9.34 Properties of Convexity
Refer back to the Price/Yield curve to illustrate some of the following characteristics (you may wish to re-draw the curve).
- Convexity is inversely related to coupon.
The price-yield curve flattens as coupons increase; a flat curve reflects less interest rate sensitivity.
Higher coupons are less sensitive.
Given a yield and term-to-maturity, the lower the coupon, the greater the convexity.
- Convexity is positively related to duration.
The convexity of a bond increases at an increasing rate as duration increases. Duration is linear; convexity is curvilinear.
- The longer the duration, the more substantial the gain from convexity.
This is because the numerator in the duration formula is a function of “P,” while in the convexity formula, it is a function of “P (P + 1).”
Given term-to-maturity, a Zero-coupon bond will have the greatest convexity.
Convexities of Various Bonds at a YTM of 10%
| 0% Coupon | 8% Coupon | 14% Coupon | |
| 5-Year Maturity | 24.79 | 19.58 | 17.44 |
| 30-Year Maturity | 768.60 | 167.56 | 148.28 |
- Convexity (like Duration and Price) is inversely related to yield.
As the yield increases (decreases), the dollar duration, i.e., Modified Duration × Initial Price, of a bond decreases (increases) – because Duration has.
The price yield curve is greater (steeper) for lower yields than for higher yields.
“Convexity is always good news.” Convexity always adds to the Duration-based price estimate – in either yield direction.
- Two bonds may have the same duration, but different convexities.
Imagine, for example, a Zero-coupon bond with a short-term duration and a longer-term positive-coupon bond with the same Duration. They would share the same point of tangency, but one curve (the Zero) would be “curvier,” i.e., more convex.
We should maximize convexity in order to capitalize on large, expected decreases in rates.
This is because the bond will go up more (or down less) in dollar price than a bond whose convexity is lower for large yield changes – due to steeper curvature.
- Noncallable (“Bullet”) bonds have positive convexity.
The actual price will always be higher than the duration-based price estimate.
(Put Bonds have very high convexity because when yields up go, convexity adds a lot to price.)
