# Fixed-income Attribution - Attribution On Mortgage-backed Securities - Key Rate Durations

Key Rate Durations

For managers who need to account for changes in the shape of the yield curve in detail, a single risk measure for interest-rate sensitivity is insufficient and a more detailed way of measuring changes across the entire term structure is required.

One of the most popular techniques to accomplish this is the use of key-rate durations (KRDs), introduced by Thomas Ho (1992). Ho defines a number of maturities on the yield curve as being the key rate durations, with typical values of 3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25 and 30 years. At each point, we define a duration that measures interest-rate sensitivity to a movement at that point only, with the effect of the duration at other maturities decreasing linearly to the neighboring points.

In other words, a key rate duration measures the effect of a change in the yield curve that is localized at a particular maturity, and restricted to the immediate vicinity of that maturity, usually by having the change drop linearly to zero at neighboring points.

Of course, the yield curve is most unlikely to behave in this way. The idea is that the actual change in the yield curve can be modeled in terms of a sum of such saw-tooth functions. At each key-rate duration, we know the change in the curveâ€™s yield, and can combine this change with the KRD to calculate the overall change in value of the portfolio. In other words,

where the sum is across all key rate maturities.

The sum of an instrumentâ€™s key rate durations is approximately equal to its modified duration. The sum may not be exact because modified duration assumes a flat yield curve, which is seldom the case.

This approach can easily be combined with the earlier decomposition into shift, twist and curvature components to give price changes due to these yield curve movement types. For instance, suppose we know the amount by which the yield curve has steepened at each key rate maturity. Then the return of the MBS due to a steepening Treasury curve is given by