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Economics
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ECON 4400
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Lecture

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CHAPTER 6DISCOUNTED CASH FLOW VALUATIONLearning ObjectivesLO1How to determine the future and present value of investments with multiple cash flowsLO2How loan payments are calculated and how to find the interest rate on a loanLO3How loans are amortized or paid offLO4How interest rates are quoted and misquotedAnswers to Concepts Review and Critical Thinking Questions1LO1 The four pieces are the present value PV the periodic cash flow C the discount rate r and the number of payments or the life of the annuity t2LO1 Assuming positive cash flows both the present and the future values will rise3LO1 Assuming positive cash flows the present value will fall and the future value will rise4LO1 Its deceptive but very common The basic concept of time value of money is that a dollar today is not worth the same as a dollar tomorrow The deception is particularly irritating given that such lotteries are usually government sponsored5LO1 If the total money is fixed you want as much as possible as soon as possible The team or more accurately the team owner wants just the opposite6LO1 The better deal is the one with equal installments51Solutions to Questions and ProblemsNOTE All end of chapter problems were solved using a spreadsheet Many problems require multiple steps Due to space and readability constraints when these intermediate steps are included in this solutions manual rounding may appear to have occurred However the final answer for each problem is found without rounding during any step in the problemBasic1LO1 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 usetPVFV1r234PV10950110104011011301101075110330637234PV181950118104011811301181075118279422234PV24950124104012411301241075124248988 2LO1 To find the PVA we use the equationtPVAC111r r At a 5 percent interest rate9X5PVA600011105 05 42646936Y5 PVA80001110505 4060554And at a 15 percent interest rate9X22 PVA600011115 15 28629506Y22PVA800011115 15 3027586Notice 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 the beginning At the higher interest rate these bigger cash flows early are more important since the cost of waiting the interest rate is so much greater 3LO1 To solve this problem we must find the FV of each cash flow and add them To find the FV of a lump sum we usetFVPV1r32FV894010810901081340108140553077132FV1194011110901111340111140555209632FV24940124109012413401241405653481Notice we are finding the value at Year 4 the cash flow at Year 4 is simply added to the FV of the other cash flows In other words we do not need to compound this cash flow524LO1 To find the PVA we use the equationtPVAC111r r 15PVA15 yrs PVA530011107 07482719440PVA40 yrs PVA530011107 07706580675PVA75 yrs PVA530011107 077524070To find the PV of a perpetuity we use the equationPVCrPV5300077571429Notice 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 473585LO1 Here we have the PVA the length of the annuity and the interest rate We want to calculate the annuity payment Using the PVA equationtPVAC111r r 15PVA34000C1110765 0765We can now solve this equation for the annuity payment Doing so we getC3400087454816523887726LO1 To find the PVA we use the equationtPVAC111r r 8PVA73000111085 085411660367LO1 Here we need to find the FVA The equation to find the FVA istFVAC1r1r20FVA for 20 years4000111211122627811640FVA for 40 years400011121112245907263Notice that because of exponential growth doubling the number of periods does not merely double the FVA8LO1 Here we have the FVA the length of the annuity and the interest rate We want to calculate the annuity payment Using the FVA equationtFVAC1r1r1090000C10681068We can now solve this equation for the annuity payment Doing so we getC90000136866163265757753

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