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Unformatted text preview: 207.14 Because interest compounds on the interest already earned, the interest earned in part c is more than twice the interest earned in part a. With compound interest, future values grow exponentially. 3. To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV = $15,451 / (1.07)6 PV = $51,557 / (1.15)9 PV = $886,073 / (1.11)18 PV = $550,164 / (1.18)23 = $10,295.65 = $14,655.72 = $135,411.60 = $12,223.79 51 4. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)1 / t – 1 FV = $307 = $242(1 + r)2; FV = $896 = $410(1 + r)9; FV = $162,181 = $51,700(1 + r)15; FV = $483,500 = $18,750(1 + r)30; r = ($307 / $242)1/2 – 1 r = ($896 / $410)1/9 – 1 r = ($162,181 / $51,700)1/15 – 1 r = ($483,500 / $18,750)1/30 – 1 = 12.63% = 9.07% = 7.92% = 11.44% 5. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for t, we get: t = ln(FV / PV) / ln(1 + r) FV = $1,284 = $625(1.06)t; FV = $4,341 = $810(1.13)t; FV = $402,662 = $18,400(1.32)t; FV = $173,439 = $21,500(1.16)t; t = ln($1,284/ $625) / ln 1.06 = 12.36 years t = ln($4,341/ $810) / ln 1.13 = 13.74 years t = ln($402,662 / $18,400) / ln 1.32 = 11.11 years t = ln($173,439 / $21,500) / ln 1.16 = 14.07 years 6. To find the length of time for money to double, triple, etc., the present value and future value are irrelevant as long as the future value is twice the present value for doubling, three times as large for tripling, etc. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for t, we get: t = ln(FV / PV) / ln(1 + r) The length of time to double your money is: FV = $2 = $1(1.09)t t = ln 2 / ln 1.09 = 8.04 years The length of time to quadruple your money is: FV = $4 = $1(1.09)t t = ln 4 / ln 1.09 = 16.09 years 52 Notice that the length of time to quadruple your money is twice as long as the time needed to double your money (the difference in these answers is due to rounding). This is an important concept of time value of money. 7. To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV = $750,000,000 / (1.082)20 = $155,065,808.54 8. To answer this question, we can use either the FV or the PV formula. Both will give the same answer since they are the inverse of each other. We will use the FV formula, that is: FV = PV(1 + r)t Solving for r, we get: r = (FV / PV)1 / t – 1 r = ($10,311,500 / $12,377,500)1/4 – 1 = – 4.46% Notice that the interest rate is negative. This occurs when the FV is less than the PV. 9. A consol is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C / r PV = $120 / .057 PV = $2,105.26 10. To find the future value with continuous compounding, we use the equation: FV = PVeRt a. b. c. d. FV = $1,900e.12(5) FV = $1,900e.10(3) FV = $1,900e.05(10) FV = $1,900e.07(8) = $3,462.03 = $2,564.73 = $3,132.57 = $3,326.28 11. To solve this problem, we must find the PV of each cash flow and add them. To find the PV of a lump sum, we use: PV = FV / (1 + r)t PV@10% = $1,200 / 1.10 + $730 / 1.102 + $965 / 1.103 + $1,590 / 1.104 = $3,505.23 PV@18% = $1,200 / 1.18 + $730 / 1.182 + $965 / 1.183 + $1,590 / 1.184 = $2,948.66 PV@24% = $1,200 / 1.24 + $730 / 1.242 + $965 / 1.243 + $1,590 / 1.244 = $2,621.17 53 12. To find the PVA, we use the equation: PVA = C({1 – [1/(1 + r)]t } / r ) At a 5 percent interest rate: X@5%: PVA = $5,500{[1 – (1/1.05)9 ] / .05 } = $39,093.02 Y@5%: PVA = $8,000{[1 – (1/1.05)5 ] / .05 } = $34,635.81 And at a 22 percent interest rate: X@22%: PVA = $5,500{[1 – (1/1.22)9 ] / .22 } = $20,824.57 Y@22%: PVA = $8,000{[1 – (1/1.22)5 ] / .22 } = $22,909.12 Notice that the PV of Cash flow X has a greater PV at a 5 percent interest rate, but a lower PV at a 22 percent interest rate. The reason is that X has greater total cash flows. At a lower interest rate, the total cash flow is more important since the cost of waiting (the interest rate) is not as great. At a higher interest rate, Y is more valuable since it has larger cash flows. At a higher interest rate, these bigger cash flows early are more important since the cost of waiting (the interest rate) is so much greater. 13. To find the PVA, we use the equation: PVA = C({1 – [1/(1 + r)]t } / r ) PVA@15 yrs: PVA@40 yrs: PVA@75 yrs: PVA = $4,300{[1 – (1/1.09)15 ] / .09} = $34,660.96 PVA = $4,300{[1 – (1/1.09)40 ] / .09} = $46,256.65 PVA = $4,300{[1 – (1/1.09)75 ] / .09} = $47,703.26 To find the PV of a perpetuity, we use the equation: PV = C / r PV = $4,300 / .09 PV = $47,777.78 Notice that as the length of the annuity payments increases, the present value of the annuity approaches the present value of the perpetuity. The...
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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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