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chapter 3

QUANTITATIVE PROBLEMS 1. Calculate the present value of a $1,000 zero-coupon bond with five years to maturity if the yield to matu- rity is 6%. 2. A lottery claims its grand prize is $10 million, payable over 20 years at $500,000 per year. If the first pay. ment is made immediately, what is this grand prize really worth? Use an interest rate of 6% 3. Consider a bond with a 7% annual coupon and a face value of $1,000. Complete the following table, What relationships do you observe between maturity and discount rate and the current price? 4. Consider a coupon bond that has a $1,000 par value and a coupon rate of 10%. The bond is currently sell- ing for $1,150 and has eight years to maturity. What is the bond's yield to maturity? 5. You are willing to pay $15,625 now to purchase a per petuity that 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? 6. What is the price of a perpetuity that has a coupon of $50 per year and a yield to maturity of 2.5%? If the yield to maturity doubles, what will happen to its price? 7. Property taxes in DeKalb Countylare 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 Years to Maturity Yield to Maturity Current Price 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 three years, with no other payments till then. The second loan is for $40 million. It requires an annual interest payment of $3.6 million. The principal of $40 million is due in three 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 increases from 8% to 8,5%? 15. Consider a bond that promises the following cash flows. The yield to maturity is 12% property tax payments? Assume that the house remains worth $100,000 forever, property tax rates never change, and that a 9% interest 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 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 matures 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 port- folio, increasing the duration of the portfolio to 12.5 years. What is the duration of the $40 million in new securities? Year Promised Payments 160 160 170 180 230 You plan to buy this bond, hold it for 25 years, and then sell the bond a. What total cash will you receive from the bond after the 2.5 years? Assume that periodie cash flows are reinvested at 12% T. b. If immediately after buying this bond all market interest rates drop to 11% (including your rein- vestment rate), what will be the impact on your total cash flow after 2.5 years? How does this com- pare to part (a)? c. Assuming all market interest rates are 12%, what is the duration of this bond? QUANTITATIVE PROBLEMS 1. Calculate the present value of a $1,000 zero-coupon bond with five years to maturity if the yield to matu- rity is 6%. 2. A lottery claims its grand prize is $10 million, payable over 20 years at $500,000 per year. If the first pay. ment is made immediately, what is this grand prize really worth? Use an interest rate of 6% 3. Consider a bond with a 7% annual coupon and a face value of $1,000. Complete the following table, What relationships do you observe between maturity and discount rate and the current price? 4. Consider a coupon bond that has a $1,000 par value and a coupon rate of 10%. The bond is currently sell- ing for $1,150 and has eight years to maturity. What is the bond's yield to maturity? 5. You are willing to pay $15,625 now to purchase a per petuity that 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? 6. What is the price of a perpetuity that has a coupon of $50 per year and a yield to maturity of 2.5%? If the yield to maturity doubles, what will happen to its price? 7. Property taxes in DeKalb Countylare 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 Years to Maturity Yield to Maturity Current Price 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 three years, with no other payments till then. The second loan is for $40 million. It requires an annual interest payment of $3.6 million. The principal of $40 million is due in three 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 increases from 8% to 8,5%? 15. Consider a bond that promises the following cash flows. The yield to maturity is 12% property tax payments? Assume that the house remains worth $100,000 forever, property tax rates never change, and that a 9% interest 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 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 matures 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 port- folio, increasing the duration of the portfolio to 12.5 years. What is the duration of the $40 million in new securities? Year Promised Payments 160 160 170 180 230 You plan to buy this bond, hold it for 25 years, and then sell the bond a. What total cash will you receive from the bond after the 2.5 years? Assume that periodie cash flows are reinvested at 12% T. b. If immediately after buying this bond all market interest rates drop to 11% (including your rein- vestment rate), what will be the impact on your total cash flow after 2.5 years? How does this com- pare to part (a)? c. Assuming all market interest rates are 12%, what is the duration of this bond

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