Estimating Market Risk Measures -...

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Estimating Market Risk Measures

© Pristine FRM – II

Estimating VaR using Historical Simulation

This is by far the easiest method for estimating VaR.

All historical observations are simply ordered from the smallest to the largest and the observation that follows the threshold loss level denotes the VaR level.

To determine the VaR for n observations at the (1- ᾳ) confidence level will be (ᾳ*n) + 1

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Identifying VaR Limit - Example

Calculate the ordered observation in a sample of 10,000 data points that corresponds to VaR at the 97% confidence level.

Solution

• VaR is to be estimated at the 97% confidence level, this means that 3% (i.e. 300) of the ordered observations will fall in the tail of the distribution.

• Therefore the 301st ordered loss observation will separate the 3% of the largest losses from the remaining 97% of returns.

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Parametric Estimation Approach for Estimating VaR

In this method we explicitly assume a distribution for the underlying observations.

There are two cases in this approach

• VaR for returns with a normal distribution

– The VaR for a given confidence level denotes the point that separates the tail losses from the remaining distribution.

– VaR cutoff will be in the left tail of the returns distribution

VaR(α%) = - μP/L + σP/L*zα

– In practice, since the population std. dev and mean are not known, the researcher will use the sample mean and std. dev.

– If instead of the profit/loss distribution the arithmetic returns are normally distributed then we use the following formula

VaR(α%) = (- μP/L + σP/L*zα)*Pt-1

• VaR for returns with a lognormal distribution

– The log-normal distribution is rightly skewed with positive outliers and bounded below by zero.

– The log-normal distribution is commonly used to eliminate the possibility of negative asset prices (Pt).

– If we assume that the geometric returns for a normal distribution, then the natural log of the asset prices follow a normal distribution and Pt follows a lognormal distribution.

VaR(α%) = Pt-1(1-eμR - σR*zα)

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Computing VaR - Example

VaR (normal distribution)Seema Malhotra has constructed a fixed income portfolio, she is submitting a report to her supervisor. She notes that the profit/loss distribution for the portfolio is normally distributed with annual mean of $27mn and a std. dev. Of $14mn. Calculate the VaR at the 97.5% and 99% confidence level using the parametric approach.

Solution

• VaR(2.5%) = -$27mn + $14mn*1.96 = $440k

• VaR(1.0%) = -$27mn + $14mn*2.326= $5.564mn

VaR (arithmetic returns)A portfolio has a beginning period value of $100. The arithmetic returns follow a normal distribution with a mean of 12% and a std. dev. of 25%. Calculate VaR at the 99.5% confidence level

Solution

• VaR(0.5%) = ( -12% + 2.576*25%)*100 = $52.4

VaR (lognormal distribution)A diversified portfolio exhibits a normal distributed geometric return with mean & std. dev. Of 8% and 20%. Calculate the 5% lognormal VaR assuming beginning period portfolio value is $100.

Solution

• Lognormal VaR(5%) = 100*(1- exp[0.08 – 0.2*1.65]) = $22.12

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Expected Shortfall given P/L or Return Data

VaR measure does not tell the investor the amount or magnitude of the actual loss.

The expected shortfall (ES) provides an estimate of the tail loss by averaging the VaRs for increasing the confidence levels in the tail.

The tail mass is divided into n-equal slices and the corresponding n-1 VaRs are computed.

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Confidence Level VaR Difference

96% 1.7507

97% 1.8808 0.1301

98% 2.0537 0.1729

99% 2.3263 0.2726

Average 2.003

Theoretical true value 2.063


Coherent Risk Measures

Coherent risk measure is the weighted average of the quantiles of the loss distribution where the weights are user-specific based on individual risk aversion.

This is a more general risk measure than either a VaR or ES.

When modeling for ES, the weighing function is set to [1/(1-confidence level)] for all tail loses.

The four properties are:

• Monotonicity

• Sub-additivity

• Positive homogeneity

• Translation invariance

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Estimating Coherent Risk Measures by Estimating Quantiles

Consider the process when n = 10.

The entire distribution is divided into nine equal probability mass slices at 10%, 20%, …. 90%

The 10% quantile relates to -1.2816, the 20% quantile relates to -0.8416, and the 90% quantile relates to 1.2816.

Each quantile is then weighted by the specific risk aversion function and then averaged to arrive at the value of the coherent risk measure.

This measure is more sensitive to the choice of n.

As the value of n increases the quantiles will be further into the tails where more extreme values of the distribution is located.

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Estimating Standard Errors for Estimators of Coherent Risk Measures

Any estimator is only as useful as it is precise, if it is not accurate then it has limited practical value.

Generally it is the best practice to compute the standard errors for all coherent risk measures.

Lets start with a sample size of n and arbitrary bin width of h around quantile, q.

A bin width is just the width of the intervals, sometimes called "bins", in a histogram.

Computing the std. error is done by realizing that the square root of the variance of the quantile is equal to the std. error of the quantile.

A confidence. level is then calculated.

[q + se(q)*zα] > VaR > [q - se(q)*zα]

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QQ Plots for Identifying the Distribution of Data

A visual inspection can be a very simple and powerful tool for determining the distribution of our observations.

A quantile-quantile (QQ) plot is a straightforward way to visualize if the empirical data fits the hypothesized theoretical distribution.

The process graphs the quantiles at regular confidence intervals for the empirical distribution against the theoretical distribution.

If both the empirical and theoretical data are drawn from the same distribution, then the median of the empirical distribution would plot very close to zero, while the median of the theoretical distribution.

We can continue in this fashion for the other quantiles.

When the two distributions are very similar then the resulting QQ plot will be linear.

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Estimating Market Risk Measures – Questions

1. Which of the following statement is least likely correct:

A. Expected shortfall gives an estimate of the magnitude of a loss for unfavorable events while VaR doesn't

B. The risk surface for Expected shortfall is concave because of the property of sub-additivity is met.

C. Expected shortfall satisfies all the properties of coherent risk measurement while VaR satisfies only for normal distributions.

D. None of the above.

B. Coherent risk measures are NOT required to satisfy one the following properties;

A. Sub-additivity

B. Homogeneity

C. Monotonicity

D. Risk averse condition

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Estimating Market Risk Measures – Solutions

1. B.The Correct answer is The risk surface for Expected shortfall is concave because of the property of sub-additivity is met. The risk surface for Expected Shortfall is convex.

2. D.They are not required to satisfy risk-free condition and not risk averse condition.

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Estimating Market Risk Measures -...

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