1 Non-Normal Distributions Elements of Financial Risk Management Chapter 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 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 â 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 σPF,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 10 Figure 6.2: QQ Plot of Daily S&P 500 Returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Estimating Tail Index Parameter, ξ 58 • 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, ξ 59 • The log-likelihood function is therefore • Taking the derivative with respect to ξ 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, ξ 60 • 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, ξ • 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 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 61 Choosing the Threshold, u 62 • 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, ξ, 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 ξ Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Choosing the Threshold, u 63 • 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 ξ • 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, ξ Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 64 Constructing the QQ Plot from EVT • Define y to be a standardized loss • The first step is to estimate ξ 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 65 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 66 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 67 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 68 Calculating VaR and ES from EVT 69 • We usually calculate the VaR taking Φ-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 ξ 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 71 • The preceding formula shows that when ξ= 0.5 then the ES to VaR ratio is • 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 72 ... the 45-degree line Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 10 Figure 6.2: QQ Plot of Daily S&P 500 Returns Elements of Financial Risk Management Second... 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... 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