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**preview**shows pages 1-3. to view the full**45 pages of the document.**CHAPTER 6

DISCOUNTED CASH FLOW VALUATION

Learning Objectives

LO1 How to determine the future and present value of investments with multiple cash flows.

LO2 How loan payments are calculated and how to find the interest rate on a loan.

LO3 How loans are amortized or paid off.

LO4 How interest rates are quoted (and misquoted).

Answers to Concepts Review and Critical Thinking Questions

1. (LO1) 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, t.

2. (LO1) Assuming positive cash flows, both the present and the future values will rise.

3. (LO1) Assuming positive cash flows, the present value will fall and the future value will rise.

4. (LO1) It’s 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 sponsored!

5. (LO1) 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 opposite.

6. (LO1) The better deal is the one with equal installments.

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Solutions to Questions and Problems

NOTE: 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 problem.

Basic

1. (LO1) 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% = $1,100 / 1.10 + $720 / 1.102 + $940 / 1.103 + $1,160 / 1.104 = $3,093.57

PV@18% = $1,100 / 1.18 + $720 / 1.182 + $940 / 1.183 + $1,160 / 1.184 = $2,619.72

PV@24% = $1,100 / 1.24 + $720 / 1.242 + $940 / 1.243 + $1,160 / 1.244 = $2,339.03

2. (LO1) 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 = $7,000{[1 – (1/1.05)8 ] / .05 } = $45,242.49

Y@5%: PVA = $9,000{[1 – (1/1.05)5 ] / .05 } = $38,965.29

And at a 22 percent interest rate:

X@22%: PVA = $7,000{[1 – (1/1.22)8 ] / .22 } = $25,334.87

Y@22%: PVA = $9,000{[1 – (1/1.22)5 ] / .22 } = $25,772.76

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 the higher interest rate, these bigger cash flows early are more

important since the cost of waiting (the interest rate) is so much greater.

3. (LO1) 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 use:

FV = PV(1 + r)t

FV@8% = $700(1.08)3 + $950(1.08)2 + $1,200(1.08) + $1,300 = $4,585.88

FV@11% = $700(1.11)3 + $950(1.11)2 + $1,200(1.11) + $1,300 = $4,759.84

FV@24% = $700(1.24)3 + $950(1.24)2 + $1,200(1.24) + $1,300 = $5,583.36

Notice 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 flow.

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4. (LO1) To find the PVA, we use the equation:

PVA = C({1 – [1/(1 + r)]t } / r )

PVA@15 yrs: PVA = $4,600{[1 – (1/1.08)15 ] / .08} = $39,373.60

PVA@40 yrs: PVA = $4,600{[1 – (1/1.08)40 ] / .08} = $54,853.22

PVA@75 yrs: PVA = $4,600{[1 – (1/1.08)75 ] / .08} = $57,320.99

To find the PV of a perpetuity, we use the equation:

PV = C / r

PV = $4,600 / .08 = $57,500.00

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 $179.01.

5. (LO1) Here we have the PVA, the length of the annuity, and the interest rate. We want to calculate the annuity

payment. Using the PVA equation:

PVA = C({1 – [1/(1 + r)]t } / r )

PVA = $28,000 = $C{[1 – (1/1.0825)15 ] / .0825}

We can now solve this equation for the annuity payment. Doing so, we get:

C = $28,000 / 8.43035 = $3,321.33

6. (LO1) To find the PVA, we use the equation:

PVA = C({1 – [1/(1 + r)]t } / r )

PVA = $65,000{[1 – (1/1.085)8 ] / .085} = $366,546.89

7. (LO1) Here we need to find the FVA. The equation to find the FVA is:

FVA = C{[(1 + r)t – 1] / r}

FVA for 20 years = $3,000[(1.10520 – 1) / .105] = $181,892.42

FVA for 40 years = $3,000[(1.10540 – 1) / .105] = $1,521,754.74

Notice that because of exponential growth, doubling the number of periods does not merely double the FVA.

8. (LO1) Here we have the FVA, the length of the annuity, and the interest rate. We want to calculate the annuity

payment. Using the FVA equation:

FVA = C{[(1 + r)t – 1] / r}

$80,000 = $C[(1.06510 – 1) / .065]

We can now solve this equation for the annuity payment. Doing so, we get:

C = $80,000 / 13.49442 = $5,928.38

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