Elements of financial risk management chapter 6

• 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 norma

Non-Normal Distributions

Elements of Financial Risk Management

Chapter 6Peter Christoffersen

• 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

2

Figure 6.1: Histogram of Daily S&P 500 Returns

and Histogram of GARCH Shocks

3

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

4

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

5

Visualising Non-normality Using

QQ Plots

• Consider a portfolio of n assets with N i,t units or shares

of asset i then the value of the portfolio today is

6

• Yesterday’s portfolio value would be

Visualising Non-normality Using

QQ Plots

• Allowing for a dynamic variance model we can say

7

• where PF,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

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

8

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

9

• If the data were normally distributed, then the scatterplot

Figure 6.2: QQ Plot of Daily S&P 500 Returns

10

Figure 6.2: QQ Plot of Daily S&P 500 GARCH

Shocks

11

Filtered Historical Simulation Approach

• We have seen the pros and cons of both

data-based and model-data-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

12

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

13

 d

t~

Filtered Historical Simulation Approach

• To fix ideas, consider again the simple example of a

GARCH(1,1) model

14

• 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

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

15

• Expected shortfall (ES) for the 1-day horizon is

• The ES is calculated from the historical shocks via

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

16

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

17

The Cornish-Fisher Approximation

to VaR

18

• 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

The Cornish-Fisher Approximation

to VaR

19

• 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

The Cornish-Fisher Approximation

to VaR

20

• 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

The Cornish-Fisher Approximation to

VaR

• Thus, for example, if skewness equals –1 and excess

kurtosis equals 4, then we get

21

• which is much higher than the VaR number from a

normal distribution, which equals 2.33PF,t+1

The Cornish-Fisher Approximation

to VaR

22

• The expected shortfall can be derived as

• Where

The Cornish-Fisher Approximation

to VaR

23

• 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

The Standardized t distribution 24

• 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

The Standardized t distribution 25

• Define Z by standardizing x so that,

• The Standardized t distribution, , is then

defined as

• where

The Standardized t distribution 26

• 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 Standardized t distribution

• The distribution is symmetric around zero, and the mean , variance 2,skewness 1, and excess kurtosis 2

of the distribution are

27

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

28

Maximum Likelihood Estimation

• We can combine dynamic volatility model such as

GARCH with the standardized t distribution to specify

our model portfolio returns as

29

• If we ignore the fact that variance is estimated with error,

we can treat standardized return as a regular random

variable, calculated as

Maximum Likelihood Estimation

• The d parameter can then be chosen to maximize

the log likelihood function

30

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

31

• We can then maximize

Maximum Likelihood Estimation

• As a simple univariate example of the difference between QMLE and MLE consider the GARCH(1,1)- model with leverage:

32

• We can estimate all the parameters in one

step using lnL2 from before, which would correspond to exact MLE

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 2 suggests calculating 2,

from the z t variable and calculating d from

33

Calculating VaR and ES

• Having estimated d we can calculate the VaR of

the portfolio return

34

• as

• Where is the pth quantile of distribution

Therefore,

Calculating VaR and ES

• The formula for ES is,

35

Figure 6.3: QQ Plot of S&P500 GARCH Shocks

Against the Standardized t Distribution

38

The Asymmetric t distribution

• If one would like to have skewness in t distribution, the asymmetric

t distribution can be used,

39

The Asymmetric t distribution

• Here d1 > 2, and -1 < d2 < 1

• Note that C(d1) = C(d) from the symmetric

Student’s t distribution

40

Figure 6.4: The Asymmetric t Distribution

41

The Asymmetric t distribution

• In order to derive the moments of the distribution

we first define,

42

The Asymmetric t distribution

• The moments of the asymmetric t distribution can

be derived as

43

Figure 6.5: Skewness and Kurtosis in the

Asymmetric t Distribution

44

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,

45

• which yields

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,

46

Estimation of d1 and d2

• Where

47

• This assumes the estimation of

is done without estimation error

Estimation of d1 and d2

• Alternatively the joint estimation of volatility and

distribution parameters can be done via,

Calculating VaR and ES

• Having estimated d1 and d2 we can calculate the

VaR of the portfolio return

49

• as

• Where is the pth percentile of the

asymmetric t distribution.

Calculating VaR and ES

• is given by

50

• The ES is given by

QQ Plots

• Knowing the CDF we can construct the QQ plot as,

51

• where z i 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

Figure 6.6: QQ Plot of S&P 500 GARCH Shocks

against the Asymmetric t Distribution

52

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

53

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

54

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

55

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

56

Estimating Tail Index Parameter, 

• If we assume that the tail parameter, , is strictly positive, then we can use the Hill estimator to approximate the

Estimating Tail Index Parameter, 

• We can get the density function of y from F(y):

• The likelihood function for all observations y i larger than

the threshold, u,

• The log-likelihood function is therefore

59

Estimating Tail Index Parameter, 

• Taking the derivative with respect to  and setting it to zero

yields the Hill estimator of tail index parameter

• We can estimate the c parameter by ensuring that the fraction

of observations beyond the threshold is accurately captured

by the density as in

60

Estimating Tail Index Parameter, 

• Cumulative density function for observations beyond u is

• Solving this equation for c yields the estimate

Estimating Tail Index Parameter, 

• 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

61

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, ,

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

62

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 

• We set T u = 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

63

Constructing the QQ Plot from EVT

• Define y to be a standardized loss

64

• The first step is to estimate  and c from the losses, y i,

using the Hill estimator

• Next, we need to compute the inverse cumulative

distribution function, which gives us the quantiles