Value at Risk: Univariate Estimation Methods

Value-at-Risk (VaR) is one of the most commonly used risk measures in the financial industry1 in part thanks to its simplicity - because VaR reduces the market risk associated with any portfolio to just one number2 - and in part due to regulatory requirements (Basel market risk frameworks34, SEC Rule 18f-45…). Nevertheless, when it comes to actual computations, the above definition is by no means constructive1 and accurately estimating VaR is a very challenging statistical problem2 for which several methods have been developed. In this blog post, I will describe some of the most well-known univariate VaR estimation methods, ranging from non-parametric methods based on empirical quantiles to semi-parametric methods involving kernel smoothing or extreme value theory and to parametric methods relying on distributional assumptions. Value-at-Risk Definition The Value-at-Risk of a portfolio of financial instruments corresponds to the maximum potential change in value of [that portfolio] with a given probability over a certain horizon2. More formally, the Value-at-Risk $VaR_{\alpha}$ of a portfolio over a time horizon $T$ (1 day, 10 days, 20 days…) and at a confidence level $\alpha$% $\in ]0,1[$ (95%, 97.5%, 99%…) can be defined6 as the opposite7 of the $1 - \alpha$ quantile of the portfolio return distribution over the time horizon $T$ [\text{VaR}{\alpha} = - \inf{x} \left{x \in \mathbb{R}, P(X \leq x) \geq 1 - \alpha \right}] , where $X$ is a random variable representing the portfolio return over the time horizon $T$. This formula is also equivalent8 to [\text{VaR}{\alpha} = - F_X^{-1}(1 - \alpha)] , where $F_X^{-1}$ is the inverse cumulative distribution function, also called the quantile function, of the random variable $X$. Graphically, this definition is illustrated in: Figure 1, for a continuous portfolio return distribution at a generic confidence level $\alpha$% and over a generic horizon. Figure 1. Graphical illustration of a portfolio VaR as a quantile of its continuous return distribution. Source: Adapted from Yamai and Yoshiba. Figure 2, for a discrete portfolio return distribution at a confidence level $\alpha$ = 99% and over a 1-month horizon, which is commented as follows in Jorion9: We need to find the loss that will not be exceeded in 99% of cases, or such that 1% of [the 624] observations - that is, 6 out of 624 occurrences - are lower. From Figure [2], this number is about -3.6%, [resulting in a portfolio VaR of 3.6%]. Figure 2. Graphical illustration of a portfolio VaR as a quantile of its discrete monthly return distribution, $\alpha$ = 99%. Source: Jorion. Univariate v.s. multivariate Value-at-Risk A portfolio can be considered as both: An asset in itself, with its own return distribution. A weighted collection of individual assets, each with their own return distribution. This raises the question of whether first to aggregate profit and loss data and proceed with a univariate [VaR] model for the aggregate, or to start with disaggregate data10 and proceed with a multivariate VaR model from the disaggregated data. In this blog post, I will only discuss univariate11 VaR models - originally suggested by Zangari12 as simple and effective approach[es] for calculating Value-at-Risk12 - in which portfolio returns are considered as a univariate time series without reference to the portfolio constituents13. Indeed, since the goal of VaR is to measure the market risk of a portfolio, it seems reasonable to model the portfolio return series directly12. Arithmetic returns v.s. logarithmic returns in Value-at-Risk calculations In VaR calculations, it is usually prefered, for a variety of reasons, to work with logarithmic returns rather than arithmetic (simple, linear) ones13, c.f. Ballotta13 and Jorion9 for more details. In that case, though, because investors are primarily interested in simple returns14, the logarithmic VaR $\text{VaR}_{\alpha}^{(l)}$ needs to be converted into an arithmetic VaR $\text{VaR}_{\alpha}^{(a)}$. Thanks to the definition of VaR as a quantile of the portfolio return distribution and the relationship between arithmetic and logarithmic returns, this is easily be done through the formula [\text{VaR}{\alpha}^{(a)} = 1 - \exp \left( - \text{VaR}{\alpha}^{(l)} - 1 \right)] While not frequently mentioned in the litterature15, it is important to be aware of this subtlety. History of Value-at-Risk Searching for the best means to represent the risk exposure of a financial institution’s trading portfolio in a single number16 is a quest that folklore attributes the inception of to Dennis Weatherstone at J.P. Morgan [in the late 1980s], who was looking for a way to convey meaningful risk exposure information to the financial institution’s board without the need for significant technical expertise on the part of the board members16. It is then Till Guldimann, head of global research at J.P. Morgan at that time, who designed what would come to be known as the J.P. Morgan’s daily VaR report17 and who thus can be viewed as the creator of the term Value-at-Risk9. The interested reader is referred to Holton18 for an historical perspective on Value-at-Risk, in which the origins of VaR as a measure of risk are even traced back as far as 1922 to capital requirements the New York Stock Exchange imposed on member firms18. Value-at-Risk estimation When a sample of portfolio returns over a given time horizon $r_1,…,r_n$ is available - like in ex post analysis -, the Value-at-Risk of that portfolio over the same horizon19 at a confidence level $\alpha \in ]0,1[$ is a textbook example of VaR calculation as the opposite of the $(1 - \alpha)$% quantile of the empirical return distribution $r_1,…,r_n$. Problem is, the discrete nature of the extreme returns of interest makes it difficult to accurately compute that quantile, as explained in Danielsson and de Vries20: In the interior, the empirical sampling distribution is very dense, with adjacent observations very close to each other. As a result the sampling distribution is very smooth in the interior and is the mean squared error consistent estimate of the true distribution. The closer one gets to the extremes, the longer the interval between adjacent returns becomes. This can be seen in [Figure 3] where the 7 largest and smallest returns on the stocks in the sample portfolio and SP-500 Index for 10 years are listed. These extreme observations are typically the most important for VaR analysis, however since these values are clearly discrete, the VaR will also be discrete, and hence be either underpredicted or overpredicted. Figure 3. Extreme daily returns for select U.S. stocks and S&P 500, 1987-1996. Source: Danielsson and de Vries. In other words, the quantile corresponding to the estimation of the Value-at-Risk […] rather depends on the realizations of the [portfolio returns] than on their probability distribution1, so that the Value-at-Risk calculated with a quantile of the empirical distribution will be highly unstable, especially when considering a Value-at-Risk with a high confidence level with only few available data1. This is why VaR estimation is a very challenging statistical problem2, sharing many similarities with the problem of estimating the frequency and/or severity of extreme events in other domains, like floods frequency estimation21 in hydrology. In order to compute a statistical estimator of a portfolio Value-at-Risk, three main approaches exist: Non-parametric approaches, that do not make any specific distributional assumptions on the portfolio return distribution and whose VaR estimators do not depend on any auxiliary parameter. Semi-parametric approaches, that do not make any specific distributional assumptions on the portfolio return distribution but whose VaR estimators depend on one or several auxiliary parameters. Parametric approaches, that make a specific distributional assumption on the portfolio return distribution and whose VaR estimators depend on one or several auxiliary parameters. To be noted that non-parametric and semi-parametric approaches might still make distributional assumptions, in particular for convergence proofs - like assuming that returns are independent and identically distributed (i.i.d.) -, but these assumptions are then generic in nature, contrary to parametric approaches which assume a very specific return distribution, like a Gaussian distribution, which is one of the most widely applied parametric probability distribution22 in finance. Non-parametric and semi-parametric Value-at-Risk estimation Chen and Yong Tang23 notes that non-parametric and semi-parametric VaR estimators have the advantages of (i) being free of distributional assumptions […] while being able to capture fat-tail and asymmetry distribution of returns automatically; and (ii) imposing much weaker assumptions on the dynamics of the return process and allowing data “speak for themselves”23. Empirical quantile of the portfolio return distribution A well-known estimator of the $(1 - \alpha)$% quantile of any probability distribution is the empirical $(1 - \alpha)$% quantile of that distribution, which relies on order statistics. In the context of VaR estimation, the underlying idea is explained in Dowd24: If we have a sample of $n$ profit and loss (P/L) observations, we can regard each observation as giving an estimate of VaR at an implied probability level. For example, if $n$ = 100, we can take the 5% VaR as the negative of the sixth25 smallest P/L observation, the 1% VaR as the negative of the second-smallest, and so on. This leads to the empirical portfolio VaR estimator, defined26 as the opposite of the $n (1 - \alpha) + 1$-th highest portfolio return27 [\text{VaR}{\alpha} = -r_{\left( n (1 - \alpha) + 1 \right)}] , where $r_{(1)} \leq r_{(2)} \leq … \leq r_{(n-1)} \leq r_{(n)}$ are the order statistics of the portfolio returns. Now, due to the discrete nature of the portfolio return distribution, there is little chance that $n (1 - \alpha) + 1$ is an integer. In that case, two26 possible choices are: Either to define the opposite of the $\lfloor n (1 - \alpha) \rfloor + 1$-th highest portfolio return27 as the empirical portfolio VaR estimator24 [\text{VaR}{\alpha} = - r{\left( \lfloor n (1 - \alpha) \rfloor + 1 \right)}] Or to define a linear interpolation28 between the opposite of the $\lfloor (n+1) \left( 1 - \alpha \right) \rfloor$-th and $\lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1$-th highest portfolio returns27 as the empirical portfolio VaR estimator293013 [\text{VaR}{\alpha} = - \left( 1 - \gamma \right) r{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor \right)} - \gamma r_{\left( \lfloor (n+1) \left( 1 - \alpha \right) \rfloor + 1 \right)}] , with $\gamma$ $= (n+1) \left( 1 - \alpha \right)$ $- \lfloor (n+1) \left( 1 - \alpha \right) \rfloor$. An interesting property of the resulting portfolio VaR estimator is that it is consistent in the presence of weak dependence31 between portfolio returns, c.f. Chen and Yong Tang23. In terms of drawbacks, the two major limitations of the empirical portfolio VaR estimator are that: It only takes into account a small part of the information contained in the [portfolio returns] distribution function1 - that is, at most two returns - which is highly inefficient, especially when the number of portfolio returns is already relatively small. It cannot generate any information about the tail of the return distribution beyond the smallest sample observation16, which might lead to severly underestimate the true risk of the portfolio. Kernel-smoothed quantile of the portfolio return distribution Another way to account for the information available in the empirical […] distribution1 than using the empirical quantile estimator discussed in the previous sub-section is to use the kernel-smoothed quantile estimator introduced in Gourieroux et al.32. Kernel-smoothing is a methodology belonging to statistics and probability theory that can be thought of as a way of generalizing a histogram constructed with the sample data16, as illustrated in Figure 4. Figure 4. Histogram v.s. kernel-smoothed density for the same sample of data. Source: Wikipedia. On Figure 4, where a histogram results in a density that is piecewise constant, a kernel[-smoothed} approximation results in a smooth density23. Coming back to the estimator of Gourieroux et al.32, it is defined as the $(1 - \alpha)$% quantile of a kernel-smoothed approximation of the portfolio return distribution, which essentially results in a weighted average of the order statistics around [$r_{\left( \lfloor n (1 - \alpha) \rfloor + 1 \right)}$] rather than […] a single order statistic23 or a linear interpolation between two order statistics. From a practical perspective, that VaR estimator is computed as follows: Select a kernel function $K$, usually23 taken as a symmetric33 probability density function. The theoretical optimal34 choice for such a kernel function is the Epanechnikov kernel, defined as $K(u) = \frac{3}{4} \left( 1 - u^2 \right) I_{|u| \leq 1}$. However, the litterature suggests that the form of the kernel has little effect on the [accuracy] of the [kernel-smoothed approximation of the return distribution]32, mainly because: The theoretical framework used to establish the optimality of the Epanechnikov kernel relies on large asymptotics, moreover in a debatable way35. The performances36 of other commonly used kernel functions are anyway very close to those of the Epanechnikov kernel37. So, in applications, the most common2332 choice for a kernel function is rather the Gaussian kernel, defined as $K(u) = \frac{1}{\sqrt{2 \pi }} e^{- \frac{u^2}{2} }$. As a side note, using a kernel function to approximate the portfolio return distribution might look like a parametric approach to VaR estimation in disguise, but Butler and Schachter16 explains why this is not the case: Note that use of a normal or Gaussian kernel estimator does not make the ultimate estimation of the VaR parametric. As the sample size grows, the net sum of all the smoothed points approaches the true [portfolio return distribution], whatever that may be, irrespective of the method of smoothing the data. This is because the influence of each point becomes arbitrarily small as the sample size grows, so the choice of kernel imposes no restrictions on the results. Select a kernel bandwidth parameter h > 0 for the kernel function. Gourieroux et al.32 describes that parameter as follows: The bandwidth parameter controls the range of data points that will be used to estimate the distribution. A small bandwidth results in a rough distribution that does not improve appreciably on the original data, while a large bandwidth over-smoothes the density curve and erases the underlying structure. This latter point is illustrated in Figure 5 and Figure 6. Figure 5. Influence of the bandwidth parameter on the kernel-smoothed approximation of a normal mixture distribution (dashed) from n = 1000 observations. Source: Adapted from Wand and Jones. Figure 6. Dynamic influence of the bandwidth parameter on the kernel-smoothed approximation of a normal distribution. Source: KDEpy. On Figure 5, it is clearly visible that: a) The estimate of the normal mixture distribution is very rough37. This corresponds to a too small bandwidth parameter $h$ that undersmoothes the observations. b) The estimate of the normal mixture distribution smoothes away its bimodality structure. This corresponds to a too big bandwidth parameter $h$ that oversmoothes the observations. c) The estimate of the normal mixture distribution is not overly noisy, yet the essential structure of the underlying density has been recovered37. This corresponds to an adequate bandwidth parameter $h$. On Figure 6, the situation is the same as in Figure 5, except that the bandwidth parameter $h$ is being dynamically increased from 0 to ~20. Figure 5 and Figure 6 empirically demonstrate that the choice of the bandwidth is of crucial importance38, although a difficult task, especially when smoothing the tails of underlying distributions with possible data scarcity38. The interested reader is refered to Wand and Jones37, Tsybakov35 and Cheng and Sun39 for the description of several methods to choose the optimal bandwidth for a kernel function. Compute the kernel-smoothed portfolio VaR estimator as [\text{VaR}{\alpha} = - \hat{F}^{-1}(1 - \alpha)] , where $\hat{F}^{-1}(1 - \alpha)$ is the solution of the quantile equation [\hat{F}(x) = \int{-\infty}^x \hat{f}(u) , du = 1 - \alpha] with $\hat{f}(x) = \frac{1}{n h} \sum_{i=1}^n K \left( \frac{x - r_i}{h} \right)$. That part is typically done with a numerical algorithm, like the Gauss–Newton algorithm mentioned in Gourieroux et al.32. Two important positive results on the kernel-smoothed VaR estimator are established in Chen and Yong Tang23: Theoretically, it is consistent in the presence of weak dependence between portfolio returns. Empirically, it produces more precise estimates40 than those obtained with the empirical VaR estimator - especially when the number of observations is small - which can translate to a large amount in financial terms23. Similar results - in a non-financial context - are reported in Cheng and Sun39: It turns out that kernel smoothed quantile estimators, with no matter which bandwidth selection method used, are more efficient than the empirical quantile estimator in most situations. And when sample size is relatively small, kernel smoothed estimators are especially more efficient than the empirical quantile estimator. In other words, the extra effort of smoothing pays of at the end23! The major limitation of the kernel-smoothed VaR estimator, though, is that if the selected kernel function does not reflect the tail features of the true portfolio return distribution, some problems may arise when the quantile to be estimated requires an extrapolation […] far beyond the range of observed data41. In the words of Danielsson and de Vries20: Almost all kernels are estimated with the entire data set, with interior observations dominating the kernel estimation. While even the most careful kernel estimation will provide good estimates for the interior, there is no reason to believe that the kernel will describe the tails adequately. Tail bumpiness is a common problem in kernel estimation. So, while the kernel-smoothed VaR estimator is capable of tail extrapolation - contrary to the empirical VaR estimator - that capability should be used with extreme caution42. A proper portfolio VaR estimator when tail extrapolation is needed thus remains elusive at this stage. Extrapolated empirical quantile of the portfolio return distribution In order to solve the problem of tail extrapolation while retaining the simplicity of the empirical quantile estimator, Hutson30 proposes to extend the linearly interpolated quantile function into a tail extrapolation quantile function35 that allows for non-parametric extrapolation beyond the observed data30. In terms of VaR estimation, Hutson’s work translates into the following extrapolated empirical portfolio VaR estimator: For 0 < \left( 1 - \alpha \right) \leq \frac{1}{n+1} [\text{VaR}{\alpha} = - r{(1)} - \left( r_{(2)} - r_{(1)} \right) \log \left( (n+1) \left( 1 - \alpha \right) \right)] For $\left( 1 - \alpha \right) \in ]\frac{1}{n+1} , \frac{n}{n+1}[$, $\text{VaR}_{\alpha}$ is defined as the standard empirical portfolio VaR estimator For \frac{n}{n+1} < \left( 1 - \alpha \right) < 1 [\text{VaR}{\alpha} = - r{(n)} + \left( r_{(n)} - r_{(n-1)} \right) \log \left( (n+1) \alpha \right)] Hutson30 establishes the consistency of his quantile estimator for i.i.d. observations and empirically demonstrates using misc. theoretical distributions that it fits well to the ideal sample for all distributions for ideal samples as small as $n = 10$30. Unfortunately for financial applications