The document discusses MaxVaR, an alternative to VaR that measures the probability of portfolio losses on or before the terminal date rather than just at the terminal date. It presents MaxVaR as a more accurate risk measurement than VaR because it accounts for interim risks rather than just terminal risk. The document provides formulas to calculate VaR and MaxVaR and applies these to sample portfolios, finding that MaxVaR is always greater than VaR and better reflects risks. It assesses the precision of VaR and MaxVaR estimates and finds MaxVaR passes proportion of failures tests while VaR does not.

1. 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.

2. RISK MANAGEMENT – MAXVAR
VAR

3. RISK MANAGEMENT – MAXVAR
MAXVAR (BOUDOUKH 2004)

4. RISK MANAGEMENT – MAXVAR
HYPOTHESES
1 LOG NORMAL DISTRIBUTION
STOCK
2 WIENER PROCESS

5. RISK MANAGEMENT – MAXVAR
COMPARISON
VAR MAXVAR
P(R<z)=C P(min R <z)=C
R daily return
C confidence
z loss level

6. RISK MANAGEMENT – MAXVAR
VAR FORMULA
Solve z in
Proof: direct derivation
cumulative standard distribution

7. RISK MANAGEMENT – MAXVAR
MAXVAR FORMULA
Solve z in
Proof: Schuster 1968 theorem applied to first
passage of time of the Wiener process

8. 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) .

9. 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”.

10. 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.

11. 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

12. RISK MANAGEMENT – MAXVAR
COMPARISON OF MAXVAR AND VAR
STI VaR and MaxVaR for α = 5%
VaR = 0.06
MaxVaR = 0.08

13. ASSESSING VAR PRECISION
In our case, T=504. Confidence level equals 95%.

14. 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

15. 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.

16. Kupiec's Test
VaR MaxVaR
Confidence Test Statistic Observed Test Test Statistic Observed Test
Level LR N Outcome LR N 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.

17. 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.