maxvar slides v0
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RISK MANAGEMENT – MAXVAR
INTRODUCTION
The standard VaR approach considers only terminal risk, completely ignoring the samplepath 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 tradingstrategies.
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
VAR
RISK MANAGEMENT – MAXVAR
MAXVAR (BOUDOUKH 2004)
RISK MANAGEMENT – MAXVAR
HYPOTHESES
LOG NORMAL DISTRIBUTION
WIENER PROCESS
STOCK
1
2
RISK MANAGEMENT – MAXVAR
COMPARISON
P(R<z)=C
VAR MAXVAR
P(min R <z)=C
R daily returnC confidencez loss level
RISK MANAGEMENT – MAXVAR
VAR FORMULA
Proof: direct derivation
Solve z in
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 Variance
EWMA
We observe returns (log price change)
over M days and the volatility estimate is calculated using moving average.
We Use EWMA to estimate time varying volatility. Where is decay factor =0.94 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”.
α, level of
significance
μ VaR MaxVaR MaxVaR/VaR
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
RISK MANAGEMENT – MAXVAR
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.
COMPARISON OF MAXVAR AND VAR
RISK MANAGEMENT – MAXVAR
COMPARISON OF MAXVAR AND VAR
VaR and MaxVaR for S&P 500 and STI
α, significance level
VaR MaxVaR MaxVaR/VaR
1% 0.1080 0.1218 1.128
2.5% 0.0895 0.1051 1.174
5% 0.0735 0.0916 1.246
α, significance level
VaR MaxVaR MaxVaR/VaR
1% 0.0955 0.1098 1.150
2.5% 0.0777 0.0941 1.211
5% 0.0623 0.0810 1.300
Table: S&P 500 VaR and MaxVaR for μ = 27.7%, σ = 25.4%, T = 10/252
Table: STI VaR and MaxVaR for μ = 47.3%, σ = 24.4%, T = 10/252
RISK MANAGEMENT – MAXVAR
COMPARISON OF MAXVAR AND VAR
STI VaR and MaxVaR for α = 5%
VaR = 0.06MaxVaR = 0.08
ASSESSING VAR PRECISION
In our case, T=504. Confidence level equals 95%.
ASSESSING VAR PRECISION
Equal Weighted Estimatiom EWMA Estimatiom
Estimated µ S.D. of µ Confidence Interval Estimated µ S.D. of µ Confidence Interval
SPX 27.73% 1.13% [25.51%,29.94%] 27.73% 0.72% [26.32%,29.14%]
STI 47.28% 1.11% [45.10%,49.46%] 47.28% 0.69% [45.92%,48.64%]
Estimated σ S.D. of σ Confidence Interval Estimated σ S.D. of σ Confidence Interval
SPX 25.40% 0.80% [23.83%,26.97%] 16.17% 0.51% [15.18%,17.17%]
STI 24.99% 0.79% [23.45%,26.53%] 15.56% 0.49% [14.60%,16.52%]
Table 5 - Assessing VaR precision
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.
Kupiec's Test
VaR MaxVaR
Confidence
Level
Test Statistic
LR
Observed
N
Test
Outcome
Test Statistic
LR
Observed
N
Test
Outcome
SPX 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 Accept
STI 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.