CHAPTER 2
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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From this equation, we can solve for the amount to be put aside each year.
PV(boat) = $20,000/(1.10)5 = $12,418
PV(savings) = Annual savings
Because PV(savings) must equal PV(boat):
Annual savings
Annual savings
Another approach is to use the future value of an annuity formula:
Annual savings = $ 3,276
22. The fact that Kangaroo Autos is offering “free credit” tells us what the cash payments are; it does not change the fact that money has time value. A 10% annual rate of interest is equivalent to a monthly rate of 0.83%: rmonthly = rannual /12 = 0.10/12 = 0.0083 = 0.83%
The present value of the payments to Kangaroo Autos is:
A car from Turtle Motors costs $9,000 cash. Therefore, Kangaroo Autos offers the better deal, i.e., the lower present value of cost.
23. The NPVs are: at 5% at 10% at 15%
The figure below shows that the project has zero NPV at about 11%.
As a check, NPV at 11% is:
24. a. This is the usual perpetuity, and hence:
b. This is worth the PV of stream (a) plus the immediate payment of $100:
PV = $100 + $1,428.57 = $1,528.57
c. The continuously compounded equivalent to a 7% annually compounded rate is approximately 6.77%, because: e0.0677 = 1.0700
Thus:
Note that the pattern of payments in part (b) is more valuable than the pattern of payments in part (c). It is preferable to receive cash flows at the start of every year than to spread the
b. What is the present value of this annuity if the opportunity cost rate is 10% annually? 10% compounded semiannually?
10. An investment of $1,000 today will grow to $1,100 in one year. What is the continuously compounded rate of return?
c) The present value of $500 to be received in one year when the opportunity cost rate is 8 percent (discounting):
C) The cash flows for an annuity due must all occur at the ends of the periods.
b. How would you estimate the appropriate discount rate to be used in valuing the cash flows?
Fishbone Corporation bought a new machine and agreed to pay for it in equal annual installments of $5,670 at the end of each of the next 10 years. Assuming that
Which of the following is the actual rate of interest paid or earned over a year's time?
With a 35% tax, the alternative of selling of the vessel with a predicted total lifetime of 25 years, yields a slightly higher NPV compared to scraping after 15 years of use. The increase in NPV due to selling is because of a higher cash flow in the beginning of 2018 when the sale of the vessel occurs.
15) A homeowner in a sunny climate has the opportunity to install a solar water heater in his home for a cost of
You want to buy a new sports car from City Toyota for $62,000. The contract is in the form of a 48-month annuity due at a 9% APR. What will your monthly payment be? $1531.39
a. Compute the rate for the current year. (Show your work for credit; 2 points)
Calculate the net present value (NPV) for the following 20 year projects. Comment on the acceptability of each. Assume that the firm has an opportunity cost of 14%.
D. Have no effect on either cash flows or risk because the cash flows are already reflected in the income statement
CALC: n = 2.5 x 2 =5 r = ? PV = -$500 PMT = 8.8% x 1000 / 2 = $44