Solutions to Questions and Problems 10. To find the future value with continuous compounding, we use the equation: FV = PVeRt a. b. c. d. FV = $1,000e.12(5) FV = $1,000e.10(3) FV = $1,000e.05(10) FV = $1,000e.07(8) = $1,822.12 = $1,349.86 = $1,648.72 = $1,750.67

23. We need to find the annuity payment in retirement. Our retirement savings ends at the same time the retirement withdrawals begin, so the PV of the retirement withdrawals will be the FV of the retirement savings. So, we find the FV of the stock account and the FV of the bond account and add the two FVs. (Monthly installment and so monthly compounding: 30 years time 12 months equal to 360months) Stock account: FVA = Rs.700[{[1 + (.11/12) ]360 – 1} / (.11/12)] = Rs.1,963,163.82 Bond account: FVA = Rs.300[{[1 + (.07/12) ]360 – 1} / (.07/12)] = Rs.365,991.30 So, the total amount saved at retirement is: Rs.1,963,163.82 + 365,991.30 = Rs.2,329,155.11 Solving for the withdrawal amount in retirement using the PVA equation gives us: PVA = Rs.2,329,155.11 = C[1 – {1 / [1 + (.09/12)]300} / (.09/12)] C = Rs.2,329,155.11 / 119.1616 = Rs.19,546.19 withdrawal per month 26. This is a growing perpetuity. The present value of a growing perpetuity is: PV = C / (r – g) PV = Rs.200,000 / (.10 – .05) PV = Rs.4,000,000 39. We are given the total PV of all four cash flows. If we find the PV of the three cash flows we know, and subtract them from the total PV, the amount left over must be the PV of the missing cash flow. So, the PV of the cash flows we know are:

PV of Year 1 CF: PHP1,000 / 1.10 PV of Year 3 CF: PHP2,000 / 1.103 PV of Year 4 CF: PHP2,000 / 1.104 So, the PV of the missing CF is:

= PHP909.09 = PHP1,502.63 = PHP1,366.03

PHP5,979 – 909.09 – 1,502.63 – 1,366.03 = PHP2,201.25 The question asks for the value of the cash flow in Year 2, so we must find the future value of this amount. The value of the missing CF is: PHP2,201.25(1.10)2 = PHP2,663.52 Calculating the PV of an ordinary annuity, we get: PVA = $525{[1 – (1/1.095)6 ] / .095} = $2,320.41 b. To calculate the PVA due, we calculate the PV of an ordinary annuity for t – 1 payments, and add the payment that occurs today. So, the PV of the annuity due is: PVA = 525 + 525{[1 – (1/1.095)5] / .095} = 2,540.85 60. The cash flows of the loan are the £20,000 you must repay in one year, and the £17,600 you borrow today. The interest rate of the loan is: £20,000 = £17,600(1 + r) r = (£20,000 / 17,600) – 1 = 13.64% Because of the discount, you only get the use of £17,600, and the interest you pay on that amount is 13.64%, not 12%. 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 = $750(1.11)5 = $1,263.79 FV2 = $750(1.11)4 = $1,138.55 FV3 = $850(1.11)3 = $1,162.49

49. a.

FV4 = $850(1.11)2 = $1,047.29 FV5 = $950(1.11)1 = $1,054.50 Value at year six = $1,263.79 + 1,138.55 + 1,162.49 + 1,047.29 + 1,054.50 + 950.00 = $6,616.62 Finding the FV of this lump sum at the child’s 65th birthday: FV = $6,616.62(1.07)59 = $358,326.50 The policy is not worth buying; the future value of the policy is $358,326.50, but the policy contract will pay off $250,000. The premiums are worth $108,326.50 more than the policy payoff. Note, we could also compare the PV of the two cash flows. The PV of the premiums is: PV = $750/1.11 + $750/1.112 + $850/1.113 + $850/1.114 + $950/1.115 + $950/1.116 = $3,537.51 And the value today of the $250,000 at age 65 is: PV = $250,000/1.0759 = $4,616.33 PV = $4,616.33/1.116 = $2,468.08 The premiums still have the higher cash flow. At time zero, the difference is $2,148.25. 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 $2,148.25, the difference in the cash...