Corporate_Finance_9th_edition_Solutions_Manual_FINAL0

Since we already know the value needed at retirement

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Unformatted text preview: (.068/12)]12 – 1 = .0702 or 7.02% 64. Be careful of interest rate quotations. The actual interest rate of a loan is determined by the cash flows. Here, we are told that the PV of the loan is $1,000, and the payments are $43.36 per month for three years, so the interest rate on the loan is: PVA = $1,000 = $43.36[ {1 – [1 / (1 + r)]36 } / r ] Solving for r with a spreadsheet, on a financial calculator, or by trial and error, gives: r = 2.64% per month APR = 12(2.64%) = 31.65% EAR = (1 + .0264)12 – 1 = .3667 or 36.67% It’s called add-on interest because the interest amount of the loan is added to the principal amount of the loan before the loan payments are calculated. 74 65. Here, we are solving a two-step time value of money problem. Each question asks for a different possible cash flow to fund the same retirement plan. Each savings possibility has the same FV, that is, the PV of the retirement spending when your friend is ready to retire. The amount needed when your friend is ready to retire is: PVA = $110,000{[1 – (1/1.09)25] / .09} = $1,080,483.76 This amount is the same for all three parts of this question. a. If your friend makes equal annual deposits into the account, this is an annuity with the FVA equal to the amount needed in retirement. The required savings each year will be: FVA = $1,080,483.76 = C[(1.0930 – 1) / .09] C = $7,926.81 b. Here we need to find a lump sum savings amount. Using the FV for a lump sum equation, we get: FV = $1,080,483.76 = PV(1.09)30 PV = $81,437.29 c. In this problem, we have a lump sum savings in addition to an annual deposit. Since we already know the value needed at retirement, we can subtract the value of the lump sum savings at retirement to find out how much your friend is short. Doing so gives us: FV of trust fund deposit = $50,000(1.09)10 = $118,368.18 So, the amount your friend still needs at retirement is: FV = $1,080,483.76 – 118,368.18 = $962,115.58 Using the FVA equation, and solving for the payment, we get: $962,115.58 = C[(1.09 30 – 1) / .09] C = $7,058.42 This is the total annual contribution, but your friend’s employer will contribute $1,500 per year, so your friend must contribute: Friend's contribution = $7,058.42 – 1,500 = $5,558.42 75 66. We will calculate the number of periods necessary to repay the balance with no fee first. We simply need to use the PVA equation and solve for the number of payments. Without fee and annual rate = 18.6%: PVA = $9,000 = $200{[1 – (1/1.0155)t ] / .0155 } where .0155 = .186/12 Solving for t, we get: t = ln{1 / [1 – ($9,000/$200)(.0155)]} / ln(1.0155) t = ln 3.3058 / ln 1.0155 t = 77.74 months Without fee and annual rate = 8.2%: PVA = $9,000 = $200{[1 – (1/1.006833)t ] / .006833 } where .006833 = .082/12 Solving for t, we get: t = ln{1 / [1 – ($9,000/$200)(.006833)]} / ln(1.006833) t = ln 1.4440 / ln 1.006833 t = 53.96 months Note that we do not need to calculate the time necessary to repay your current credit card with a fee since no fee will be incurred. The time to repay the new card with a transfer fee is: With fee and annual rate = 8.20%: PVA = $9,180 = $200{ [1 – (1/1.006833)t ] / .006833 } where .006833 = .092/12 Solving for t, we get: t = ln{1 / [1 – ($9,180/$200)(.006833)]} / ln(1.006833) t = ln 1.45698 / ln 1.006833 t = 55.27 months 67. We need to find the FV of the premiums to compare with the cash payment promised at age 65. We have to find the value of the premiums at year 6 first since the interest rate changes at that time. So: FV1 = $800(1.11)5 = $1,348.05 FV2 = $800(1.11)4 = $1,214.46 FV3 = $900(1.11)3 = $1,230.87 FV4 = $900(1.11)2 = $1,108.89 FV5 = $1,000(1.11)1 = $1,110.00 76 Value at year six = $1,348.05 + 1,214.46 + 1,230.87 + 1,108.89 + 1,110.00 + 1,000.00 = $7,012.26 Finding the FV of this lump sum at the child’s 65th birthday: FV = $7,012.26(1.07)59 = $379,752.76 The policy is not worth buying; the future value of the policy is $379,752.76, but the policy contract will pay off $350,000. The premiums are worth $29,752.76 more than the policy payoff. Note, we could also compare the PV of the two cash flows. The PV of the premiums is: PV = $800/1.11 + $800/1.112 + $900/1.113 + $900/1.114 + $1,000/1.115 + $1,000/1.116 = $3,749.04 And the value today of the $350,000 at age 65 is: PV = $350,000/1.0759 = $6,462.87 PV = $6,462.87/1.116 = $3,455.31 The premiums still have the higher cash flow. At time zero, the difference is $293.73. Whenever you are comparing two or more cash flow streams, the cash flow with the highest value at one time will have the highest value at any other time. Here is a question for you: Suppose you invest $293.73, the difference in the cash flows at time zero, for six years at an 11 percent interest rate, and then for 59 years at a seven percent interest rate. How much will it be worth? Without doing calculations, you know it will be worth $29,752.76, the difference in the cash flows at time 65! 68. Since the payments occur at six month intervals, we need to get the effective six-month interest rate. We can calculate...
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This note was uploaded on 07/10/2010 for the course FIN 6301 taught by Professor Eshmalwi during the Spring '10 term at University of Texas-Tyler.

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