Convexity and Convexity Adjustment

Duration provides a linear approximation of the change in a bond’s price with respect to changes in yield. On the other hand, convexity measures the non-linear, second-order effect of yield changes on a bond’s price. It captures the curvature of the price-yield relationship.

While duration estimates price changes linearly, the true bond price-yield relationship is convex. Convexity becomes particularly crucial when considering significant yield changes and for bonds with longer maturities.

Calculating Convexity

Convexity can be calculated using the formula:

\[\%\Delta P_{VFull} \approx ( – \text{AnnModDur} \times \Delta\text{Yield}) + \left\lbrack \frac{1}{2} \times \text{AnnConvexity} \times (\Delta\text{Yield})^{2} \right\rbrack\]

The first term captures the effect from modified duration. The second term represents the convexity adjustment.

Convexity can also be approximated using the following formula:

\[ApproxCon\ = \frac{\left( PV_{-} \right) + \left( PV_{+} \right) – \left\lbrack 2 \times \left( PV_{0} \right) \right\rbrack}{(\Delta\text{Yield})^{2} \times \left( PV_{0} \right)}\]

Factors Affecting Convexity

1. Maturity: Longer maturity increases convexity.
2. Coupon rate: Lower coupon rate increases convexity.
3. YTM: Lower YTM increases convexity.
4. Cash Flow Dispersion: For two bonds with the same duration, the one with more dispersed cash flows will have greater convexity.

Benefits of Convexity

Bonds with greater convexity perform better in both rising and falling yield scenarios, making them less risky for investors. This assumes that the difference in convexity is not reflected in the bond’s price. For large yield changes, a bond’s price will rise more with a decrease in yield and fall less with an increase in yield if it has higher convexity.

Example: Calculating Convexity

Consider a bond that has a term to maturity of 3 years, an annual coupon rate of 2%, a yield-to-maturity (YTM) of 2%, and is priced at 100 per 100 par value.

1. Calculate the modified duration and convexity for the bond at issuance.
2. Calculate ApproxModDur and ApproxCon for the bond using a 10 bp increase and decrease in the yield-to-maturity.

Calculating Modified Duration and Convexity

\[ \begin{array}{c|c|c|c|c|c|c} \textbf{Period} & \textbf{Time to Receipt} & \textbf{Cashflow Amount} & \textbf{Present Value} & \textbf{Weights} & \textbf{Time to Receipt*Weight} & \textbf{Convexity of Cashflows} \\ \hline 1 & 1.0000 & 2 & 1.9608 & 0.01960 & 0.02 & 0.04 \\ \hline 2 & 2.0000 & 2 & 1.9223 & 0.01922 & 0.04 & 0.11 \\ \hline 3 & 3.0000 & 102 & 96.1169 & 0.96118 & 2.88 & 11.31 \\ \hline \textbf{Total} & & & \textbf{100.0000} & \textbf{1.0000} & \textbf{2.94} & \textbf{11.46} \\ \end{array} \]

Annualized Macaulay Duration = 2.94

Annualized convexity = 11.46

Convexity for each period has been calculated as:

\[\text{Convexity}= \text{Time to receipt of cashflows} \times \left(\text{Time to receipt of cashflows} + 1\right) \times \text{Weight} \times \left( 1 + \frac{YTM}{m} \right)^{- m}\]

Where m is the periodicity.

Calculating ApproxModDur and ApproxCon

\[ApproxCon = \frac{\left( PV_{-} \right) + \left( PV_{+} \right) – \left\lbrack 2 \times \left( PV_{0} \right) \right\rbrack}{(\Delta\text{Yield})^{2} \times \left( PV_{0} \right)}\]

\[{PV}_{0} = \frac{2}{1.02} + \frac{2}{{1.02}^{2}} + \frac{102}{{1.02}^{3}} = 100\]

\[{PV}_{-} = \frac{2}{1.019} + \frac{2}{{1.019}^{2}} + \frac{102}{{1.019}^{3}} = 100.288951\]

\[{PV}_{+} = \frac{2}{1.021} + \frac{2}{{1.021}^{2}} + \frac{102}{{1.021}^{3}} = 99.71217249\]

\[\text{ApproxCon} = \frac{100.288951\ + 99.71217249\ – \lbrack 2 \times 100\rbrack}{(0.001)^{2} \times (100)} = 11.2349\]

Question

Which of the following factors most likely increases the convexity of a bond?

1. Higher coupon rate
2. Shorter maturity
3. Lower yield-to-maturity (YTM)

Solution

The correct answer is C:

Lower YTM increases convexity.

A is incorrect: Lower coupon rates, not higher, increase convexity.

B is incorrect: Longer maturities, not shorter, increase convexity.