Abstract
VaR and expected-shortfall are extremely useful portfolio risk measures. They are, however, reasonably difficult to compute and often challenging to interpret. Effective use and communication of one’s risk measures nonetheless requires significant insight into their underlying structure. In particular, it is inordinately useful to understand how individual obligors contribute to one’s risk estimates. This is referred to as risk attribution and is, it must be admitted, a non-trivial undertaking. To address this important area and do justice to its complexity, we allocate the entire chapter to this task. We begin with a surprising relationship between risk attribution and conditional expectation, which subsequently motivates development of a general-purpose numerical algorithm. To offer alternatives to this computationally intensive and often noisy approach, we examine two analytical techniques. The first, termed the normal approximation, provides insight into the underlying problem, but is unfortunately not a robust solution. The saddlepoint approximation, conversely, offers an accurate and fast alternative in the one-factor setting; it also enhances our technical understanding of the underlying risk measures. As always, analysis of all approaches are accompanied by detailed derivations and concrete examples.
Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different.
(Johann Wolfgang von Goethe)
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Notes
- 1.
A detailed description of the main conditions for the risk-factor decomposition of the parametric VaR measure are found in Bolder (2015).
- 2.
See Loomis and Sternberg (2014) for more discussion on homogeneous functions.
- 3.
We further assume that both L and X n are absolutely continuous random variables with associated density functions, f L(l) and \(f_{X_{n}}(x)\), respectively.
- 4.
To reconcile with equation 7.9, merely note that the weight in front of each X n is equal to one. This is because we are working in currency units and not return space.
- 5.
This is also referred to as the ceiling function, which maps a real number into the largest adjacent integer.
- 6.
The getY function is used to capture the state variables. As indicated in previous chapters, this is one of those situations where it is convenient to simulate state variables independent of the risk-metric calculation.
- 7.
In Chap. 8 we will discuss, in detail, how these interval estimates are constructed.
- 8.
The coherence of expected shortfall—as described by Artzner et al. (2001)—is a further argument for moving to this risk measure.
- 9.
The choice of p = 2 leads to the L 2 notion of distance, which essentially amounts to a least-squares estimator. Other choices of p are, of course, possible, but this option is numerically robust and easily understood.
- 10.
Naturally, for a quadratic function, the approximation will be perfect.
- 11.
This is the reason we require positivity of m(x, t).
- 12.
- 13.
The moment-generating function is the statistical analogue of the Laplace transform. The density function is recovered through the inverse Laplace transform. See Billingsley (1995, Section 26) for much more detail and rigour.
- 14.
See Daniels (1987) for a useful exposition of the derivation.
- 15.
Both the numerator and denominator of \(K^{\prime }_L(t)\) have a \(e^{t c_n}\) term, but the coefficient on the numerator, c n p n, will, for positive exposures, dominate the p n coefficient in the denominator.
- 16.
In both cases, Python is merely a wrapper around MINPACK’s hybrd algorithms.
- 17.
These require, as one would expect, higher-order derivatives of the cumulant generating function.
- 18.
It is, however, an extremely useful model, given its simplicity, for the purposes of comparison, benchmarking, and model diagnostics.
- 19.
- 20.
The expected-shortfall code is, however, found in the varContributions library. See Appendix D for more details on the library structure.
- 21.
The differences are even smaller if we were to construct confidence intervals for the Monte Carlo estimates. This important question is addressed in the following chapter.
- 22.
One could construct a simple grid to find ℓ α in the t-threshold model and estimate one’s desired loss threshold with interpolation. A 100-point grid could be constructed in approximately five minutes. If one seeks a single loss threshold, it is still probably better to use the optimization approach.
- 23.
This is precisely what makes the saddlepoint approximation inappropriate for a multi-factor model.
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Bolder, D.J. (2018). Risk Attribution. In: Credit-Risk Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-94688-7_7
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