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Unformatted text preview: the daily interest rate since we have an APR compounded daily, so the effective sixmonth interest rate is: Effective sixmonth rate = (1 + Daily rate)180 – 1 Effective sixmonth rate = (1 + .09/360)180 – 1 Effective sixmonth rate = .0460 or 4.60% Now, we can use the PVA equation to find the present value of the semiannual payments. Doing so, we find: PVA = C({1 – [1/(1 + r)]t } / r ) PVA = $750,000({1 – [1/(1 + .0460]40 } / .0460) PVA = $13,602,152.32 This is the value six months from today, which is one period (six months) prior to the first payment. So, the value today is: PV = $13,602,152.32 / (1 + .0460) PV = $13,003,696.50 77 This means the total value of the lottery winnings today is: Value of winnings today = $13,003,696.50 + 2,000,000 Value of winnings today = $15,003,696.50 You should not take the offer since the value of the offer is less than the present value of the payments. 69. Here, we need to find the interest rate that makes the PVA, the college costs, equal to the FVA, the savings. The PV of the college costs are: PVA = $20,000[{1 – [1 / (1 + r)]4 } / r ] And the FV of the savings is: FVA = $8,000{[(1 + r)6 – 1 ] / r } Setting these two equations equal to each other, we get: $20,000[{1 – [1 / (1 + r)]4 } / r ] = $8,000{[ (1 + r)6 – 1 ] / r } Reducing the equation gives us: (1 + r)10 – 4.00(1 + r)4 + 40.00 = 0 Now, we need to find the roots of this equation. We can solve using trial and error, a rootsolving calculator routine, or a spreadsheet. Using a spreadsheet, we find: r = 10.57% 70. Here, we need to find the interest rate that makes us indifferent between an annuity and a perpetuity. To solve this problem, we need to find the PV of the two options and set them equal to each other. The PV of the perpetuity is: PV = $20,000 / r And the PV of the annuity is: PVA = $35,000[{1 – [1 / (1 + r)]10 } / r ] Setting them equal and solving for r, we get: $20,000 / r = $35,000[{1 – [1 / (1 + r)]10 } / r ] $20,000 / $35,000 = 1 – [1 / (1 + r)]10 .57141/10 = 1 / (1 + r) r = 1 / .57141/10 – 1 r = .0576 or 5.76% 78 71. The cash flows in this problem occur every two years, so we need to find the effective two year rate. One way to find the effective two year rate is to use an equation similar to the EAR, except use the number of days in two years as the exponent. (We use the number of days in two years since it is daily compounding; if monthly compounding was assumed, we would use the number of months in two years.) So, the effective twoyear interest rate is: Effective 2year rate = [1 + (.13/365)]365(2) – 1 = 29.69% We can use this interest rate to find the PV of the perpetuity. Doing so, we find: PV = $8,500 /.2969 = $28,632.06 79 This is an important point: Remember that the PV equation for a perpetuity (and an ordinary annuity) tells you the PV one period before the first cash flow. In this problem, since the cash flows are two years apart, we have found the value of the perpetuity one period (two years) before the first payment, which is one year ago. We need to compound this value for one year to find the value today. The value of the cash flows today is: PV = $28,632.06(1 + .13/365)365 = $32,606.24 The second part of the question assumes the perpetuity cash flows begin in four years. In this case, when we use the PV of a perpetuity equation, we find the value of the perpetuity two years from today. So, the value of these cash flows today is: PV = $28,632.06 / (1 + .13/365)2(365) = $22,077.81 72. To solve for the PVA due: C C C + + .... + 2 (1 + r ) (1 + r ) (1 + r ) t C C + .... + PVAdue = C + (1 + r ) (1 + r ) t  1 PVA = C C C PVAdue = (1 + r ) (1 + r ) + (1 + r ) 2 + .... + (1 + r ) t PVAdue = (1 + r) PVA And the FVA due is: FVA = C + C(1 + r) + C(1 + r)2 + …. + C(1 + r)t – 1 FVAdue = C(1 + r) + C(1 + r)2 + …. + C(1 + r)t FVAdue = (1 + r)[C + C(1 + r) + …. + C(1 + r)t – 1] FVAdue = (1 + r)FVA 73. a. The APR is the interest rate per week times 52 weeks in a year, so: APR = 52(9%) = 468% EAR = (1 + .09)52 – 1 = 87.3442 or 8,734.42% b. In a discount loan, the amount you receive is lowered by the discount, and you repay the full principal. With a 9 percent discount, you would receive $9.10 for every $10 in principal, so the weekly interest rate would be: $10 = $9.10(1 + r) r = ($10 / $9.10) – 1 = .0989 or 9.89% 80 Note the dollar amount we use is irrelevant. In other words, we could use $0.91 and $1, $91 and $100, or any other combination and we would get the same interest rate. Now we can find the APR and the EAR: APR = 52(9.89%) = 514.29% EAR = (1 + .0989)52 – 1 = 133.8490 or 13,384.90% c. Using the cash flows from the loan, we have the PVA and the annuity payments and need to find the interest rate, so: PVA = $58.84 = $25[{1 – [1 / (1 + r)]4}/ r ] Using a spreadsheet, trial and error, or a financial calculator, we find: r = 25.19% per week APR = 52(25.19%) = 1,309.92% EAR = 1.251952 – 1 = 118,515.0194 or 11,851,501.94% 74. To answer this, we can diagram the perpetuity cash flows, which are: (Note...
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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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