On the White Board - March 2010

Mar 15, 2010

Fixed Income Performance Attribution: Curve Change Returns

Performance attribution has become an integral part of the investment process. It serves several purposes, one of which is to ascertain that the realized sources of returns correspond to those set forth in the investment strategy. Our approach for fixed income attribution achieves this objective as we construct a multiple common factor model, where the factors are chosen to reflect active decisions made by portfolio managers. There will always be a residual return that we cannot attribute to common factors. This residual is treated using the Brinson asset grouping approach. Hence, we have a hybrid performance attribution model. In this column, we discuss the return contribution due to changes in the government yield curve, highlighted in blue color as the first factor in Figure 1 below.

Figure 1: Hybrid Performance Attribution Model

Traditionally, returns due to curve changes have been captured by effective duration. While effective duration is an informative statistic, it is important to recognize that it only measures the price sensitivity to a parallel shift in interest rates and that actual curve changes are in general non-parallel. To remedy this shortcoming, we use key rate durations to measure sensitivity to non-parallel shifts in the yield curve. Durations are based on a first-order Taylor series expansion of price sensitivities due to interest rate changes. Since the relation between bond prices and interest rates is non-linear, we include convexity which is captured by the second-order term. While convexity only makes a small contribution in most instances, some portfolio strategies, such as bullet versus barbell, are explicit convexity strategies.

Our model for returns due to yield curve changes thus consists of two parts: a return due to a parallel shift and a return due to a reshaping of the curve. The parallel shift return is computed as the sum of the return from duration and convexity exposure

where D is effective duration, C is convexity, and the average change in interest rates is taken over our chosen key rates. To compute the contribution of the reshaping of the curve over and beyond the parallel shift, we use key rate durations. We model the return due to reshaping as

where KRD_i is the key rate duration for node i and K is the number of key rates. The total return contribution from movements in the government yield curve is the sum of the two equations above. (Note that the contribution from effective duration cancels.)

As a concrete example, we can consider the return on a U.S. Treasury benchmark between October 30 and November 30, 2009. The top panel in Figure 2 below shows the yield curves that prevailed as of the two month-ends. The bottom panel shows the difference between the two yield curves.

Figure 2: Yield Curves and Yield Curve Changes

Although in general yields went down, the shift is far from parallel. This means that we may make a significant error by approximating returns due to yield curve changes by effective duration only. Figure 3 shows both the effective duration and the key rate durations of this specific Treasury benchmark.

Figure 3: Effective Duration and Key Rate Durations

The blue bars display the key rate durations at each key rate node. The key rate nodes are 0.5, 2, 5, 10, 20, and 30 years. The red bar plots the effective duration on both axes. The total return on the benchmark was 1.40%. From our formulas above, we find that the part of the return that can be explained by effective duration is 0.66%. Using key rate durations and convexity in addition, the return due to Treasury curve changes is 1.03%, and we explain 26% more of the benchmark return. When we include key rate durations, convexity, and the remaining factors in our fixed income factor model, such as the curve carry component, we can explain almost all of the total benchmark return. Failing to take non-parallel yield curve movements into account, leaves a significant residual.