Popular attribution models
In this chapter I describe a number of widely used fixed income attribution models. Attribution approaches are often described in terms of one or other of these models.
23.1 THE CAMPISI MODEL
The Campisi model was first proposed in a paper published by Stephen Campisi (Campisi, 2000). It is perhaps the simplest possible security-level model for fixed income attribution (see Figure 23.1).
Figure 23.1 Breakdown of effects in the Campisi model
The Campisi model can be applied at the portfolio, sector or security levels.
- Income return: Campisi’s original paper describes income return as generated by coupons; specifically, as the average annual coupon divided by the average beginning market price. This is actually running yield, as it omits any pull-to-par effects.
- Treasury effect combines the returns from parallel and non-parallel curve movements into one source of return, given by − MDδytreasury. The change in treasury yield δytreasury is given by the change in the treasury curve at the tenor point equal to the duration of the portfolio.
- Spread effect is the effect of changes in the average spread δyspread, given by − MDδyspread. Treasury and spread durations are assumed to be equivalent.
- Selection or security-specific return is given by total return, minus the sum of the previous three effects.
There is no residual term. Any unexplained returns are aggregated into selection effect.
23.2 DURATION ATTRIBUTION
The duration model provides slightly more information than the Campisi model. Treasury effects, which are returns generated by changes in the level of the sovereign curve, are decomposed into returns made by parallel movements in the curve (duration return), and those from non-parallel movements (curve return) (see Figure 23.2). Modified duration measures sensitivity to both types of yield movement, so this effect gives its name to the model.
As noted in Chapter 10, the definition of how parallel curve movements should be calculated varies widely between practitioners.
There is no formal definition of the duration model in the attribution literature, but it is one of the most widely used.
Figure 23.2 Duration attribution
23.3 THE TIM LORD MODEL
The Tim Lord model is similar to the Campisi and Duration models, but includes a number of more detailed effects (Lord, 1997; Figure 23.3).
- Income return is given by change in accrued interest and other income cash flows. As with the Campisi model, this is more correctly described as the return due to running yield. Unlike the Campisi model, pull-to-par and roll-down effects are measured here as calendar returns (see below).
- Duration return is defined as price return due to changes in the treasury par curve, measured as the product of the negative modified duration of each security times the change in yield of its duration matched treasury (a synthetic treasury bond with the same modified duration as the target bond).
- Shift return is generated by parallel shifts in the treasury curve. The shift is measured at the five-year maturity point on the curve.
- Twist return is the portion of return generated by non-parallel shifts in the treasury curve.
- Spread return is further decomposed into sector return and issuer-specific return.
- Sector return is calculated by recording the change in spread for the security’s defined sector, which is the average OAS change for all benchmark securities in that category.
- Issuer-specific return is the difference between spread return and sector return.
- Residual return is any remaining return that is unexplained by the sum of the previous effects.
Lord notes that the model can generate substantial residuals if a security’s pricing model is wrongly specified. However, this is easily addressed if the attribution system can use specialised pricing models, rather than a one-size-fits-all approach.
Most security-based models are variants of the Tim Lord approach.
23.4 KEY RATE ATTRIBUTION
Modified duration measures the sensitivity of price to changes in overall levels of interest rates. For instance, if the yield curve moves downwards by 10 basis points, a bond with a 10-year modified duration will generate a return of 100 basis points, but a bond with a two-year modified duration will generate only 20 basis points.
A key rate duration measures the price sensitivity of a security to a change in its yield curve at a single maturity, rather than to movements in the curve as a whole. It is therefore well suited to detailed analysis of the return of securities with cash flows spread over a range of maturities, such as amortising bonds and MBS, since the returns of these securities are affected by movements at many points on the curve, rather than being dominated by the curve level governing the value of the main principal payment.
Key rate duration attribution usually follows the same pattern as Campisi and Tim Lord models. The difference is that returns from changes in the yield curve are represented as returns from different maturities along the yield curve, rather than from global movements such as shift and twist (see Figure 23.4).
Figure 23.4 Key rate attribution
23.5 TOP-DOWN ATTRIBUTION
Top-down (or mixed, or balanced) attribution forms a useful approach to attribution on fixed income portfolios where over- or underweighting of sectors forms part of the investment process, in addition to fixed income investment decisions (see Figure 23.5). Top-down attribution forms a combination of the successive portfolio methodology, of which Brinson attribution is the simplest example, and the successive spread methodology, which underlies the various models shown earlier. The approach is described in the GRAP paper (Groupe de Recherche en Attribution de Performance, 1997).
In practice, the results from a top-down attribution are the same as for one of the previous analyses, but with an additional asset allocation term. Results are usually presented down to the sector level only.