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Unformatted text preview: and then bring the lump sum PV back to today. The annuity has 17 payments, so the PV of the annuity is: PVA = $4,000{[1 – (1/1.07)17] / .07} = $39,052.89 Since this is an ordinary annuity equation, this is the PV one period before the first payment, so it is the PV at t = 8. To find the value today, we find the PV of this lump sum. The value today is: PV = $39,052.89 / 1.078 = $22,729.14 44. This question is asking for the present value of an annuity, but the interest rate changes during the life of the annuity. We need to find the present value of the cash flows for the last eight years first. The PV of these cash flows is: PVA2 = $1,500 [{1 – 1 / [1 + (.09/12)]96} / (.09/12)] = $102,387.66 Note that this is the PV of this annuity exactly seven years from today. Now, we can discount this lump sum to today. The value of this cash flow today is: PV = $102,387.66 / [1 + (.13/12)]84 = $41,415.70 Now, we need to find the PV of the annuity for the first seven years. The value of these cash flows today is: PVA1 = $1,500 [{1 – 1 / [1 + (.13/12)]84} / (.13/12)] = $82,453.99 The value of the cash flows today is the sum of these two cash flows, so: PV = $82,453.99 + 41,415.70 = $123,869.99 45. Here, we are trying to find the dollar amount invested today that will equal the FVA with a known interest rate, and payments. First, we need to determine how much we would have in the annuity account. Finding the FV of the annuity, we get: FVA = $1,200 [{[ 1 + (.098/12)]180 – 1} / (.098/12)] = $488,328.61 Now, we need to find the PV of a lump sum that will give us the same FV. So, using the FV of a lump sum with continuous compounding, we get: FV = $488,328.61 = PVe.09(15) PV = $488,328.61e–1.35 = $126,594.44 64 46. To find the value of the perpetuity at t = 7, we first need to use the PV of a perpetuity equation. Using this equation we find: PV = $2,100 / .073 = $28,767.12 Remember that the PV of a perpetuity (and annuity) equations give the PV one period before the first payment, so, this is the value of the perpetuity at t = 14. To find the value at t = 7, we find the PV of this lump sum as: PV = $28,767.12 / 1.0737 = $17,567.03 47. To find the APR and EAR, we need to use the actual cash flows of the loan. In other words, the interest rate quoted in the problem is only relevant to determine the total interest under the terms given. The interest rate for the cash flows of the loan is: PVA = $26,000 = $2,491.67{(1 – [1 / (1 + r)]12 ) / r } Again, we cannot solve this equation for r, so we need to solve this equation on a financial calculator, using a spreadsheet, or by trial and error. Using a spreadsheet, we find: r = 2.219% per month So the APR is: APR = 12(2.219%) = 26.62% And the EAR is: EAR = (1.02219)12 – 1 = 30.12% 48. The cash flows in this problem are semiannual, so we need the effective semiannual rate. The interest rate given is the APR, so the monthly interest rate is: Monthly rate = .12 / 12 = .01 To get the semiannual interest rate, we can use the EAR equation, but instead of using 12 months as the exponent, we will use 6 months. The effective semiannual rate is: Semiannual rate = (1.01)6 – 1 = 6.15% We can now use this rate to find the PV of the annuity. The PV of the annuity is: PVA @ t = 9: $4,500{[1 – (1 / 1.0615)10] / .0615} = $32,883.16 Note, that this is the value one period (six months) before the first payment, so it is the value at t = 9. So, the value at the various times the questions asked for uses this value 9 years from now. PV @ t = 5: $32,883.16 / 1.06158 = $20,396.12 65 Note, that you can also calculate this present value (as well as the remaining present values) using the number of years. To do this, you need the EAR. The EAR is: EAR = (1 + .01)12 – 1 = 12.68% So, we can find the PV at t = 5 using the following method as well: PV @ t = 5: $32,883.16 / 1.12684 = $20,396.12 The value of the annuity at the other times in the problem is: PV @ t = 3: $32,883.16 / 1.061512 = $16,063.29 PV @ t = 3: $32,883.16 / 1.12686 = $16,063.29 PV @ t = 0: $32,883.16 / 1.061518 = $11,227.04 PV @ t = 0: $32,883.16 / 1.12689 = $11,227.04 49. a. If the payments are in the form of an ordinary annuity, the present value will be: PVA = C({1 – [1/(1 + r)t]} / r )) PVA = $10,000[{1 – [1 / (1 + .11)]5}/ .11] PVA = $36,958.97 If the payments are an annuity due, the present value will be: PVAdue = (1 + r) PVA PVAdue = (1 + .11)$36,958.97 PVAdue = $41,024.46 b. We can find the future value of the ordinary annuity as: FVA = C{[(1 + r)t – 1] / r} FVA = $10,000{[(1 + .11)5 – 1] / .11} FVA = $62,278.01 If the payments are an annuity due, the future value will be: FVAdue = (1 + r) FVA FVAdue = (1 + .11)$62,278.01 FVAdue = $69,128.60 c. Assuming a positive interest rate, the present value of an annuity due will always be larger than the present value of an ordinary annuity. Each cash flow in an annuity due is received one period earlier, which means there is one period less to discount each cash flow. Assuming a positive interest rate, the future value of an...
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 Spring '10
 eshmalwi
 Finance, Corporate Finance

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