Non-Normal Distributions Elements of Financial Risk Management Chapter 6 Peter Christoffersen Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Overview Third part of the Stepwise Distribution Modeling (SDM) approach: accounting for conditional nonnormality in portfolio returns. Returns are conditionally normal if the dynamically standardized returns are normally distributed. Fig.6.1 illustrates how histograms from standardized returns typically do not conform to normal density 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 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.1: Histogram of Daily S&P 500 Returns and Histogram of GARCH Shocks Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Learning Objectives We introduce the quantile-quantile (QQ) plot, which is a graphical tool better at describing tails of distributions than the histogram. We define the Filtered Historical Simulation approach which combines GARCH with historical simulation. We introduce the simple Cornish-Fisher approximation to VaR in non-normal distributions. We consider the standardized Student’s t distribution and discuss the estimation of it. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Learning Objectives We extend the Student’s t distribution to a more flexible asymmetric version. We consider extreme value theory for modeling the tail of the conditional distribution For each of these methods we will consider the Value-at-Risk and the expected shortfall formulas Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Visualising Non-normality Using QQ Plots Consider a portfolio of n assets with Ni,t units or shares of asset i then the value of the portfolio today is Yesterday’s portfolio value would be The log return can now be defined as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Visualising Non-normality Using QQ Plots Allowing for a dynamic variance model we can say where sPF,t is the conditional volatility forecast 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 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Visualising Non-normality Using QQ Plots QQ (Quantile-Quantile) plot: Plot the quantiles of the calculated returns against the quantiles of the normal distribution. Systematic deviations from the 45 degree angle signals that the returns are not well described by normal distribution. QQ Plots are particularly relevant for risk managers who care about VaR, which itself is essentially a quantile. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Visualising Non-normality Using QQ Plots 1) Sort all standardized returns in ascending order and call them zi 2) Calculate the empirical probability of getting a value below the value i as (i-.5)/T 3) Calculate the standard normal quantiles as 4) Finally draw scatter plot If the data were normally distributed, then the scatterplot should conform to the 45-degree line. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.2: QQ Plot of Daily S&P 500 Returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.2: QQ Plot of Daily S&P 500 GARCH Shocks Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Filtered Historical Simulation Approach We have seen the pros and cons of both data-based and model-based approaches. The Filtered Historical Simulation (FHS) 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 variance with model-free method of distribution in the following fashion. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Filtered Historical Simulation Approach 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 or a distribution. Instead we would like the past returns data to tell us about the distribution directly without making further assumptions. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Filtered Historical Simulation Approach 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. Next we calculate past standardized returns from the observed returns and from the estimated standard deviations as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Filtered Historical Simulation Approach We will refer to the set of standardized returns as To calculate the 1-day VaR using the percentile of the database of standardized residuals Expected shortfall (ES) for the 1-day horizon is The ES is calculated from the historical shocks via Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Filtered Historical Simulation Approach where the indicator function 1(*) returns a 1 if the argument is true and zero if not FHS can generate large losses in the forecast period even without having observed a large loss in the recorded past returns FHS deserves serious consideration by any risk management team Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR We consider a simple alternative way of calculating Value at Risk, which has certain advantages: First, it allows for skewness and excess kurtosis. Second, it is easily calculated from the empirical skewness and excess kurtosis estimates from the standardized returns. Third, it can be viewed as an approximation to the VaR from a wide range of conditionally nonnormal distributions. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR Standardized portfolio returns is defined by where D(0,1) denotes a distribution with a mean equal to 0 and a variance equal to 1 i.i.d. denotes independently and identically distributed The Cornish-Fisher VaR with coverage rate, p, can be calculated as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR Where Where is the skewness and is the excess kurtosis of the standardized returns If we have neither skewness nor excess kurtosis so that . , then we get the quantile of the normal distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR Consider now for example the one percent VaR, where Allowing for skewness and kurtosis we can calculate the Cornish-Fisher 1% quantile as and the portfolio VaR can be calculated as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR 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 2.33sPF,t+1 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR The expected shortfall can be derived as Where Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Cornish-Fisher Approximation to VaR Recall that the ES for the normal case is Which can be derived by setting in the equation for . The CF approach is easy to implement and we avoid having to make an assumption about exactly which distribution fits the data best Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Standardized t distribution The Student’s t distribution is defined by (*) notation refers to the gamma function the distribution has only one parameter d In the Student’s t distribution we have the following first two moments Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Standardized t distribution Define Z by standardizing x so that, The Standardized t distribution, , is then defined as where Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Standardized t distribution In standardized t distribution random variable z has mean equal to zero and a variance equal to 1 Note also that the parameter d must be larger than two for standardized distribution to be well defined In distribution, 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 distribution to have fatter tails than the normal Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Standardized t distribution The distribution is symmetric around zero, and the mean m, variance s2, skewness z1, and excess kurtosis z2 of the distribution are Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Standardized t distribution Note that d must be higher than 4 for the kurtosis to be well defined. Note 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. For values of d above 50, the distribution is difficult to distinguish from the standard normal distribution. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Maximum Likelihood Estimation We can combine dynamic volatility model such as GARCH with the standardized t distribution to specify our model portfolio returns as If we ignore the fact that variance is estimated with error, we can treat standardized return as a regular random variable, calculated as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Maximum Likelihood Estimation The d parameter can then be chosen to maximize the log likelihood function Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Maximum Likelihood Estimation If we want to jointly maximize over the parameter d and we should adjust the distribution to take into account the variance We can then maximize Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Maximum Likelihood Estimation As a simple univariate example of the difference between QMLE and MLE consider the GARCH(1,1)- model with leverage: We can estimate all the parameters in one step using lnL2 from before, which would correspond to exact MLE. Exact MLE is clearly feasible as the total number of parameters is only five Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

An Easy Estimate of d There is a simple alternative estimation procedure to the QMLE estimation procedure above. If the conditional variance model has already been estimated, then we are only estimating one parameter, namely d. The simple closed-form relationship between d and the excess kurtosis z2 suggests calculating z2, from the zt variable and calculating d from Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES Having estimated d we can calculate the VaR of the portfolio return as Where is the pth quantile of distribution. Therefore, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES The formula for ES is, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

QQ Plots We can relate the t(d) to the standardized t(d) by where is the pth quantile of the distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

QQ Plots We are now ready to construct the QQ plots, where zi denotes the ith sorted standardized return Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.3: QQ Plot of S&P500 GARCH Shocks Against the Standardized t Distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Asymmetric t distribution If one would like to have skewness in t distribution, the asymmetric t distribution can be used, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Asymmetric t distribution Here d1 > 2, and -1 < d2 < 1. Note that C(d1) = C(d) from the symmetric Student’s t distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.4: The Asymmetric t Distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Asymmetric t distribution In order to derive the moments of the distribution we first define, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Asymmetric t distribution The moments of the asymmetric t distribution can be derived as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Asymmetric t distribution Notice that the symmetric t distribution is nested in the asymmetric t distribution and can be derived by setting d1 = d, d2 = 0 which implies A = 0 and B = 1 Therefore, which yields Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimation of d1 and d2 MLE can be used to estimate the parameters of the asymmetric distribution, d1 and d2 The only complication is that the shape of the distribution on each day depends on zt As before the likelihood function for the shock can be defined as, Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimation of d1 and d2 Where This assumes the estimation of is done without estimation error. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimation of d1 and d2 Alternatively the joint estimation of volatility and distribution parameters can be done via, We can also estimate d1 and d2 using the sample moments. Equations below can be solved numerically Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES Having estimated d1 and d2 we can calculate the VaR of the portfolio return as Where is the pth percentile of the asymmetric t distribution. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES is given by The ES is given by Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

QQ Plots Knowing the CDF we can construct the QQ plot as, where zi denotes the ith sorted standardized return The asymmetric t distribution is cumbersome to estimate and implement but it is capable of fitting GARCH shocks from daily asset returns quite well The t distributions attempt to fit the entire range of outcomes using all the data available Consequently, the estimated parameters in the distribution (for example d1 and d2 ) may be influenced excessively by data values close to zero Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.6: QQ Plot of S&P 500 GARCH Shocks against the Asymmetric t Distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Extreme Value Theory (EVT) Typically, the biggest risks to a portfolio is the sudden occurrence of a single large negative return. Having an as precise as possible knowledge of the probabilities of such extremes is therefore at the essence of financial risk management. Consequently, risk managers should focus attention explicitly on modeling the tails of the returns distribution. Fortunately, a branch of statistics is devoted exactly to the modeling of these extreme values. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Extreme Value Theory (EVT) The central result in EVTstates 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. Virtually all results in Extreme Value Theory assumes that returns are i.i.d. and are therefore 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. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Extreme Value Theory (EVT) Unfortunately, the i.i.d assumption is the least appropriate at short horizons due to the time-varying variance patterns. We therefore need to get rid of the variance dynamics before applying EVT. Consider therefore again the standardized portfolio returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

The Distribution of Extremes Define a threshold value u on the horizontal axis of the histogram in Figure 6.1 As you let the threshold u go to infinity, the distribution of observations beyond the threshold (y) converge to the Generalized Pareto Distribution, where With b>0 and y≥u. Tail-index parameter x controls the shape of the distribution tail Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimating Tail Index Parameter, x If we assume that the tail parameter, x, is strictly positive, then we can use the Hill estimator to approximate the GPD distribution for y > u and x> 0. Recall now the definition of a conditional distribution Note that from the definition of F(y) we have Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimating Tail Index Parameter, x We can get the density function of y from F(y): The likelihood function for all observations yi larger than the threshold, u, where Tu is the number of observations y larger than u Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimating Tail Index Parameter, x The log-likelihood function is therefore Taking the derivative with respect to x and setting it to zero yields the Hill estimator of tail index parameter Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimating Tail Index Parameter, x We can estimate the c parameter by ensuring that the fraction of observations beyond the threshold is accurately captured by the density as in Solving this equation for c yields the estimate Cumulative density function for observations beyond u is Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Estimating Tail Index Parameter, x Notice that our estimates are available in closed form So far, we have implicitly referred to extreme returns as being large gains. As risk managers, we are more interested in extreme negative returns corresponding to large losses We can simply do the EVT analysis on the negative of returns (i.e. the losses) instead of returns themselves. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Choosing the Threshold, u When choosing u we must balance two evils: bias and variance. If u is set too large, then only very few observations are left in the tail and the estimate of the tail parameter, x, will be very noisy. If on the other hand u is set too small, then the data to the right of the threshold does not conform sufficiently well to the Generalized Pareto Distribution to generate unbiased estimates of x. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Choosing the Threshold, u Simulation studies have shown that in typical data sets with daily asset returns, a good rule of thumb is to set the threshold so as to keep the largest 50 observations for estimating x We set Tu = 50. Visually gauging the QQ plot can provide useful guidance as well. Only those observations in the tail that are clearly deviating from the 45-degree line indicating the normal distribution should be used in the estimation of the tail index parameter, x Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Constructing the QQ Plot from EVT Define y to be a standardized loss The first step is to estimate x and c from the losses, yi, using the Hill estimator Next, we need to compute the inverse cumulative distribution function, which gives us the quantiles Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Constructing the QQ Plot from EVT We now set the estimated cumulative probability function equal to 1-p so that there is only a p probability of getting a standardized loss worse than the quantile, F-11-p From the definition of F(*), we can solve for the quantile to get Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Constructing the QQ Plot from EVT We are now ready to construct the QQ plot from EVT using the relationship where yi is the ith sorted standardized loss Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.7: QQ Plot of Daily S&P 500 Tail Shocks against the EVT Distribution Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES from EVT We are ultimately interested not in QQ plots but rather in portfolio risk measures such as VaR. Using the loss quantile F-11-p defined above by The VaR from the EVT combined with the variance model is now easily calculated as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES from EVT We usually calculate the VaR taking -1p to be the pth quantile from the standardized return so that But we now take F-11-p to be the (1-p)th quantile of the standardized loss so that The expected shortfall can be computed using Where when x<1 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES from EVT In general, the ratio of ES to VaR for fat-tailed distribution will be higher than that of the normal. When using the Hill approximation of the EVT tail the previous formulas for VaR and ES show that we have a particularly simple relationship, namely so that for fat-tailed distributions where x > 0, the fatter the tail, the larger the ratio of ES to VaR: Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Calculating VaR and ES from EVT The preceding formula shows that when x= 0.5 then the ES to VaR ratio is 2 Thus even though the 1% VaR is the same in the two distributions by construction, the ES measure reveals the differences in the risk profiles of the two distributions, which arises from one being fat-tailed The VaR does not reveal this difference unless the VaR is reported for several extreme coverage probabilities, p. Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

Figure 6.8: Tail Shapes of the Normal Distribution (blue) and EVT (red) Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen