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Unformatted text preview: ordinary due will always higher than the future value of an ordinary annuity. Since each cash flow is made one period sooner, each cash flow receives one extra period of compounding. 66 50. We need to use the PVA due equation, that is: PVAdue = (1 + r) PVA Using this equation: PVAdue = $65,000 = [1 + (.0645/12)] × C[{1 – 1 / [1 + (.0645/12)]48} / (.0645/12) C = $1,531.74 Notice, to find the payment for the PVA due we simply compound the payment for an ordinary annuity forward one period. Challenge 51. The monthly interest rate is the annual interest rate divided by 12, or: Monthly interest rate = .104 / 12 Monthly interest rate = .00867 Now we can set the present value of the lease payments equal to the cost of the equipment, or $3,500. The lease payments are in the form of an annuity due, so: PVAdue = (1 + r) C({1 – [1/(1 + r)]t } / r ) $3,500 = (1 + .00867) C({1 – [1/(1 + .00867)]24 } / .00867 ) C = $160.76 52. First, we will calculate the present value of the college expenses for each child. The expenses are an annuity, so the present value of the college expenses is: PVA = C({1 – [1/(1 + r)]t } / r ) PVA = $35,000({1 – [1/(1 + .085)]4 } / .085) PVA = $114,645.88 This is the cost of each child’s college expenses one year before they enter college. So, the cost of the oldest child’s college expenses today will be: PV = FV/(1 + r)t PV = $114,645.88/(1 + .085)14 PV = $36,588.29 And the cost of the youngest child’s college expenses today will be: PV = FV/(1 + r)t PV = $114,645.88/(1 + .085)16 PV = $31,080.12 Therefore, the total cost today of your children’s college expenses is: Cost today = $36,588.29 + 31,080.12 Cost today = $67,668.41 67 This is the present value of your annual savings, which are an annuity. So, the amount you must save each year will be: PVA = C({1 – [1/(1 + r)]t } / r ) $67,668.41 = C({1 – [1/(1 + .085)]15 } / .085) C = $8,148.66 53. The salary is a growing annuity, so using the equation for the present value of a growing annuity. The salary growth rate is 3.5 percent and the discount rate is 12 percent, so the value of the salary offer today is: PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} PV = $45,000{[1/(.12 – .035)] – [1/(.12 – .035)] × [(1 + .035)/(1 + .12)]25} PV = $455,816.18 The yearly bonuses are 10 percent of the annual salary. This means that next year’s bonus will be: Next year’s bonus = .10($45,000) Next year’s bonus = $4,500 Since the salary grows at 3.5 percent, the bonus will grow at 3.5 percent as well. Using the growing annuity equation, with a 3.5 percent growth rate and a 12 percent discount rate, the present value of the annual bonuses is: PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} PV = $4,500{[1/(.12 – .035)] – [1/(.12 – .035)] × [(1 + .035)/(1 + .12)]25} PV = $45,581.62 Notice the present value of the bonus is 10 percent of the present value of the salary. The present value of the bonus will always be the same percentage of the present value of the salary as the bonus percentage. So, the total value of the offer is: PV = PV(Salary) + PV(Bonus) + Bonus paid today PV = $455,816.18 + 45,581.62 + 10,000 PV = $511,397.80 54. Here, we need to compare to options. In order to do so, we must get the value of the two cash flow streams to the same time, so we will find the value of each today. We must also make sure to use the aftertax cash flows, since it is more relevant. For Option A, the aftertax cash flows are: Aftertax cash flows = Pretax cash flows(1 – tax rate) Aftertax cash flows = $175,000(1 – .28) Aftertax cash flows = $126,000 The aftertax cash flows from Option A are in the form of an annuity due, so the present value of the cash flow today is: PVAdue = (1 + r) C({1 – [1/(1 + r)]t } / r ) PVAdue = (1 + .10)$126,000({1 – [1/(1 + .10)]31 } / .10 ) PVAdue = $1,313,791.22 68 For Option B, the aftertax cash flows are: Aftertax cash flows = Pretax cash flows(1 – tax rate) Aftertax cash flows = $125,000(1 – .28) Aftertax cash flows = $90,000 The aftertax cash flows from Option B are an ordinary annuity, plus the cash flow today, so the present value: PV = C({1 – [1/(1 + r)]t } / r ) + CF0 PV = $90,000{1 – [1/(1 + .10)]30 } / .10 ) + $530,000 PV = $1,378,422.30 You should choose Option B because it has a higher present value on an aftertax basis. 55. We need to find the first payment into the retirement account. The present value of the desired amount at retirement is: PV = FV/(1 + r)t PV = $1,500,000/(1 + .10)30 PV = $85,962.83 This is the value today. Since the savings are in the form of a growing annuity, we can use the growing annuity equation and solve for the payment. Doing so, we get: PV = C {[1/(r – g)] – [1/(r – g)] × [(1 + g)/(1 + r)]t} $85,962.83 = C{[1/(.10 – .03)] – [1/(.10 – .03)] × [(1 + .03)/(1 + .10)]30} C = $6,989.68 This is the amount you need to save next year. So, the percentage of your salary is: Percentage of salary = $6,989.68/$70,000 Percentage of salary = .0999 or 9.99% Note that this is the percentage of...
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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 TexasTyler.
 Spring '10
 eshmalwi
 Finance, Corporate Finance

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