How to Calculate Present Values

Answers to Problem Sets

1. If the discount factor is .507, then .507*1.126 = $1

2. 125/139 = .899

3. PV = 374/(1.09)9 = 172.20

4. PV = 432/1.15 + 137/(1.152) + 797/(1.153) = 376 + 104 + 524 = $1,003

5. FV = 100*1.158 = $305.90

6. NPV = -1,548 + 138/.09 = -14.67 (cost today plus the present value of the perpetuity) 7. PV = 4/(.14-.04) = $40

8. a. PV = 1/.10 = $10

b. Since the perpetuity will be worth $10 in year 7, and since that is roughly double the present value, the approximate PV equals $5. PV = (1 / .10)/(1.10)7 = 10/2= $5 (approximately)

c. A perpetuity paying $1 starting now would be worth $10, whereas a

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b. PV = $1 billion/(0.08 – 0.04) = $25.0 billion

c.

d. The continuously compounded equivalent to an 8% annually compounded rate is approximately 7.7% , because: e0.0770 = 1.0800

Thus:

This result is greater than the answer in Part (c) because the endowment is now earning interest during the entire year.

26. With annual compounding: FV = $100 (1.15)20 = $1,636.65

With continuous compounding: FV = $100 e(0.15×20) = $2,008.55

27. One way to approach this problem is to solve for the present value of:

(1) $100 per year for 10 years, and

(2) $100 per year in perpetuity, with the first cash flow at year 11.

If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate (r).

The present value of $100 per year for 10 years is:

The present value, as of year 10, of $100 per year forever, with the first payment in year 11, is: PV10 = $100/r

At t = 0, the present value of PV10 is:

Equating these two expressions for present value, we have:

Using trial and error or algebraic solution, we find that r = 7.18%.

28. Assume the amount invested is one dollar.

Let A represent the investment at 12%, compounded annually.

Let B represent the investment at 11.7%, compounded semiannually.

Let C represent the investment at 11.5%, compounded continuously.

After one year:

FVA = $1 (1 + 0.12)1 = $1.1200

FVB = $1 (1 + 0.0585)2 = $1.1204

FVC = $1 e(0.115