Asset Prices and
Interest Rates
1. Suppose you win the lottery. You have a choice between receiving $100,000 a
year for 20 years or an immediate payment of $1,200,000.
a. Which should you choose if the interest rate is 3 percent? If it is 6 percent?
ANSWER: For both interest rates you need to figure out the present value of the 20
annual payments of $100,000. Assuming that each annual payment is received at
the end of the year (so that the first payment is received after one year), the equa-
tion for the discounted present value of the 20 annual payments reads:
Present value = $100,000/(1 + i) + $100,000/(1 + i)2+ $100,000/(1 + i)3+ … +
$100,000/(1 + i)20
For the 3 percent interest rate (i = 0.03), the present value is: PV = 1,487,747.49.
For the 6 percent interest rate (i = 0.06), the present value is: PV = 1,146,992.12.
Note that the present value of fixed future cash flows is lower the higher the interest
rate.
With an interest rate of 3 percent, you should take the 20 annual payments of
$100,000. With an interest rate of 6 percent, you are better off to take an immediate
payment of $1,200,000.
b. For what range of interest rates should you take the immediate payment?
ANSWER: From your answer in part (a), it is clear that you should take immediate
payment for any interest rate above 6 percent. Even with an interest rate just below
6 percent (e.g., 5.8 percent), you will be better off in present-value terms with the im-
mediate payment. In fact, you can solve for the interest rate at which the present
value of the 20 annual payments exactly equals $1,200,000. This is difficult without
a financial calculator, but you can check that an interest rate of 5.45 percent would
make you indifferent between the two options. You should take immediate payment
for any interest rate above 5.45 percent.
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2. Suppose a bond has a maturity of 3 years, annual coupon payments of $5, and a
face value of $100.
a. If the interest rate is 4 percent, is the price of the bond higher or lower than the
face value? What if the interest rate is 6 percent?
ANSWER:
Bond price = $5/(1 + 0.04) + $5/(1 + 0.04)2+ $105/(1 + 0.04)3= $4.81 + $4.62 +
$93.34 = $102. 78
At an interest rate of 4 percent, the bond price is higher than the face value.
Bond price = $5/(1 + 0.06) + $5/(1 + 0.06)2+ $105/(1 + 0.06)3= $4.72 + $4.45 +
$88.16 = $97.33
At an interest rate of 6 percent, the bond price is lower than the face value.
b. For what range of interest rates does the price exceed the face value? Can you
explain the answer?
ANSWER: Note that the $5 coupon payments on a bond with face value of $100
translate into a 5 percent interest rate when the bond is priced at its face value. For
any interest rate above 5 percent, the present value of future payments will fall below
$100.
3. Suppose that people expect a company’s earnings to grow in the future at the
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