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CHAPTER 6 THE TIME VALUE OF MONEY FOCUS This chapter develops and applies time value formulas for amounts and annuities. The focus is on using time value concepts to solve business problems. PEDAGOGY Students learn and retain discounted cash flow concepts best when they become comfortable with formulas and tables before relying on financial calculators. For that pedagogical reason the formula approach is presented in the text even though calculators are used almost universally in practice. Switching to a calculator after one understands what's going on is a trivial matter. A large number of illustrative problems are included and instructors can supplement solution techniques with calculators if they're so inclined. Calculator solutions are included in the margins next to all illustrative examples. The chapter's illustrative examples serve two purposes. In addition to demonstrating technique, several are designed to teach practice in specific areas. For example, by the time readers finish with annuity problems, they're familiar with some of the intricacies of mortgage loans. That means they're relatively knowledgeable about real estate finance. TEACHING OBJECTIVES Students should develop an understanding of discounted cash flow concepts and a facility for solving problems. Their problem solving ability should extend to relatively complex applications. OUTLINE I. OUTLINE OF APPROACH A brief explanation of the fact that money promised in the future is worth less than money in hand today. Four kinds of problems and using time lines. II. AMOUNT PROBLEMS A. The Future Value of an Amount The expression relating future and present values of a single amount is developed in terms of the future value factor and the associated table. Problem solving technique is introduced. Applications include deferred payment terms as the equivalent of cash discounts and the opportunity cost rate. Financial calculators are introduced and instruction is provided on their use in solving time value problems. B. The Expression for The Present Value of an Amount An alternate formulation emphasizing the present value in terms of the future value. More on technique. III. ANNUITY PROBLEMS A. Annuities The concept of an annuity and its present and future values. 128 The Time Value of Money 129 B. The Future Value of an Annuity - Developing a Formula The FVA expression, factor and tables are developed. C. The Future Value of an Annuity - Solving Problems Problem solving technique, the sinking fund concept, using a financial calculator for annuity problems. D. Compound Interest and Non-Annual Compounding Compound interest concepts and how to handle non-annual compounding in problems. The EAR and APR. E. The Present Value of an Annuity The formula, factor, and table are developed. Computer spreadsheet techniques for solving time value problems are introduced and explained. F. The Present Value of an Annuity - Solving Problems Applications include discounting a stream of payments, amortized loans and amortization schedules, and working with mortgages. G. The Annuity Due The concept and formula are developed and applied. H. Perpetuities The idea of a never-ending stream with a finite present value is developed intuitively. Applications include the capitalization of earnings and preferred stock. I. Multi-Part Problems Dealing with situations in which the solutions to problems become inputs to other problems. Visualizing and time lining complex problems. J. Uneven Streams and Imbedded Annuities Recognizing and dealing with annuities imbedded in uneven payment streams. QUESTIONS 1. Why are time value concepts important in ordinary business dealings, especially those involving contracts? ANSWER: Business contracts and agreements generally specify payments that are due at future times. If such payments are more than a few months into the future, the correct analysis of the value of the agreement depends on a recognition of the time value of money. 2. Why are time value concepts crucial in determining what a bond or a share of stock should be worth? ANSWER: All securities derive their value solely from the future cash flows that come from owning them. The only way to value a future cash flow today is through the present value concept. Therefore, the value of a security depends entirely on time value ideas. 3. In a retail store a discount is a price reduction. What's a discount in finance? Are the two ideas related? ANSWER: Discounting in finance means taking the present value of a sum promised in the future. The present value process always results in an figure that's less than the future amount, so in a sense, present valuing reduces the price of the future payment. 4. Calculate the present value of one dollar 30 years in the future at 10% interest. What does the result tell you about very long-term contracts? 130 Chapter 6 ANSWER: PV = FV30 [PVF10,30] = $1 [.0573] = $0.0573 = A little less than six cents. Money promised in a very long-term contract isn't worth much today, even if its receipt is certain. Therefore, we should be careful about what we give up today for a commitment in the distant future. 5. Write a brief, verbal description of the logic behind the development of the time value formulas for annuities. ANSWER: To develop a time value formula for an annuity we take an annuity with a finite number of payments and develop its present or future value by treating each payment as an amount. We then examine the resulting expression looking for a pattern that can be extended to longer streams of payments. Recognizing such a pattern allows us to generalize the expression to an arbitrary number of annuity payments, n. 6. Deferred payment terms are equivalent to a cash discount. Discuss and explain this idea. ANSWER: Deferred payment means the seller will accept the promise of a future payment instead of the full price today (with no interest charged on the amount deferred). Since the present value of the deferred amount is less than its nominal value, the transaction is actually being conducted at a price (in present value terms) that's less than the stated price of the article. In essence this is a sale at a discount. Looking at it another way, the buyer could put the deferred amount in the bank until it was due. At that time she could withdraw the amount deposited, pay the bill, and keep the interest that would be approximately the amount of the discount. 7. What's an opportunity cost interest rate? ANSWER: An opportunity rate is the rate that would be earned on money given up for some purpose. For example, in the previous question, if the seller finances his business at 12%, his cost of deferring payment is 12%, because he could have paid off some 12% debt if he'd have received the money at the time of the sale. 8. Discuss the idea of a sinking fund. How is it related to time value? ANSWER: Lenders are often concerned that borrowers won't be able to raise the cash to pay off loan principals even though they're able make interest payments during the lives of loans. A sinking fund is an arrangement in which a borrower is required to put aside money periodically during the life of a loan, so that at maturity funds are available to pay off the principal. In one such arrangement, funds are deposited at interest to accumulate into the principal amount by the loan's maturity. Time value is involved because it takes a future value of an annuity problem to calculate the periodic payment that will just pay off the loan at maturity. 9. The amount formulas share a closer relationship than the annuity formulas. Explain and interpret this statement. ANSWER: The two amount formulas are really the same expression written differently for convenience. Both expressions involve the present value and the future value of the same money, and either can be used to solve any amount problem. The Time Value of Money 131 The annuity formulas are two distinct, separately derived expressions. They deal with two different things, the present and future value of the annuity in question. Annuity problems must be classified as either present or future value, and the correct expression has to be used to solve either. 10. Describe the underlying meaning of compounding and compounding periods. How does it relate to time value? Include the idea of an effective annual rate (EAR). What is the annual percentage rate (APR)? Is the APR related to the EAR? ANSWER: Compounding relates to earning interest on previously earned interest. Money initially deposited at interest earns interest on that amount until the end of a compounding period. At that time the interest is "credited" to the account, and future interest is earned on the sum of the original deposit plus the interest earned in the first period. Interest earned in the second period is credited at its end, so interest earned in the third period is based on the original deposit plus the first and second period's interest. And so on. The more frequently (shorter compounding periods) interest is compounded, the more interest is earned. Interest rates are generally stated in annual terms (the nominal rate) and must be adjusted to reflect the compounding periods in use. Time value assumes compound interest. In problems, compounding periods and rates must be consistently specified. The Effective Annual Rate (EAR) is the rate of annually compounded interest that is just equivalent to a nominal rate compounded more frequently. The APR is the nominal rate. The APR and the EAR are not directly related. 11. What information are we likely to be interested in that’s contained in a loan amortization schedule? ANSWER: A loan amortization schedule details the interest and principal components of every loan payment as well as the beginning and ending loan balance for every payment period. 12. Discuss mortgage loans in terms of the time value of money and loan amortization. What important points should every homeowner know about how mortgages work? (Hint: Think about taxes and getting the mortgage paid off.) ANSWER: A mortgage is an amortized loan, generally of fairly long term. 30 years is common. Payments are made monthly so there are 360 payments in a 30-year mortgage. Amortized loan payments are generally constant in amount, but the split between interest and principal within the payments varies during the life of the loan. Early payments contain relatively more interest and later payments relatively more principal repayment. This phenomenon is extreme in long term loans like mortgages. Early payments are nearly all interest while later payments are nearly all principal repayment. This leads to two important facts for homeowners. First, because interest is tax deductible, early mortgage payments save a lot on taxes while later payments save only a little. Second, a loan is not reduced by much during the early years of its life. That is, half way through the loan’s life, a great deal more than half of the original loan balance is still unpaid. 13. Discuss the idea of capitalizing a stream of earnings in perpetuity. Where is this idea useful? Is there a financial asset that makes use of this idea? ANSWER: A constant stream of earnings that can be expected to go on forever has a finite present value, which is known as the capitalized value of the stream. The idea is useful in valuing businesses. Essentially, any firm is worth the capitalized value of its expected future earnings. Where the best estimate of future earnings is simply a continuation of current earnings, capitalizing a stream of that magnitude gives an estimate of value. 132 Chapter 6 Preferred stock is a security that pays a constant dividend indefinitely. Its value is the capitalization of the stream of its dividends. 14. When an annuity begins several time periods into the future, how do we calculate its present value today? Describe the procedure in a few words. ANSWER: The formula for the present value of an annuity gives a value at the beginning of the annuity. If that time is in the future, the "present value" of the annuity has to be brought back in time to the true present as an amount. Hence two consecutive calculations are required. First take the present value of the annuity, then treat that figure as an amount and take its present value. BUSINESS ANALYSIS 1. A business can be valued by capitalizing it’s earnings stream (see example 6.15). How might you use the same idea to value securities, especially the stock of large publicly held companies? Is there a way to calculate a value that could be compared to the stock’s market price that would tell an investor whether it’s a good buy? (If the market price is lower than the calculated value, the stock is a bargain.) What financial figures associated with shares of stock might be used in the calculation. Consider the per share figures and ratios discussed in chapter 3 including EPS, dividends, book value per share etc. Does one measure make more sense than the others? What factors would make a stock worth more or less than your calculated value. Answer: A privately or closely held company is valued by capitalizing a stream of earnings (net income) because all of a firm’s earnings are available to its owners. The analogous figure for the stocks of publicly held companies is dividends, because they represent cash received by stockholders. Although all earnings technically belong to owners, the stockholders of larger companies generally can’t influence how much of those earnings they receive. Hence dividends, which stockholders do receive, are the best measure for mathematically calculating value. (However, practitioners do use EPS regularly for less precise estimates.) The starting point should be the capitalized value of the current dividend, essentially assuming it will go on forever. The rate used for capitalization should be consistent with the riskiness of the company involved. This starting estimated value should be factored up or down based on expectations about future dividends. If the stream is expected to grow or shrink, value will be higher or lower respectively. Further, a stable stream should be worth more than one that varies substantially from year to year. Expectations about future performance usually come from performance in the recent past so a stock with a record of consistent dividend (or EPS) growth should be worth more than one whose dividends have been constant for some time or erratic. PROBLEMS NOTE: FOR ALL PROBLEMS INVOLVING NON-ANNUAL COMPOUNDING, SEE CONCEPT CONNECTION EXAMPLES 6.7 AND 6.8, PAGES 263 – 264 AS WELL AS THE PROBLEM REFERENCED. Amount Problems Present Value of an Amount – Example 6.2 (page 246) 1. The Lexington Property Development Company has a $10,000 note receivable from a customer due in three years. How much is the note worth today if the interest rate is The Time Value of Money 133 a. 9%? b. 12% compounded monthly? c. 8% compounded quarterly? d. 18% compounded monthly? e. 7% compounded continuously? SOLUTION: PV = FV [PVFk,n ] a. PV = $10,000 [PVF9,3 ] = $10,000 (.7722) = $7,722 b. PV = $10,000 [PVF1,36 ] = $10,000 (.6989) = $6,989 c. PV = $10,000 [PVF2,12] = $10,000 (.7885) = $7,885 d. PV = $10,000 [PVF1.5,36] = $10,000 (.5851) = $5,851 e. FV = PV (e kn ) $10,000 = PV [e .07(3)] $10,000 = PV [1.2337] PV = $8,105.70 Future Value of an Amount – Example 6.1 (page 246) 2. What will a deposit of $4,500 left in the bank be worth under the following conditions: a. Left for nine years at 7% interest? b. Left for six years at 10% compounded semiannually? c. Left for five years at 8% compounded quarterly? d. Left for 10 years at 12% compounded monthly? SOLUTION: FV = PV [FVFk,n ) a. FV = $4,500 [FVF7,9 ] = $4,500 (1.8385) = $8,273.25 b. FV = $4,500 [FVF5,12 ] = $4,500 (1.7959) = $8,081.55 c. FV = $4,500 [FVF2,20 ] = $4,500 (1.4859) = $6,686.55 d. FV = $4,500 [FVF1,120 ] = $4,500 (3.3004) = $14,851.80 Finding the Interest Rate – Example 6.3, (page 250) 3. What interest rates are implied by the following lending arrangements? a. You borrow $500 and repay $555 in one year. b. You lend $1,850 and are repaid $2,078.66 in two years. c. You lend $750 and are repaid $1,114.46 in five years with quarterly compounding. d. You borrow $12,500 and repay $21,364.24 in three years under monthly compounding. (Note: In c and d, be sure to give your answer as the annual nominal rate.) 134 Chapter 6 SOLUTION: FV = PV [FVFk,n ] a. $555 = $500 [FVFk,1 ] FVFk,1 = 1.1100 k = 11% b. $2,078.66 = $1,850.00 [FVFk,2 ] FVFk,2 = 1.1236 k = 6% c. $1,114.46 = $750.00 [FVFk,20 ] FVFk,20 = 1.4859 k = 2% knom = 8% d. $21,364.24 = $12,500.00 [FVFk,36 ] FVFk,36 = 1.7091 k = 1.5% knom = 18% Solving for the Number of Periods – Example 6.4 (page 251) 4. How long does it take for the following to happen? a. $856 grows into $1,122 at 7%. b. $450 grows into $725.50 at 12% compounded monthly. c. $5,000 grows into $6724.44 at 10% compounded quarterly. SOLUTION: PV = FV [PVFk,n ] a. $856 = $1,122 [PVF7,n ] PVF7,n = .7629 n = 4 years b. $450.00 = $725.50 [PVF1,n ] PVF1,n = .6203 n = 48 months = 4 years c. $5,000 = $6,724.44 [PVF2.5,n ] PVF2.5,n = 0.7436 n = 12 quarters = 3 years 5. Sally Guthrie is looking for an investment vehicle that will double her money in five years. a. What interest rate, to the nearest whole percentage, does she have to receive? b. At that rate, how long will it take the money to triple? c. If she can't find anything that pays more than 11%, approximately how long will it take to double her investment? d. What kind of financial instruments do you think Sally is looking at? Are they risky? What could happen to Sally's investment? The Time Value of Money 135 SOLUTION: FV = PV [FVFk,n ] a. 2 = 1 [FVFk,5 ] FVFk,5 = 2 k= 15% b. FVF15,n = 3 n = 7.9 years (approximate with 8 years) c. FVF11,n = 2 n = 6.6 years (approximate with 7 years) d. Investments with anticipated returns like these are probably growth-oriented stocks with considerable risk. She could lose money. Deferred Payment is Equivalent to a Cash Discount – Example 6.2 (page 246) 6. Branson Inc. has sold product to the Brandywine Company, a major customer, for $20,000. As a courtesy to Brandywine, Branson has agreed to take a note due in two years for half of the amount due, and half in cash. a. What is the effective price of the transaction to Branson if the interest rate is: (1) 6%, (2) 8%, (3) 10%, (4) 12%? b. Under what conditions might the effective price be even less as viewed by Brandywine? SOLUTION: a. 1) PV = FV [PVF6,2 ] = $10,000 (.8900) = $8,900 $8,900 + $10,000 = $18,900 Effective Discount = 5.5% 2) PV = FV [PVF8,2 ] = $10,000 (.8573) = $8,573 $8,573 + $10,000 = $18,573 Effective Discount = 7.1% 3) PV = FV [PVF10,2 ] = $10,000 (.8264) = $8,264 $8,264 + $10,000 = $18,264 Effective Discount = 8.7% 4) PV = FV [PVF12,2 ] = $10,000 (.7972) = $7,972 $7,972 + $10,000 = $17,972 Effective Discount = 10.1% b. The discount from Brandywine's perspective is calculated as in part a), but using the interest rate at which that firm borrows. If Brandywine's rate is higher than Branson's, it will perceive a greater discount. 7. John Cleaver's grandfather died in 2007 and left him a trunk that had been locked in his attic for years. At the bottom of the trunk John found a packet of 50 World War I "liberty bonds" that had never been cashed in. The bonds were purchased for $11.50 each in 1918, and pay 3% interest as long as they're held. (Government savings bonds like these accumulate and compound their interest unlike corporate bonds, which regularly pay out interest to bondholders.) a. How much are the bonds worth in 2007? (Hint: [FVFk,a+b ] = [FVFk,a][FVFk,b ] b. How much would they have been worth if they paid interest at a rate more like that paid during the 1970s and 80s, say 7%? c. Comment on the difference between the answers to parts a and b. 136 Chapter 6 SOLUTION: First notice that [FVFk,a+b ] = [FVFk,a] [FVFk,b ] because (1+k)a+b = (1+k)a (1+k)b , and FVFk,n = (1+k)n Therefore, [FVFk,92 ] = [FVFk,60 ] [FVFk,30 ] [FVFk,2] Hence, [FVF3,92 ] = [FVF3,60 ] [FVF3,30 ] [FVF3,9] = (5.8916)(2.4273)(1.0609) = 15.1716 and [FVF7,92 ] = [FVF7,60 ] [FVF7,30 ] [FVF7,2] = (57.9464)(7.6123)(1.1449) = 505.022 a. FV = PV [FVF3,92 ] = $11.50 (15.1716) = $174.47 per bond b. FV = PV [FVF7,92 ] = $11.50 (505.22) = $5,810.03 per bond c. Over a long period the interest rate makes an enormous difference in investment results! Deferred Payment is Equivalent to a Cash Discount – Example 6.2 (page 246) 8. Paladin Enterprises manufactures printing presses for small-town newspapers that are often short of cash. To accommodate these customers, Paladin offers the following payment terms: 1/3 on delivery 1/3 after six months 1/3 after 18 months The Littleton Sentinel is a typically cash-poor newspaper considering one of Paladin's presses a. What discount is implied by the terms from Paladin's point of view if it can invest excess funds at 8% compounded quarterly? b. The Sentinel can borrow limited amounts of money at 12% compounded monthly. What discount do the payment terms imply to the Sentinel? c. Reconcile these different views of the same thing in terms of opportunity cost. SOLUTION: Assume a price of $300. a. PV = $100 + $100 [PVF2,2 ] + $100 [PVF2,6 ] = $100 + $100 (.9612)+ $100 (.8880) = $284.92 Discount = $15.08/$300 = 5% b. PV = $100 + $100 [PVF1,6 ] + $100 [PVF1,18 ] = $100 + $100 (.9420)+ $100 (.8360) = $277.80 Discount = $22.20/$300 = 7.4% The Time Value of Money 137 c. The buyer is avoiding more financing cost than the seller is giving up because funds are available to both of them at different opportunity rates. 9. Charlie owes Joe $8,000 on a note which is due in five years with accumulated interest at 6%. Joe has an investment opportunity now that he thinks will earn 18%. There’s a chance, however, that it will earn as little as 4%. A bank has offered to discount the note at 14% and give Joe cash that he can invest today. a. How much ahead will Joe be if he takes the bank’s offer and the investment does turn out to yield 18%? b. How much behind will he be if the investment turns out to yield only 4%? Solution: a. First calculate the amount Charlie’s note will pay in five years. FV = PV[FVF6,5] = $8,000(1.3382) = $10,705.60 If the bank discounts that at 14%, Joe will receive PV = FV[PVF14,5 ] = $10,705.60(.5194) = $5,560.49 This amount invested at 18% for five years will grow into FV = PV[FVF18,5 ] = $5,560.49(2.2878) = $12,721.29. And Joe will be ahead by the difference $12,721.29 - $10,705.60 = $2,015.69 b. If the investment yie lds only 4%, the last time value calculation will be FV = PV[FVF4,5] = $5,560.49(1.2167) = $6,765.45, And Joe will be behind by $10,705.60 - $6,765.45 = $3,940.15 Annuity Problems Future Value of an Annuity – Example 6.5 (page 256) 10. How much will $650 per year be worth in eight years at interest rates of a. 12% b. 8% c. 6% SOLUTION: FVA = PMT [FVFAk,n] a. FVA = $650 [FVFA12,8 ] = $650 (12.2997) = $7,994.81 b. FVA = $650 [FVFA8,8 ] = $650 (10.6366) = $6,913.79 c. FVA = $650 [FVFA6,8 ] = $650 (9.8975) = $6,433.38 11. The Wintergreens are planning ahead for their son's education. He's eight now and will start college in 10 years. How much will they have to set aside each year to have $65,000 when he starts if the interest rate is 7%? SOLUTION: FVA = PMT [FVFAk,n ] $65,000 = PMT [FVFA7,10 ] = PMT (13.8164) PMT = $4,704.55 12. What interest rate would you need to get to have an annuity of $7,500 per year accumulate to $279,600 in 15 years? 138 Chapter 6 SOLUTION: FVA = PMT [FVFAk,n] $279,600 = $7,500 [FVFA k,15 ] FVFAk,15 = 37.28 k = 12% 13. How many years will it take for $850 per year to amount to $20,000 if the interest rate is 8%? Interpolate and give an answer to the nearest month. SOLUTION: FVA = PMT [FVFAk,n ] $20,000 = $850 [FVFA8,n ] FVFA8,n = 23.529 n = 13.75 yrs = 13 yrs 9 mos Present Value of an Annuity – Example 6.9 (page 266) 14. What would you pay for an annuity of $2,000 paid every six months for 12 years if you could invest your money elsewhere at 10% compounded semiannually? SOLUTION: PVA = PMT [PVFAk,n ] PVA = $2,000 [PVFA5,24 ] = $2,000 (13.7986) PVA = $27,597.20 Amortized Loan Finding Payment – Example 6.11 (page 271) 15. Sam Rothstein wants borrow $15,500 to be repaid in quarterly installments over five years at 16% compounded quarterly. How much will his payment be? SOLUTION: For quarterly compounding we have k = knom/12 = 16%/4 = 4% n = 5 years x 12 months/year = 60 months. Write equation 6.19 and substitute. PVA = PMT[PVFAk,n ] $15,500 = PMT[PVFA4,60 ] Using Appendix A-4 $15,500 = PMT[22.6235] PMT = $685 Amortized Loan Finding Amount Borrowed – Example 6.12 (page 271) 16. Harry Clements would like to buy a new car. He can afford payments of $650 a month. The bank makes four-year car loans at 12% compounded monthly. How much can Harry borrow toward a new car? SOLUTION: First calculate k and n for monthly compounding, k = knom/12 = 12%/12 = 1% n = 4 years x 12 months/year = 48 months. Write equation 6.19 and substitute. The Time Value of Money 139 PVA = PMT[PVFAk,n ] PVA = $650[PVFA1,48 ] Using Appendix A-4 PVA = $650[37.9740] PVA = $24,683 17. Construct an amortization schedule for a four-year, $10,000 loan at 6% interest compounded annually. SOLUTION: PVA = PMT [PVFAk,n ] $10,000 = PMT [PVFA6,4] PMT = $2,885.92 Year Beg Bal PMT INT Prin Red End Bal 1 $10,000.00 $2,885.92 $600.00 $2,285.92 $7,714.08 2 7,714.08 2,885.92 462.84 2,423.08 5,291.00 3 5,291.00 2,885.92 317.46 2,568.46 2,722.54 4 2,722.54 2,885.92 163.35 2,722.54 0.00 18. A $10,000 car loan has payments of $361.52 per month for three years. What is the interest rate? Assume monthly compounding and give the answer in terms of an annual rate. SOLUTION: PVA = PMT [PVFAk,n ] $10,000 = $361.52 [PVFAk,36 ] PVFAk,36 = 27.661 k = 1.5% knom = 18% EAR (page 261) 19. Ralph Renner just borrowed $30,000 to pay for a new sports car. He took out a 60 month loan and his car payments are $761.80 per month. What is the effective annual interest rate (EAR) on Ralph’s loan? SOLUTION: First, calculate the periodic interest rate n = 60; PV = 30,000; PMT = (761.80); FV = 0 CPT I/Y = 1.5% EAR = (1 + .015)12 – 1 = .1956 or 19.56% Annuity Due – Example 6.17 (page 277) 20. Joe Ferro's uncle is going to give him $250 a month for the next two years starting today. If Joe banks every payment in an account paying 6% compounded monthly, how much will he have at the end of three years? SOLUTION: FVAd = PMT [FVFAk,n ](1+k) = $250 [FVFA.5,24 ](1.005) = $250 (25.4320) (1.005) = $6,389.79 140 Chapter 6 which stays in the bank for another year: FV = $6,389.79 [FVF.5,12 ] = $6,389.79 (1.0617) = $6,784.04 21. How long will it take a payment of $500 per quarter to amortize a loan of $8,000 at 16% compounded quarterly? Approximate your answer in terms of years and months. How much less time will it take if loan payments are made at the beginning of each month rather than at the end? SOLUTION: PVA = PMT [PVFAk,n ] $8,000 = $500 [PVFA4,n ] PVFA4,n = 16 n = 26 quarters = 6.5 years = 6 years 6 months PVAd = PMT [PVFAk,n ](1+k) $8,000 = $500 [PVFA4,n ](1.04) PVFA4,n = 15.3846 n = 24 1/3 quarters = 6 years 1 month So it will take about 5 months less time. Mortgage Loans – Examples 6.13- 6.15 (pages 272-274) 22. Ryan and Laurie Middleton just purchased their first home with a traditional (monthly compounding and payments) 6% 30-year mortgage loan of $178,000. a. How much is their monthly payment? b. How much interest will they pay the first month? c. If they make all their payments on time over the 30-year period, how much interest will they have paid? d. If Ryan and Laurie decide to move after 7 years what will the balance of their loan be at that time? e. If they finance their home over 15 rather than 30 years at the same interest rate, how much less interest will they pay over the life of the loan? SOLUTION (using a financial calculator): a. PV = $178,000; n = 360, I/Y = .5; CPT PMT = $1,067.20 b. Interest in the first payment is the opening loan balance times the monthly rate $178,000 x .005 = $890 c. Total interest is total payments minus the amount borrowed Total payments are $1067.20 x 360 = $384,192 And total interest is $384,192 - $178,000 = $206,192 d. The loan balance at any time is the PV of the remaining payments. After 7 years, the Middletons will have made 7 x 12 = 84 monthly payments which leaves 360-84 = 276 remaining. n = 276, I/Y = .5, PMT = $1,067.20; CPT PV = $159,558.11 e. The payment on a 15 year loan is n = 180, I/Y = .5, PV = $178,000; CPT PMT = $1,502.07 Total interest is total payments minus the amount borrowed Total payments are $1,502.07 x 180 = $270,372.60. And total interest is $270,372.60 - $178,000 = $92,372.60 The difference is $113,819.40 The Time Value of Money 141 23. What are the monthly mortgage payments on a 30-year loan for $150,000 at 12%? Construct an amortization table for the first six months of the loan. SOLUTION: PVA = PMT [PVFAk,n ] $150,000 = PMT [PVFA1,360 ] = PMT(97.2183) PMT = $1,542.92 Month Beg Bal PMT INT Prin Red End Bal 1 $150,000.00 $1,542.92 $1,500.00 $42.92 $149,957.08 2 149,957.08 1,542.92 1,499.57 43.35 149,913.73 3 149,913.73 1,542.92 1,499.13 43.79 149,869.94 4 149,869.94 1,542.92 1,498.70 44.22 149,825.72 5 149,825.72 1,542.92 1,498.26 44.66 149,781.06 6 149,781.06 1,542.92 1,497.81 45.11 149,735.95 24. Construct an amortization schedule for the last six months of the loan in Problem 21. (Hint: What is the unpaid balance at the end of 29 ½ years?) SOLUTION: PVA = PMT [PVFA1,6 ] = $1,542.92 (5.7955) = $8,941.99 Month Beg Bal PMT INT Prin Red End Bal 355 $8,941.99 $1,542.92 $89.42 $1,453.50 $7,488.49 356 7,488.49 1,542.92 74.88 1,468.04 6,020.45 357 6,020.45 1,542.92 60.20 1,482.72 4,537.73 358 4,537.73 1,542.92 45.38 1,497.54 3,040.19 359 3,040.19 1,542.92 30.40 1,512.52 1,527.67 360 1,527.67 1,542.92 15.25 1,527.67 0.00 25. How soon would the loan in Problem 21 pay off if the borrower made a single additional payment of $17,936.29 to reduce principal at the end of the fifth year? SOLUTION: 360 60 = 300 PVA = PMT [PVFA1,300 ] PVA = $1,542.92 (94.9466) = $146,495.01 Less additional payment 17,936.29 New Balance = $128,558.72 PVA = PMT [PVFA1,n ] $128,558.72 = $1,542.92 [PVFA1,n ] PVFA1,n = 83.3217 n = 180 months = 15 years So the loan would pay off in a total of 20 years. 26. What are the payments to interest and principal during the twenty-fifth year of the loan in problem 21? SOLUTION: Calculate the loan balance after 25 and 26 years (five and four years remaining). The difference is the amount paid into principal during the twenty-fifth year. 142 Chapter 6 PVA = PMT [PVFA1,60] = $1,542.92 (44.9550) = $69,361.97 PVA = PMT [PVFA1,48] = $1,542.92 (37.9740) = $58,590.84 Payments to principal: $69,361.97 58,590.84 $10,771.13 Total payments: $1,542.92 12 = $18,515.04 Interest payments: $18,515.04 10,771.13 $ 7,743.91 27. Adam Wilson just purchased a home and took out a $250,000 mortgage for 30 years at 8%, compounded monthly. a. How much is Adam’s monthly mortgage payment? b. How much sooner would Adam pay off his mortgage if he made an additional $100 payment each month? c. Assume Adam makes his normal mortgage payments and at the end of five years, he refinances the balance of his loan at 6%. If he continues to make the same mortgage payments, how soon after the first five years will he pay off his mortgage? d. How much interest will Adam pay in the tenth year of the loan i. if he does not refinance? ii. if he does refinance? SOLUTION: a. n = 360; I/Y = 8/12 = .666667; PV = 250,000; FV = 0 CPT PMT = $1,834.41 b. Add $100.00 to the PMT calculated above and recompute n: I/Y = .666667; PV = 250,000; PMT = (1,934.41); FV = 0 CPT n = 297.62 Adam would eliminate the last 62 months of payments by paying an additional $100/month c. First calculate the mortgage balance at the end of 5 years n = 60; I/Y = .666667; PV = 250,000; PMT = (1,834.4193) CPT FV = $237,674.75 Then solve for n using the original PMT, the new I/Y, and the mortgage balance as the new PV I/Y = 6.0/12 = .50; PV = 237,674.64; PMT = (1,834.41); FV = 0 CPT n = 209.25 The mortgage would pay off about 90 months earlier. d. i. FV after the 108th payment n = 108; I/Y = 8/12 = .666667; PV = 250,000 PMT = ($1834.41); CPT FV = $223,592.02 FV after the 120th payment = $219,311.76 Principal payments = $223,592.02 - $219,311.76 = $4,280.26 Interest payments = 12 x $1834.41 = $22,012.92 - $4,280.26 = $17,732.66 ii. FV after the 60th payment (when the refinancing would occur) The Time Value of Money 143 n = 60; I/Y = 8/12 = .666667; PV = $250,000; PMT = ($1834.41); CPT FV = $237,674.75; Then find the balance after 48 payments of the refinanced loan n = 48; I/Y = 6/12 = .50; PV = $237,674.64; PMT = ($1834.41); CPT FV = $202,725.34. FV after the 60th payment = $192,600.63 Principal payments = $202,725.34 - $192,600.63 = $10,124.71 Interest payments = 12 x $1834.41 = $22,012.92 - $10,124.71 = $11,888.21 Estimating an ARM Reset – Example 16.6 (page 275) 28. Harrison Conway is choosing between a fix rate and an adjustable rate mortgage (ARM) for $300,000. Both are 30 year mortgages with monthly payments and compounding. The fixed rate is offered at 8% while the initial rate on the ARM is 6%. Harrison is concerned that inflation may be a problem within ten years and that rates may return to levels not seen since the mid 1980s, i.e., in the neighborhood of 12%. Compare the payment on the fixed rate loan to what Harrison would have to pay on the ARM if it reset to 12% after ten years. For simplicity assume just that one resetting occurs. Round all calculations to the nearest dollar. Solution: First calculate the initial ARM payment. k = 6 / 12 = .5 n = 30 x 12 = 360 PVA = PMT [PVFAk,n ] $300,000 = PMT [PVFA.5,360] $300,000 = PMT [166.792] PMT = $1,799 Next calculate the projected unpaid ARM balance after ten years (20 years or 240 months remaining). PVA = PMT [PVFAk,n ] = $1,799 [PVFA.5,240 ] = $1,799 [139.581] = $251,106 Next calculate the payment required to amortize $251,106 over the remaining 20 years at 12%. k = 12 / 12 = 1 n = 20 x 12 = 240 PVA = PMT [PVFAk,n ] $251,106 = PMT [PVFA1,240 ] $251,106 = PMT [90.8194] PMT = $2,765 The fixed rate payment at 8% is k = 8 / 12 = .67 n = 30 x 12 = 360 PVA = PMT [PVFAk,n ] $300,000 = PMT [PVFA.67,360 ] $300,000 = PMT [136.283] PMT = $2,201 Hence the ARM would be $564 per month higher 144 Chapter 6 29. Lee Childs is negotiating a contract to do some work for Haas Corp. over the next five years. Haas proposes to pay Lee $10,000 at the end of each of the third, fourth and fifth years. No payments will be received prior to that time. If Lee discounts these payments at 8%, what is the contract worth to him today? SOLUTION: This is an annuity problem, but the annuity doesn’t begin today. Since the payments occur at the end of the 3rd through 5th years, we can solve for the value of the annuity at the end of the second year using the following values: n = 3; I/Y = 8; PMT = 10,000; FV = 0 CPT PV = $25,770.97 This amount needs to be discounted back two periods to find the value today n = 2; I/Y = 8; PMT = 0; FV = 25,770.97 CPT PV = $22,094.45 30. Referring to the previous problem, Lee wants to receive the payments for his work sooner than Haas proposes to make them. He has counter proposed that the payments be made at the beginning of the third, fourth and fifth years rather than at the end. What will the contract be worth to Lee if Haas accepts his counter proposal? SOLUTION: We want to retain the PV of the contract from the previous problem at $22,094.45. Since payments will occur at the beginning of the 3rd through 5th years under this proposal, we first need to determine the future value of the contract at the beginning of the 2nd period. Then we can treat the payments as an ordinary annuity starting at that time. n = 1; I/Y = 8; PV = 22.094.45; PMT = 0 CPT FV = $23,862.01 This amount becomes the PV of an ordinary annuity three years long n = 3; I/Y = 8; PV = 23,862.01; FV = 0 CPT PMT = $9,259.26 31. The Orion Corp. is evaluating a proposal for a new project. It will cost $50,000 to get the undertaking started. The project will then generate cash inflows of $20,000 in its first year and $16,000 per year in the next five years after which it will end. Orion uses an interest rate of 15% compounded annually for such evaluations. a. Calculate the “Net Present Value” (NPV) of the project by treating the initial cost as a cash outflow (a negative) in the present, and adding the present value of the subsequent cash inflows as positives. b. What is the implication of a positive NPV? (Words only.) c. Suppose the inflows were somewhat lower, and the NPV turned out to be negative. What would be the implication of that result? (Words only.) (This problem is a preview of a technique called capital budgeting, which we’ll study in detail in Chapters 10, 11, and 12.) Solution: a. First take the present value of the first inflow PV = FV[PVF15,1 ] = $20,000(.8696) = $17,392 Next take the present value of the annuity in two steps PVA = PMT[PVFA15,5 ] = $16,000(3.3522) = $53,635.20 Next bring that figure backward one year into the present as an amount. PV = FV[PVF15,1 ] = $53,635.20(.8696) = $46,641.17 Hence the NPV of the project is NPV = $50,000 + $17,392 + $46,641.17 = $14,033.17 The Time Value of Money 145 b. A positive NPV means that on a present value basis the project’s cash inflows exceed its outflows. That implies it’s a good deal for the company. In essence it is expected to increase the firm’s wealth and value to stockholders. Since that’s what management is supposed to do, they should accept the project. c. A negative means just the opposite. It implies that, on balance, the project will cost the firm money. Hence it should be rejected. Perpetuities – Capitalization of Earnings – Example 6.19 (page 280) 32. Roper Metals Inc. is in negotiations to acquire the Hanson Sheet Metal Company. Hanson’s after tax earnings have averaged $19 million per year for the last four years without much variation around that average figure. So far discussions have been about the business fit of the two firms and pricing has been conspicuously ignored. Roper’s CEO feels the venture is risky and needs to pay a price that would yield his firm a return of about 20% if no operating improvements came out of the merger. a. What offering price should he put on the table to open negotiations? b. Hanson’s management is sure to want a higher price. Would that imply capitalizing earnings at a higher or lower rate. Why. c. What arguments is Hanson likely to use. SOLUTION: a. Roper’s CEO should start by capitalizing earnings as a perpetuity at the rate he feels he needs to see as a return. This yields Earnings $19 M PHanson $95 M Rateof Re turn .20 b. Paying a higher price would imply a lower percentage return because Roper would be paying more for the same earnings stream. Mathematically, a lower denominator in the equation above implies a higher price. c. Hanson is likely to argue that the acquisition is not very risky because the firm is stable and has a solid ongoing business. He also may argue that there’s a good deal of growth potential which would increase the earnings stream into the future above the value of a perpetuity. Multi-Part Problems Simple Multipart – Example 6.21 (page 282) 33. The Tower family wants to make a home improvement that is expected to cost $60,000. They want to fund as much of the cost as possible with a home equity loan, but can afford payments of only $600 per month. Their bank offers equity loans at 12% compounded monthly for a maximum term of 10 years. a. How much cash do they need as a down payment? b. Their bank account pays 8% compounded quarterly. If they delay starting the project for two years, how much would they have to save each quarter to make the required down payment if the loan rate and estimated cost remains the same? SOLUTION: Loan: PVA = PMT [PVFAk,n] = $600 [PVFA1,120 ] = $600 (69.7005) = $41,820.30 146 Chapter 6 a. Savings req: $60,000 $41,820.30 = $18,179.70 b. Save-up: FVA = PMT [FVFA k,n ] = PMT [PVFA2,8 ] $18,179.70 = PMT (8.5830) PMT = $2,118.10 Complex Multipart – Example 6.22 (page 284) 34. The Stein family wants to buy a small vacation house in a year and a half. They expect it to cost $75,000 at that time. They have the following sources of money 1. They currently have $10,000 in a bank account that pays 6% compounded monthly. 2. Uncle Murray has promised to give them $1,000 a month for 18 months starting today. 3. At the time of purchase, they'll take out a mortgage. They anticipate being able to make payments of about $300 a month on a 15-year, 12% loan. In addition, they plan to make quarterly deposits to an investment account to save up any shortfall in the amount required. How much must those additions be if the investment account pays 8% compounded quarterly? SOLUTION: Current bank account: FV = PV [FVFk,n ] = $10,000 [FVF.5,18 ] = $10,000 (1.0939) = $10,939 Uncle Murray: FVAd = PMT [FVFAk,n ](1+k) = $1,000 [FVFA.5,18 ](1+k) = $1,000 (18.7858) (1.005) = $18,879.73 Loan: PVA = PMT [PVFAk,n ] = $300 [PVFA1,180 ] = $300 (83.3217) = $24,996.51 Sources: $10,939.00 Shortfall: $75,000.00 18,879.73 54,815.24 24,996.51 $20,184.76 $54,815.24 Save-up: FVA = PMT [FVFA k,n ] = PMT [FVFA2,6 ] $20,184.76 = PMT (6.3081) PMT = $3,199.82 35. Clyde Atherton wants to buy a car when he graduates college in two years. He has the following sources of money: 1. He has $5,000 now in the bank in an account paying 8% compounded quarterly. 2. He will receive $2,000 in one year from a trust. 3. He'll take out a car loan at the time of purchase on which he'll make $500 monthly payments at 18% compounded monthly over four years. 4. Clyde's uncle is going to give him $1,500 a quarter starting today for one year. In addition Clyde will save up money in a credit union through monthly payroll deductions at his part-time job. The credit union pays 12% compounded monthly. If the car is expected to cost $40,000 (Clyde has expensive taste!), how much must he save each month? The Time Value of Money 147 SOLUTION: Current bank account: FV = PV [FVFk,n ] = $5,000 [FVF2,8 ] = $5,000 (1.1717) = $5,858.50 Trust: FV = PV [FVFk,n ] = $2,000 [FVF2,4 ] = $2,000 (1.0824) = $2,164.80 Loan: PVA = PMT [PVFAk,n ] = $500 [PVFA1.5,48 ] = $500 (34.0426) = $17,021.30 Uncle: FVAd = PMT [FVFAk,n](1+k) = $1,500 [FVFA2,4 ](1.02) = $1,500 (4.1216) (1.02) = $6,306.05 Bring this forward as an amount FV = PV [FVFk,n ] = $6,306.05 [FVF2,4 ] = $6,306.05 (1.0824) = $6,825.67 Sources: $ 5,858.50 Shortfall: $40,000.00 2,164.80 31,870.27 17,021.30 $ 8,129.73 6,825.67 $31,870.27 Save-up: FVA = PMT [FVFAk,n ] = PMT [FVFA1,24 ] $8,129.73 = PMT (26.9735) PMT = $301.40 36. Joe Trenton expects to retire in 15 years and has suddenly realized that he hasn’t saved anything toward that goal. After giving the matter some thought, he has decided that he would like to retire with enough money in savings to withdraw $85,000 per year for 25 years after he retires. Assume a 6% return on investment before and after retirement and that all payments into and withdrawals from savings are at year end. a. How much does Joe have to save in each year for the next 15 years to reach this goal? b. How much would Joe have needed to save each year if he had started when retirement was 25 years away? c. Comment on the difference between the results of parts a and b. SOLUTION: a. This is a 2-step solution. First, calculate the amount needed at retirement to fund 25 annual withdrawals of $85,000. This is the amount that will be needed in savings at retirement. Then calculate how much Joe will need to save each year until retirement to have that much. PVA = PMT[PVFAk,n ] = $85,000[FVA6,25 ] = $85,000 (12.7834) = $1,086,589 This PVA becomes the FVA of the required savings over the next 15 years. FVA = PMT[FVFA6,15 ] $1,086,589 = PMT[23.2760] PMT = $46,683 148 Chapter 6 b. The first calculation is the same, $1,086,589 is required at retirement. The only change in the second calculation is n becomes 25 rather than 15. FVA = PMT[FVFA6,25 ] $1,086,589 = PMT[54.8645] PMT = $19,805 c. For most people saving more than $46,000 a year is impossible, while about putting aside $20,000 or less than half of that amount would be very difficult but might be possible. There is no substitute for starting retirement planning early. 37. Janet Elliott just turned 20, and received a gift of $20,000 from her rich uncle. Janet plans ahead and would like to retire on her 55th birthday. She thinks she’ll need to have about $2 million saved by that time in order to maintain her lavish lifestyle. She wants to make a payment at the end of each year until she’s 50 into an account she’ll open with her uncle’s gift. After that she’d like to stop making payments and let the money grow at interest until it reaches $2 million when she turns 55. Assume she can invest at 7% compounded annually. Ignore the effect of taxes. a. How much will she have to invest each year in order to achieve her objective? b. What percent of the $2 million will have been contributed by Janet (including the $20,000 she got from her uncle)? SOLUTION: a. First we need to know how much she will need to accumulate by the time she reaches 50 so that amount can grow to $2 million by the time she reaches 55. n = 5; I/Y = 7; PMT = 0; FV = 2,000,000 CPT PV = $1,425,972.36 That number becomes the FV in an ordinary annuity with a $20,000 PV n = 30; I/Y = 7; PV = 20,000; FV = (1,425,972.36) CPT PMT = $13,484.19 b. Janet will have contributed $20,000 + 30 x $13,484.19 = $424,525.70, and 424,525.70/2,000,000 = 21.23% 38. Merritt Manufacturing needs to accumulate $20 million to retire a bond issue that matures in 13 years. The firm’s manufacturing division can contribute $100,000 per quarter to an account that will pay 8%, compounded quarterly. How much will the remaining divisions have to contribute every month to a second account that pays 6% compounded monthly in order to reach the $20 million goal? SOLUTION: First calculate how much the known source of funding ($100,000 per quarter) will grow to by the end of 13 years n = 13 x 4 = 52; I/Y = 8/4 = 2; PV = 0; PMT = 100,000 CPT FV = $9,001,640.93 The remainder $10,998,359.07 will have to be raised from the other divisions and becomes the FV of the second annuity problem n = 156; I/Y = .5; PV = 0; FV = 10,998,359.07 CPT PMT = $46,712.61 39. Carol Pasca just had her fifth birthday. As a birthday present, her uncle promised to contribute $300 per month to her education fund until she turns 18 and starts college. Carol’s parents estimate college will cost $2,500 per month for four years, but don’t think they’ll be able to save anything toward The Time Value of Money 149 it for five years. How much will Carol’s parents need to contribute to the fund each month starting on her tenth birthday to pay for her college education? Assume the fund earns 6% compounded monthly? SOLUTION: First, we need to calculate how much money Carol will need when she starts college. That’s the present value of $2,500 per month for four years. n = 48; I/Y = .5; PMT = 2,500; FV = 0 CPT PV = $106,450.79 The money from her uncle will contribute the future value of an annuity of $300 per month for 13 years: n = 156; I/Y = .5; PV = 0; PMT = 300 CPT FV = $70,634.20 The shortfall is the difference of $35,816.59 which must be made up by her parents contributions over five years n = 60; I/Y = .5; FV = 35;816.59; PV=0; CPT PMT = $513.35 40. Joan Colby is approaching retirement and plans to purchase a condominium in Florida in three years. She now has $40,000 saved toward the purchase in a bank account that pays 8% compounded quarterly. She also has five $1,000 face value corporate bonds that mature in two years. She plans to deposit the bonds’ principal repayments in the same account when they’re paid. Joan also receives $1,200 per month alimony from her ex-husband which will continue for two more years until he retires (24 checks including one that arrived today). She’s decided to put her remaining alimony money toward her condo depositing it as received in a credit union account that pays 8% compounded monthly. She’ll make the first deposit today with the check she already has. Joan anticipates buying a $200,000 property. What will her monthly payment be on a 15 year mortgage at 6%? What would the payment be on a 30 year loan at the same interest rate? SOLUTION: Calculate the value of money Joan will have in the bank and credit union in three years $40,000 already on deposit FV = PV[FVFk,n] = $40,000[FVF2,12 ] = $40,000(1.2682) = $50,728.00 Bond principal repayment FV = PV[FVFk,n] = $5,000[FVF2,4 ] = $5,000(1.0824) = $5,412.00 Alimony: First calculate the future value of the stream at its end in two years recognizing that the deposits will be an annuity due. Then project at sum forward one year to the date of purchase. FVA = PMT[FVFAk,n ](1+k) = $1,200[FVFA.67,24 ](1.0067) = $1,200(25.9332)(1.0067) = $31,119.84(1.0067) = $31,328.34 FV = PV[FVFk,n] = $31,328.34[FVF.67,12 ] = $31,328.34 (1.0830) = $33,928.59 Funds saved $50,728.00 + $5,412 + $33,928.59 = $90,068.59 Next calculate the mortgage required $200,000 - $90,068.59 = $109,931.41 150 Chapter 6 Finally calculate the mortgage payment PVA = PMT[PVFAk,n ] = PMT[PVFA.5,180 ] $109,931.41 = PMT(118.504) PMT = $927.66 If Joan chooses a 30 year mortgage at the same rate, her payments will be as follows: PVA = PMT[PVFAk,n ] = PMT[PVFA.5,360 ] $109,931.41 = PMT(166.792) PMT = $659.09 Imbedded Annuity – Example 6.24 (page 283) 41. Amy’s uncle died recently and left her some money in a trust that will pay her $500 per month for five years starting on her twenty fifth birthday. Amy is getting married soon, and would like to use this money as a down payment on a house now. If the trust allows her to assign its future payments to a bank, and her bank is willing to discount them at 9% compounded monthly, how much will she have toward her down payment on home ownership? Amy just turned 23. SOLUTION: This problem requires a two-step calculation. The payments are an annuity, so we need to take its present value. That will give us a result at the beginning of the annuity, which is two years in the future. That PVA then has to be brought to the present as an amount. First calculate the “present” value of the annuity at its beginning. k = 9/12 = .75 n = 5 12 = 60 PVA = PMT[PVFA.75,60 ] = $500(48.1734) = $24,086.70 Next bring that figure backward two years into the present as an amount. k = 9/12 = .75 n = 2 12 = 24 PV = FV[PVF.75,24 ] = $24,086.70(.8358) = $20,131.66 COMPUTER PROBLEMS 42. At any particular time, home mortgage rates are determined by market forces and individual borrowers can't do much about them. The length of time required to pay off a mortgage loan, however, varies a great deal with the size of the monthly payment made, which is under the borrower's control. You're a junior loan officer for a large metropolitan bank. The head of the mortgage department is concerned that customers don't fully appreciate that a relatively small increase in the size of mortgage payments can make a big difference in how long the payments have to be made. She feels homeowners may be passing up an opportunity to make their lives better in the long run by not choosing shorter-term mortgages that they can readily afford. To explain the phenomenon to customers she's asked you to put together a chart that displays the variation in term with payment size at typical interest rates. The starting point for the charts should be the term for a typical thirty-year (360-month) loan. Use the TIMEVAL program to construct the following chart. The Time Value of Money 151 Mortgage Payments per $100,000 Borrowed as Term Decreases MORTGAGE TERM IN YEARS 30 25 20 15 6% RATES 8% 10% 12% Write a paragraph using the chart to explain the point. What happens to the effect as interest rates rise? Why? SOLUTION: Mortgage Payments per $100,000 Borrowed as Term Decreases MORTGAGE TERM IN YEARS 30 25 20 15 RATES 6% $600 $644 $836 8% $878 $909 $965 $1,075 10% $1,029 $1,053 $1,101 $1,200 12% $1,185 $1,204 $1,244 $1,332 Notice how much faster the homeowner's mortgage is paid off if a little extra money is devoted to house payment each month. At 8% an extra $38 a month pays off the typical mortgage loan a full five years sooner, while the term is cut in half by paying $222 a month extra. That means a 30% increase (or 222/734) in payment brings a 50% reduction in the time mortgage payments have to be made. At higher interest rates the effect more significant. At 12%, five years of payments can be avoided by paying only an additional $24 dollars a month, and the term can be cut in half with a 17% increase in the monthly payment. The term shortening effect of an extra dollar paid increases with the interest rate, because the additional payment reduces principal ahead of schedule, and early pay-down avoids more interest at higher rates. 43. Amitron Inc. is considering an engineering project that requires an investment of $250,000 and is expected to generate the following stream of payments (income) in the future. Use the TIMEVAL program to determine if the project is a good idea in a present value sense. That is, does the present value of expected cash inflows exceed the value of the investment that has to be made today? Year Payment 1 $63,000 2 $69,500 3 $32,700 4 $79,750 5 $62,400 6 $38,250 152 Chapter 6 a. Answer the question if the relevant interest rate for taking present values is 9%, 10%, 11% and 12%. In the program, notice that period zero represents a cash flow made at the present time, which isn't discounted. The program will do the entire calculation for if you input the initial investment as a negative number in this cell. b. Use trial and error in the program to find the interest rate (to the nearest hundredth of a percent) at which Amitron would be just indifferent to the project. This problem is a preview of an important method of evaluating projects known as capital budgeting. We'll study the topic in detail in chapters 10, 11, and 12. In part a of this problem, we found the net present value (NPV) of the project's cash flows at various interest rates, and reasoned intuitively that the project was a good idea if that figure was positive. In part b we found the return inherent in the project itself, which is called the internal rate of return (IRR). We'll learn how to use that in Chapter 10. SOLUTION: a. k NPV 9% $11,405 good idea 10% 4,086 good idea 11% (2,910) bad idea 12% (9,601) bad idea b. Indifference implies NPV = 0 which occurs at k = 10.58% 44. The Centurion Corp. is putting together a financial plan for the company covering the next three years, and needs to forecast its interest expense and the related tax savings. The firm's most significant liability is a fully amortized mortgage loan on its real estate. The loan was made exactly ten and one half years ago for $3.2 million at 11% compounded monthly for a term of thirty years. Use the AMORTIZ program to predict the interest expense associated with the real estate mortgage over the next three years. (Hint: Run AMORTIZ from the loan's beginning and add up the months in each of the next three years.) SOLUTION: In the AMORTIZ output, the next three years are months 127-138, 139-150, and 151- 162. Interest each year is: 1 $320,213.11 2 $314,950.35 3 $309,078.59 DEVELOPING SOFTWARE 1. Write your own program to amortize a ten-year, $20,000 loan at 10% compounded annually. Input the loan amount, the payment, and the interest rate. Set up your spreadsheet just like Table 6-4, and write your program to duplicate the calculation procedure described. SOLUTION: A sample solution is in the Instructor's Material on the text website.

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