Tuition
Interest rate
New rate ( c 1 )
New rate ( c 2 )
Solution
Year
Tuition PV (CF) % CF Yr x %(CF)
1 –$ $0.00 #DIV/0! #DIV/0!
You will be paying $10,000 a year in tuition expenses at the end of the next two years. Bonds
currently yield 8%.
a. What is the present value and duration of your obligation?
b. What maturity zero-coupon bond would immunize your obligation?
c. Suppose you buy a zero-coupon bond with value and duration equal to your obligation. Now
suppose that rates immediately increase to 9%. What happens to your net position, that is, to
the difference between the value of the bond and that of your tuition obligation? What if rates
fall to 7%?
Perpetuity CF mil
Bond 1 matruity
Bond 2 maturity
Perpetuity YTM
Solution
a.
#DIV/0!
Duration of perpetuity
You manage a pension fund that will provide retired workers with lifetime
annuities. You determine that the payouts of the fund are essentially going to
resemble level perpetuities of $1 million per year. The interest rate is 10%. You
plan to fully fund the obligation using 5-year and 20-year maturity zero-coupon
bonds.
a. How much market value of each of the zeros will be necessary to fund the
plan if you desire an immunized position?
b. What must be the face value of the two zeros to fund the plan?
Maturity of bond (years)
Coupon rate (annual pmts)
Duration
Convexity
Base YTM
New YTM 1
New YTM 2
Solution
Using PV Calculations
Price at Based YTM $1,000.00
Price at YTM 1 $1,000.00
A 30-year maturity bond making annual coupon payments with a coupon
rate of 12% has duration of 11.54 years and convexity of 192.4. The bond
currently sells at a yield to maturity of 8%. Use this spreadsheet to find the
price of the bond if its yield to maturity falls to 7% or rises to 9%. What
prices for the bond at these new yields would be predicted by the duration
rule and the duration-with-convexity rule? What is the percent error for
each rule? What do you conclude about the accuracy of the two rules?
Maturity of bond (years)
Coupon rate (annual pmts)
Modified Duration
Convexity
Base YTM
New YTM 1
New YTM 2
Solution
Using PV Calculations
Price at Based YTM $1,000.00 $1,000.00
Price at YTM 1 $1,000.00 $1,000.00
A 12.75-year maturity zero-coupon bond selling at a yield to maturity of 8% (effective
annual yield) has convexity of 150.3 and modified duration of 11.81 years. A 30-year
maturity 6% coupon bond making annual coupon payments also selling at a yield to
maturity of 8% has nearly identical modified duration—11.79 years—but considerably
higher convexity of 231.2.
a. Suppose the yield to maturity on both bonds increases to 9%. What will be the actual
percentage capital loss on each bond? What percentage capital loss would be
predicted by the duration-with-convexity rule?
b. Repeat part (a), but this time assume the yield to maturity decreases to 7%.
c. Compare the performance of the two bonds in the two scenarios, one involving an
increase in rates, the other a decrease. Based on their comparative investment
performance, explain the attraction of convexity.
d. In view of your answer to (c), do you think it would be possible for two bonds with
equal duration, but different convexity, to be priced initially at the same yield to
maturity if the yields on both bonds always increased or decreased by equal amounts,
as in this example? Would anyone be willing to buy the bond with lower convexity
under these circumstances?