The modified duration measures the effect on a bond’s percentage price change given a change in the yield-to-maturity. A secondary effect is measured by the convexity statistic.

## Approximate Convexity

The true relationship between the bond price and the yield-to-maturity (YTM) is a curved (convex) line. The duration (in particular, money duration) estimates the change in bond price along with the straight line that is tangent to the curved line. For small yield-to-maturity changes, there is little difference between the lines but as the change in YTM grows larger, the difference becomes significant.

A convexity measure is used to improve the estimate of the percentage price change.

$$\%ΔPV^{FULL}≈(-AnnModDur×ΔYield)+(\frac{1}{2}×AnnConvexity×(ΔYield)^2)$$

The second bracketed expression is the convexity adjustment:

$$ Convexity \quad effect ≈ \frac{1}{2}×AnnConvexity×(ΔYield)^2 $$

The convexity adjustment is the annual convexity statistic, AnnConvexity, times one-half, multiplied by the change in the yield-to-maturity squared. This amount adds to the linear estimate provided by the duration alone, which brings the adjusted estimate very close to the actual price on the curved line. The sign “≈” is used because the resulting adjusted price is just an estimate.

## Effective Convexity

The effective convexity of a bond is a curve convexity statistic that measures the secondary effect of a change in a benchmark yield curve. Recall that for bonds with somewhat unpredictable cash flows, we use effective duration to measure interest rate risk. In the same way, we use the effective convexity to measure the change in price for a change in the benchmark yield curve for securities with uncertain cash flows. In fact, effective convexity is a secondary curve convexity statistic that measures the secondary effect of a change in the benchmark yield curve.

$$Effective \quad convexity=\frac{PV_{-}+PV_{+}-2PV_{0}}{(ΔCurve)^2PV_{0}}$$

Where:

PV_{–} = Price if yield curve declines by Yield (parallel shift)

PV_{+} = Price if yield curve increases by Yield (parallel shift)

PV_{0} = Initial bond price

ΔCurve = Change in yield on the benchmark curve

Note that the difference between the approximate convexity and effective convexity is in the denominator where the former includes the change in the yield-to-maturity squared, ΔYield, while the latter includes the change in the benchmark yield curve squared, ΔCurve.

Here are a few quick facts about bonds with embedded options:

- When the benchmark yield is high, prices of callable and non-callable change almost similarly for interest rate changes
- When the benchmark yield is low, the value of the call option (to the issuer) increases
- Putable bonds always have positive convexity
- Callable bonds exhibit negative convexity

## Zero-coupon Bonds

For a zero-coupon bond, the exact convexity statistic in terms of periods is given by:

$$Convexity_{zero-coupon \quad bond}=\frac{[N-\frac{t}{T}]×[N+1-\frac{t}{T}]}{(1+r)^2}$$

Where:

*N *= number of periods to maturity as of the beginning of the current period;

*t/T *= the fraction of the period that has gone by; and

*r *= the yield-to-maturity per period.

QuestionA bond has a convexity of 120. What is the convexity effect if there is a sudden interest rate change of 50 basis points?

A. 0.015

B. 0.3

C. 0.003

SolutionThe correct answer is A.

\(Convexity \quad effect ≈ \frac{1}{2}×AnnConvexity×(ΔYield)^2\)

\(≈ \frac{1}{2}×120×(0.005)^2\)

*Reading 54 LOS 54h:*

*Calculate and interpret approximate convexity and distinguish between approximate and effective convexity*