Abstract
Performance and risk attribution are the heart of this book. After several chapters of preparation, we are finally in a position to begin with the performance dimension. Performance attribution centres around one principal question: where did the return on my portfolio come from? This is an easy question to ask, but it turns out that it is surprisingly difficult to answer. It is difficult because it essentially involves breaking down, or decomposing, the return of a portfolio over a given period into different buckets. Each of these buckets must describe an alternative dimension of the return.
Measure what is measurable, and make measurable what is not so.
Galileo Galilei
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Notes
- 1.
The additive risk-factor-based decomposition developed in a previous chapter provides the starting point for such a decomposition. As we will see in the following discussion, there are numerous alternative directions that can be taken within this framework.
- 2.
It is even closer when the overall return is adjusted for the approximately 0.5 basis points of 1-day carry return.
- 3.
In a subsequent chapter, we will introduce a version of this idea and apply it to long and short duration positions relative to the benchmark.
- 4.
Our treatment of external cash flows is, more or less, illustrative. For a more exhaustive discussion, see Bacon [1].
- 5.
Without an adjustment, one could fabricate any desired portfolio return by merely arranging for the necessary injection or withdrawal of funds into the portfolio.
- 6.
It is important to note that within each approach, there are a number of different variations and sub-methods.
- 7.
The value-weighted approach that we will consider is called the modified-Dietz method.
- 8.
Moreover, more information is required for the time-weighted approach. This may seem like a trivial point, but some external managers provide only monthly valuations. In such a situation, computation of time-weighting returns is not an option.
- 9.
As a general remark, the differences between the two approaches will tend to be larger when the injection or withdrawal is large relative to the size of the underlying portfolio.
- 10.
There are naturally other possible directions one may take to attributing return—here we show you a general approach with a number of alternatives. Alternative approaches are legion. A few possibilities may be found in Campisi and Spaulding [4], Menchero and Hu [14], Hansen and Andersen [10], Knight and Satchell [12] or, when faced with limited data, Colin [6].
- 11.
See Colin [5] for another, complementary perspective on fixed-income performance attribution.
- 12.
If we use the precise exact carry term from the Taylor expansion derived in the previous chapters, we arrive at
$$\displaystyle\begin{array}{rcl} \text{Carry return}& =& \ln (1 + y)\varDelta t, \\ & =& \ln (1 + 2.61\,\%) \cdot \left ( \frac{30} {365}\right ), \\ & =& 21.18\text{ bps.} {}\end{array}$$(7.25)We see, therefore, that the difference associated with our approximation is only a fraction of a basis point.
- 13.
This is assuming, as we do, that time increments are always positive!
- 14.
OA spread stands for option-adjusted spread. There need not be any optionality in the underlying bond, although the term “option-adjusted” is used because the concept originated from instruments with embedded optionality such as mortgage-backed securities and callable bonds—in the presence of optionality, a model is required to compute the OA spread.
- 15.
Of course, there may be some additional compensation for favourable movements in the credit spread. This compensation may also be unfavourable. The credit-carry compensation, in contrast, is always positive.
- 16.
The notion of a risk-free issuer has come increasingly under fire and, thus, one could imagine situations where one might even be tempted to use the swap or Overnight Interest-Rate Swap (OIS) curve as the reference curve in a given economy.
- 17.
There is no clear correct way to accomplish this task. Instead, there exists a number of reasonable alternatives.
- 18.
The following discussion is a bit involved, but recall that one of our basic principles was to avoid the use of black boxes.
- 19.
We can see from Eq. (7.33) that the Nelson-Siegel model is constructed with smooth mathematical functions. As a consequence, it will never be sufficiently flexible to capture the uneven form of the true US Treasury curve—this is typical and it is to be expected that all models will lead to some level of estimation error.
- 20.
This is not such a strong assumption for relatively small time intervals as these are smooth functions. Indeed, for f 0(t, T), there is no assumption involved as it is a constant function: f 0(t, T) = 1 for all value of t and T. Moreover, since the first factor accounts for roughly 90 % of yield-curve movements in the Diebold and Li [7] dynamic Nelson-Siegel setting, we probably shouldn’t be too worried.
- 21.
With a bond duration of 5 and a yield error of four basis points, the discrepancy will be about 20 basis points.
- 22.
Again, the solution to this problem descends into a high level of detail, but we promised you no black boxes.
- 23.
Imagine a situation where the curve is essentially flat across all sectors except for a slight increase of 10 basis points in the 10-year sector—such a yield curve movement is unlikely, but entirely possible. The contribution of all other key-rate durations will be essentially zero as the yield change is zero. Only the 10-year sector will contribute to the curve return leading to an under- or overestimate of the overall curve return.
- 24.
- 25.
- 26.
Note that although one can easily use the modified and spread duration to separate the curve and credit return, such a decomposition is a bit trickier with convexity. This is due to the fact that the yield movement enters non-linearly in the convexity computation.
- 27.
A good source for the general discussion on attributing foreign-exchange rate movements can be found in Karnosky and Singer [11].
- 28.
The multiplicative form of Eq. (7.44) is not terribly practical for performance attribution purposes. It simply does not permit one to isolate the return contribution from foreign-exchange movements from the other risk factors.
- 29.
Recalling the precise, but perhaps less intuitive, expression from Chap. 3, we have,
$$\displaystyle\begin{array}{rcl} r_{\text{CAD}}& \approx &\underbrace{\mathop{\frac{110.48-111.72} {111.72} }}\limits _{-110.99\text{bps.}} +\underbrace{\mathop{ \frac{1.2296-1.2188} {1.2188} }}\limits _{88.61\text{bps.}} \\ & & +\underbrace{\mathop{ \left (\frac{110.48-111.72} {111.72} \right ) + \left (\frac{1.2296-1.2188} {1.2188} \right )}}\limits _{0.98\text{bps}}, \\ & \approx &-23.36\text{ bps}, {}\end{array}$$(7.46)which coincides exactly with Eq. (7.44). The additional interaction term is typically ignored because it is typically small and generally hard to interpret.
- 30.
The optimization approach is based on a linear-programming framework described in Appendix B.
- 31.
The error on the benchmark approximation is almost zero. This is quite often, although of course not always, the case due to the large number of securities in the benchmark. With so many securities of different types, the various approximation errors often act to cancel one another out. This cancellation effect is smaller for a typical portfolio, as it is generally constructed with a relatively smaller number of securities.
- 32.
A curve flattening typically involves increases in the yield curve at the short end offset by decreases in the long end. The short duration position, at the short end, will outperform the benchmark in the face of yield increases associated with a curve flattening. Simultaneously, the long duration position at the long end of the yield curve will outperform the benchmark as rates fall.
- 33.
This is the reason that it is useful to understand the overall spread duration, or spread sensitivity, in one’s portfolio. Spread exposure to short-duration bonds is not equivalent to similar exposure to long-duration bonds.
- 34.
The ex-ante tracking error of this portfolio would, quite likely, be quite sizeable given the large spread exposure not only to Austria, but also to the other un-invested sovereigns in the benchmark.
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Bolder, D.J. (2015). Basic Performance Attribution. In: Fixed-Income Portfolio Analytics. Springer, Cham. https://doi.org/10.1007/978-3-319-12667-8_7
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