This question, motivated by a forum query, is a variation on an example given in the FRM Handbook (page 11); the spreadsheet contains solutions for both. I thought this would give practice on several bond ideas.
Question:
Consider a 10-year zero-coupon Treasury bond with face value of $100 trading at a yield of 5%. Questions:
a. What is the bond’s price under (i) annual, (ii) semi-annual and (iii) continuous compounding?
b. For each of these compound frequencies, what is modified duration?
c. What is the dollar duration and how can we describe it in calculus terms?
d. For each of these compound frequencies, what is convexity?
e. What are the units of modified duration and convexity?
f. For each of these compound frequencies, what is the estimated bond price change associated with a 20 basis point increase in yield?
g. Would a 20 basis point decrease imply a symmetrical change in bond price (i.e., higher price but same change)? If not, why not?
Answer:
a. (i) $61.39, (ii) $61.03, and (iii) $60.65
b. 9.52, 9.76 and 10
c. $584, $595 and $606; dollar duration is the negative first derivative
d. 99.77, 99.94, 110
e. Modified duration is unitless; i.e., first derivative $/% multiplied by 1/$ cancels the dollars leaving a semi-elasticity. Convexity is time^2; e.g., about 99 years squared
f. For an increase in 20 basis points: -$1.157 (annual), -$1.179 (semi-annual) and -$1.2 (continuous)
g. No, the convexity adjustment is always positive. So, for a 20 basis point decrease, the corresponding estimated increase will be higher; e.g., under continuous, the estimate price change given a -20 bps shock will produce an estimate greater than (>) $1.2
