ARE143_CHAP_6b_Time_Value_KEY

# ARE143_CHAP_6b_Time_Value_KEY - 1 Managerial Economics(ARE...

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1 Managerial Economics (ARE) 143 University of California, Davis Instructor: John H. Constantine KEY—Chapter 6—Time Value of Money Problem 1: (a) One Year Future Value What is the future value of \$400 deposited for one year earning an interest rate of 11 percent per year? FV N = PV × (1 + i ) N FV 1 = 400 × (1 + 0.11) 1 = 400 × 1.11 = 444 (b) Multi-Year Future Value How much would be in your savings account in 8 years after depositing \$150 today if the bank pays 7 percent per year? FV N = PV × (1 + i ) N FV 8 = 150 × (1 + 0.07) 8 = 150 × 1.71819 = 257.73 (c) A deposit of \$350 earns the following interest rates: 8 percent in the first year, 7 percent in the second year, and 5 percent in the third year. What would be the third year future value? FV = PV × (1 + i ) (1 + j ) (1 + k ) FV = 350 × (1 + 0.08) (1 + 0.07) (1 + 0.05) = 350 × 1.08 × 1.07 × 1.05 = 424.68

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2 Problem 2: (a) What is the present value of a \$400 payment in one year when the discount rate is 7 percent? PV = F V /(1+ i ) PV = 400/(1+0.07) = 400/1.07 = 373.83 (b) What is the present value of a \$1,500 payment made in 5 years when the discount rate is 8 percent? PV = F V /(1+ i ) N PV = 1500/(1+0.08) 5 = 1500/1.46933 = 1020.87 (c) Compute the present value of a \$850 payment made in 10 years when the discount rate is 12 percent. PV = F V /(1+ i ) N PV = 850/(1+0.12) 10 = 850/3.10585 = 273.68 (d) Compute the present value of \$1,000 paid in three years using the following discount rates; 6 percent in the first year, 7 percent in the second year, and 6 percent in the third year. PV = F V/ [(1 + i ) (1 + j ) (1 + k )] PV = 1000 / [(1 + 0.06) (1 +0.07) (1 + 0.06)] = 1000/[1.06 × 1.07 × 1.06] = 1000/1.2023 = 831.77 (e) What would be more valuable, receiving \$500 today or receiving \$625 in three years when interest rates are 9 percent? Why? PV = F V /(1+ i ) N PV = 625/(1+0.09) 3 = 625/1.295029 = 482.61 The present value of \$625 to be paid in three years at 9% interest is \$482.61. This amount is worth less than \$500 received today. Therefore the \$500 payment made today is more valuable. Problem 3: How long will it take \$2,000 to reach \$6,000 when it grows at 10 percent per year? FV N = PV × (1 + i ) N 6,000 = 2,000 × (1 + 0.10) N (1.10) N = 3 ln (1.10) N = ln 3 N = 11 years, 6.3 months Problem 4: You invested \$3,000 in the stock market one year ago. Today, the investment is valued at \$3,500. What return did you earn? What return would you suffer next year for your investment to be valued at the original \$3,000? FV N = PV × (1 + i ) N 3,500 = 3,000 × (1 + i ) 1 (1 + i ) = 3,500/3,000 i = 0.1667 or 16.67% (first year return is positive) 3,000 = 3,500 × (1 + i ) 1 (1 + i ) = 3,000/3,500 i = -0.143 or 14.3% (second year return is negative)
3 Problem 5: People can become millionaires in their retirement years quite easily if they start saving early in employer 401(k) or 403(b) programs (or even if their employers don’t offer such programs). Demonstrate the growth of a \$250 monthly contribution for 40 years earning 9 percent APR.  480 40 10 . 0 9 / 1 2 1 FVA 250 \$1,170,330.07 0.09/12   Problem 6: (a) Compute the future value in year 8 of a \$1,000 deposit in year 1 and another \$1,500 deposit at the end of year 3 using a 10% interest rate. FV = \$1,000 × (1 + 0.10) 7 + \$1,500× (1 + 0.10) 5 = \$1,948.72 + \$2,415.77 = \$4,364.48 (b) Compute the future value in year 7 of a \$2,000 deposit in year 1 and another \$2,500 deposit at the end of year 4 using a 8% interest rate. .

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ARE143_CHAP_6b_Time_Value_KEY - 1 Managerial Economics(ARE...

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