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## South African Journal of Economic and Management Sciences

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*On-line version* ISSN 1015-8812

### S. Afr. j. econ. manag. sci. vol.13 n.3 Pretoria Jan. 2010

**ARTICLES**

**Value at Risk in the South African equity market: A view from the tails**

**C Milwidsky; E Maré**

Department of Mathematics and Applied Mathematics, University of Pretoria

**ABSTRACT**

Traditional parametric Value at Risk (VaR) estimates assume normality in financial returns data. However, it is well known that this distribution, while convenient and simple to implement, underestimates the kurtosis demonstrated in most financial returns. Huisman, Koedijk and Pownall (1998) replace the normal distribution with the Student's *t* distribution in modelling financial returns for the calculation of VaR. In this paper we extend their approach to the Monte Carlo simulation of VaR on both linear and non-linear instruments with application to the South African equity market. We show, via backtesting, that the *t* distribution produces superior results to the normal one.

**JEL: **G32

**“Full text available only in PDF format”**

**References**

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EVANS, M., HASTINGS, N. & PEACOCK, B. 1993.*Statistical distributions* (2^{nd} ed.) New York: John Wiley and Sons Inc. [ Links ]

HENDRICKS, D. 1996. Evaluation of value-at-risk models using historical data, *Economic Policy Review,* 2(1). Available at SSRN: http://ssrn.com/abstract=1028807. [ Links ]

HILL, B. 1975. A simple approach to inference about the tail of a distribution. *The Annals of Statistics,* 3(5): 1163-1174. [ Links ]

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Accepted March 2010

**Endnotes**

1 Formally, *VaRx* = *inf*{*l* Є *R:P*(L > *l* __<__ 1-*x*)}

= *inf*{*l* Є *R:F _{L}*(

*l*)≥

*x*} (McNeil et al., 2005).

2 The kurtosis inherent in financial returns is not the only way in which the distribution of returns differs from a normal distribution. Another example of the imperfections of the normal distribution when applied to financial data is the skewness present in many financial time series. This factor is not addressed in this paper since the Student's

*t*distribution, presented here as an alternative to the normal distribution, is also symmetrical.

3 In the sequel the terms returns and log returns will be used interchangeably following standard practice in finance and risk management (McNeil et al., 2005).

4 Matlab's 'dfittool' was used to fit the distribution to the data as well as to plot the graphs in Figures 1, 2, 3 and 4

5 The test statistic is where

*O*is the observed frequency for bin

_{i}*i*and

*E*is the expected frequency for bin

_{i}*i.*The decision is to reject the null hypothesis that the data follows a specific distribution if where

*k*is the number of bins into which we divide the data,

*p*is the number of parameters in the distribution against which we're testing. For more information on this and other goodness-of-fit tests, see for example Steyn et al. (1994).

6 The value of the test statistic in the first test (for normality) was 161 (due mostly to the underestimation of tail events) and that of the second test (for the Student's

*t*test) was 5.

7 In this paper, we took the volatility to be the standard deviation of the most recent 250 days' returns. More sophisticated volatility estimation methods could be used such as volatility updating (JP Morgan/Reuters, 1996), but the aim of the paper is to compare results generated from the

*t*distribution to those generated from the normal distribution.

8 Note that since the standard statistical tables provide

*t*random numbers for a distribution with variance equal to dof/(dof-2), the numbers sampled have to be scaled by dividing by sqrt(dof/(dof-2)) (Evans et al., 1993, Huisman et al., 1998 and Mc Neil et al., 2005).

9 For alternative methods to the Hill estimator of specifying the degrees of freedom of the

*t*Distribution see for example Resnick, 2007 and Van den Goorbergh, 1999.

10 Since there is no set rule for determining whether or not an observation constitutes a tail event, this is not necessarily the decision that has to be applied. It was decided to use the 3 standard deviation rule in this paper as it consistently produced good results and it was necessary to make a decision on what observations to use in parameterising the

*t*distribution so that results could be compared with one another and a consistent rule was applied ensuring that comparisons were not biased.

11 The details of the call options are given in the table below.

12 The difference between the 97 per cent VaR-X value with a DoF parameter of 4.5 and 5.46 for this stock is only R11 per point and both values are more conservative than the VaR-N estimate.

13 The daily VaR or DVaR is the VaR for a holding period of one day.

14 Cornish-Fisher VaR is calculated as follows: where *q _{p}* is the

*p*'th percentile of the standard normal distribution, is the mean return and

*S*is the skewness and

*K*the kurtosis of the returns series (see VaR 101 reference).

15 While we don't wish to underestimate risk, overestimation would result in a waste of capital.