Essay on Normal Distribution

1

Non-Normal Distributions
Reading: Christoffersen, Elements of Financial
Risk Management, Chapter 6

2

Overview
• Returns are conditionally normal if the dynamically standardized returns are normally distributed.
A standardized return is zt = Rt/t, where t is the (estimate) of the standard deviation of the return Rt. Typically t comes from a variance forecasting model, e.g. a GARCH model.
• Fig.6.1 illustrates how histograms from returns and standardized returns typicallydo not conform to the 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 Figure 6.1: Histogram of Daily S&P 500 Returns and
Histogram of GARCH Shocks

Question: Why does normal distribution provide a better approximation of standardized returns than returns?

3

4

Plan for Today
• 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 consider combining GARCH models with the standardized Student’s t distribution, and discuss the estimation of it.
• We extend the Student’s t distribution to a more flexible asymmetric version

5

Visualising Non-normality Using QQ Plots
• Consider a portfolio of n assets with Ni,t units or shares of asset i. Portfolio value at time t is

•

Yesterday’s portfolio value would be

• The log return can now be defined as

Note that this is equivalent to the return computed using historical simulation.

6

Visualising Non-normality Using QQ Plots

• Allowing for a dynamic variance model we can say

where PF,t is the conditional volatility (standard deviation) 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

7

Visualising Non-normality Using QQ Plots
• QQ (Quantile-Quantile) plot: Plot the quantiles of the calculated (standardized) returns against the quantiles of the (standard) 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.

8

Visualising Non-normality Using QQ Plots
(1) Sort the standardized returns in ascending order; call the ith standardized return zi
(2) Calculate the empirical probability of getting a value below the value i as (i  0.5)/T
(3) Calculate the standard normal quantiles as
(4) Finally draw scatter plot

•

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

9

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

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Visualising Non-normality Using QQ Plots
This slide says

(1) Sort the standardized returns in ascending order; call the ith
“standardized returns”; what standardized return zi do we mean by that? That is, standardized by what?

(2) Calculate the empirical probability of getting a value below the value i as (i  0.5)/T
(3) Calculate the standard normal quantiles as
Now we consider two standardizations:
(4) Finally draw scatter plot standard deviation , i.e. zt = Rt/
(a)Standardize
by unconditional
(b)Standardize by conditional standard deviation t from a variance forecasting model, e.g. a GARCH model. In this case zt = Rt/t are are asking whether zt = Rt/ is described by a normalshould
• InIf(a) thewedata normally distributed, then the scatterplot distribution. conform to the 45-degree line.
In (b) we are asking whether zt = Rt/t is described by a normal distribution. 11

Figure 6.2: QQ Plot of Daily S&P 500 Returns
This QQ plot addresses whether zt = Rt/ are described by normal distribution. We might hypothesize that standard deviation is constant, i.e. all returns have the same standard

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