Extreme Losses
Measures of extreme risk inform investors about possible losses that could be experienced during difficult times. These measures are widely used in risk management as they provide a straightforward answer to one of the most crucial questions in asset management: how much can I lose.
Extreme risk focuses on the losses that are less frequent but have the largest impact on a portfolio's value. In general, measures of extreme risk approach the notion of loss from the angle of probabilities. They are thus associated with a so-called confidence threshold that is expressed in the form of a percentage. A confidence threshold of 5% for instance corresponds to the losses an investor could experience during the worst 5% of all trading days. The point about confidence thresholds is that they are generally small, but not vanishingly so. This is why measures of extreme risk are so useful in practice: they tell investors about losses that will not occur very often, but still losses that they must be capable of withstanding.
The SP platform provides state-of-the-art estimates of a well-known measure of extreme risk: Conditional Value at Risk (CVaR). This measure of extreme risk corresponds to the average of the losses that one would experience during the worst 5% of all weekly returns. In other words, it measures what a really bad week could look like. The CVaR is an extension of Value at Risk (see [1] and references therein). The originality of the method we employ, which is an extension of Boudt et al. [2], is that it allows for an accurate estimate of CVaR as well as the ability to decompose it into contributions attributable to an instrument's exposure to different risk factors. This makes CVaR not only a mere attribute of the instrument, but also a risk that can be addressed by controlling risk exposures.
Like any measure of extreme risk, CVaR is difficult to estimate since it is based on the average of a small fraction of available returns. This problem is illustrated in the figure below. If one considers all returns of an instrument (the blue curve) over a given period of time, the estimation of CVaR only uses a small fraction of the returns at the extreme left of the distribution. Because most instruments have a limited history, the returns available for the estimation of CVaR are often sparse. This makes the evaluation unstable and sensitive to the period chosen. This problem is shared by all measures of extreme risk and consequently, has received much attention from the academic community.
The solutions that have been developed to mitigate the issue of data scarcity use the probability distributions of the instrument returns to generate more data points. These distributions are parametrized to mimic the distributions of actual instruments. Because there is no limit to the number of returns that can be generated using these distributions, the precision of the extreme risk measures evaluated from them is no longer problematic. These so-called "parametric" methods, if correctly specified, are shown to produce better out-of-sample estimates of extreme risk than those obtained from historical data alone. The problem with parametric methods is that the distribution itself must be capable of mimicking the true distribution of returns. Parametric methods somewhat replace the problem of data scarcity by the problem of modelling and parametrizing the distribution of returns.
Scientific Portfolio has developed a parametric method based on the Cornish-Fisher expansion that has been shown to produce particularly accurate returns distributions for a wide range of financial instruments. Because this distribution can reproduce the first four moments of returns as parameters, it is more accurate at describing extreme returns than older methods that use distributions with the first two moments only. These older methods, also referred to as the variance-covariance methods, commonly assume that returns are normally distributed and use a simple normal distribution to mimic an instrument's returns. However, empirical evidence shows that financial returns often exhibit significant skewness and kurtosis; consequently, assuming a distribution with heavier tails can significantly improve the estimation of extreme events, a crucial feature in the present context.
The problem with a more accurate distribution is the evaluation of the parameters. The evaluation of four moments requires more data than the evaluation of the mean and variance only. If the objective of the parametric method is to mitigate the problem of data scarcity, it is not exactly ideal if the parameters to be fitted require a lot of data. In fact, it was shown that for the same amount of data, fitting the parameters of a Cornish-Fisher distribution yields better estimates of CVaR than historical data alone, so the incentive to use a more accurate distribution remains intact. Also, in the context of the Scientific Portfolio platform, we benefit from our risk model which makes it possible to produce accurate simulations of actual returns over long periods of time, as we explain in a section below. This further improves the quality of the fits, which eventually improves the precision of the CVaR estimates.
Scientific Portfolio takes a two-step approach for the evaluation of a portfolio's CVaR. First, we use the long-term factors of the Scientific Portfolio risk model to create a longer history based on the instrument's exposure to these factors. We then use these returns to estimate the parameters of the distribution of returns. This is done by using a parametric method that characterizes portfolio returns with an inverse Cornish-Fisher distribution, allowing to reliably capture the skewness and excess kurtosis found in the distribution of returns. The method we use is based on Maillard [3]. Once the parameters of the distribution are fitted, we use the method from Boudt et al. [2] to decompose CVaR into contributions attributable to the instruments' exposures to risk factors.
Methodology
The Inverse Cornish-Fisher expansion
The inverse quantile function is a key element in the estimation of extreme risk measures. For a given instrument
where
The definition of the inverse quantile function is sometimes very involved. This is where the Cornish-Fisher approximation is very useful: it makes it possible to approximate the inverse quantile function of any distribution using a simple linear combination (see [4]):
where denotes the standard normal distribution function and the variables and correspond to the mean and standard deviation of the returns. Here, the coefficients are all functions of two simple parameters and where and . The definition of , and as functions of and are those from [5]:
The parameters and are also known as the cumulants of the distribution . They constitute the key ingredient of the Cornish-Fisher expansion.
One key observation made in [3] regarding cumulants is that unless the distribution of the returns is known, they must be fitted from the distribution of returns. Following the treatment in Lamb et al. [5], the parameters and are fitted by solving the following optimization problem:
where and denote the sample skew and kurtosis, respectively. The skew and kurtosis are indeed the only parameters coming from the sample. Once fitted, the parameters can be re-injected into the above formulas to obtain the inverse quantile function of the distributions of returns.
At this point, it is important to remark that the Cornish-Fisher approximation applies to many, but not all distributions of returns. The conditions for which the approximation is applicable depend on the sample skewness and kurtosis. Broadly speaking, the kurtosis should be smaller than twenty and the absolute value of the skewness smaller than three. A particularly good treatment of the conditions for which the Cornish-Fisher expansion applies is given in [5]. Rather than using the historical return distribution which may be severely limited and thus negatively impact the estimation, we leverage Scientific Portfolio's proprietary risk engine. Accordingly, we separate the portfolio return into its systematic and idiosyncratic components
where using the notation of [6] , where represents the portfolio weights, is the time-dependent instrument characteristics, specifies the mapping from instrument to loadings (target of the estimation procedure) and designates time-dependent statistical factors. Using the systematic return allows us to significantly extend the history of the newer instruments by approximating their return with a linear combination of their characteristics – which usually feature a significantly longer history. To adjust for the idiosyncratic return, we approximate it with the first empirical moment. By combining these two components, we obtain a long history of simulated portfolio returns, which is then suitable for VaR and CVaR.
Another key point is concerned with the convergence of the above optimization program. There is in fact no guarantee that the solution of the above function exists or is unique, as the invertibility of the above function has never been demonstrated. Only empirical evidence suggests that it is actually the case (see [3]).
CVaR and the Cornish-Fisher expansion
As we have explained, given the weekly returns of a portfolio, and setting the confidence threshold to , the estimation of the inverse quantile function associated with the distribution of returns indicates that a loss greater than is to be expected once a year. In other words, it provides the entry-point of the worst possible losses that could happen with a probability of .
From an investor's point of view, this definition is a little bit troubling as it only provides a lower bound for the loss. CVaR actually removes part of the uncertainty around this definition by providing the average of the worst % losses. It thus not only provides information on the lower bound of the losses to expect, but the entire distribution of losses past the chosen confidence threshold.
Mathematically, the average of the worst αα% is also obtained from the inverse quantile function
Using the definition of the Cornish-Fisher expansion in the above formula yields
where denotes the lower partial moment integral of the standard normal distribution. Because the values of do not depend on returns, the only input required to compute CVaR are indeed the parameters and .
Decomposition of extreme risk
When a portfolio contains NN assets, its returns become a weighted superposition of the returns from each individual asset:
where is the individual asset returns. Note that the above formula also applies to the case where corresponds to risk factors and the weights to the portfolio's exposure to risk factors.
When the returns are defined by the above formula, the parameters associated with the distribution of the portfolio returns become functions of the weights:
where corresponds to the mean of and is the covariance matrix between the . Note that the parameters and are also functions of the weights, although they need to be fitted from :
To measure the contributions of each instrument (or risk factor) to the extreme risk of an instrument, we must decompose extreme risk using the Euler formula. This decomposition requires the derivative of the risk measure with respect to a portfolio's weights. For a risk measure , such as CVaR, the Euler decomposition is:
Following the notation of Boudt et al. [2], the contribution of asset to the risk measure equals
The percentage contribution, , can then be shown as
We can apply these formulas to calculate the decomposition of CVaR with respect to the weights. Using the above formulas, the partial CVaR contribution is given as
where . All terms are analytically computable using the equations above, with the exception of and that appear when evaluating and . Indeed, we have:
To approximate and , we proceed in two steps. First, we perturb the weights of the component to obtain a slightly modified portfolio and associated return . The skew and kurtosis of this modified series of returns is then used in the above optimization program to evaluate a pair of parameters and . We then repeat this calculation for another modified portfolio to obtain another set of parameters and . Finally, the partial derivatives of and with respect to the weights are recovered as a central difference:
Once evaluated, the above derivatives can be injected into the above formulas to obtain the contributions to CVaR coming from each weight.
Comparison of the Cornish-Fisher method to the historical method
To illustrate the benefits of our methodology, we create synthetic portfolios whose returns have known distributions and hence for which the CVaR is also known. In this way, we can measure the performance of various CVaR estimations against the actual realized value and demonstrate which method is superior.
We consider portfolios with a gradually increasing number of assets and compute the extreme risk decomposition for each portfolio using both the historical and Cornish-Fisher methods.
Our results suggest that the Cornish-Fisher reliably delivers a lower relative error with a lower variance between the estimates. Additionally, while the estimation results that arise from using the historical method deteriorate as the number of assets within the portfolio increases, Cornish-Fisher does not exhibit sensitivity and maintains its performance.
References
1 Carlo Acerbi and Dirk Tasche. (2002). Expected shortfall: A natural coherent alternative to value at risk. Economic Notes. 31(2). 10.1111/1468-0300.00091.
2 Kris Boudt and Brian Peterson and Christophe Croux. (2008). Estimation and decomposition of downside risk for portfolios with non-normal returns. The Journal of Risk. 11(2): 79-103. 10.21314/jor.2008.188.
3 Didier Maillard. (2018). A User’s Guide to the Cornish Fisher Expansion. SSRN pre-print. 10.2139/ssrn.1997178.
4 Edmund A. Cornish and Ronald A. Fisher. (1938). Moments and Cumulants in the Specification of Distributions. Revue de l'Institut International de Statistique. 5(4): 307-320. 10.2307/1400905.
5 John D. Lamb and Maura E. Monville and Kai-Hong Tee. (2016). Making Cornish-Fisher Distributions Fit. SSRN pre-print. 10.2139/ssrn.2845873.
6 Bryan T. Kelly and Seth Pruitt and Yinan Su. (2019). Characteristics are covariances: A unified model of risk and return. Journal of Financial Economics. 134(3): 501-524. 10.1016/j.jfineco.2019.05.001.