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# Bond Price, Future Value of Investment, Annuity

1. Today is t=0. Consider the following bonds in which the first coupon payment (if any) will begin in t=1.

Bond.....................Coupon Rate (Annual payment).................Maturity (years)
A..............................................0%....................................................3
B..............................................7%....................................................3

a. Derive the t=0 price of each bond when the yield-to-maturity is 6%.
b. Derive the t=0 price of the bond when the yield-to-maturity is 5%.
c. Which of the bonds is most sensitive to the decline in the interest rate from 6% to 5%? Why?

2. Today is t=0. David is considering investing in a two-year zero-coupon Treasury note with a face amount of \$10,000. The price is \$8,900 today. David's financial adviser tells him that the rate of return on the Treasury note is relatively low compared to other investment opportunities. He offers David the alternative investment of a \$10,000 two-year IBM bond with a 5% coupon rate (annual payment) and a price of \$8,800 today. At the same time, a friend is opening a new business and asks David to lend him \$10,000. The friend promises to pay David \$6,000 at t=1 and \$7,000 at t=2. Assume that David will be able to reinvest any cash received at t=1 at a reinvestment rate of 6%.

a. Calculate the future value of the three proposed investments at t=2.
b. Derive the expected rate of return of the three investments.
c. Discuss the risk of each proposed investment.
d. If David abhors risk, which investment should he undertake? Why?

3. Today is t=0. Sarah's three children have finally graduated from college and she decides to buy a new car whose price is \$24,000. The car dealer offers her the following payment plan: \$4,800 down payment today and a three-year car loan. The amount financed is \$19,200 and the payments are \$7,900 in t=1, t=2 and t=3. Sarah's rich uncle also offers to pay the full price of the car today but will require her to pay him back \$5,500 in each of the next five years (t=1 through t=5). Sarah's alternative borrowing instrument is a home equity loan from her bank.

a. How should Sarah finance her car purchase if the home equity loan rate is 8%?
b. How should Sarah finance her car purchase if the home equity loan rate is 14%?

4. Today is t=0 and you are thinking of retiring. Your retirement plan will pay you either \$300,000 immediately, or 4 yearly payments of \$90,000 starting at t=1.

a. Which alternative would you chose if the interest rate is 5% per year?
b. Suppose your financial adviser tells you that you can invest the \$300,000 cash at t=0 in an IBM zero-coupon bond with a face amount of \$420,000 at t=4. Will this potential investment opportunity change your decision in (a)? Explain.

#### Solution Preview

Please view the attached file for proper formatting of the solution.

1. Today is t=0. Consider the following bonds in which the first coupon payment (if any) will begin in t=1.

Bond Coupon Rate (Annual payment) Maturity (years)
A 0% 3
B 7% 3

To calculate the price of the bond we need to calculate / read from tables the values of
PVIF= Present Value Interest Factor
PVIFA= Present Value Interest Factor for an Annuity
Price of bond= PVIF * Redemption value + PVIFA * interest payment per period

PVIFA( n, r%)= =[1-1/(1+r%)^n]/r%
PVIF( n, r%)= =1/(1+r%)^n

a. Derive the t=0 price of each bond when the yield-to-maturity is 6%.

YTM= 6%
Price of bonds
Bond A
Coupon rate= 0.00%
Face value= \$1,000
No of Periods=n= 3
Discount rate annually= 6.00%
Price of bond=PVIF X Redemption value ( as there are no interest payments)
Redemption value= \$1,000 =Face Value

PVIF (3 periods, 6.% rate)= 0.839619

PVIF X Redemption value= \$839.62 =0.839619*1000

Price of Bond A = \$839.62

Bond B
Coupon rate= 7.00%
Face value= \$1,000
No of Periods=n= 3
Discount rate annually= 6.00%
Price of bond=PVIFA X Interest Payment per period +PVIF X Redemption value
Interest payment per year= \$70.00 =7.% x 1000
Redemption value= \$1,000 =Face Value

PVIF (3 periods, 6.% rate)= 0.839619
PVIFA (3 periods, 6.% rate)= 2.673012

PVIFA X Interest Payment= \$187.11 =2.673012*70
PVIF X Redemption value= \$839.62 =0.839619*1000
Total= \$1,026.73 =Price of bond

Price of Bond B = \$1,026.73

b. Derive the t=0 price of the bond when the yield-to-maturity is 5%.

YTM= 5%
Price of bonds
Bond A
Coupon rate= 0.00%
Face value= \$1,000
No of Periods=n= 3
Discount rate annually= 5.00%
Price of bond=PVIF X Redemption value ( as there are no interest payments)
Redemption value= \$1,000 =Face Value

PVIF (3 periods, 5.% rate)= 0.863838

PVIF ...

#### Solution Summary

This solution illustrates how to solve for 4 problems regarding bond price, future value of investment, annuity, evaluation of alternative investments and the time value of money. An Excel file is attached which contains proper formatting of this solution.

\$2.19