Chapter 3 - Time Value of Money (Kèm đáp án)

TIME VALUE OF MONEY PROBLEMS Exercise 1 (FV) Sales of the P.J. Cramer Company were $500,000 this year, and they are expected to grow at a compound rate of 20 percent for the next six years. What will be the sales figure at the end of each of the next six years? Solution: Year 1 2 3 4 5 6 Sales $ 600,000 720,000 864,000 1,036,800 1,244,160 1,492,992 Exercise 2 (PV) Suppose you were to receive $1,000 at the end of 10 years. If your opportunity rate is 10 percent, what is the present value of this amount if interest is compounded (a) annually? (b) quarterly? (c) continuously? Solution: (a) $386 (b) $372 (c) $368 Exercise 3 (Interest rate) Vernal Equinox wishes to borrow $10,000 for three years. A group of individuals agrees to lend him this amount if he contracts to pay them $16,000 at the end of the three years. What is the implicit compound annual interest rate implied by this contract (to the nearest whole percent)? Solution: 16,000 = 10,000 x (1+r)3 => (1+r)3 = 1.6 => r = 16.96% Exercise 4 (Interest rate) In connection with the United States Bicentennial, the Treasury once contemplated offering a savings bond for $1,000 that would be worth $1 million in 100 years. Approximately what compound annual interest rate is implied by these terms? Solution: (1 + x%)100 = $1,000,000/$1,000 = 1,000 => x = 7.15% Exercise 5 (EAR, APR) Suppose that an investment promises to pay a nominal 9.6 percent annual rate of interest. What is the effective annual interest rate on this investment assuming that interest is compounded (a) annually? (b) semiannually? (c) quarterly? (d) monthly? (e) daily (365 days)? (f ) continuously? (Note: Report your answers accurate to four decimal places – e.g., 0.0987 or 9.87%.) Solution: Effective annual interest rate = (1 + [APR/m])m – 1 (a) EAR = .0960 (b) EAR = .0983 (c) EAR = .0995 (d) EAR = .1003 (e) EAR = .1007 (f ) EAR = .1008 Exercise 6 (FV, EAR, Different compounding) The following are exercises in future (terminal) values: a. At the end of three years, how much is an initial deposit of $100 worth, assuming a compound annual interest rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent? b. At the end of five years, how much is an initial $500 deposit followed by five year-end, annual $100 payments worth, assuming a compound annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) 0 percent? c. At the end of six years, how much is an initial $500 deposit followed by five year-end, annual $100 payments worth, assuming a compound annual interest rate of (i) 10 percent? (ii) 5 percent? (iii) 0 percent? d. At the end of three years, how much is an initial $100 deposit worth, assuming a quarterly compounded annual interest rate of (i) 100 percent? (ii) 10 percent? e. At the end of 10 years, how much is a $100 initial deposit worth, assuming an annual interest rate of 10 percent compounded (i) annually? (ii) semiannually? (iii) quarterly? (iv) continuously? Solution: a) FVn = PV0 x (1 + i)n (i) FV3 = $100 x (2.0)3 = $100 x (8) = $800 (ii) FV3 = $100 x (1.10)3 = $100 x (1.331) = $133.10 (iii) FV3 = $100 x (1.0)3 = $100 x (1) = $100 b) FVn = PV0 x (1 + i)n; FVn(A) = A x [([1 + i]n - 1)/i] (i) FV5 = $500 x (1.10)5 = $500 x (1.611) = $805.50; FV5(A) = $100 x [([1.10]5 - 1)/(.10)] = $100 x (6.105) = 610.50 => Total = $1,416.00 (ii) FV5 = $638.00; FV5(A) = $552.60 => Total = $1,190.60 (iii) FV5 = $500.00; FV5(A) = $500.00 => Total = $1,000.00 c) FVn = PV0 x (1 + i)n; FVn(AD) = A x [([1 + i]n - 1)/i] x [1 + i] (i) FV6 = $500 x (1.10)6 = $500 x (1.772) = $886.00; FV5 (AD) = $100 x [([1.10]5 - 1)/(.10)] x [1.10] = $100 x (6.105) x (1.10) = 671.55 => Total = $1,557.55 (ii) FV6 = $670.00; FV5 (AD) = 580.23 => Total = $1,250.23 (iii) FV6 = $500.00; FV5 (AD) = 500.00 => Total = $1,000.00 d) FVn = PV0 x (1 + [APR/m])mn (i) FV3 = $100 x (1 + [1/4])12 = $100 x (14.552) = $1,455.20 (ii) FV3 = $100 x (1 + [.10/4])12 = $100 x (1.345) = $134.50 f) FVn = PV0 x (1 + [i/m])mn; FVn = PV0(e)in (i) $100 x (1 + [.10/1])10 = $100 x (2.594) = $259.40 (ii) $100 x (1 + [.10/2])20 = $100 x (2.653) = $265.30 (iii) $100 x (1 + [.10/4])40 = $100 x (2.685) = $268.50 (iv) $100 x (2.71828)1 = $271.83 Exercise 7 (PV, One CF, EAR, Different compounding) The following are exercises in present values: a. $100 at the end of three years is worth how much today, assuming a discount rate of (i) 100 percent? (ii) 10 percent? (iii) 0 percent? b. What is the aggregate present value of $500 received at the end of each of the next three years, assuming a discount rate of (i) 4 percent? (ii) 25 percent? c. $100 is received at the end of one year, $500 at the end of two years, and $1,000 at the end of three years. What is the aggregate present value of these receipts, assuming a discount rate of (i) 4 percent? (ii) 25 percent? d. $1,000 is to be received at the end of one year, $500 at the end of two years, and $100 at the end of three years. What is the aggregate present value of these receipts assuming a discount rate of (i) 4 percent? (ii) 25 percent? Solution: a) PV0 = FVn x [1/(1 + i)n] (i) $100 x [1/(2)3] = $100 x (.125) = $12.50 (ii) $100 x [1/(1.10)3] = $100 x (.751) = $75.10 (iii) $100 x [1/(1.0)3] = $100 x (1) = $100 b) PVn (A) = A x [(1 -[1/(1 + i)n])/i] (i) $500 x [(1 - [1/(1 + .04)3])/.04] = $500 x (2.775) = $1,387.50 (ii) $500[(1 - [1/(1 + .25)3])/.25] = $500 x (1.952) = $ 976.00 c) PV0 = FVn x [1/(1 + i)n] (i) $100 x [1/(1.04)1] = $ 100(.962) = $ 96.20; 500 x [1/(1.04)2] = 500 x (.925) = 462.50 1,000 x [1/(1.04)3] = 1,000 x (.889) = 889.00 => Total = $1,447.70 (ii) $ 80.00 + 320.00 + 512.00 = $ 912.00 d) (i) 962.00 + 462.50 + 88.90 = $1,513.40 (ii) $ 800.00 + 320.00 + 51.20 = $1,171.20 Exercise 8 (EAR, One CF) A bank offers you a seven-month certificate of deposit (CD) at a 7.06 percent annual rate that would provide a 7.25 percent effective annual yield. For the seven-month CD, is interest being compounded daily, weekly, monthly, or quarterly? And, by the way, having invested $10,000 in this CD, how much money would you receive when your CD matures in seven months? Solution: Effective annual interest rate = (1 + [0.0706/4])4 – 1 = 0.07249 (approximately 7.25%) Therefore, we have quarterly compounding. And, investing $10,000 at 7.06 percent compounded quarterly for seven months (Note: Seven months equals 2 and 1⁄3 quarter periods), we get: $10,000 x (1 + [0.0706/4])2.33 = $10,000 x (1.041669) = $10,416.69 Exercise 9 (PV) Lost Dutchman Mines, Inc., is considering investing in Peru. It makes a bid to the government to participate in the development of a mine, the profits of which will be realized at the end of five years. The mine is expected to produce $5 million in cash to Lost Dutchman Mines at that time. Other than the bid at the outset, no other cash flows will occur, as the government will reimburse the company for all costs. If Lost Dutchman requires a nominal annual return of 20 percent (ignoring any tax consequences), what is the maximum bid it should make for the participation right if interest is compounded (a) annually? (b) semiannually? (c) quarterly? (d) continuously? Solution: (a) PV = $5,000,000/2.488 = $2,009,646 (b) PV = $5,000,000/2.594 = $1,927,525 (c) PV = $5,000,000/2.653 = $1,884,659 (d) PV = $5,000,000/2.71828 = $1,839,398 Exercise 10 (Uneven stream of CFs) The following cash-flow streams need to be analyzed: Cash Flow Stream W X Y Z 1 $100 600 200 2 $200 - End of Year 3 $200 500 4 $300 - 5 $300 1,200 300 a. Calculate the future (terminal) value of each stream at the end of year 5 with a compound annual interest rate of 10 percent. b. Compute the present value of each stream if the discount rate is 14 percent. Solution: a. r = 10% Cash Flow Stream W X Y Z FV5 for Individual Cash Flows received at End of Year 1 2 3 4 5 $146.40 $266.20 $242.00 $330.00 $300 878.40 1,200 292.80 605.00 300 Total Future Value $1,284.60 878.40 1,200.00 1,197.80 PV0 for Individual Cash Flows received at End of Year 1 2 3 4 5 $87.70 $153.80 $135.00 $177.60 $155.70 526.20 622.80 175.40 337.50 155.70 Total Future Value $709.80 526.20 622.80 668.60 b. r = 14% Cash Flow Stream W X Y Z Exercise 11 (Annuity) You have been offered a note with four years to maturity, which will pay $3,000 at the end of each of the four years. The price of the note to you is $10,200. What is the implicit compound annual interest rate you will receive (to the nearest whole percent)? Solution: Implicit rate = 6.83% (using financial calculator/excel) Exercise 12 (Annuity) The Happy Hang Glide Company is purchasing a building and has obtained a $190,000 mortgage loan for 20 years. The loan bears a compound annual interest rate of 17 percent and calls for equal annual installment payments at the end of each of the 20 years. What is the amount of the annual payment? Solution: A = $190,000/5.628 = $33,760 Exercise 13 (Annuity) You have borrowed $14,300 at a compound annual interest rate of 15 percent. You feel that you will be able to make annual payments of $3,000 per year on your loan. (Payments include both principal and interest). How long will it be before the loan is entirely paid off (to the nearest year)? Solution: [1 – 1/(1 + 15%)n]/15% = $14,300/$3,000 = 4.767 => n = 9 Exercise 14 (Uneven CFs) The H & L Bark Company is considering the purchase of a debarking machine that is expected to provide cash flows as follows: End of Year 1 2 3 4 5 6 7 8 9 10 Cash Flow 1,200 2,000 2,400 1,900 1,600 1,400 1,400 1,400 1,400 1,400 If the appropriate annual discount rate is 14 percent, what is the present value of this cash-flow stream? Solution: Cash Flow PV0 for Individual Cash Flows received at End of Year 1 2 3 4 5 6 - 10 $1,052.40 1,538.00 1,620.00 1,124.80 830.40 $2,496.20 Total $8,661.80 Exercise 15 (Annuity) Emerson Cammack wishes to purchase an annuity contract that will pay him $7,000 a year for the rest of his life. The Philo Life Insurance Company figures that his life expectancy is 20 years, based on its actuary tables. The company imputes a compound annual interest rate of 6 percent in its annuity contracts. a. How much will Cammack have to pay for the annuity? b. How much would he have to pay if the interest rate were 8 percent? Solution: a. PV0 = $7,000 x (11.470) = $80,290 b. PV0 = $7,000 x (9.818) = $68,726 Exercise 16 (Annuity) On a contract you have a choice of receiving $25,000 six years from now or $50,000 twelve years from now. At what implied compound annual interest rate should you be indifferent between the two contracts? Solution: Indifference implies that you could reinvest the $25,000 receipt for 6 years at X% to provide an equivalent $50,000 cash flow in year 12. In short, $25,000 would double in 6 years. Using the “Rule of 72,” 72/6 = 12%. Alternatively, note that $50,000 = $25,000 x (1+X%)6. Therefore, (1+X%)6 = $50,000/$25,000 = 2. => (1 + X%) = 21/6 = 20.1667 = 1.1225 => X% = 12.25% Exercise 16 (Annuity) Joe Hernandez has inherited $25,000 and wishes to purchase an annuity that will provide him with a steady income over the next 12 years. He has heard that the local savings and loan association is currently paying 6 percent compound interest on an annual basis. If he were to deposit his funds, what year-end equal-dollar amount (to the nearest dollar) would he be able to withdraw annually such that he would have a zero balance after his last withdrawal 12 years from now? Solution: A = $25,000/8.384 = $2,982 Exercise 17 (Annuity) You need to have $50,000 at the end of 10 years. To accumulate this sum, you have decided to save a certain amount at the end of each of the next 10 years and deposit it in the bank. The bank pays 8 percent interest compounded annually for long-term deposits. How much will you have to save each year (to the nearest dollar)? If you deposit a certain amount at the beginning of each of the next 10 years, how much will you have to save each year (to the nearest dollar)? Solution: A = $50,000/14.486 = $3,452; AD = $50,000/15.645 = $3,196 Exercise 18