Quantitative Problem Chapter 3

1. Calculate the present value of $1,000 zero-coupon bond with 5 years to maturity if the required annual interest rate is 6%.

2-A lottery claims its grand prize is $10 million, payable over 20 years at $500,000 per year. If the first payment is made immediately, what is this grand prize really worth? Use a discount rate of 6%.

3. Consider a bond with a 7% annual coupon and a face value of $1,000. Complete the following table:

Years to Maturity

Discount Rate

Current Price

3

5

3

7

6

7

9

7

9

9

What relationship do you observe between yield to maturity and the current market value?

4. Consider a coupon bond that has a $1,000 per value and a coupon rate of 10%. The bond is currently selling for $1,150 and has 8 years to maturity. What is the bond’s yield to maturity?

5. You are willing to pay $15,625 now to purchase a perpetuity which will pay you and your heirs $1,250 each year, forever, starting at the end of this year. If your required rate of return does not change, how much would you be willing to pay if this were a 20-year, annual payment, ordinary annuity instead of a perpetuity?

7. Property taxes in DeKalbCounty are roughly 2.66% of the purchase price every year. If you just bought a $100,000 home, what is the PV of all the future property tax payments? Assume that the house remains worth $100,000 forever, property tax rates never change, and that a 9% discount rate is used for discounting.

8-Assume you just deposited $1,000 into a bank account. The current real interest rate is 2% and inflation is expected to be 6% over the next year. What nominal interest rate would you require from the bank over the next year? How much money will you have at the end of one year? If you are saving to buy a stereo that currently sells for $1,050, will you have enough to buy it?

9. A 10-year, 7% coupon bond with a face value of $1,000 is currently selling for $871.65. Compute your rate of return if you sell the bond next year for $880.10.

10. You have paid $980.30 for an 8% coupon bond with a face value of $1,000 that mature in five years. You plan on holding the bond for one year. If you want to earn a 9% rate of return on this investment, what price must you sell the bond for? Is this realistic?

11. Calculate the duration of a $1,000 6% coupon bond with three years to maturity. Assume that all market interest rates are 7%.

12. Consider the bond in the previous question. Calculate the expected price change if interest rates drop to 6.75% using the duration approximation. Calculate the actual price change using discounted cash flow.

13. The duration of a $100 million portfolio is 10 years. $40 million in new securities are added to the portfolio, increasing the duration of the portfolio to 12.5 years. What is the duration of the $40 million in new securities?

14. A bank has two, 3-year commercial loans with a present value of $70 million. The first is a $30 million loan that requires a single payment of $37.8 million in 3 years, with no other payments till then. The second is for $40 million. It requires an annual interest payment of $3.6 million. The principal of $40 million is due in 3 years.

(a) What is the duration of the bank’s commercial loan portfolio?

(b) What will happen to the value of its portfolio if the general level of interest rates increased from 8% to 8.5%?

15. Consider a bond that promises the following cash flows. The required discount rate is 12%.

Year

0

1

2

3

4

Promised Payments

160

170

180

230

You plan to buy this bond, hold it for 2½ years, and then sell the bond.

(a) What total cash will you receive from the bond after the 2½ years? Assume that periodic cash flows are reinvested at 12%.

(b) If immediately after buying this bond, all market interest rates drop to 11% (including your reinvestment rate), what will be the impact on your total cash flow after 2½ years? How does this compare to part (a)?

(c) Assuming all market interest rates are 12%, what is the duration of this bond?

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