An evaluation and comparison of Value at Risk and Expected Shortfall

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As a risk measure, Value at Risk (VaR) is neither sub-additive nor coherent. These drawbacks have coerced regulatory authorities to introduce and mandate Expected Shortfall (ES) as a mainstream regulatory risk management metric. VaR is, however, still needed to estimate the tail conditional expectation (the ES): the average of losses that are greater than the VaR at a significance level These two risk measures behave quite differently during growth and recession periods in developed and emerging economies. Using equity portfolios assembled from securities of the banking and retail sectors in the UK and South Africa, historical, variance-covariance and Monte Carlo approaches are used to determine VaR (and hence ES). The results are back-tested and compared, and normality assumptions are tested. Key findings are that the results of the variance covariance and the Monte Carlo approach are more consistent in all environments in comparison to the historical outcomes regardless of the equity portfolio regarded. The industries and periods analysed influenced the accuracy of the risk measures; the different economies did not.

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    • Figure 1. Example of a risk aversion function ϕ (x)for ES
    • Figure 2. Comparison of South Africa’s and the UK’s real GDP growth rates from 2000 to 2017
    • Figure 3. Comparison of South Arica’s and the UK’s unemployment rates from 2000 to 2017
    • Figure 4. Comparison of South Arica’s and the UK’s M2 money supply rates from 2000 to 2017
    • Figure 5. Relative portfolio development from 2003 to 2006. Portfolios rebased to 100 on January 1, 2003
    • Figure 6. Relative portfolio development from 2008 to 2011. Portfolios rebased to 100 on January 1, 2008
    • Figure 7. Historical simulation ES/VaR ratios for (a) SA 03-05, (b) SA 08-10, (c) UK 03-05 and (d) UK 08-10
    • Figure 8. Variance-covariance ES/VaR ratios for (a) SA 03-05, (b) SA 08-10, (c) UK 03-05 and (d) UK 08-10
    • Figure 9. Monte Carlo simulation ES/VaR ratios for (a) SA 03-05, (b) SA 08-10, (c) UK 03-05 and (d) UK 08-10
    • Table 1. BCBS back-testing color table for a 99% confidence level
    • Table 2. Correlation between all securities (a) from 2003 to 2005 and (b) from 2008 to 2010 using daily returns. Demarcated areas (boxed) indicate SA securities, remainder UK
    • Table 3. JB test results during (a) 2003–2005 and (b) 2008–2010, 99% confidence level, daily returns. Demarcated areas are for SA, remainder UK
    • Table 4. Results of a two-sided means test at a 99% confidence level (daily returns)
    • Table 5. Historical simulation results of 99% VaR and ES (daily returns)
    • Table 6. Variance-covariance method results of 99% VaR and ES (daily returns)
    • Table 7. Monte Carlo simulation results of 99% VaR and ES (daily returns)
    • Table 8. BCBS back-testing results for VaR at a 99% confidence level
    • Table 9. BCBS back-testing results for ES at a 99% confidence level