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%.

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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.

*2.55 If the interest rate is 5% per...