P1.T3.22.30. More Interest Rate Properties

Nicole Seaman

Director of FRM Operations
Staff member
Learning objectives: Calculate the Macaulay duration, modified duration, and dollar duration of a bond. Evaluate the limitations of duration and explain how convexity addresses some of them. Calculate the change in a bond’s price given its duration, its convexity, and a change in interest rates.


22.30.1. Sally and her team are discussing the interest rate risk of one of the firm's fixed-income portfolios. Sally reports to her team that "while the current yield is an impressive 15.0% with semi-annual compound frequency, this coupon-bearing bond portfolio has a yield-based duration of 4.50 years." Sally did not report the portfolio's convexity. Assuming the bond portfolio consists entirely of long positions, each of the following statements is true EXCEPT which is false?

a. The portfolio's modified duration is about 4.1860 years
b. For this portfolio, an increase in yield will be associated with a decrease in the modified duration
c. If the yield increases by 10 basis points, they expect the portfolio's value to decrease by 0.450% as a first-order approximation
d. If they omit convexity in a first-order (i.e., duration only) approximation, the interest rate risk will be non-conservatively underestimated

22.30.2. Assume the zero-rate term structure is upward-sloping as follows: 3.00% at six months, 3.50% at 1.0 year, 3.90% at 1.5 years, 4.20% at 2.0 years, 4.40% at 2.5 years and 4.50% at 3.0 years. These zero rates are given per annum with continuous compounding. Under this term structure, the three-year annuity factor, A(3.0) = $5.5830; i.e., the present value of $1.0 received every six months over the next three years is $5.5830.

Which of the following is nearest to the semi-annual par yield, i.e., the per annum par yield for a bond that pays semi-annual coupons?

a. 2.9270%
b. 3.7580%
c. 4.5240%
d. Need more information (bond price)

22.30.3. A bond with three years to maturity pays a semi-annual coupon at a rate of 8.0% per annum. The bond's current price is $88.00 which implies a semi-annual yield of about 12.954%. The bond's Macaulay duration is 2.7050 years, and its (Macaulay) convexity is 7.7830 squared years. If the yield jumps +2.00%, according to a second-order approximation, which of the following is nearest to the price change?

a. Decrease by ~$0.37
b. Decrease by ~$4.35
c. Decrease by ~$7.76
d. Increase by ~$2.73

Answers here:
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