Ross5eChap05sm

# Ross5eChap05sm - Chapter 5 The Time Value of Money 5.1 The...

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Chapter 5: The Time Value of Money 5.1 The simple interest per year is: \$5,000 × 0.07 = \$350 So, after 10 years, you will have: \$350 × 10 = \$3,500 in interest. The total balance will be \$5,000 + 3,500 = \$8,500 With compound interest, we use the future value formula: FV = PV(1 +r) t FV = \$5,000(1.07) 10 = \$9,835.76 The difference is: \$9,835.76 – 8,500 = \$1,335.76 5.2 To find the FV of a lump sum, we use: FV = PV(1 + r) t a. FV = \$1,000(1.05) 10 = \$1,628.89 b. FV = \$1,000(1.07) 10 = \$1,967.15 c. FV = \$1,000(1.05) 20 = \$2,653.30 d. Because interest compounds on the interest already earned, the future value in part c is more than twice the future value in part a. With compound interest, future values grow exponentially. 5.3 To find the PV of a lump sum, we use: PV = FV / (1 + r) t PV = \$15,451 / (1.05) 6 = \$11,529.77 PV = \$51,557 / (1.11) 9 = \$20,154.91 PV = \$886,073 / (1.16) 18 = \$61,266.87 PV = \$550,164 / (1.19) 23 = \$10,067.28 5.4 To find the future value with continuous compounding, we use the equation: FV = PV e rt a. FV = \$1,000 e .12(5) = \$1,822.12 b. FV = \$1,000 e .10(3) = \$1,349.86 c. FV = \$1,000 e .05(10) = \$1,648.72 d. FV = \$1,000 e .07(8) = \$1,750.67 Answers to End–of–Chapter Problems B–21

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5.5 For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)] m – 1 EAR = [1 + (.11 / 4)] 4 – 1 = .1146 or 11.46% EAR = [1 + (.07 / 12)] 12 – 1 = .0723 or 7.23% EAR = [1 + (.09 / 365)] 365 – 1 = .0942 or 9.42% To find the EAR with continuous compounding, we use the equation: EAR = e q – 1 EAR = e 0.17 – 1 = 0.1853 or 18.53% 5.6. Here, we are given the EAR and need to find the APR. Using the equation for discrete compounding: EAR = [1 + (APR / m )] m – 1 We can now solve for the APR. Doing so, we get: APR = m[(1 + EAR) 1/m – 1] EAR = 0.081 = [1 + (APR / 2)] 2 – 1 APR = 2[(1.081) 1/2 – 1] = 0.0794 or 7.94% EAR = 0.076 = [1 + (APR / 12)] 12 – 1 APR = 12[(1.076) 1/12 – 1] = 0.0735 or 7.35% EAR = 0.168 = [1 + (APR / 52)] 52 – 1 APR = 52[(1.168) 1/52 – 1] = 0.1555 or 15.55% Solving the continuous compounding EAR equation: EAR = e q – 1 We get: APR = ln(1 + EAR) APR = ln(1 + 0.262) APR = 0.2327 or 23.27% 5.7. For discrete compounding, to find the EAR, we use the equation: EAR = [1 + (APR / m)] m – 1 So, for each bank, the EAR is: IntraCanada Bank: EAR = [1 + (0.122 / 12)] 12 – 1 = 0.1291 or 12.91% Bank Depot: EAR = [1 + (0.124 / 2)] 2 – 1 = 0.1278 or 12.78% Notice that the higher APR does not necessarily mean the higher EAR. The number of compounding periods within a year will also affect the EAR. Answers to End–of–Chapter Problems B–22
5.8 Here, we are trying to find the interest rate when we know the PV and FV. Using the FV equation: FV = PV(1 + r ) \$4 = \$3(1 + r ) r = 4/3 – 1 = 0.3333 or 33.33% per week The interest rate is 33.33% per week. To find the APR, we multiply this rate by the number of weeks in a year, so: APR = (52)33.33% = 1,733.33% And using the equation to find the EAR: EAR = [1 + (APR / m )] m – 1 EAR = [1 + 0.3333] 52 – 1 = 3,135,086.84 or 313,508,684.1% Friendly’s operations are not legal since its rate is extremely high, compared to the legal charge of 60% per annum. 5.9 The cost of a case of wine is 10 percent less than the cost of 12 individual bottles, so the cost of a case will be: Cost of case = (12)(\$50)(1 – 0.10) Cost of case = \$540 Now, we need to find the interest rate. The cash flows are an annuity due, so: PV = (1 + r) C({1 – [1/(1 + r) t ]} / r) \$540 = (1 + r) \$50 ({1 – [1 / (1 + r) 12 ] / r ) Solving for the interest rate, we get: r = 0.0166 or 1.66% per week So, the APR of this investment is: APR = 0.0166(52) APR = 0.8632 or 86.32% And the EAR is: EAR = (1 +0 .0166) 52 – 1 EAR = 1.3539 or 135.39% The analysis appears to be correct. He really can earn about 135.39percent buying wine by the case.

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Ross5eChap05sm - Chapter 5 The Time Value of Money 5.1 The...

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