Fixed-income attribution is the process of measuring returns generated by various sources of risk in a fixed income portfolio, particularly when multiple sources of return are active at the same time.
Importance The risks affecting the return on a bond portfolio, as an example, include the overall level of the yield curve, the slope of the yield curve, and the credit spreads of the bonds in the portfolio. A portfolio manager may hold firm views on the ways in which these factors will change in the near future, so in three separate risk decisions he positions the assets in the portfolio to take advantage of these expected market movements. If all views subsequently prove to be correct, then each decision will generate a profit. If one view is wrong, it will generate a loss, but the effect of the other bets may compensate. The overall performance will then be the sum of the performance contributions from each source of risk.
Attribution is therefore an extremely useful tool in verifying a fund manager's claims to possessing particular investment skills. If a fund is marketed as being interest-rate neutral while providing consistent returns from superior credit research, then an attribution report will confirm this claim. Conversely, if the attribution report shows that this same manager is making non-zero returns from interest rate movements, then his exposure to interest rate risk is clearly not zero and his investment process clearly differs from his stated position.
Fixed-income attribution therefore provides a much deeper level of information than is available from a simple portfolio performance report. Typically, such a report only shows returns at an aggregated level, and provides no feedback as to where the investor's true skills lie. For these reasons, fixed-income attribution is rapidly growing in importance in the investment industry; see .
Sector-based attribution
Among the simplest fixed income attribution techniques is sector-based attribution. This is based on the standard Brinson-Fachler attribution scheme, where the securities in the portfolio and benchmark are divided up into buckets based on their modified duration.
This scheme has the advantage that it is readily understandable, particularly by managers who have an equity background. However, it does not provide a very deep analysis. The overall effects of a parallel change in the yield curve are supplied but there is none of the more detailed analysis supplied by a true fixed-income decomposition.
A useful account of sector-based attribution, with worked examples, is provided in Dynkin et al. (1998).
Yield curve attribution A more widely used approach to fixed-income attribution is to decompose the returns of individual securities by source of risk, and then to aggregate these risk-specific returns over an entire portfolio. Typical sources of risk include yield return, return due to yield curve movements, and credit spread shifts. These sub-returns can then be aggregated over time and sector to give the overall portfolio return, attributed by source of risk.
Sources of return Over a given interval, the return of each security will be made up of return from various sub-returns (see below for explanations)
return due to yield (equivalently coupon, or accrued interest, or running yield); return due to rolling down the yield curve; return due to movements in the reference yield curve; return due to credit shifts; other sources of return, such as option-adjusted spread (OAS), liquidity, inflation, paydown, etc.
First principles versus perturbational attribution To calculate the return arising from each effect, we can reprice the security from first principles by using a pricing formula, or some other algorithm, before and after each source of return is considered. For instance, in calculating yield return, we might calculate the price of the security at the start and end of the calculation interval, but using the yield at the beginning of the interval. Then the difference between the two prices may be used to calculate the security's return due to the passage of time.
This approach is simple in principle but can lead to operational difficulties. It requires accurate pricing formulae including, where relevant, ex-coupon, settlement, and country-specific conventions; security-specific data, such as day-count conventions and whether a bond has a non-standard first and last coupons; accurate inputs to these formulae, including market yields and other variable quantities such as the 90-day bank bill swap rate (BBSW) and consumer price index (CPI) factors for floating rate notes and inflation-linked securities, and regular updates for these quantities; a reconciliation function between existing performance measurement systems and the attribution system
For these reasons, a pricing model-based approach to attribution may not be the right one where data sourcing or reconciliation is an issue. An alternative solution is to perform a Taylor expansion on the price of a security and remove higher-order terms, which gives
Writing the return of the security as
,
this leads to the perturbation equation
where the last term denotes higher-order corrections that may be ignored, and
The terms and measure first- and second-order interest rate sensitivity. These are conventionally referred to as the modified duration and convexity of the security, and are often called risk numbers.
The data requirements for this approach to attribution are less onerous than for the first-principle approach. The perturbation equation does require externally calculated risk numbers, but this may not be a major obstacle, since these quantities are readily available from the same sources as yields and prices. There may also be inherent advantages in this approach with its ability to work with user-supplied risk numbers, since it allows the user to use sensitivity measures from in-house models, which is particularly useful where (for instance) the user has custom repayment models for mortgage-backed securities.
The approach is also self-checking, in that the size of the residual returns should be very low. If this is not the case, there will be presumably be an error in the calculated return or the risk numbers, or some other source of risk will be distorting the returns.
Conveniently, the perturbational approach may be extended to new asset types without requiring any new pricing code or types of data, and it also works for benchmark sectors as well as individual securities, which is useful if benchmark data is only available at sector level.
Modeling the yield curve
Historically, one of the most important drivers of return in fixed-income portfolios has been the yield curve, and many investment strategies are expressed in terms of changes in the curve. Any discussion of fixed-income attribution therefore requires an appreciation of how changes in the curve are described, and their effect on the performance of a portfolio.
If one is only interested in gross changes in the yield curve at a particular maturity, then one can read yields off the various datasets, using interpolation where necessary, and there is no need to model any part of the curve.
If, on the other hand, one wants to describe curve movements in terms used by traders (or to extrapolate), then some form of parameterization is required. The most widely used nomenclature for describing yield curve changes uses the terms "shift", "twist" and "butterfly". Briefly:
shift measures the degree to which a curve has moved upwards or downwards, in parallel, across all maturities twist measures the degree to which the curve has steepened or flattened. For instance, one might measure the steepness of the Australian yield curve as the difference between the 10-year bond future yield and the 3-year bond future yield. curvature (or butterfly, or curve reshaping) measures the degree to which the term structure has become more or less curved. For instance, a yield curve that can be fitted to a straight line exhibits no curvature at all.
Describing these movements in numerical terms typically requires fitting a model to the observed yield curve with a limited number of parameters. These parameters can then be translated into shift, twist, and butterfly movements – or whatever other interpretation the trader chooses to use. This model is often also used for extrapolating credit default swaps (CDS).
Two of the most widely used models are polynomial functions and Nelson-Siegel functions (Nelson and Siegel (1987)).
Here, polynomial functions are usually of the form
where is maturity, are parameters to be fitted, and is the yield of the curve at maturity .
Nelson-Siegel functions take the form
where and are as above, and , , and , are parameters to be fitted Raphael Franco Chaves (2018). Accessible approach to estimation of the Nelson-Siegel Yield Curve, BTRM Working Paper via a least-squares or similar algorithm (see Diebold and Li ; Bolder and Stréliski ): is interpreted as the long run levels of interest rates (the loading is 1, it is a constant that does not decay); is the short-term component (it starts at 1, and decays monotonically and quickly to 0); is the medium-term component (it starts at 0, increases, then decays to zero); is the decay factor: small values produce slow decay and can better fit the curve at long maturities, while large values produce fast decay and can better fit the curve at short maturities; also governs where achieves its maximum.
Svensson (1994) adds a "second hump" term; this is the Nelson–Siegel–Svensson (NSS) model. The additional term is: ,
and the interpretation is similar to and above.
Another generalization of Nelson-Siegel is the family of Exponential Polynomial Model ("EPM(n)") where the number of linear coefficients is free.
Once a curve has been fitted, the user can then define various measures of shift, twist and butterfly, and calculate their values from the calculated parameters. For instance, the amount of shift in a curve modeled by a polynomial function can be modeled as the difference between the polynomial parameters at successive dates. In practice, the Nelson-Siegel function has the advantages that it is well-behaved at long maturities, and that its parameters can be set to model virtually any yield curve (see Nelson and Siegel ).
Factor-based attribution A factor-based model of yield curve movements is calculated by deriving the covariance matrix of yield shifts at predefined maturities, and calculating the eigenvectors and eigenvalues of this matrix. Each eigenvector corresponds to a fundamental model of the yield curve, and each eigenvector is orthogonal, so that the curve movement on any given day is a linear combination of the basis eigenvectors. The eigenvalues of this matrix then give the relative weights, or importance, of these curve shifts. [Phoa (1998)].
Factor models use a large sample of historical yield curve data and construct a set of basis functions that can be linearly combined to represent these curve movements in the most economical way. The algorithm always attributes as much of the curve movement to the first basis function, then as much as possible to the second, and so on. Since these functions roughly correspond to our shift and twist motions, this approach attributes almost all of the curve change to these two modes, leaving a very small contribution from higher modes. Typical results attribute 90% of curve movements to shift changes, 8% to twist, and 2% to curvature (or butterfly) movements. However, the issue that these basis functions may be different from those in which the risk decisions were expressed is not widely appreciated.