Note:

This topic has more questions than can be covered in a  1 hour tutorial session. The questions to be covered by your tutor are indicated by an asterisk (*); the rest should be viewed as extra practice problems.

Question 1*.   What are the two different general interpretations of the concept of duration, and what is the technical definition of this term? How does duration differ from maturity?

Question 2.     A one-year, $100,000 loan carries a coupon rate and a market interest rate of 12  percent.  The loan requires payment of accrued interest and one-half of the principal at the end of six months. The remaining principal and accrued interest are due at the end of the

year.

a.   What will be the cash flows at the end of six months and at the end of the year?

b.   What is the present value of each cash flow discounted at the market rate? What is the total present value?

c.   What proportion of the total present value of cash flows occurs at the end of six months? What proportion occurs at the end of the year?

d.   What is the duration of this loan?

Question 3. Two bonds are available for purchase in the financial markets. The first bond is a two-year, $1,000 bond that pays an annual coupon of 10 percent. The second bond is a two- year, $1,000, zero-coupon bond.

a.   What is the duration of the coupon bond if the current yield to maturity (R) is 8 percent? 10 percent? 12 percent? (Hint: You may wish to create a spreadsheet program to assist in the calculations.)

b.   How does the change in the yield to maturity affect the duration of this coupon bond?

c.   Calculate the duration of the zero-coupon bond with a yield to maturity of 8 percent, 10 percent, and 12 percent.

d.   How does the change in the yield to maturity affect the duration of the zero-coupon bond?

e.   Why does the change in the yield to maturity affect the coupon bond differently than it affects the zero-coupon bond?

Question 4. Consider three Treasury bonds each of which has a 10 percent semiannual coupon

and trades at par (that is, the yield to maturity is equal the coupon rate)

a.   Calculate the duration for a bond that has a maturity of four years, three years, and two years.

b.   What conclusions can you reach about the relationship between duration and the time to maturity? Plot the relationship.

Question 5. An insurance company is analyzing three bonds and is using duration as the measure of interest rate risk. All three bonds trade at a yield to maturity of 10 percent, have $10,000 par   values, and have five years to maturity. The bonds differ only in the amount of annual coupon     interest that they pay: 8, 10, and 12 percent.

a.   What is the duration for each five-year bond?

b. What is the relationship between duration and the amount of coupon interest that is paid?

Question 6*. How is duration related to the interest elasticity of a fixed-income security? What is the relationship between duration and the price of the fixed-income security?

Question 7. You have discovered that the price of a bond rose from $975 to $995 when the yield

to maturity fell from 9.75 percent to 9.25 percent. What is the duration of the bond?

Question 8*. A 10-year, 10 percent annual coupon, $1,000 bond trades at a yield to maturity of 8 percent. The bond has a duration of 6.994 years. What is the modified duration of this          bond? What is the practical value of calculating modified duration? Does modified duration

change the result of using the duration relationship to estimate price sensitivity? Question 9*. What is dollar duration? How is dollar duration different from duration?

Question 10.    Calculate the duration of a two-year, $1,000 bond that pays an annual coupon of

10 percent and trades at a yield of 14 percent. What is the expected change in the price of the bond if interest rates fall by 0.50 percent (50 basis points)?

Question 11*. The duration of an 11-year, $1,000 Treasury bond paying a 10 percent semiannual coupon and selling at par has been estimated at 6.9106 years.

a.   What is the modified duration of the bond? What is the dollar duration of the bond?

b.   What will be the estimated price change on the bond if interest rates increase 0.10 percent (10 basis points)?  If rates decrease 0.20 percent (20 basis points)?

Question 12*. Suppose you purchase a six-year, 8 percent coupon bond (paid annually) that is priced to yield 9 percent. The face value of the bond is $1,000.

a.   Show that the duration of this bond is equal to five years.

b.   Show that if interest rates rise to 10 percent within the next year and your investment horizon is five years from today, you will still earn a 9 percent yield on your investment.

c.   Show that a 9 percent yield also will be earned if interest rates fall next year to 8 percent.

Question 13.   Suppose you purchase a five-year, 15 percent coupon bond (paid annually) that is priced to yield of 9 percent. The face value of the bond is $1,000.

a.   Show that the duration of this bond is equal to four years.

b.   Show that if interest rates rise to 10 percent within the next year and your investment horizon is four years from today, you will still earn a 9 percent yield on your investment.

c.   Show that a 9 percent yield also will be earned if interest rates fall next year to 8 percent.

Question 14*. If an FI uses only duration to immunize its portfolio, what three factors affect changes in the net worth of the FI when interest rates change?

Question 15. Financial Institution XY has assets of $1 million invested in a 30-year, 10 percent  semiannual coupon Treasury bond selling at par. The duration of this bond has been estimated at 9.94 years. The assets are financed with equity and a $900,000, two-year, 7.25 percent semiannual coupon capital note selling at par.

a.   What is the leverage adjusted duration gap of Financial Institution XY?

b.   What is the impact on equity value if the relative change in all market interest rates is a decrease of 20 basis points? Note: The relative change in interest rates is ∆R/(1+R/2) = -0.0020.

c.   Using the information calculated in parts (a) and (b), what can be said about the desired duration gap for the financial institution if interest rates are expected to increase or decrease.

d.   Verify your answer to part (c) by calculating the change in the market value of equity assuming that the relative change in all market interest rates is an increase of 30 basis points.

e.   What would the duration of the assets need to be to immunize the equity from changes in market interest rates?