Another approach to assessing interest rate risk of a bond is to estimate the percentage change in price against a change in the benchmark yield curve such as the government par curve. The effective duration is defined as the sensitivity of a bond’s price against a change in a benchmark yield curve. The formula for calculation of effective duration (EffDur):

$$EffDur=\frac { { PV }_{ – }-{ { P }V_{ + } } }{ 2\times \Delta curve\times { PV }_{ 0 } } $$

Note: Although they appear similar, the approximate modified duration and effective duration are different. The modified duration is a *yield duration *statistic that measures interest rate risk in terms of a change in the bond’s own yield-to-maturity (ΔYield). On the other hand, effective duration is a *curve duration *statistic that measures interest rate risk in terms of a parallel shift in the benchmark yield curve (ΔCurve).

## Bonds with Call, Put and Convertible options

The effective duration is an essential measure of interest rate risk for complex bonds, such as bonds with embedded call, put or convertible options. The duration of a callable bond is not the sensitivity of the bond price to a change in yield-to-worst such as the lowest of the yield-to-maturity, yield-to-first-call, yield-to-second-call, and so forth.

The problem is that future cash flows are uncertain because of the volatility of future interest rates. Therefore, callable bonds do not have a well-defined internal rate of return (yield-to-maturity), and yield durations statistics such as Modified and Macaulay durations do not apply. Then, the effective duration is the appropriate duration measure.

## Mortgage-backed Bonds

Another fixed income security whose yield duration statistics such as Modified and Macaulay durations are not relevant is a mortgage-backed bond. As seen in the previous reading, the cash flows of a mortgage-backed bond are contingent on homeowners’ ability to refinance debt at a lower rate.

## Question

For which of the following bonds should we

least likelyuse effective duration?A. A floating-rate bond

B. A convertible bond

C. A residential mortgage-backed security

SolutionThe correct answer is A.

Effective duration is essential for the measurement of interest rate risk for complex bonds, such as bonds with embedded call, put or convertible options but also for mortgage-backed securities. However, it is not as useful for floating-rate bonds.

*Reading 55 LOS 55c:*

*Explain why effective duration is the most appropriate measure of interest rate risk for bonds with embedded options*