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Chapter 17: Q2. (page 368)

What is compound interest? How does it relate to the formula Xt = (1 + i)t X0? What is present value? How does it relate to the formula Xt/(1 + i)t = X0?

Short Answer

Expert verified

By the given formula, compound interest calculates the present value of money in the future when the interest is compounded.

The present value refers to the present-day value of the future returns or costs by the given formula.

Step by step solution

01

Step 1. Compound interest

Compound interest explains how quickly an investment increases in value when interest is compounded, not only on the original amount invested but also on all interest payments that have been previously made. In the given formula, Xt is the value of the initial investment after t years when the annual compound rate of interest is i.

02

Step 2. Present value 

Present value is the present-day value of returns or costs that are expected to arrive in the future. Present value is especially useful when investors wish to determine the proper current price to pay for an asset. The given equation says that Xt dollars in t years convert into exactly Xt/(1 + i)t dollars today.

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Most popular questions from this chapter

If we compare the betas of various investment opportunities, why do the assets that have higher betas also have higher average expected rates of return?

Suppose that a risk-free investment will make three future payments of \(100 in one year, \)100 in two years, and $100 in three years. If the Federal Reserve has set the risk-free interest rate at 8 percent, what is the proper current price of this investment? What is the price of this investment if the Federal Reserve raises the risk-free interest rate to 10 percent?

Suppose that you desire to get a lump sum payment of $100,000 two years from now. Rounded to full dollars, how many current dollars will you have to invest today at a 10 percent interest to accomplish your goal?

Identify each of the following investments as either an economic investment or a financial investment.

a. A company builds a new factory.

b. A pension plan buys some Google stock.

c. A mining company sets up a new gold mine.

d. A woman buys a 100-year-old farmhouse in the countryside.

e. A man buys a newly built home in the city.

f. A company buys an old factory.

Next, consider another pair of assets, C and D. Asset C will make a single payment of \(150 in one year while D will make a single payment of \)200 in one year. Assume that the current price of C is \(120 and that the current price of D is \)180.

c. What are the rates of return of assets C and D at their current prices? Given these rates of return, which asset should investors buy and which asset should they sell?

d. Assume that arbitrage continues until C and D have the same expected rate of return. When arbitrage ends, will C and D have the same price?

Compare your answers to questions a through d before answering question e.

e. We know that arbitrage will equalize rates of return. Does it also guarantee to equalize prices? In what situations will it equalize prices?
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