RISK MANAGEMENT – MAXVAR INTRODUCTION The standard VaR approach considers only terminal risk, completely ignoring the sample path of portfolio values. In reality interim risk may be critical in a mark-to- market environment. Sharp declines in value may generate margin calls and affect trading strategies. Thus here we explain the concept of MaxVaR, analogous to VaR in every way except it quantifies the probability of seeing a given loss on or before the terminal date rather than at the terminal date.
RISK MANAGEMENT – MAXVAR HYPOTHESES 1 LOG NORMAL DISTRIBUTION STOCK 2 WIENER PROCESS
RISK MANAGEMENT – MAXVAR COMPARISON VAR MAXVAR P(R<z)=C P(min R <z)=C R daily return C confidence z loss level
RISK MANAGEMENT – MAXVAR VAR FORMULA Solve z in Proof: direct derivation cumulative standard distribution
RISK MANAGEMENT – MAXVAR MAXVAR FORMULA Solve z in Proof: Schuster 1968 theorem applied to first passage of time of the Wiener process
RISK MANAGEMENT – MAXVAR Volatility Estimation Equally Weighted EWMA Variance We observe returns (log price change) We Use EWMA to estimate time varying over M days and the volatility estimate volatility. Where is decay factor =0.94 is calculated using moving average. as per RiskMetrics) .
RISK MANAGEMENT – MAXVAR COMPARISON OF MAXVAR AND VAR Considering a Portfolio with σ = 15%, T = 1, and μ = 10% and 15%. MaxVaR and VaR are calculated using an Excel spreadsheet. VaR is calculated using “ -NORMINV(B8,B10,B11)” MaxVaR is calculated using “Solver”.
RISK MANAGEMENT – MAXVAR COMPARISON OF MAXVAR AND VAR α, level of μ VaR MaxVaR MaxVaR/VaR significance 1% 10% 0.2603 0.3100 1.191 2.5% 10% 0.2052 0.2628 1.281 5% 10% 0.1580 0.2240 1.418 1% 15% 0.2102 0.2702 1.285 2.5% 15% 0.1553 0.2256 1.453 5% 15% 0.1080 0.1893 1.753 General Observations: 1. MaxVaR is always greater than the VaR. 2. MaxVaR and VaR are not linearly correlated as the ratios are not constant. 3. MaxVaR/VaR ratio decreases as α (level of significance) decreases. 4. MaxVaR/VaR ratio increases as the drift (m = μ – σ2/2) increases.
RISK MANAGEMENT – MAXVAR COMPARISON OF MAXVAR AND VAR VaR and MaxVaR for S&P 500 and STI Table: S&P 500 VaR and MaxVaR for μ = 27.7%, σ = 25.4%, T = 10/252 α, significance VaR MaxVaR MaxVaR/VaR level 1% 0.1080 0.1218 1.128 2.5% 0.0895 0.1051 1.174 5% 0.0735 0.0916 1.246 Table: STI VaR and MaxVaR for μ = 47.3%, σ = 24.4%, T = 10/252 α, significance VaR MaxVaR MaxVaR/VaR level 1% 0.0955 0.1098 1.150 2.5% 0.0777 0.0941 1.211 5% 0.0623 0.0810 1.300
RISK MANAGEMENT – MAXVAR COMPARISON OF MAXVAR AND VAR STI VaR and MaxVaR for α = 5% VaR = 0.06 MaxVaR = 0.08
ASSESSING VAR PRECISION In our case, T=504. Confidence level equals 95%.
KUPIEC(1995) PROPORTION OF FAILURES TEST •The number of exceptions follows the binomial Bernoulli trails. •the LR is asymptotically chi-square distributed with one degree of freedom.
Kupiecs Test VaR MaxVaR Confidence Test Statistic Observed Test Test Statistic Observed Test Level LR N Outcome LR N OutcomeSPX 99.0% N/A 0 Reject 0.428 1 Accept 97.5% 1.792 2 Accept 0.580 3 Accept 95.0% 0.121 8 Accept 0.505 7 AcceptSTI 99.0% N/A 0 Reject N/A 0 Reject 97.5% 1.792 2 Accept 4.061 1 Reject 95.0% 5.616 3 Reject 3.657 4 Accept The MaxVaR is statistically better at indicating the failure rate than the VaR.
Extension• Fat tailed, skewed distribution could be implemented.• The GARCH model is a better alternative to get the volatility of the stock return.• Estimates VaR and daily to include the new trading activities.