a different emphasis. Compounding in these long range settings refers to the exponential growth idea — that if something grows at a certain rate from year to year, over enough years its size will end up being two or more times larger than what you started with. For instance, if you start with a population of, say, 10,000 persons in a town and its population grows 6 percent per year, its population will double to 20,000 in 12 years. You just invested your $10,000 year-end bonus in a 401(k) plan (a qualified tax-deferred retirement account). You don’t pay income tax on the $10,000 or on the earnings in your retirement account until you withdraw money from the account sometime in the future. You plan to retire in 20 years. Being conservative, you put the money in a savings account that pays 5 percent annual effective interest. (The interest rate could change in the future, but assume that the interest rate remains the same over all 20 years.) Q. Assuming that your retirement account earns 5 percent annual interest for 20 years, what will be the balance in your account when you retire in 20 years? A. In order to answer the question for the 20year lifespan of the investment, you need to understand how year-to-year compounding works. Compounding means that you don’t withdraw your interest earnings each year. Instead, you reinvest the annual earnings. The result of compounding for, say, the first four years is shown as follows: Retirement Account Year at Start of Year Interest Earnings at Retirement Account 5.0% at End of Year 1 $10,000.00 $500.00 $10,500.00 2 $10,500.00 $525.00 $11,025.00 3 $11,025.00 $551.25 $11,576.25 4 $11,576.25 $578.81 $12,155.06 The total amount of your earnings over the first four years is $2,155.06. Someone may think that, at 5 percent, you earn $500.00 each year on your $10,000.00 investment, and over four years, you will have earned $2,000.00. But as you can see in the schedule, you earn $2,155.06. You reinvest your annual earnings, which means that, year-to-year, you have more money invested in your retirement account. Over 20 years, your retirement account balance grows to $26,532.98. This amount assumes an annual 5 percent annual earnings rate and assumes that the financial institution you have your retirement investment account with doesn’t go belly up. (The FDIC may insure your account, but that still doesn’t guarantee that you’ll get all your earnings.) Your total amount of earnings over 20 years is $16,532.98 (the future value less the $10,000.00 you started with). It may be useful to think of your total earnings as follows: At $500.00 per year interest, based on your initial investment, you earned $10,000.00 ($500.00 per year × 20 years = $10,000.00). The other $6,532.98 of earnings over the 20 years comes from compounding (reinvesting) your earnings every year. How did I get the answer? To prepare the four-year schedule, I grabbed my trusty HP calculator: I entered 20 for N, 5 for I/YR, negative 10,000 for PV, and then I punched the FV key to get the answer. (By the way, be sure that the PMT [payment] key has zero entered.) Before computer spreadsheet programs and hand-held business/financial calculators came along, you had to use a table look-up method to solve problems like this one. Some of the biggest disadvantages of this method are that tables of future values and present values don’t cover every situation, they’re clumsy to use, and they require you to do pencil and paper calculations by hand. Surprisingly, many college accounting and finance textbooks still include these tables. For the life of me, I don’t know why — it’s like teaching the Morse code when everyone has a telephone. 267 More free books @ www.BingEbook.com 268 Part III: Managerial, Manufacturing, and Capital Accounting 9. Refer to the preceding example question, the retirement savings account example in which you invest $10,000 today and earn 5 percent annual interest (compounded annually) for 20 years. That example assumes that the 5 percent annual interest remains the same over all years. Instead, assume that you earn 4.5 percent annual interest during the first ten years and 5.5 percent annual interest during the last ten years. Are you better off in this situation? Solve It 10. Suppose you just received your $25,000 year-end bonus. Instead of buying a new car, you decide to put the entire $25,000 in a qualified tax-deferred retirement investment account. You’re 55 years old and plan to retire when you’re 65 years old. You’ve done some research and have come up with two options for where to put your money: One is a safe, conservative investment vehicle that should earn an annual 4.5 percent interest rate (compounded annually), and the other is a more risky investment that has a good chance of being worth $45,000 ten years later, but there’s some chance that it could be worth less than this amount. Compare your two options. Solve It Borrowing and Investing in Installments Borrowing and investing are most commonly done in installments. With this method, payments are made regularly to pay off a loan or to build an investment. In this section, I stick with interest-based investments and examples of fixed income investments and loans. (In the section “Measuring Return on Investment (ROI)” later in the chapter, I cover investments in which changes in the market value of the investment are an important part of the return [earnings or loss] on the investment.) Paying off a loan Your business borrows $100,000 from a bank. You and the bank negotiate an installment loan in which you will pay off the loan over four years. The effective annual interest rate is 6 percent. The bank wants your business to amortize one-fourth of the principal amount each year. Amortize means to pay down the principal value of the loan. At the end of the first year, for instance, your business has to pay $25,000 on the principal balance of the loan plus interest for that year, and so on for the following three years. You sign the note to the bank and receive $100,000, which is deposited in your business’s checking account. More free books @ www.BingEbook.com Chapter 12: Figuring Out Interest and Return on Investment Q. How much is each annual payment to the bank? A. Probably the best approach to answering this question is to prepare an Excel spreadsheet to do the year-by-year calculations. (Of course you could do the calculations with pencil and paper by hand, but the Excel program is much faster and less prone to calculation mistakes.) The loan payment schedule is as follows: Year Loan Balance at Start of Year Interest at 6.0% Principal Payment Q. Using the basic premise of the preceding question, suppose the bank wants equal payments at the end of each year. (In the preceding answer, the total payment varies year to year.) What is the annual payment on the loan under these terms? A. A question like this shows the value of a hand-held business/financial calculator, which is designed for the express purpose of solving problems like this one. You enter 4 for N (the number of periods); 6 in INT (the I/YR key on HP calculators); 100,000 in PV (the present value of the loan, or the amount borrowed); and 0 in FV (future value). The reason for entering 0 in FV is that you want the loan completely paid off and reduced to a zero balance at the end of the fourth year. Press the PMT (payment) key, and the answer pops up on the screen — $28,859.15. Each payment to the bank should be $28,859.15. Total Payment Loan Balance at to Bank End of Year 1 $100,000.00 $6,000.00 $25,000.00 $31,000.00 $75,000.00 2 $75,000.00 $4,500.00 $25,000.00 $29,500.00 $50,000.00 3 $50,000.00 $3,000.00 $25,000.00 $28,000.00 $25,000.00 4 $25,000.00 $1,500.00 $25,000.00 $26,500.00 $0.00 The following schedule shows the proof of this answer. Compared with the schedule in the answer to the preceding question note that the annual payments are equal in this schedule: Year Loan Balance at Start of Year Interest at 6.0% Principal Payment Total Payment Loan Balance at to Bank End of Year 1 $100,000.00 $6,000.00 $22,859.15 $28,859.15 $77,140.85 2 $77,140.85 $4,628.45 $24,230.70 $28,859.15 $52,910.15 3 $52,910.15 $3,174.61 $25,684.54 $28,859.15 $27,225.61 4 $27,225.61 $1,633.54 $27,225.61 $28,859.15 $0.00 You can see that the principal balance reduces to zero at the end of the fourth quarter. In this schedule, as the amount of interest goes down each quarter, the amount of principal amortization goes up. 269 More free books @ www.BingEbook.com 270 Part III: Managerial, Manufacturing, and Capital Accounting 11. Year 1 Suppose a business borrows $1,000,000 from a bank. The annual interest rate is 7.5 percent and the loan is for four years. The bank wants the business to make payments at the end of each year such that the principal of the loan is amortized in four equal amounts. Determine the annual payments required under the terms of this loan. You may find the following form helpful: Loan Balance at Start of Year $100,000.00 Interest at 7.5% Principal Payment $75,000.00 $250,000.00 Suppose a business borrows $1,000,000 from a bank. The annual interest rate is 7.5 percent and the loan is for four years. The bank wants the business to make equal payments at the end of each year such that the principal of the loan is completely amortized (paid off) by the end of the fourth year. Determine the amount of annual payment required under the terms of the loan. You may find the following form helpful: Total Payment Loan Balance at to Bank End of Year $325,000.00 $750,000.00 2 Year 1 3 4 12. Loan Balance at Start of Year $100,000.00 Interest at 7.5% Total Payment Loan Balance at to Bank End of Year 2 $0.00 3 4 Solve It Principal Payment $75,000.00 $0.00 Solve It Investing in a retirement account Many people invest in tax-deferred retirement accounts on the installment, or serial basis. They put some money in their retirement accounts at the end of each pay period, and their employers may or may not make matching payments. The federal income tax law encourages setting aside money from wages and other sources of steady income to build up a retirement fund, such as a 401(k), IRA, and many other plans. Assume that your employer encourages employees to invest money from their monthly salaries in retirement accounts, and you’ve decided to do so. Each month, you put $250 into your retirement account, and your employer adds $150, so $400 is invested each month. More free books @ www.BingEbook.com Chapter 12: Figuring Out Interest and Return on Investment Q. To be conservative, assume that your retirement account will earn 4.8 percent income per year, compounded monthly (because you make monthly contributions). Although you may increase your monthly contributions in the future if your salary increases, at the present time, you can’t forecast an increase. So you assume that $400 will be contributed into your retirement account at the end of each month. Determine the balance in your retirement account at the end of 20 years. A. One way you can do this computation is to go to one of many Web sites that have retirement calculators. You enter your monthly contribution, the assumed rate of income per period, the number of years, and presto — the answer comes up on the screen. You can also use a business/ financial calculator or the Excel spreadsheet program. In Excel, the FV function asks you to enter the same variables as a Web site retirement calculator and a business/financial calculator. The balance in your retirement account after 20 years is $160,670. How do you know this answer is correct? Does it pass the common sense test? You invest $4,800 per year for 20 years, which is a total investment of $96,000. If the answer is correct, you will earn more than $64,000 income over the 20 years. Does this amount seem reasonable? Your intuition isn’t particularly helpful here. To be reasonably certain that $160,670 is correct, you could program an Excel spreadsheet to see how your retirement balance accumulates month by month for 20 years. Or you could do the calculation a second time, to see if you come up with the same answer. Frankly, there’s no easy way to prove the calculation is correct. I’m 99.9 percent sure that my answer here is correct, but if you come up with a different answer please let me know as soon as possible! 13. Each month, you put $250 into your retirement account, and your employer matches $150, so $400 is invested each month. Looking ahead, you wonder how much your retirement account will be worth when you retire in 20 years. You assume that your annual rate of income will be 5.4 percent over the next 20 years. Determine the future value of your retirement account 20 years from now. Solve It 14. Assume that your goal is to retire 20 years from today with $500,000 in your retirement account. At this time, you have $50,000 in your retirement account. You would like to know how much you need to put into your retirement account each year from now until retirement to meet your goal, assuming that your retirement investment account will earn 6 percent interest per year. Assume that you make one payment into your retirement account at the end of each year (although in actual practice, it’s more likely that you make monthly contributions during the course of the year). Determine the annual contributions you need to make into your retirement account over the next 20 years to end up with a $500,000 retirement nest egg. Solve It 271 More free books @ www.BingEbook.com 272 Part III: Managerial, Manufacturing, and Capital Accounting Measuring Return on Investment (ROI) There are many kinds of investments — precious metals, real estate, farms and ranches, art, small businesses, corporate bonds, United States Treasury debt securities, municipal bonds, life insurance policies, retirement annuities, stocks, mutual funds, hedge funds, and so on. One thing all these different investment alternatives have in common is that the investor wants to take more money out of the investment than the amount of money put into the investment. Measuring investment performance can be as simple as reading a comic strip or as perplexing as reading a book on nuclear physics. The primary measure of investment performance is the annual rate of return on investment, or ROI. “Return” in ROI refers to the earnings, income, profit, or gain, depending on the type of investment. A fundamental point in measuring investment performance is that you have to recover, or recoup, the amount of capital you put in the investment venture. Only the excess over and above recovery of capital is return on the investment. Another fundamental point is that calculating return on investment focuses on cash flows into and out of the investment — unless changes in the market value of the investment are an integral and important part of the investment, such as investments in marketable securities (stocks and bonds, for example). In just a few pages, I couldn’t possibly do justice to even one of the investment alternatives open to you. Each type of investment requires at least a full chapter to explain its nature, risks, and procedures. So in the remainder of this chapter, I focus on how to calculate ROI for generic investment examples. I begin with a simple investment as far as calculating its ROI goes. Then I move on to more complicated examples. Example 1: Steady income flow; liquidation value equals entry cost In this scenario, you invest $100,000 today, you receive $6,000 at the end of each year for four years, and at the end of the fourth year, the investment is liquidated (converted back into cash) for $100,000. This is about as simple an investment as you can find. Q. What is the annual rate of return on investment in this scenario? A. You can eyeball this example scenario and see that the annual ROI rate is 6 percent. You don’t really need to do any calculations — the annual cash flow is $6,000, or 6 percent of the amount invested. And you get your $100,000 entry cost back in full — no more, no less — at the termination of the investment. Because I’m using generic investment examples, Figure 12-1 uses a generic template to analyze this particular investment. Cash Flow at End of Year Year Figure 12-1: Investment analysis template. Investment Balance Total Amount at Start of Year Earnings at 6.0% Capital Recovery Investment Balance at End of Year 1 $100,000.00 $6,000.00 $6,000.00 $0.00 $100,000.00 2 $100,000.00 $6,000.00 $6,000.00 $0.00 $100,000.00 3 $100,000.00 $6,000.00 $6,000.00 $0.00 $100,000.00 4 $100,000.00 $106,000.00 $6,000.00 $100,000.00 $0.00 More free books @ www.BingEbook.com Chapter 12: Figuring Out Interest and Return on Investment In Figure 12-1, turn your attention to the three columns under Cash Flow at End of Year. The first column is the total cash flow for the period (one year, in this example); the second column is for the earnings on the investment for the period (based on the ROI for the investment); and the third column is for capital recovery for the period. Capital recovery equals the excess of the total cash flow over the amount earnings for the period. In the first three years, capital recovery is zero because the cash flows equal earnings for the year. But in the fourth and final year, the investment generates $106,000 total cash flow; the first $6,000 of this is the amount of earnings for the year, and the remainder is capital recovery. The $100,000 capital recovery in the final year exhausts the investment project; the investment venture is completed at this point. The entry cost of the investment has been fully recovered, and there are no more future cash flows. The template presented in Figure 12-1 can handle just about every investment problem you can think of — it’s a very powerful tool of analysis. You can alter it for any number of periods. I use four periods in these examples because that’s all I need to demonstrate the key points for return on investment analysis. (Of course, an investment could run for 20 or more years and therefore have many more periods.) In Examples 2, 3, 4, and 5, I present the solutions using the Figure 12-1 template. The template offers one key advantage: It shows the capital recovery and investment balance year-to-year. 15. You invested $1,000,000 today and receive $50,000 at the end of each year for four years. At the end of the fourth year, you liquidate the investment for $1,000,000. Determine the annual ROI for the investment, and prove your answer. You may find the following form helpful: 16. Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year ?% 1 $1,000,000 Capital Recovery You invested $250,000 four years ago. Unfortunately, you made a bad decision. At the end of each year for four years, you received no income. But the good news is that you liquidated the investment for $250,000 at the end of the fourth year. Determine the annual ROI for the investment. You may find the following form helpful: Investment Balance at End of Year Cash Flow at End of Year $50,000 2 $50,000 3 $50,000 4 $1,050,000 Investment Balance Earnings at Year Total Amount at Start of Year ?% $0 1 $250,000 2 Solve It Investment Balance at End of Year $0 $0 3 $0 4 $250,000 Solve It Capital Recovery $0 273 More free books @ www.BingEbook.com 274 Part III: Managerial, Manufacturing, and Capital Accounting Example 2: Substantial cash flows each year You invest $100,000 today. The year-end cash flows from the investment are as follows: year 1 = $31,000; year 2 = $29,500; year 3 = $28,000; and year 4 = $26,500. Q. What is the annual rate of return on investment for this scenario? A. The annual rate of return on investment is 6 percent. Check out the following schedule, which uses the Figure 12-1 template. There’s $25,000 capital recovery every year, so the investment balance decreases year-to-year. This decrease may or may not be attractive to you as an investor because you recover your capital quicker, but you earn less on the investment year-to-year. Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year 6.0% Capital Recovery Investment Balance at End of Year 1 $100,000.00 $31,000.00 $6,000.00 $25,000.00 $75,000.00 2 $75,000.00 $29,500.00 $4,500.00 $25,000.00 $50,000.00 3 $50,000.00 $28,000.00 $3,000.00 $25,000.00 $25,000.00 4 $25,000.00 $26,500.00 $1,500.00 $25,000.00 $0.00 Earnings are taken out of cash flow for the period first, before capital recovery is determined. In other words, the amount of capital recovery each year is subordinate to earnings. Earnings come first, and capital recovery is second. 17. You invest $100,000 today. You receive $29,656.22 at the end of each year for four years. Determine the annual ROI on the investment, and prove your answer. You may find the following form helpful: 18. Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year ?% 1 $100,000 $29,656 3 $29,656 Solve It Investment Balance at End of Year Cash Flow at End of Year $29,656 2 4 Capital Recovery You invest $1,000,000 today. You would like to earn 6.5 percent ROI each year for four years and recover $250,000 capital each year. Determine the annual amounts of return for each year you need to meet your objectives. You may find the following form helpful: $29,656 Investment Balance Earnings at Year Total Amount at Start of Year 6.5% $0 1 $1,000,000 Capital Recovery 2 $250,000 3 $250,000 4 $250,000 Solve It Investment Balance at End of Year $250,000 $0 More free books @ www.BingEbook.com Chapter 12: Figuring Out Interest and Return on Investment Example 3: Zero cash flow until final year You invest $100,000 today. The year-end cash flows from the investment are as follows: year 1 = $0; year 2 = $0; year 3 = $0; and year 4 = $136,048.90. Q. A. What is the annual rate of return on investment for this scenario? The annual rate of return on investment is 8 percent. Check out the following schedule, which uses the Figure 12-1 template. Even though no cash flow is received in the first three years, earnings are assigned to each of the first three years at the rate of 8 percent per year. The negative numbers for the first three years in the capital recovery column mean that the imputed earnings are, in effect, compounded or added into the investment balance. For instance, at the start of year 2, the investment balance includes the amount of non-received earnings for the first year. Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year 8.0% Capital Recovery Investment Balance at End of Year 1 $100,000.00 $0.00 $8,000.00 ($8,000.00) $108,000.00 2 $108,000.00 $0.00 $8,640.00 ($8,640.00) $116,640.00 3 $116,640.00 $0.00 $9,331.20 ($9,331.20) $125,971.20 4 $125,971.20 $136,048.90 $10,077.70 $125,971.20 ($0.00) This example brings out an exceedingly important point: There’s no cash flow until the end of the investment, so from a strict cash flow point of view, you can argue that the annual earnings for the first three years are zero. Then in the fourth year, the investment earns 36.05 percent ROI ($36,048.90 cash flow in excess of the entry cost ÷ $100,000.00 entry cost = 36.05 percent). In summary, you can make the case that the annual ROI is as follows: year 1 = 0.0 percent; year 2 = 0.0 percent; year 3 = 0.0 percent; and, year 4 = 36.05 percent. Well . . . you could argue this point of view, but you’d be lonely because no one does it this way. In the world of finance, the ROI for Example 3 is measured at 8 percent per year. The standard method for determining the ROI on the investment assumes that the annual earnings are theoretically received in cash but then immediately reinvested. Thus, you see the compounding effect from year to year; the investment balance increases year-to-year by the amount of reinvested earnings. Don’t think that because the investment earns 8 percent ROI each year, you actually have any cash flow from the investment. You don’t. In short, the 8 percent ROI solution is a convenient way to express the annual rate of growth in the value of the investment. It’s rather arbitrary, but it’s the way things are done. 275 More free books @ www.BingEbook.com 276 Part III: Managerial, Manufacturing, and Capital Accounting 19. You invest $100,000 today. The year-end cash flows from the investment are as follows: year 1 = $0; year 2 = $0; year 3 = $0; and year 4 = $128,646.64. Determine the annual ROI for this investment. You may find the following form helpful: 20. You invest $100,000 today. The year-end cash flows from the investment are as follows: year 1 = $0; year 2 = $0; year 3 = $0; and year 4 = $90,368.79. Determine the annual ROI for this investment. Does this answer make sense? You may find the following form helpful: Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year ?% 1 $100,000 Capital Recovery Investment Balance Earnings at Year Total Amount at Start of Year ?% $0 2 3 $0 4 $128,647 Cash Flow at End of Year Investment Balance at End of Year $0 1 $100,000 Solve It Investment Balance at End of Year $0 2 $0 Capital Recovery $0 3 $0 4 $90,369 $0 Solve It Example 4: Irregular cash flows, both positive and negative You invest $100,000 today. The year-end cash flows from the investment are as follows: year 1 = negative $15,000; year 2 = negative $25,000; year 3 = $50,000; and year 4 = $141.625. Q. What is the annual rate of return on investment for this scenario? A. The annual rate of return on investment is 10 percent. Check out the following schedule, which uses the Figure 12-1 template. This sort of investment may not be for you because you put $100,000 in the investment to get it started, and then at the end of the first and second years, you put additional money in the investment. These additional payments into the investment are called negative cash flows. Cash Flow at End of Year Investment Balance Earnings at Year Total Amount at Start of Year 10.0% Capital Recovery Investment Balance at End of Year 1 $100,000.00 ($15,000.00) $10,000.00 ($25,000.00) $125,000.00 2 $125,000.00 ($25,000.00) $12,500.00 ($37,500.00) $162,500.00 3 $162,500.00 $50,000.00 $16,250.00 $33,750.00 $128,750.00 4 $128,750.00 $141,625.00 $12,875.00 $128,750.00 $0.00 Would you make this investment? You would have to pump $140,000 into the project before you see any positive cash flow at the end of year 3. You may or may not be in a position to do this. On the other hand, the investment yields 10 percent annual ROI, which is pretty good. Generally, most investments involving negative cash flows are riskier and, therefore, demand a higher than average ROI to justify taking on the risks.