In the article entitled "Simple Risk Minimising Techniques for Equity Portfolios" published in the Business Express on November 16, 2011, three simple risk reduction techniques - portfolio limits, stock correlations and stop loss limits - were discussed. This article continues the discussion and focuses specifically on Value at Risk (VaR).

VaR is a measure of market risk, which is the maximum (expected) loss that can occur with X% confidence over a holding period of t days according to Choudhry (2006). Hull (2007) defines it in the context of a statement: "We are X% certain that we will not lose more than $V in the next t days." There are three main methodologies that are commonly used to estimate VaR: (1) Historical Simulation (2) Analytical/Parametric Method (3) Monte Carlo Simulation. This article will focus on simple VaR models using the historical simulation and parametric methods. More sophisticated VaR models such as those using Monte Carlo simulation and incorporating time-varying volatility will be discussed in future articles.

The effectiveness of estimated VaR is evaluated by a process commonly referred to as "back-testing". This process uses various tools to verify that losses estimated by VaR are close to actual losses. For example, if we estimate a VaR with using a 95% confidence level, we would expect that on 95 out of 100 trading days, the actual portfolio losses will be less than or equal to the losses estimated. However, on five (5) (or less) trading days, the actual loss may be more than the estimate. If the VaR meets this criterion, it can be considered effective.

The Historical Simulation (HS) method uses the recent empirical return distribution. It is based on the underlying assumption that the near future will be like the recent past, and therefore data from the recent past can be used to estimate market risk over the near future. To apply this method, a hypothetical profit/loss (P/L) time series is constructed for the current portfolio. Daily P/Ls are normally used in practice. The Basel (1996) rules require a minimum of one year of daily data be used to estimate VaR. After obtaining the hypothetical P/L dataset, the HS VaR is estimated by plotting the data on a simple histogram and taking the appropriate percentile. For example, the HS VaR with a confidence level of 95%, is the 5th percentile point of the P/L histogram.

Chart 1 shows a comparison of actual daily gains/losses on the TTSE Composite Index (TTCI) versus the daily maximum estimated loss estimated by the historical VaR. The 95% historical VaR was calculated as the 5th percentile of the TTCI actual [TTD] daily gains/losses time series for the period for the January 2009 to December 2010. Its value of TTD 3.57 suggests that on 95 out of 100 trading days, it is expected that the TTCI will increase or decrease in value, but will decrease by no more than TTD 3.57. On the other 5 days, gains, or losses in excess of the aforementioned amount are anticipated. The chart shows that on 5 out of 258 trading days (1.94%) in 2011, the actual losses were in excess of TTD 3.57. This VaR can be considered effective because the number of days in which the actual losses exceeded the VaR was less than 5%.

VaR calculated using the HS method is intuitive and conceptually simple, providing results that are easy to communicate to investors. It is fairly easy to calculate using a spreadsheet and can accommodate almost any type of investment, including derivatives. It uses data that are (often) easily accessible. Notwithstanding these positives, there are some disadvantages associated with this method. For example, if the data period was unusually quiet (or unusually volatile) and conditions have recently changed, historical simulation will tend to produce VaR estimates that are too low for the risks that the investor is currently facing. In general, historical simulation estimates of VaR make no allowance for plausible events that might occur but did not actually occur in the sample period - PRMIA (2005).

The Analytical/Parametric method assumes that holding period returns (such as daily stock price returns) are normally distributed. It can be calculated using the formula:

VaR = - (Z?? + µ) S

Where Z? is the lower ? percentile of the standard normal distribution (for example, if ? = 5%, the confidence level is 95% and Z? = 1.65 from the standard normal [Z] table), µ and ? are the mean and standard deviation, respectively, of the holding period returns and S is the current market value of the portfolio. Unlike with the historical simulation method, the VaR calculated using the parametric method changes as the value of the portfolio changes.

Chart 2 shows a comparison of actual daily gains/losses on the TTSE Composite Index (TTCI) versus the daily maximum estimated loss as derived from the parametric VaR. The 95% parametric VaR was calculated using daily TTCI index values and the standard deviation (approximately 0.2683%) and mean (approximately -0.0018%) of the TTCI daily returns time series for the period for the January 2009 to December 2010. The value of the parametric VaR for 2011 ranged from TTD 3.70 to TTD 4.50 with an average of TTD 4.16. The VaR fluctuated on a daily basis with changes in the value of the TTCI. The interpretation of this VaR is similar to that outlined for the HS VaR method that was addressed before. In Chart 2, it is observed that on 4 out of 258 trading days (1.55%) in 2011, the actual losses were in excess of TTD3.57. Based on this, it can be concluded that the 95% parametric VaR was effective and also more effective than the 95% historical VaR for the TTCI in 2011.

The parametric method provides the simplest and most easily implemented methods to estimate VaR. It relies on parameter estimates based on market data histories that can be easily obtained. Notwithstanding their ease of computation and practicality as rough approximations, these VaR estimates also have shortcomings. The major shortcoming of this method is that it is based on the assumption that market data changes are normally distributed, and this assumption seldom holds in practice. As a result VaR will be underestimated at relatively high confidence levels and overestimated at relatively low confidence levels - PRMIA (2005).

The simple VaR models presented in this article can be used by an investor to quantify and manage the price risk of equity portfolios. Portfolio VaR may be reduced by selling individual securities with high individual VaR and low diversification (correlation) benefits. When properly utilised, it can help to preserve the investor's capital.

References and further reading

Bollerslev, T. (1986): "Generalised autoregressive heteroskedasticity," Journal of Econometrics,

(31), 307-327.

Choudhry, M. An Introduction to Value-At-Risk, 4th ed. (West Sussex, England: John Wiley & Sons, 2006)

Engle, R. 2001. GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics, Journal of Economic Perspectives, Vol. 15(4), pages 157-168

Hull, J. C., Risk Management and Financial Institutions (New Jersey, USA: Pearson Prentice-Hall, 2007)

PRMIA Professional Risk Managers' International Association, The Professional Risk Managers' Handbook: A Comprehensive Current Theory and Best Practices, (New York, USA: PRMIA Publications, 2005), Volume III, A2 and A3.

Varma, J. R. 1999. Value at Risk Models in the Indian Stock Market, IIMA Working Paper, No. 99-07-05

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