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
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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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