The intent of the question is fine enough, but we may as well use this to reflect on different durations - David
Question:
[source: sample FRM Full exam Question 3]. The following table shows the composition of the GARP Bond Fund.
E1.03 [source] What are the portfolio duration and portfolio yield of the fund?
a. 14 years, 46.1%
b. 2.3 years, 7.5%
c. 2.3 years, 7.7%
d. 4.4 years, 15.4%%
E1.03b [my adds] What duration is implied by “Duration in years”?”
E1.03c. Why is the question incorrect to solve for portfolio duration as a weighted average of the durations listed?
E1.04c. [hard] Under what restrictive condition would the answer be correct?
E1.04d. The question assumes that portfolio duration is a weighted average of component durations; is this also true of convexity?
E1.04e. Could you tweak the durations to (i) maintain the portfolio duration yet (ii) increase/decrease the portfolio convexity? If so, how?
Answers:
E1.03 [source] What are the portfolio duration and portfolio yield of the fund?
Correct: B.
E1.03b [my adds] What duration is implied by “Duration in years”?”
When duration is represented in years, it refers to Macaulay duration. Macaulay duration equals the time-weighted present value of cash flows divided by price, where the weights are years to receipt. For example, for a 10 year zero coupon bond, Macaulay duration = 30 (years) but Modified duration is less.
E1.03c. Why is the question incorrect to solve for portfolio duration as a weighted average of the durations listed?
Porfolio duration is weighted modified duration of components. Modified duration is the sensitivity measure. Note above, in the final column, I solved for portfolio duration (final column) assuming semi-annual compounding.
E1.04c. [hard] Under what restrictive condition would the answer be correct?
If returns are continuous! Note Modified Duration = Macaulay duration / (1+yield/k) where k = periods per year. Only under the special case of continuous compounding do the durations equal each other as yield/k approaches a limit of 0 as k -> infinity.
E1.04d. The question assumes that portfolio duration is a weighted average of component durations; is this also true of convexity?
Yes, both portfolio duration and convexity are weighted average of the components.
E1.04e. Could you tweak the durations to (i) maintain the portfolio duration yet (ii) increase/decrease the portfolio convexity? If so, how?
Yes, because convexity scales approximately (not exactly) with the square of maturity; a barbell has greater convexity than a bullet portfolio. To maintain duration but increase convexity, we could move the portfolio more toward a barbell portfolio (i.e., increase longer maturities and decrease shorter maturities while keeping average duration the same). To maintain duration but decrease convexity, we could “tighten” the portfolio more toward a bullet portfolio (i.e., increase shorter maturities and decrease longer maturities in order to “converge them” on the average duration).
