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

2. 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 © by Peter Christoffersen

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

4. 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 © by Peter Christoffersen

5. 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 © by Peter Christoffersen

6. 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 © by Peter Christoffersen

7. 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 © by Peter Christoffersen

8. 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 © by Peter Christoffersen

9. 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 © by Peter Christoffersen

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

12. 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.
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13. 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 © by Peter Christoffersen

14. 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 © by Peter Christoffersen

15. 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 © by Peter Christoffersen

16. 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 © by Peter Christoffersen

17. 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 © by Peter Christoffersen

18. 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 © by Peter Christoffersen

19. 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 © by Peter Christoffersen

20. 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 © by Peter Christoffersen

21. 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 © by Peter Christoffersen

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

23. 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
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24. 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 © by Peter Christoffersen

25. 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 © by Peter Christoffersen

26. 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 © by Peter Christoffersen

27. 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 © by Peter Christoffersen

28. 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.
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29. 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
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30. Maximum Likelihood Estimation
The d parameter can then be chosen to maximize the log likelihood function
Elements of Financial Risk Management Second Edition © by Peter Christoffersen

31. 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 © by Peter Christoffersen

32. 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
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33. 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 © by Peter Christoffersen

34. 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 © by Peter Christoffersen

35. Calculating VaR and ES The formula for ES is,
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36. QQ Plots We can relate the t(d) to the standardized t(d) by
where is the pth quantile of the distribution
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37. 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 © by Peter Christoffersen

38. Figure 6.3: QQ Plot of S&P500 GARCH Shocks Against the Standardized t Distribution
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39. The Asymmetric t distribution
If one would like to have skewness in t distribution, the asymmetric t distribution can be used,
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40. 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 © by Peter Christoffersen

41. Figure 6.4: The Asymmetric t Distribution
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42. The Asymmetric t distribution
In order to derive the moments of the distribution we first define,
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43. The Asymmetric t distribution
The moments of the asymmetric t distribution can be derived as
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44. Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution
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45. 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
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46. 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,
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47. Estimation of d1 and d2 Where
This assumes the estimation of is done without estimation error.
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48. 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
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49. 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.
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50. Calculating VaR and ES is given by The ES is given by
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51. 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 © by Peter Christoffersen

52. Figure 6.6: QQ Plot of S&P 500 GARCH Shocks against the Asymmetric t Distribution
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53. 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.
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54. 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 © by Peter Christoffersen

55. 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
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56. 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
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57. 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
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58. 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
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59. 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
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60. 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
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61. 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.
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62. 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.
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63. 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
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64. 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
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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
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66. 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
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67. Figure 6.7: QQ Plot of Daily S&P 500 Tail Shocks against the EVT Distribution
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68. 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
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69. 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
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70. 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:
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71. 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.
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72. Figure 6.8: Tail Shapes of the Normal Distribution (blue) and EVT (red)
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