We now turn to the final part of the stepwise univariate distribution modeling approach, namely accounting for conditional nonnormality in portfolio returns. In Chapter 1, we saw that asset returns are not normally distributed. If we construct a simple histogram of past returns on the S&P 500 index, then it will not conform to the density of the normal distribution: The tails of the histogram are fatter than the tails of the normal distribution, and the histogram is more peaked around zero. From a risk management perspective, the fat tails, which are driven by relatively few but very extreme observations, are of most interest. These extreme observations can be symptoms of liquidity risk or event risk as defined in Chapter 1.

One motivation for the time-varying variance models discussed in Chapter 4 and Chapter 5 is that they are capable of accounting for some of the nonnormality in the daily returns. For example a GARCH(1,1) model with normally distributed shocks, will imply a nonnormal distribution of returns R_t because the distribution of returns is a function of all the past return variances , .

GARCH models with normal shocks by definition do not capture what we call conditional nonnormality in the returns. Returns are conditionally normal if the shocks z_t are normally distributed. Histograms from shocks, (i.e. standardized returns) typically do not conform to the normal density. Figure 6.1 illustrates this point.

The top panel shows the histogram of the raw returns superimposed on the normal distribution and the bottom panel shows the histogram of the standardized returns superimposed on the normal distribution as well. The volatility model used to standardize the returns is the NGARCH(1,1) model, which includes a leverage effect. Notice that while the bottom histogram conforms more closely to the normal distribution than does the top histogram, there are still some systematic deviations, including fat tails and a more pronounced peak around zero.

We will analyze the conditional nonnormality in several ways:

For each of these methods we will provide the Value-at-Risk and the expected shortfall formulas.

Throughout this chapter, we will assume that we are working with a time series of portfolio returns using today's portfolio weights and past returns on the underlying assets in the portfolio. Therefore, we are modeling a univariate time series. We will assume that the portfolio variance has already been modeled using the methods presented in Chapter 4 and Chapter 5.

Working with the univariate time series of portfolio returns is convenient from a modeling perspective but it has the disadvantage of being conditional on exactly the current set of portfolio weights. If the weights are changed, then the portfolio distribution modeling will have to be redone. Multivariate risk models will be studied in Chapter 7, Chapter 8 and Chapter 9.

As in Chapter 2, consider a portfolio of n assets. If we today own N_i,t units or shares of asset i then the value of the portfolio today is

Using today's portfolio holdings but historical asset prices we can compute the history of (pseudo) portfolio values. For example, yesterday's portfolio value is

The log return can now be defined as

Allowing for a dynamic variance model we can write

where is the conditional volatility forecast constructed using the methods in the previous two chapters.

The focus in this chapter is on modeling the distribution of the innovations, D(0, 1), which has a mean of zero and a standard deviation of 1. So far, we have relied on setting D(0, 1) to N(0, 1), but we now want to assess the problems of the normality assumption in risk management, and we want to suggest viable alternatives.

Before we venture into the particular formulas for suitable nonnormal distributions, let us first introduce a valuable visual tool for assessing nonnormality, which we will also use later as a diagnostic check on nonnormal alternatives. The tool is commonly known as a quantile-quantile (QQ) plot, and the idea is to plot the empirical quantiles of the calculated returns, which is simply the returns ordered by size, against the corresponding quantiles of the normal distribution. If the returns are truly normal, then the graph should look like a straight line at a 45-degree angle. Systematic deviations from the 45-degree line signal that the returns are not well described by the normal distribution. QQ plots are, of course, particularly relevant to risk managers who care about Value-at-Risk, which itself is a quantile.

The QQ plot is constructed as follows: First, sort all standardized returns in ascending order, and call the ith sorted value z_i. Second, calculate the empirical probability of getting a value below the actual as where T is the total number of observations. The subtraction of 0.5 is an adjustment for using a continuous distribution on discrete data.

Calculate the standard normal quantiles as , where denotes the inverse of the standard normal density as before. We can then scatter plot the standardized and sorted returns on the Y-axis against the standard normal quantiles on the X-axis as follows:

If the data were normally distributed, then the scatterplot should conform roughly to the 45-degree line.

Figure 6.2 shows a QQ plot of the daily S&P 500 returns from Chapter 1. The top panel uses standardized returns from the unconditional standard deviation, , so that , and the bottom panel uses returns standardized by an NGARCH(1,1) with a leverage effect, .

Notice that the GARCH model does capture some of the nonnormality in the returns, but some still remains. The patterns of deviations from the 45-degree line indicate that large positive returns are captured remarkably well by the normal GARCH model but that the model does not allow for a sufficiently fat left tail as compared with the data.

The Filtered Historical Simulation approach (FHS), which we present next, attempts to combine the best of the model-based with the best of the model-free approaches in a very intuitive fashion. FHS combines model-based methods of dynamic variance, such as GARCH, with model-free methods of distribution in the following way.

Assume we have estimated a GARCH-type model of our portfolio variance. Although we are comfortable with our variance model, we are not comfortable making a specific distributional assumption about the standardized returns, such as a normal distribution. Instead, we would like the past returns data to tell us about the distribution directly without making further assumptions.

To fix ideas, consider again the simple example of a GARCH(1,1) model:

where

Given a sequence of past returns, , we can estimate the GARCH model and calculate past standardized returns from the observed returns and from the estimated standard deviations as

We will refer to the set of standardized returns as

We can simply calculate the 1-day VaR using the percentile of the database of standardized residuals as in

At the end of Chapter 2, we introduced expected shortfall (ES) as an alternative risk measure to VaR. ES is defined as the expected return given that the return falls below the VaR. For the 1-day horizon, we have

The ES measure can be calculated from the historical shocks via

where the indicator function returns a 1 if the argument is true and zero if not.

An interesting and useful feature of FHS as compared with the simple Historical Simulation approach introduced in Chapter 2 is that it can generate large losses in the forecast period, even without having observed a large loss in the recorded past returns. Consider the case where we have a relatively large negative z in our database, which occurred on a relatively low variance day. If this z gets combined with a high variance day in the simulation period then the resulting hypothetical loss will be large.

We close this section by reemphasizing that the FHS method suggested here combines a conditional model for variance with a Historical Simulation method for the standardized returns. FHS thus retains the key conditionality feature through but saves us from having to make assumptions beyond that the sample of historical z s provides a good description of the distribution of future z s. Note that this is very different from the standard Historical Simulation approach in which the sample of historical R s is assumed to provide a good description of the distribution of future R s.

Filtered Historical Simulation offers a nice model-free approach to the conditional distribution. But FHS relies heavily on the recent series of observed shocks, z_t. If these shocks are interesting from a risk perspective (that is, they contain sufficiently many large negative values) then the FHS will deliver accurate results; if not, FHS may suffer.

We now consider a simple alternative way of calculating VaR, which has certain advantages. First, it does allow for skewness as well as excess kurtosis. Second, it is easily calculated from the empirical skewness and excess kurtosis estimates from the standardized returns, z_t. Third, it can be viewed as an approximation to the VaR from a wide range of conditionally nonnormal distributions.

We again start by defining standardized portfolio returns by

where D(0, 1) denotes a distribution with a mean equal to 0 and a variance equal to 1. As in Chapter 4, i.i.d. denotes independently and identically distributed.

The Cornish-Fisher VaR with coverage rate p can then be calculated as

where

where ζ₁ is the skewness and ζ₂ is the excess kurtosis of the standardized returns, z_t. The Cornish-Fisher quantile can be viewed as a Taylor expansion around the normal distribution. Notice that if we have neither skewness nor excess kurtosis so that , then we simply get the quantile of the normal distribution

Consider now for example the 1% VaR, where . Allowing for skewness and kurtosis we can calculate the Cornish-Fisher 1% quantile as

and the portfolio VaR can be calculated as

Thus, for example, if skewness equals −1 and excess kurtosis equals 4, then we get

which is much higher than the VaR number from a normal distribution, which equals .

The expected shortfall can be derived as

where

This derivation can be found in Appendix B. Recall from Chapter 2 that the ES for the normal distribution is

which is also a special case of when .

The CF approach is easy to implement and we avoid having to make an assumption about exactly which distribution fits the data best. However, exact distributions have advantages too. Perhaps most importantly for risk management, exact distributions allow us to compute VaR and ES for extreme probabilities (as we did in Chapter 2) for which the approximative CF may not be well-defined. Exact distributions also enable Monte Carlo simulation, which we will discuss in Chapter 8. We therefore consider useful examples of exact distributions next.

Perhaps the most important deviations from normality we have seen are the fatter tails and the more pronounced peak in the distribution of z_t as compared with the normal distribution. The Student's t distribution captures these features. It is defined by

The notation refers to the gamma function, which can be found in most quantitative software packages. Conveniently, the distribution has only one parameter, namely d. In the Student's t distribution we have the following first two moments:

We have already modeled variance using GARCH and other models and so we are interested in a distribution that has a variance equal to 1. The standardized t distribution—call it the distribution—is derived from the Student's t to achieve this goal.

Define z by standardizing x so that

The standardized density is then defined by

where

Note that the standardized t distribution is defined so that the random variable z has mean equal to zero and a variance (and standard deviation) equal to 1. Note also that the parameter d must be larger than two for the standardized distribution to be well defined.

The key feature of the distribution is that the random variable, z, is taken to a power, rather than an exponential, which is the case in the standard normal distribution where

The power function driven by d will allow for the distribution to have fatter tails than the normal; that is, higher values of when z is far from zero.

The distribution is symmetric around zero, and the mean (μ), variance , skewness , and excess kurtosis of the distribution are

Thus, notice that d must be higher than 4 for the kurtosis to be well defined. Notice also that for large values of d the distribution will have an excess kurtosis of zero, and we can show that it converges to the standard normal distribution as d goes to infinity. Indeed, for values of d above 50, the distribution is difficult to distinguish from the standard normal distribution.

The Student's t distribution can allow for kurtosis in the conditional distribution but not for skewness. It is possible, however, to develop a generalized, asymmetric version of the Student's t distribution. It is defined by pasting together two distributions at a point −A/B on the horizontal axis. The density function is defined by

where

and where , and . Note that from the symmetric Student's t distribution. Figure 6.4 shows the asymmetric t distribution for in blue, and in red.

In order to derive the moments of the distribution we first define

With these in hand, we can derive the first four moments of the asymmetric t distribution to be

Note from the formulas that although skewness is zero if d₂ is zero, skewness and kurtosis are generally highly nonlinear functions of d₁ and d₂.

Consider again the two distributions in Figure 6.4. The red line corresponds to a skewness of +1 and an excess kurtosis of 2.6; the blue line corresponds to a skewness of −1 and an excess kurtosis of 2.6.

Skewness and kurtosis are both functions of d₁ as well as d₂. The upper panel of Figure 6.5 shows skewness plotted as a function of d₂ on the horizontal axis. The blue line uses (high kurtosis) and the red line uses (moderate kurtosis). The lower panel of Figure 6.5 shows kurtosis plotted as a function of d₁ on the horizontal axis. The red line uses (no skewness) and the blue line uses (positive skewness). The asymmetric t distribution is capable of generating a wide range of skewness and kurtosis levels.

Notice that the symmetric standardized Student's t is a special case of the asymmetric t where , , which implies A = 0 and B = 1, so we get

which yields

as in the previous section.

Typically, the biggest risks to a portfolio is the sudden occurrence of a single large negative return. Having explicit knowledge of the probabilities of such extremes is, therefore, at the essence of financial risk management. Consequently, risk managers ought to focus on modeling the tails of the returns distribution. Fortunately, a branch of statistics is devoted exactly to the modeling of such extreme values.

The central result in extreme value theory states that the extreme tail of a wide range of distributions can approximately be described by a relatively simple distribution, the so-called Generalized Pareto Distribution (GPD).

Virtually all results in extreme value theory (EVT) assume that returns are i.i.d. and therefore are not very useful unless modified to the asset return environment. Asset returns appear to approach normality at long horizons, thus EVT is more important at short horizons, such as daily. Unfortunately, the i.i.d. assumption is the least appropriate at short horizons due to the time-varying variance patterns. Therefore we need to get rid of the variance dynamics before applying EVT. Consider again, therefore, the standardized portfolio returns

Fortunately, it is typically reasonable to assume that these standardized returns are i.i.d. Thus, we will proceed to apply EVT to the standardized returns and then combine EVT with the variance models estimated in Chapter 4 and Chapter 5 in order to calculate VaR s.

Time-varying variance models help explain nonnormal features of financial returns data. However, the distribution of returns standardized by a dynamic variance tends to be fat-tailed and may be skewed. This chapter has considered methods for modeling the nonnormality of portfolio returns by building on the variance and correlation models established in earlier chapters and using the same maximum likelihood estimation techniques.

We have introduced a graphical tool for visualizing nonnormality in the data, the so-called QQ plot. This tool was used to assess the appropriateness of alternative distributions.

Several alternative approaches were considered for capturing nonnormality in the portfolio risk distribution.

This chapter has focused on one-day-ahead distribution modeling. The multiday distribution requires Monte Carlo simulation, which will be covered in Chapter 8.

We end this chapter by stressing that in Part II of the book we have analyzed the conditional distribution of the aggregate portfolio return only. Thus, the distribution is dependent on the particular set of current portfolio weights, and the distribution must be reestimated when the weights change. Part III of the book presents multivariate risk models where portfolio weights can be rebalanced without requiring reestimation of the model.

In this appendix we derive the expected shortfall (ES) measure for the asymmetric t distribution. The ES for the symmetric case will be given as a special case at the end.

We want to compute . Let us assume for simplicity that p is such that , then

We use the change of variable

which yields

The first integral can be solved to get

and the second integral can be related to the regular symmetric Student's t distribution by

where is the CDF of a Student's t distribution with d₁ degrees of freedom.

Therefore,

In the symmetric case we have d₁ = d, d₂ = 0, A = 0, and B = 1 and so we get

where now .

The Cornish-Fisher approach assumes an approximate distribution of the form

The expected shortfall is again defined as

where Solving the integral we get

Expected shortfall in the Hill approximation to EVT can be derived as

Details on the asymmetric t distribution considered here can be found in Hansen (1994), Fernandez and Steel (1998) and Jondeau and Rockinger (2003). Hansen (1994) and Jondeau and Rockinger (2003) also discuss time-varying skewness and kurtosis models. The GARCH- model was introduced by Bollerslev (1987).

Applications of extreme value theory to financial risk management is discussed in McNeil (1999). The choice of threshold value in the GARCH-EVT model is discussed in McNeil and Frey (2000). Huisman et al. (2001) explore improvements to the simple Hill estimator considered here. McNeil (1997) and McNeil and Saladin (1997) discuss the use of QQ plots in deciding on the threshold parameter, u. Brooks et al. (2005) compare various EVT approaches.

Multivariate extensions to the univariate EVT analysis considered here can be found in Longin (2000), Longin and Solnik (2001) and Poon et al. (2003).

The expected shortfall measure for the Cornish-Fisher approximation is developed in Giamouridis (2006). In the spirit of the Cornish-Fisher approach, Jondeau and Rockinger (2001) develop a Gram-Charlier approach to return distribution modeling.

Many alternative conditional distribution approaches exist. Kuerster et al. (2006) perform a large-scale empirical study.

GARCH and RV models can also be combined with jump processes. See Maheu and McCurdy (2004), Ornthanalai (2010) and Christoffersen et al. (2010).

Artzner et al. (1999) define the concept of a coherent risk measure and showed that expected shortfall (ES) is coherent whereas VaR is not. Studying dynamic portfolio management based on ES and VaR, Basak and Shapiro (2001) found that when a large loss does occur, ES risk management leads to lower losses than VaR risk management. Cuoco et al. (2008) argued instead that VaR and ES risk management lead to equivalent results as long as the VaR and ES risk measures are recalculated often. Both Basak and Shapiro (2001) and Cuoco et al. (2008) assumed that returns are normally distributed. Chen (2008) and Taylor (2008) consider nonparametric ES methods.

For analyses of GARCH-based risk models more generally see Bali et al. (2008), Mancini and Trojani (2011) and Jalal and Rockinger (2008).