Chapter 05 - The Time Value of Money Chapter 05 The Time Value of Money True / False Questions Compound interest pays interest for each time period on the original investment plus the accumulated interest True False When money is invested at compound interest, the growth rate is the interest rate True False The present value of an annuity due equals the present value of an ordinary annuity times the discount rate True False The more frequent the compounding, the higher the future value, other things equal True False A dollar tomorrow is worth more than a dollar today True False The Excel function for future value is FV (rate, nper, pmt, PV) True False For a given amount, the lower the discount rate, the less the present value True False 5-1 Chapter 05 - The Time Value of Money Comparing the values of undiscounted cash flows is analogous to comparing apples to oranges True False To calculate present value, we discount the future value by some interest rate r, the discount rate True False 10 The discount factor is used to calculate the present value of $1 received in year t True False 11 You should never compare cash flows occurring at different times without first discounting them to a common date True False 12 The Excel function for present value is PV (rate, nper, pmt, FV) True False 13 A perpetuity is a special form of an annuity True False 14 An annuity factor represents the future value of $1 that is deposited today True False 15 Accrued interest declines with each payment on an amortizing loan True False 5-2 Chapter 05 - The Time Value of Money 16 Converting an annuity to an annuity due decreases the present value True False 17 The term "constant dollars" refers to equal payments for amortizing a loan True False 18 An annuity due must have a present value at least as large as an equivalent ordinary annuity True False 19 Any sequence of equally spaced, level cash flows is called an annuity An annuity is also known as a perpetuity True False 20 A mortgage loan is an example of an amortizing loan "Amortizing" means that part of the monthly payment is used to pay interest on the loan and part is used to reduce the amount of the loan True False 21 The Excel function for interest rate is RATE (nper, pmt, PV, FV) True False 22 An effective annual rate must be greater than an annual percentage rate True False 5-3 Chapter 05 - The Time Value of Money 23 An annual percentage rate (APR) is determined by annualizing the rate using compound interest True False 24 In 2002, the U.S inflation rate was below 2% and a few countries were even experiencing deflation True False 25 Nominal dollars refer to the amount of purchasing power True False 26 The appropriate manner of adjusting for inflationary effects is to discount nominal cash flows with real interest rates True False Multiple Choice Questions 27 What is the future value of $10,000 on deposit for years at 6% simple interest? A $7,472.58 B $10,303.62 C $13,000.00 D $13,382.26 28 Under which of the following conditions will a future value calculated with simple interest exceed a future value calculated with compound interest at the same rate? A The interest rate is very high B The investment period is very long C The compounding is annually D This is not possible with positive interest rates 5-4 Chapter 05 - The Time Value of Money 29 How much interest is earned in just the third year on a $1,000 deposit that earns 7% interest compounded annually? A $70.00 B $80.14 C $105.62 D $140.00 30 How much interest will be earned in the next year on an investment paying 12% compounded annually if $100 was just credited to the account for interest? A $88 B $100 C $112 D $200 31 The concept of compound interest refers to: A earning interest on the original investment B payment of interest on previously earned interest C investing for a multiyear period of time D determining the APR of the investment 32 When an investment pays only simple interest, this means: A the interest rate is lower than on comparable investments B the future value of the investment will be low C the earned interest is nontaxable to the investor D interest is earned only on the original investment 33 Approximately how long must one wait (to the nearest year) for an initial investment of $1,000 to triple in value if the investment earns 8% compounded annually? A years B 14 years C 22 years D 25 years 5-5 Chapter 05 - The Time Value of Money 34 How much will accumulate in an account with an initial deposit of $100, and which earns 10% interest compounded quarterly for years? A $107.69 B $133.10 C $134.49 D $313.84 35 What will be the approximate population of the United States, if its current population of 300 million grows at a compound rate of 2% annually for 25 years? A 413 million B 430 million C 488 million D 492 million 36 How much interest can be accumulated during one year on a $1,000 deposit paying continuously compounded interest at an APR of 10%? A $100.00 B $105.17 C $110.50 D $115.70 37 How much interest will be earned in an account into which $1,000 is deposited for one year with continuous compounding at a 13% rate? A $130.00 B $138.83 C $169.00 D $353.34 38 What is the discount factor for $1 to be received in years at a discount rate of 8%? A .4693 B .5500 C .6000 D .6806 5-6 Chapter 05 - The Time Value of Money 39 Assume the total expense for your current year in college equals $20,000 Approximately how much would your parents have needed to invest 21 years ago in an account paying 8% compounded annually to cover this amount? A $952.00 B $1,600.00 C $1,728.00 D $3,973.00 40 How much must be deposited today in an account earning 6% annually to accumulate a 20% down payment to use in purchasing a car one year from now, assuming that the car's current price is $20,000, and inflation will be 4%? A $3,774 B $3,782 C $3,925 D $4,080 41 Given a set future value, which of the following will contribute to a lower present value? A Higher discount rate B Fewer time periods C Less frequent discounting D Lower discount factor 42 Cash flows occurring in different periods should not be compared unless: A interest rates are expected to be stable B the flows occur no more than one year from each other C high rates of interest can be earned on the flows D the flows have been discounted to a common date 5-7 Chapter 05 - The Time Value of Money 43 A corporation has promised to pay $1,000 20 years from today for each bond sold now No interest will be paid on the bonds during the 20 years, and the bonds are discounted at a 7% interest rate Approximately how much should an investor pay for each bond? A $70.00 B $258.42 C $629.56 D $857.43 44 What is the present value of your trust fund if it promises to pay you $50,000 on your 30th birthday (7 years from today) and earns 10% compounded annually? A $25,000.00 B $25,657.91 C $28,223.70 D $29,411.76 45 How much more would you be willing to pay today for an investment offering $10,000 in years rather than the normally advertised 5-year period? Your discount rate is 8% A $544.47 B $681.48 C $740.74 D $800.00 46 What is the present value of $100 to be deposited today into an account paying 8%, compounded semiannually for years? A $85.48 B $100.00 C $116.00 D $116.99 5-8 Chapter 05 - The Time Value of Money 47 How much must be invested today in order to generate a 5-year annuity of $1,000 per year, with the first payment year from today, at an interest rate of 12%? A $3,604.78 B $3,746.25 C $4,037.35 D $4,604.78 48 The salesperson offers, "Buy this new car for $25,000 cash or, with appropriate down payment, pay $500 per month for 48 months at 8% interest." Assuming that the salesperson does not offer a free lunch, calculate the "appropriate" down payment A $1,000.00 B $4,520.64 C $5,127.24 D $8,000.00 49 What is the present value of the following payment stream, discounted at 8% annually: $1,000 at the end of year 1, $2,000 at the end of year 2, and $3,000 at the end of year 3? A $5,022.11 B $5,144.03 C $5,423.87 D $5,520.00 50 What is the present value of the following set of cash flows at an interest rate of 7%: $1,000 today, $2,000 at end of year 1, $4,000 at end of year 3, and $6,000 at end of year 5? A $9,731 B $10,412 C $10,524 D $11,524 5-9 Chapter 05 - The Time Value of Money 51 A cash-strapped young professional offers to buy your car with four, equal annual payments of $3,000, beginning years from today Assuming you're indifferent to cash versus credit, that you can invest at 10%, and that you want to receive $9,000 for the car, should you accept? A Yes; present value is $9,510 B Yes; present value is $11,372 C No; present value is $8,645 D No; present value is $7,461 52 How much more is a perpetuity of $1,000 worth than an annuity of the same amount for 20 years? Assume a 10% interest rate and cash flows at end of period A $297.29 B $1,486.44 C $1,635.08 D $2,000.00 53 A stream of equal cash payments lasting forever is termed: A an annuity B an annuity due C an installment plan D a perpetuity 54 Which of the following factors is fixed and thus cannot change for a specific perpetuity? A PV of a perpetuity B Cash payment of a perpetuity C Interest rate on a perpetuity D Discount rate of a perpetuity 55 The present value of a perpetuity can be determined by: A Multiplying the payment by the interest rate B Dividing the interest rate by the payment C Multiplying the payment by the number of payments to be made D Dividing the payment by the interest rate 5-10 Chapter 05 - The Time Value of Money 92 What is the annually compounded rate of interest on an account with an APR of 10% and monthly compounding? A 10.00% B 10.47% C 10.52% D 11.05% (1.00833) 12 - = 10.47% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 93 What is the APR on a loan with an effective annual rate of 15.01% and weekly compounding of interest? A 12.00% B 12.50% C 13.00% D 14.00% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Hard Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 5-68 Chapter 05 - The Time Value of Money 94 What is the effective annual interest rate on a 9% APR automobile loan that has monthly payments? A 9.00% B 9.38% C 9.81% D 10.94% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 95 Other things being equal, the more frequent the compounding period, the: A higher the APR B lower the APR C higher the effective annual interest rate D lower the effective annual interest rate AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Easy Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 5-69 Chapter 05 - The Time Value of Money 96 An APR will be equal to an effective annual rate if: A compounding occurs monthly B compounding occurs continuously C compounding occurs annually D an error has occurred; these terms cannot be equal AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Easy Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 97 A credit card account that charges interest at the rate of 1.25% per month would have an annually compounded rate of _ and an APR of _ A 16.08%; 15.00% B 14.55%; 16.08% C 12.68%; 15.00% D 15.00%; 14.55% Annually compounded rate = (1.0125)12 - = 16.08% APR = 1.25%  12 = 15.00% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 5-70 Chapter 05 - The Time Value of Money 98 If inflation in Wonderland averaged about 20% per month in 2000, what was the approximate annual inflation rate? A 20% B 240% C 790% D 890% (1.20)12 - = 7.916 = 791.6% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Hard Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 99 Assume your uncle recorded his salary history during a 40-year career and found that it had increased 10-fold If inflation averaged 4% annually during the period, how would you describe his purchasing power, on average? A His purchasing power remained on par with inflation B He "beat" inflation by nearly 1% annually C He "beat" inflation by slightly below 2% annually D He "beat" inflation by 5% annually 10 = 1(1 + i)40, i = 5.925% by financial calculator AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 5-71 Chapter 05 - The Time Value of Money 100 Which of the following statements best describes the real interest rate? A Real interest rates exceed inflation rates B Real interest rates can decline only to zero C Real interest rates can be negative, zero, or positive D Real interest rates traditionally exceed nominal rates AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 101 What is the expected real rate of interest for an account that offers a 12% nominal rate of return when the rate of inflation is 6% annually? A 5.00% B 5.66% C 6.00% D 9.46% + real interest rate = (1 + nominal interest rate)/(1 + inflation) + real interest rate = 1.12/1.06 Real interest rate = 5.66% AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Easy Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 102 What happens over time to the real cost of purchasing a home if the mortgage payments are fixed in nominal terms and inflation is in existence? A The real cost is constant B The real cost is increasing C The real cost is decreasing D The price index must be known to answer this question AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 5-72 Chapter 05 - The Time Value of Money 103 What is the minimum nominal rate of return that you should accept if you require a 4% real rate of return and the rate of inflation is expected to average 3.5% during the investment period? A 7.36% B 7.50% C 7.64% D 8.01% 7.64% = nominal rate AACSB: Reflective Thinking Skills Blooms: Application Difficulty: Easy Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation Essay Questions 104 Discuss the statement, "Money has a time value." Money has a time value due to the concept of opportunity cost In other words, if receipt of funds is forgone until a later period, you lose the opportunity to earn a return on the funds in the interim Thus, cash flows that occur in different periods cannot be directly compared without adjusting for these opportunity costs Discounting cash flows to a common period adjusts for the time value, and makes cash flows comparable AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Medium Learning Objective: 05-01 Calculate the future value to which money invested at a given interest rate will grow Topic: Present and Future Value 5-73 Chapter 05 - The Time Value of Money 105 Would you prefer a savings account that paid 7% interest, compounded quarterly, over an account that paid 7.5% with annual compounding if you had $1,000 to deposit? Would the answer change if you had $100,000 to deposit? FV = (1 + i)n for simple interest FV = (1 + i/m)nxm for compound interest Then, FV = (1 + 07/4)1 x = 1.0719 Versus FV = (1 + 0.75)1 = 1.075 Thus, the 7.5% account will earn 31% more in the first year than the 7% account with quarterly compounding The amount to be deposited will not change your preference: In this case the compounding is not enough to overcome the difference in APR AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-01 Calculate the future value to which money invested at a given interest rate will grow Topic: Present and Future Value 106 If years of college are expected to cost $150,000 18 years from now, how much must be deposited now into an account that will average 8% annually in order to save the $150,000? By how much would your answer change if you expected 11% annually? FV = PV (1 + i)n $150,000 = PV (1.08)18 $150,000 = PV  3.996 $37,537.35 = PV If the interest rate increases to 11%, the necessary deposit is reduced to $22,923.33 AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-02 Calculate the present value of a future payment Topic: Present and Future Value 5-74 Chapter 05 - The Time Value of Money 107 Prizes are often not "worth" as much as claimed Place a value on a prize of $5,000,000 which is to be received in equal payments over 20 years, with the first payment beginning today Assume an interest rate of 7% over the 20 years AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-03 Calculate present and future values of a series of cash payments Topic: Perpetuities and Annuities 5-75 Chapter 05 - The Time Value of Money 108 Show numerically that a savings account with a current balance of $1,000 that earns interest at 9% annually is precisely sufficient to make the payments on a 3-year loan of $1,000 that carries equal annual payments at 9% interest The loan payments are: After the first year's addition of interest, the account has $1,090.00 and $395.06 is withdrawn to make the first payment The balance of $694.94 grows to $757.48 at the end of the second year After making the second payment of $395.06, $362.42 is left in the account This amount grows to $395.04 by the end of the third year, which is within a 2-cent rounding error of making the final payment AACSB: Analytical Skills Blooms: Evaluation Difficulty: Hard Learning Objective: 05-03 Calculate present and future values of a series of cash payments Topic: Perpetuities and Annuities 5-76 Chapter 05 - The Time Value of Money 109 A loan officer states, "Thousands of dollars can be saved by switching to a 15-year mortgage from a 30-year mortgage." Calculate the difference in payments on a 30-year mortgage at 9% interest versus a 15-year mortgage with 8.5% interest Both mortgages are for $100,000 and have monthly payments What is the difference in total dollars that will be paid to the lender under each loan? Difference in total dollars = (804.62  360) - (984.69  180) = 289,663.20 - 177,244.20 = $112,419 AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-03 Calculate present and future values of a series of cash payments Topic: Perpetuities and Annuities 5-77 Chapter 05 - The Time Value of Money 110 Some home loans involve "points," which are fees charged by the lender Each point charged means that the borrower must pay 1% of the loan amount as a fee For example, if 0.5 point is charged on a $100,000 loan, the loan repayment schedule is calculated on the $100,000 loan, but the net amount the borrower receives is only $99,500 What is the effective annual interest rate charged on such a loan, assuming that loan repayment occurs over 360 months, and that the interest rate is 1% per month? Since the monthly payment is based on a $100,000 loan: Mortgage payment  annuity factor(1%, 360) = 100,000 monthly mortgage payment = $1,028.61 The net amount received is $99,500 Therefore: $1,028.61  annuity factor(r, 360) = $99,500 r = 1.006% per month The effective annual rate is: (1.01006)12 - = 0.1276 = 12.76% AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-04 Find the interest rate implied by present and future values Topic: EAR and APR 111 In 1973 Gordon Moore, one of Intel's founders, predicted that the number of transistors that could be placed on a single silicon chip would double every 18 months, equivalent to an annual growth of 59% (i.e., 1.591.5 = 2.0) The first microprocessor was built in 1971 and had 2,250 transistors By 2003 Intel chips contained 410 million transistors, over 182,000 times the number 32 years earlier What has been the annual compound rate of growth in processing power? How does it compare with the prediction of Moore's law? Call g the annual growth rate of transistors over the 32-year period between 1971 and 2003 Then 2,250  (1 + g)32 = 410,000,000 (1 + g)32 = 182,222 + g = 182,2221/32 = 1.46 So the actual growth rate has been g = 46, or 46%, not quite as high as Moore's prediction, but not so shabby either AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-04 Find the interest rate implied by present and future values Topic: Present and Future Value 5-78 Chapter 05 - The Time Value of Money 112 How should we compare interest rates quoted over different time intervals—for example, monthly versus annual rates? Interest rates for short time periods are often quoted as annual rates by multiplying the period rate by the number of periods in a year These annual percentage rates (APRs) not recognize the effect of compound interest; that is, they annualize assuming simple interest The effective annual rate annualizes using compound interest It equals the rate of interest per period compounded for the number of periods in a year AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 113 Discuss the statement, "It is always preferred to select an account that offers compound interest over an account that offers simple interest." This statement is true if the APRs are equal on the different accounts, and if the compounding occurs more frequently than annually The statement may be false if the APRs are not equal, however A point is reached where the benefit of more frequent compounding is overshadowed by the reduction in APR AACSB: Analytical Skills Blooms: Analysis Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: Present and Future Value 5-79 Chapter 05 - The Time Value of Money 114 After reading the fine print in your credit card agreement, you find that the "low" interest rate is actually an 18% APR, or 1.5% per month Now, to make you feel even worse, calculate the effective annual interest rate AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 115 Why is it difficult and perhaps risky to evaluate financial projects based on APR alone? Evaluating a project by APR alone ignores the potential significant effects that accrue as a result of compounding on a more frequent than annual basis For example, over a long period of time there is a significant difference to the value of an account that carries monthly as opposed to annual compounding In a similar manner, the cost of a loan can best be evaluated through the effective annual rate that considers the cost of payments occurring more frequently than annually AACSB: Analytical Skills Blooms: Analysis Difficulty: Medium Learning Objective: 05-05 Compare interest rates quoted over different time intervals—for example; monthly versus annual rates Topic: EAR and APR 5-80 Chapter 05 - The Time Value of Money 116 What is the difference between real and nominal cash flows and between real and nominal interest rates? A dollar is a dollar, but the amount of goods that a dollar can buy is eroded by inflation If prices double, the real value of a dollar halves Financial managers and economists often find it helpful to re-express future cash flows in terms of real dollars—that is, dollars of constant purchasing power Be careful to distinguish the nominal interest rate and the real interest rate—the rate at which the real value of the investment grows Discount nominal cash flows (that is, cash flows measured in current dollars) at nominal interest rates; discount real cash flows (cash flows measured in constant dollars) at real interest rates Never mix and match nominal and real AACSB: Communication Abilities Blooms: Knowledge Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 117 What problem can be caused by "mixing" real and nominal cash flows in discounting exercises? One of the primary components of a nominal interest rate is a premium for the rate of inflation that is expected during the time period that the interest rate is in effect On the other hand, the adjustment that takes a nominal rate to a real rate is typically a downward adjustment that "backs out" the expected impact of inflation Thus, to discount real flows with a nominal rate would be to overcompensate for the effects of inflation Alternatively, to discount nominal flows with a real rate would be to under compensate for inflationary impact The only safe, correct method is to discount nominal flows with nominal rates, or discount real flows with real rates AACSB: Reflective Thinking Skills Blooms: Understanding Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 5-81 Chapter 05 - The Time Value of Money 118 In 2004 there was widespread dismay as the price of unleaded gasoline climbed to $2.03 a gallon Motorists looked back longingly to 20 years earlier when they were paying just $1.19 a gallon But how much had the real price of gasoline changed over this period, if the consumer price index was 1.81 times itself in 1984? In 2004 the consumer price index was 1.81 times its level in 1984 If the price of gasoline had risen in line with inflation, it would have cost 1.81  $1.19 = $2.15 a gallon in 2004 That was the cost of gasoline 20 years ago but measured in terms of 2004 dollars rather than 1984 dollars Thus over the 20 years the real price of gasoline declined from $2.15 a gallon to $2.03, a fall of 6% AACSB: Analytical Skills Blooms: Evaluation Difficulty: Medium Learning Objective: 05-06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 5-82 ... rate of 1. 25% per month would have an annually compounded rate of _ and an APR of _ A 16.08%; 15. 00% B 14 .55 %; 16.08% C 12.68%; 15. 00% D 15. 00%; 14 .55 % 98 If inflation in Wonderland averaged... at end of year 1, $4,000 at end of year 3, and $6,000 at end of year 5? A $9,731 B $10,412 C $10 ,52 4 D $11 ,52 4 5- 9 Chapter 05 - The Time Value of Money 51 A cash-strapped young professional offers... Understanding Difficulty: Medium Learning Objective: 05- 06 Understand the difference between real and nominal cash flows and between real and nominal interest rates Topic: Inflation 5- 33 Chapter 05