Convexity Adjustments

Since bond prices are not linear, the duration estimate requires a convexity adjustment to make it more accurate. An exact adjustment is equal to the actual price change minus the duration estimate price change. The adjustment is not too difficult and greatly increases the accuracy of the duration estimate. Convexity adjustments are most important for a large increase in yields, when convexity is highest.

What can you conclude about the sign of the convexity adjustment for non-callable bonds?

Correct! The convexity adjustment is positive, since duration underestimates bond price changes for non-callable bonds.

Incorrect. It cannot be negative for non-callable bonds.

Incorrect. Keep in mind duration underestimates price changes for non-callable bonds.

Adding the convexity adjustment to the duration estimate results in the adjusted estimate for the percentage change in the bond price. The price change estimate as a percentage is equal to: $$\displaystyle \% \Delta PV^{Full} \approx -(\text{AnnModDur} \times \Delta \text{Yield}) + \left[ \frac{1}{2} \times \text{AnnConvexity} \times (\Delta \text{Yield})^2 \right] $$

In addition to bonds, what other type of investment will exhibit convexity?

Incorrect. Common stocks are not fixed income securities.

Incorrect. Although preferred stocks generally pay regular dividends, they are not priced like bonds because the dividends can be skipped. They also do not pay principal.

Correct! All fixed income securities exhibit some degree of convexity. Mortgage-backed securities are fixed income securities.

To summarize: [[summary]]

The convexity adjustment (CA) is computed with the annual convexity measure and the change in yield as follows: $$\displaystyle CA = \left[ \frac{1}{2} \times \text{AnnConvexity} \times ({\Delta \text{Yield}})^{2} \right]$$ Convexity is a second order measure, while duration is a first order measure. Therefore, the actual convexity numbers are a bit unique. Be careful not to confuse the convexity measure with the convexity adjustment. Convexity as a measure is comparable to duration, however, the convexity adjustment is comparable to the duration estimate. In other words, the estimated change in price from convexity is estimated by the convexity adjustment.

As an example, consider a 4.00%, 30-year non-callable bond that is currently priced at par and has a modified duration of 17.380. The convexity for this bond is 421.50. An investor is interested in the estimated change in price for a 50 bps decrease in interest rates. The estimated percentage change in the bond price is equal to: $$\displaystyle \% \Delta PV^{Full} \approx -(17.380 \times -0.005) + \left[ \frac{1}{2} \times 421.50 \times (0.005)^2 \right] \approx 0.0922 $$ The estimated price change based on duration and convexity is 9.22%. As a comparison, the _actual_ price change of this bond for a 50 bps decrease in rates is 9.24%. The two are very close!

It is always positive

It is always negative

It can be positive or negative

Common stock

Preferred stock

Mortgage-backed securities

Continue

Continue

Continue

Continue

Continue

Continue