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Chapter 4 Solutions

Chapter 4 Solutions - Chapter 4 Solutions 2 To find the FV...

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Chapter 4 Solutions 2. To find the FV of a lump sum, we use: FV = PV(1 + r ) t a. FV = $1,000(1.06) 10 = $1,790.85 b. FV = $1,000(1.09) 10 = $2,367.36 c. FV = $1,000(1.06) 20 = $3,207.14 d. 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 = $10,295.65 PV = $51,557 / (1.15) 9 = $14,655.72 PV = $886,073 / (1.11) 18 = $135,411.60 PV = $550,164 / (1.18) 23 = $12,223.79 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 ; r = ($307 / $242) 1/2 – 1 = 12.63% FV = $896 = $410(1 + r ) 9 ; r = ($896 / $410) 1/9 – 1 = 9.07% FV = $162,181 = $51,700(1 + r ) 15 ; r = ($162,181 / $51,700) 1/15 – 1 = 7.92% FV = $483,500 = $18,750(1 + r ) 30 ; r = ($483,500 / $18,750) 1/30 – 1 = 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 ; t = ln($1,284/ $625) / ln 1.06 = 12.36 years FV = $4,341 = $810(1.13) t ; t = ln($4,341/ $810) / ln 1.13 = 13.74 years FV = $402,662 = $18,400(1.32) t ; t = ln($402,662 / $18,400) / ln 1.32 = 11.11 years FV = $173,439 = $21,500(1.16) t ; t = ln($173,439 / $21,500) / ln 1.16 = 14.07 years 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 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 [email protected]% = $1,200 / 1.10 + $730 / 1.10 2 + $965 / 1.10 3 + $1,590 / 1.10 4 = $3,505.23 [email protected]% = $1,200 / 1.18 + $730 / 1.18 2 + $965 / 1.18 3 + $1,590 / 1.18 4 = $2,948.66 [email protected]% = $1,200 / 1.24 + $730 / 1.24 2 + $965 / 1.24 3 + $1,590 / 1.24 4 = $2,621.17 13. To find the PVA, we use the equation:
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PVA = C ({1 – [1/(1 + r) ] t } / r ) [email protected] yrs: PVA = $4,300{[1 – (1/1.09) 15 ] / .09} = $34,660.96 [email protected] yrs: PVA = $4,300{[1 – (1/1.09) 40 ] / .09} = $46,256.65 [email protected] yrs: 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 present value of the 75-year annuity and the present value of the perpetuity imply that the value today of all perpetuity payments beyond 75 years is only $74.51. 14. This cash flow is a perpetuity. To find the PV of a perpetuity, we use the equation: PV = C / r PV = $20,000 / .065 = $307,692.31 To find the interest rate that equates the perpetuity cash flows with the PV of the cash flows.
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