Solutions to Questions 2.1 to 2.43 Appear in the Text

Solutions to Questions 2.1 to 2.43 Appear in the Text

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Chapter 2
The Time Value of Money
and Net Present Value

Solutions to Questions 2.1 to 2.43 appear in the text.

2.44What is a perfect market? What were the assumptions made in this chapter that were not part of the perfect market scenario?

Answer:A perfect market is one with no taxes, no transaction costs, no differences in opinion, and many buyers and sellers. In this chapter, we also are assuming no uncertainty and no inflation.

2.45What is the difference between a bond and a loan?

Answer:No difference really. A bond is a loan.

2.46In the text, I assumed you received the dividend at the end of the period. In the real world, if you received the dividend at the beginning of the period instead of the end of the period, could this change your effective rate of return? Why?

Answer:Yes, because dividends could then be reinvested and earn extra returns themselves.

2.47Your stock costs $100 today, pays $5 in dividends at the end of the period, and then sells for $98. What is your rate of return?

Answer:($98  $5)/$100  1  3%.

2.48The interest rate has just increased from 6% to 8%. How many basis points is this?

Answer:200 basis points.

2.49Assume an interest rate of 10% per year. How much would you lose over 5 years if you had to give up interest on the interest—that is, if you received 50% instead of compounded interest?

Answer:You would lose 11.1%.

2.50Over 20 years, would you prefer 10% per annum, with interest compounding, or 15% per annum but without interest compounding? (That is, you receive the interest, but it is put into an account that earns no interest, which is what we call simple interest.)

Answer:Over 20 years, you would receive a rate of return of 1.120  1  573%. The uncompounded rate earns 15%  20  300%. You would prefer the compounded lower interest rate.

2.51A project returned 30%, then 30%. Thus, its arithmetic average rate of return was 0%. If you invested $25,000, how much did you end up with? Is your rate of return positive or negative? How would your overall rate of return have been different if you first earned 30% and then 30%?

Answer:The rate of return is (1  30%)  (1  30%)  1  9%. You would end up with $25,000  0.91  $22,750. This turns out to be more general—the total compounded annual rate of return is below the arithmetic average rate of return. The rate of return would not have changed if you had first lost 30% and then gained 30%. The calculations would turn out the same.

2.52A project returned 50%, then 40%. Thus, its arithmetic average rate of return was 5%. Is your rate of return positive or negative?

Answer:The rate of return is (1  50%)  (1  40%)  1  10%. You would lose money with this negative rate of return.

2.53An investment for $50,000 earns a rate of return of 1% in each month of a full year. How much money will you have at year’s end?

Answer:$50,000  1.0112  $56,341.

2.54There is always disagreement about what stocks are good purchases. The typical degree of disagreement is whether a particular stock is likely to offer, say, a 10% (pessimistic) or a 20% (optimistic) annualized rate of return. For a $30 stock today, what does the difference in belief between these two opinions mean for the expected stock price from today to tomorrow? (Assume that there are 365 days in the year. Reflect on your answer for a moment, and recognize that a
$30 stock typically moves about $1 on a typical day. This unexplainable up-and-down volatility
is often called noise.)

Answer:The daily interest rate is either 1.101/365  1  0.026% or 1.201/365  1  0.05%. Thus, the pessimist expects a stock price of $30.008 tomorrow; the optimist expects a stock price of $30.015 tomorrow. Note that the 0.7 cent or so expected increase is dwarfed by the typical $1 day-to-day noise in stock prices.

2.55If the interest rate is 5% per annum, how long will it take to double your money? How long will it take to triple it?

Answer:To find out how long it will take to double your money, solve for x:

(1  5%)x  (1  100%)  x  log(2)/log(1.05)  14.2.

Thus, it will take about 14 years and 3 months. To find out how long it will take to triple your money, solve for x:

(1  5%)x  (1  200%)  x  22.5.

Tripling will take 22.5 years, or 22 years and 6 months.

2.56If the interest rate is 8% per annum, how long will it take to double your money?

Answer:(1  0.08)x  (1  100%)  x  log(2)/log(1.08)  9. Thus, it will take just about 9 years.

2.57From Fibonacci’s Liber Abaci, written in the year 1202: “A certain man gave 1 denaro at interest so that in 5 years he must receive double the denari, and in another 5, he must have double 2 of the denari and thus forever. How many denari from this 1 denaro must he have in 100 years?”

Answer:First, solve for the interest rate:

1d  (1  r)5  2d  r  14.87%.

Therefore, in 100 years, he will have (1  r)100  1,048,576 denari. Of course, you can solve this in a simpler way: You have twenty 5-year periods, in each of which the holdings double. The answer is 220 denari.

2.58A bank quotes you a loan interest rate of 14% on your credit card. If you charge $15,000 at the beginning of the year, how much will you have to repay at the end of the year?

Answer:The effective interest rate is (1  14%/365)365  1  15%. Thus, you will have to repay $15,000  1.15  $17,250.

2.59Go to the website of a bank of your choice. What kind of quote does your bank post for a CD, and what kind of quote does your bank post for a mortgage? Why?

Answer:For the banks I looked up on the Web, they quoted the higher APY for their CDs, and the lower APRs for their mortgages. The higher APY looks attractive to investors; the lower APR looks attractive to borrowers.

2.60What is the 1-year discount factor if the interest rate is 33.33%?

Answer:The discount factor is 1/(1  33.33%)  0.75.

2.61You can choose between the following rent payments:

a.A lump sum cash payment of $100,000;

b.10 annual payments of $12,000 each, the first occurring immediately;

c.120 monthly payments of $1,200 each, the first occurring immediately. (Friendly suggestion: This is a lot easier to calculate on a computer spreadsheet.)

d.Which rental payment scheme would you choose if the interest rate was an effective 5% per year?

e.Spreadsheet question: At what interest rate would you be indifferent between the first and the second choice (10 annual payments) above? (Hint: Graph the NPV of the second project as a function of the interest rate.)

Answer:

a.$100,000

b.$12,000  $12,000/1.05  $12,000/1.052      $12,000/1.059  $97,294

c.The monthly discount rate is (1  5%)1/12  0.4074%. With 120 monthly payments, $1,200  $1,200/1.004074  $1,200/1.0040742      $1,200/1.004074119  $1,141,828.

d.About 4.3%. The incremental cash flow associated with paying the rent in its entirety is
$100,000  $12,000  $88,000. So, $88,000  $12,000/(1  r)  $12,000/(1  r)2      $12,000/(1  r)10.

e.As tenant, you would prefer the second choice, as landlord, the third choice (120 monthly payments of $1,200 each). (If you are wondering why, note that a month has 4.5 weeks!)

2.62A project has cash flows of $15,000, $10,000, and $5,000 in 1, 2, and 3 years, respectively. If the prevailing interest rate is 15%, would you buy the project if it costs $25,000?

Answer:No, because the PV is $15,000/1.15  $10,000/1.152  $5,000/1.153  $23,892.50. Therefore, the NPV is $25,000  $23,892.50  $1,107.50.

2.63Consider the same project that costs $25,000 with cash flows of $15,000, $10,000, and $5,000.
At what prevailing interest rate would this project be profitable? Try different interest rates, and
plot the NPV on the y-axis, and the interest rate on the x-axis.

Answer:The determining equation is $15,000/(1  r)  $10,000/(1  r)2  $5,000/(1  r)3  $25,000. At r  11.79%, the NPV is zero. Thus, at lower interest rates, the project is profitable.
At higher interest rates, it is unprofitable.

2.64On April 12, 2006, Microsoft stock traded for $27.11 and claimed to pay an annual dividend of $0.36. Assume that the first dividend will be paid in 1 year, and that it then grows by 5% each year for the next 5 years. Further, assume that the prevailing interest rate is 6% per year. At what price would you have to sell Microsoft stock in 5 years in order to break even?

Answer:The cash flows are:

Time
0 / 1 / 2 / 3 / 4 / 5
Cash Flow / –$27.11 / $0.36 / $0.378 / $0.3969 / $0.4167 / $0.4376
Discount Factor / 1.0000 / 0.9434 / 0.8900 / 0.8396 / 0.7921 / 0.7473
Present Value / –$27.11 / $0.340 / $0.336 / $0.333 / $0.330 / $0.327

The dividend stream is worth $1.67, so the PV so far is $25.44. This means that if you were to receive $25.44  (1.06)5  $34.04 in year 5, you would break even.

2.65Assume you are 25 years old. The IAW insurance company is offering you the following retirement contract (called an annuity): Contribute $2,000 per year for the next 40 years. When you reach
65 years of age, you will receive $30,000 per year for as long as you live. Assume that you believe that the chance that you will die is 10% per year after you will have reached 65 years of age. In other words, you will receive the first payment with probability 90%, the second payment with probability 81%, and so on. Assume the prevailing interest rate is 5% per year, all payments occur at year end, and it is January 1 now. Is this annuity a good deal? (Use a spreadsheet.)

Answer:The stream of 40 payments of $2,000 has a PV of

Now assume that you are 65 years old. You receive your first payment with a probability of 90%, and it is worth $30,000/1.05  $28,571. You receive your next payment with a probability of (1  10%)  (1  10%)  81%, and it is worth $30,000/1.052  $27,211. Iterate this forward in a spreadsheet and add up your expected payments: You expect to receive about $179,622 (well, assuming you will finally die for sure at age 105). This is standing at age 65. What is the value today? The stream has a present value of $179,622/ (1.05)40  $25,515. Thus, it is not a good deal because the insurance company is doing better than you are.

Year / Payment / Discount Factor / PV of $30,000 Payment / Probability / Expected
Cash Flow
1 / $30,000 / 0.9524 / $28,571.43 / 0.9% / $ 25,714.29
2 / $30,000 / 0.9070 / $27,210.88 / 0.8% / $ 22,040.82
3 / $30,000 / 0.8638 / $25,915.13 / 0.7% / $ 18,892.13
4 / $30,000 / 0.8227 / $24,681.07 / 0.6% / $ 16,193.25
5 / $30,000 / 0.7835 / $23,505.78 / 0.5% / $ 13,879.93
6 / $30,000 / 0.7462 / $22,386.46 / 0.5% / $ 11,897.08
7 / $30,000 / 0.7107 / $21,320.44 / 0.4% / $ 10,197.50
8 / $30,000 / 0.6768 / $20,305.18 / 0.4% / $ 8,740.71

Continued

Year / Payment / Discount Factor / PV of $30,000 Payment / Probability / Expected
Cash Flow
9 / $30,000 / 0.6446 / $19,338.27 / 0.3% / $ 7,492.04
10 / $30,000 / 0.6139 / $18,417.40 / 0.3% / $ 6,421.75
11 / $30,000 / 0.5847 / $17,540.38 / 0.3% / $ 5,504.36
12 / $30,000 / 0.5568 / $16,705.12 / 0.2% / $ 4,718.02
13 / $30,000 / 0.5303 / $15,909.64 / 0.2% / $ 4,044.02
14 / $30,000 / 0.5051 / $15,152.04 / 0.2% / $ 3,466.30
15 / $30,000 / 0.4810 / $14,430.51 / 0.2% / $ 2,971.11
16 / $30,000 / 0.4581 / $13,743.35 / 0.1% / $ 2,546.67
17 / $30,000 / 0.4363 / $13,088.90 / 0.1% / $ 2,182.86
18 / $30,000 / 0.4155 / $12,465.62 / 0.1% / $ 1,871.02
19 / $30,000 / 0.3957 / $11,872.02 / 0.1% / $ 1,603.73
20 / $30,000 / 0.3769 / $11,306.68 / 0.1% / $ 1,374.63
21 / $30,000 / 0.3589 / $10,768.27 / 0.1% / $ 1,178.25
22 / $30,000 / 0.3418 / $10,255.50 / 0.0% / $ 1,009.93
23 / $30,000 / 0.3256 / $ 9,767.14 / 0.0% / $ 865.66
24 / $30,000 / 0.3101 / $ 9,302.04 / 0.0% / $ 741.99
25 / $30,000 / 0.2953 / $ 8,859.08 / 0.0% / $ 635.99
26 / $30,000 / 0.2812 / $ 8,437.22 / 0.0% / $ 545.14
27 / $30,000 / 0.2678 / $ 8,035.45 / 0.0% / $ 467.26
28 / $30,000 / 0.2551 / $ 7,652.81 / 0.0% / $ 400.51
29 / $30,000 / 0.2429 / $ 7,288.39 / 0.0% / $ 343.29
30 / $30,000 / 0.2314 / $ 6,941.32 / 0.0% / $ 294.25
31 / $30,000 / 0.2204 / $ 6,610.78 / 0.0% / $ 252.21
32 / $30,000 / 0.2099 / $ 6,295.98 / 0.0% / $ 216.18
33 / $30,000 / 0.1999 / $ 5,996.18 / 0.0% / $ 185.30
34 / $30,000 / 0.1904 / $ 5,710.64 / 0.0% / $ 158.83
35 / $30,000 / 0.1813 / $ 5,438.71 / 0.0% / $ 136.14
36 / $30,000 / 0.1727 / $ 5,179.72 / 0.0% / $ 116.69
37 / $30,000 / 0.1644 / $ 4,933.07 / 0.0% / $ 100.02
38 / $30,000 / 0.1566 / $ 4,698.16 / 0.0% / $ 85.73
39 / $30,000 / 0.1491 / $ 4,474.44 / 0.0% / $ 73.48
40 / $30,000 / 0.1420 / $ 4,261.37 / 0.0% / $ 62.99
Total: / $179,622.08

2.66A project has the following cash flows in periods 1 through 4: $200, $200, $200, $200. If the prevailing interest rate is 3%, would you accept this project if you were offered an upfront payment of $10 to do so?

Answer:The PV of the cash flows is $200/1.03  $200/1.032  $200/1.033  $200/1.034  $10.99. Therefore the NPV is $10  $10.99  $0.99. You should not take this project.

2.67Assume you are a real estate broker with an exclusive contract—the condo association rules state that everyone selling their condominiums must go through you or a broker designated by you. A typical condo costs $500,000 today and sells again every 5 years. This will last for 50 years, and then all bets are off. Your commission will be 3%. Condos appreciate in value at a rate of 2% per year. The interest rate is 10% per annum.

a.What is the value of this exclusivity rule? In other words, at what price should you be willing to sell the privilege of exclusive condo representation to another broker?

b.If free Internet advertising was equally effective and if it could replace all real estate brokers so that buyers’ and sellers’ agents would no longer earn the traditional 6% (3% each), what would happen to the value gain of the condo?

Answer:

a.Every 5 years, the condo will be worth 1.025  1  10.4% more. Thus, the condo will sell for $552,040 in 5 years, and your commission will be $16,561. The discount rate is 61.1%, and the discount factor is 0.62. Thus, this PV is $10,283 to you. Your next commission will be $18,285 in 10 years, and the PV is $7,050. Continuing, the numbers are $4,833, $3,313, $2,271, $1,557, $1,067, $732, $502, and $344. Therefore, the present value of your commissions (the value of your exclusivity rule) is $31,952.

Year / Condo Value / Commission / Discount Factor / PV of Cash Flows
5 / $ 552,040 / $16,561.21 / 0.6209 / $10,283.21
10 / $ 609,497 / $18,284.92 / 0.3855 / $ 7,049.63
15 / $ 672,934 / $20,188.03 / 0.2394 / $ 4,832.85
20 / $ 742,974 / $22,289.21 / 0.1486 / $ 3,313.15
25 / $ 820,303 / $24,609.09 / 0.0923 / $ 2,271.32
30 / $ 905,681 / $27,170.42 / 0.0573 / $ 1,557.10
35 / $ 999,945 / $29,998.34 / 0.0356 / $ 1,067.46
40 / $1,104,020 / $33,120.59 / 0.0221 / $ 731.80
45 / $1,218,927 / $36,567.81 / 0.0137 / $ 501.68
50 / $1,345,794 / $40,373.82 / 0.0085 / $ 343.93
Total: / $31,952.13

b.The condo would immediately increase in value today by $63,904 (at 6% commissions). This is twice the $31,952 computed from 3% commissions.

2.68If the interest rate is 5%, what would be the equivalent annual cost (see Question 2.39) of a $2,000 lease payment up front, followed by $800 for three more years?

Answer:The net present cost is about

The EAC is

Thus, EAC  $1,122.41.

2.69The prevailing discount rate is 15% per annum. Firm F’s cash flows start with $500 and grow at 20% per annum for 3 years. Firm S’s cash flows also start with $500 in year 1 but shrink at 20% per annum for 3 years. What are the prices of these two firms? Which one is the better “buy”?

Answer:F’s cash flows are $500, $600, and $720. Its value is therefore $500/1.15  $600/1.152  $720/1.153  $1,361.88. Firm S’s cash flows are $500, $400, and $320. Its value is therefore $500/1.15  $400/1.152  $320/1.153  $947.65. Both firms offer your investment dollar a 15% rate of return, and neither is a better buy.