Case 030Fixed income and creditWarm up

A 20-year bond has duration 13 and convexity 220. Estimate its price change for yield moves of plus and minus 150 basis points with duration alone and with convexity, and say which error hurts someone who is short the bond.

The situation

Dhanvikam Bond Fund holds Rs 500 crore of a 20-year government bond with a modified duration of 13 and a convexity of 220. The risk report estimates price moves from duration alone. A hedge fund on the other side of a trade is short the same bond.

Your task

Estimate the percentage and rupee price change for yields up 150 basis points and down 150 basis points, first with duration only and then adding convexity, and explain who is hurt by ignoring convexity.

Quick check

Yields fall 150 basis points. Compared with duration alone, the true gain on the bond is:

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Duration alone says 19.5% either way; with convexity the bond gains 22.0% if yields fall 150 basis points and loses 17.0% if they rise. The convexity term, half of 220 times 0.015 squared, adds 2.47 points in both directions. On Rs 500 crore, duration understates the gain by Rs 12.4 crore and overstates the loss by the same. The short holder carries the mirror image: larger losses than duration shows.

Step 1What do duration and convexity each measure?

Duration is the slope: a 1 percentage point rise in yield cuts the price by about 13%. Convexity is the bend. Think of a car's speedometer and how it changes: speed tells you distance in the next second, but over a minute you need the acceleration too. For small moves duration is enough; for 150 basis points on a bond this long, the bend adds about 2.5 percentage points, which is too big to ignore. ConvexityHow much the price-yield relationship curves. Positive convexity means gains grow faster than losses as yields move. of 220 means the second-order term is half of 220 times the yield change squared.

The relationship

\frac{\Delta P}{P} \approx -D\,\Delta y + \tfrac{1}{2}\,C\,(\Delta y)^2 = -13(0.015) + \tfrac{1}{2}(220)(0.015)^2 = -19.5\% + 2.475\%

D	modified duration, 13
C	convexity, 220
\Delta y	yield change in decimals, plus or minus 0.015

What it says in wordsThe price change is the duration line plus a correction that is always positive and grows with the square of the move.

Step 2How big is the error in each direction?

Work both sides. Up 150: duration says minus 19.5%, the curve says minus 17.025%. Down 150: duration says plus 19.5%, the curve says plus 21.975%. The correction is the same 2.475 points both ways, so duration understates the gain and overstates the loss for anyone long the bond. On Rs 500 crore that is Rs 12.37 crore in each direction: the fund loses Rs 85.1 crore rather than Rs 97.5 crore if yields rise, and gains Rs 109.9 crore rather than Rs 97.5 crore if they fall.

For a bond with duration 13 and convexity 220, the duration line predicts 19.5% either way for a 150 basis point move, while the convex price curve gives plus 22.0% when yields fall and minus 17.0% when they rise.

Yield move	Duration only	With convexity	Long: error	Short: error
+150 bps	-19.50%	-17.03%	loss overstated	gain overstated
-150 bps	+19.50%	+21.98%	gain understated	loss understated

Duration alone misstates the move by 2.47 percentage points in each direction; the long holder's errors are on the safe side, and the short holder's are on the dangerous side.

Step 3Why is the short holder the one who should worry?

A short position has the opposite sign on every term, including convexity. So for the short, duration overstates what it earns when yields rise and understates what it loses when yields fall: a 150 basis point rally costs about 22.0% of face, not the 19.5% its risk report shows. This is negative convexity, the same shape an option seller has, and it gets worse the bigger the move. A risk limit set on duration alone will be breached by more than it expected in exactly the scenario it fears most.

State the limit of the method. The two-term formula is itself an approximation; for moves larger than about 200 basis points on a long bond, higher-order terms start to matter, and a full repricing of the cash flows is the honest check. Bonds with embedded calls behave differently again, because their convexity can turn negative when yields fall.

Where candidates lose it

The usual loss is a sign error on convexity: subtracting the term when yields fall because the price is going up. The convexity term is squared, so it is positive whichever way yields move, and it always helps a long bond.

The second is forgetting the half. Using 220 times 0.015 squared gives about 4.95 points of correction instead of 2.475, doubling the convexity effect.

What the interviewer asks next

What yield move makes the convexity term equal to 10% of the duration term?
How would you hedge the short's convexity exposure?
Why does a callable bond show negative convexity when yields fall?

Company names and figures are illustrative.