Convexity
Convexity means a measure of the curvature in the relationship between bond prices and bond yields that demonstrates how the duration of a bond changes as the interest rate changes. Convexity is used as a risk-management tool, and helps to measure and manage the amount of market risk to which a portfolio of bonds is exposed.
In the example above, Bond A has a higher convexity than Bond B, which means that all else being equal, Bond A will always have a higher price than Bond B as interest rates rise or fall.
As convexity increases, the systemic risk to which the portfolio is exposed increases. As convexity decreases, the exposure to market interest rates decreases and the bond portfolio can be considered hedged. In general, the higher the coupon rate, the lower the convexity (or market risk) of a bond. This is because market rates would have to increase greatly to surpass the coupon on the bond, meaning there is less risk to the investor.
Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function, and is calculated by the following equation:
Convexity Formula
P = bond price
y = yield to maturity in decimal form
T = maturity in years
CFt= cash flow at time t
The equation for duration can be improved by adding the convexity term:
Calculating the Change in Bond Prices with Interest Rates Using Duration + Convexity Adjustment
∆y = yield change
∆P = bond price change
Convexity can also be estimated with a simpler formula, similar to the approximation formula for duration:
Convexity Approximation Formula
Convexity = P+ + P- – 2P0
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2 x P0(Δy)2
P0 = bond price
P- = bond price when interest rate is incremented
P+ = bond price when interest rate is decremented
∆y = change in interest rate in decimal form
However, that this convexity approximation formula must be used with this convexity adjustment formula, then added to the duration adjustment:
Convexity Adjustment Formula
Convexity Adjustment = Convexity x 100 x (Δy)2
∆y = change in interest rate in decimal form
Hence:
Bond Price Change Formula
Bond Price Change = Duration x Yield Change + Convexity Adjustment
Convexity is usually a positive term regardless of whether the yield is rising or falling, hence, it is positive convexity. However, sometimes the convexity term is negative, such as occurs when a callable bond is nearing its call price. Below the call price, the price-yield curve follows the same positive convexity as an option-free bond, but as the yield falls and the bond price rises to near the call price, the positive convexity becomes negative convexity, where the bond price is limited at the top by the call price. Hence, similar to the terms for modified and effective duration, there is also modified convexity , which is the measured convexity when there is no expected change in future cash flows, and effective convexity , which is the convexity measure for a bond for which future cash flows are expected to change.