###### Matrix Pricing

When fixed-rate bonds are not actively traded, or there is no market price... **Read More**

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 the price of a bond against a change in a benchmark yield curve. Below is 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 with reference to a change in the yield-to-maturity (ΔYield) of a bond. 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).

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 the uncertainty that shrouds future cash flows 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 duration statistics such as Modified and Macaulay durations do not apply. This, then, makes the effective duration the appropriate duration measure.

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.

QuestionAn analyst should

least likelyuse effective duration for a:

- Convertible bond.
- Floating-rate bond.
- Residential mortgage-backed security.

SolutionThe correct answer is

B.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.