40 Part One Value
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financial markets to determine how much to discount for time and risk. By calculating the present
value of an asset, we are estimating how much people will pay for it if they have the alternative of
investing in the capital markets.
You can always work out any present value using the basic formula, but shortcut formulas can
reduce the tedium. We showed how to value an investment that makes a level stream of cash flows
forever (a perpetuity) and one that produces a level stream for a limited period (an annuity). We
also showed how to value investments that produce growing streams of cash flows.
When someone offers to lend you a dollar at a quoted interest rate, you should always check
how frequently the interest is to be paid. For example, suppose that a $100 loan requires six-month
payments of $3. The total yearly interest payment is $6 and the interest will be quoted as a rate of
6% compounded semiannually. The equivalent annually compounded rate is (1.03)^2 – 1 = .061, or
6.1%. Sometimes it is convenient to assume that interest is paid evenly over the year, so that inter-
est is quoted as a continuously compounded rate.Select problems are available in McGraw-Hill’s Connect.
Please see the preface for more information.BASIC- Future values If you invest $100 at an interest rate of 15%, how much will you have at the
end of eight years? - Discount factors If the PV of $139 is $125, what is the discount factor?
- Present values If the cost of capital is 9%, what is the PV of $374 paid in year 9?
- Present values A project produces a cash flow of $432 in year 1, $137 in year 2, and $797
in year 3. If the cost of capital is 15%, what is the project’s PV? If the project requires an
investment of $1,200, what is its NPV? - Opportunity cost of capital Which of the following statements are true? The opportunity
cost of capital:
a. Equals the interest rate at which the company can borrow.
b. Depends on the risk of the cash flows to be valued.
c. Depends on the rates of return that shareholders can expect to earn by investing on their
own.
d. Equals zero if the firm has excess cash in its bank account and the bank account pays no
interest. - Perpetuities An investment costs $1,548 and pays $138 in perpetuity. If the interest rate is
9%, what is the NPV? - Growing perpetuities A common stock will pay a cash dividend of $4 next year. After
that, the dividends are expected to increase indefinitely at 4% per year. If the discount rate is
14%, what is the PV of the stream of dividend payments? - Perpetuities and annuities The interest rate is 10%.
a. What is the PV of an asset that pays $1 a year in perpetuity?
b. The value of an asset that appreciates at 10% per annum approximately doubles in seven
years. What is the approximate PV of an asset that pays $1 a year in perpetuity beginning
in year 8?
c. What is the approximate PV of an asset that pays $1 a year for each of the next seven
yea rs?
d. A piece of land produces an income that grows by 5% per annum. If the first year’s income
is $10,000, what is the value of the land?
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SETS