Present Value problems

Present Values. Compute the present value of a $100 cash flow for the following combinations
of discount rates and times:
a. r = 8 percent. t = 10 years.
b. r = 8 percent. t = 20 years.
c. r = 4 percent. t = 10 years.
d. r = 4 percent. t = 20 years.

Future Values. Compute the future value of a $100 cash flow for the same combinations of
rates and times as in problem above.

Calculating Interest Rate. Find the interest rate implied by the following combinations of
present and future values:
The Time Value of Money

Present Value Years Future Value
$400 11 $684
$183 4 $249
$300 7 $300

Excel form

Loan Payments. If you take out an $8,000 car loan that calls for 48 monthly payments at an
APR of 10 percent, what is your monthly payment? What is the effective annual interest rate on
the loan?

Amortizing Loan. You take out a 30-year $100,000 mortgage loan with an APR of 6 percent
and monthly payments. In 12 years you decide to sell your house and pay off the mortgage.
What is the principal balance on the loan?

Amortizing Loan. Consider a 4-year amortizing loan. You borrow $1,000 initially, and repay
it in four equal annual year-end payments.
a. If the interest rate is 8 percent, show that the annual payment is $301.92.
b. Fill in the following table, which shows how much of each payment is interest versus
principal repayment (that is, amortization), and the outstanding balance on the loan at each
date.
Loan Year-End Interest Year-End Amortization
Time Balance Due on Balance Payment of Loan
0 $1,000 $80 $301.92 $221.92
1 --- --- 301.92 ---
2 --- --- 301.92 ---
3 --- --- 301.92 ---
4 0 0 - -
c. Show that the loan balance after 1 year is equal to the year-end payment of $301.92 times
the 3-year annuity factor.

4. Bond Pricing. A 6-year Circular File bond pays interest of $80 annually and sells for $950.
What is its coupon rate, current yield, and yield to maturity?

5. Bond Pricing. If Circular File (see question 4) wants to issue a new 6-year bond at face value,
what coupon rate must the bond offer?

6. Bond Yields. An AT&T bond has 10 years until maturity, a coupon rate of 8 percent, and sells
for $1,100.
a. What is the current yield on the bond?
b. What is the yield to maturity?

Bond Prices and Yields.
a. Several years ago, Castles in the Sand, Inc., issued bonds at face value at a yield to maturity
of 7 percent. Now, with 8 years left until the maturity of the bonds, the company has run into
hard times and the yield to maturity on the bonds has increased to 15 percent. What has happened
to the price of the bond?

b. Suppose that investors believe that Castles can make good on the promised coupon payments,
but that the company will go bankrupt when the bond matures and the principal
comes due. The expectation is that investors will receive only 80 percent of face value at maturity.
If they buy the bond today, what yield to maturity do they expect to receive?