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.