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Market Risk VaR: Historical Simulation Approach N. Gershun

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Market Risk VaR: Historical Simulation Approach

N. Gershun

Collect data on the daily movements in all market variables.

The first simulation trial assumes that the percentage changes in all market variables are as on the first day

The second simulation trial assumes that the percentage changes in all market variables are as on the second day

and so on

Suppose we use n days of historical data with today being day n

Let vibe the value of a variable on day i

There are n-1 simulation trials

Translate the historical experience of the market factors into percentage changes

The ith trial assumes that the value of the market variable tomorrow (i.e., on day n+1) is

Historical Simulation continued

- Rank the n-1 resulting values
- VaR is the required percentile rank

- Assume a one-day holding period and 5% probability
- Suppose that a portfolio has two assets, a one-year T-bill and a 30-year T-bond
- First, gather the 100 days of market info

- Apply all changes to the current value of assets in the portfolio
- T-bond value = 102 x % change
T-bill value = 97 x % change

- Rank the resulting 100 portfolio values
- The 5th lowest portfolio value is the VaR

Rank

1

2

3

4

5

:

:

99

100

Date

11/12/10

12/1/10

10/17/10

10/13/10

9/11/10

:

:

12/8/10

9/25/10

Value

195.45

196.24

197.13

197.60

198.00

:

:

202.15

203.00

- Historical simulation is relatively easy to do: Only requires knowing the market factors and having the historical information

Suppose that x is the qth quantile of the loss distribution when it is estimated from n observations. The standard error of x is

where f(x) is an estimate of the probability density of the loss at the qth quantile calculated by assuming a probability distribution for the loss

We are interested in estimating the 99 percentile from 500 observations

We estimated f(x) by approximating the actual empirical distribution with a normal distribution mean zero and standard deviation $10 million

Using Excel, the 99 percentile of the approximating distribution is NORMINV(0.99,0,10) = 23.26 and the value of f(x) is NORMDIST(23.26,0,10,FALSE)=0.0027

The estimate of the standard error is therefore

- Suppose that we estimated the 99th percentile using historical simulation as $25M
- Using our estimate of standard error, the 95% confidence interval is:
25-1.96×1.67<VaR<25+1.96×1.67

That is:

Prob($21.7<VaR>$28.3) = 95%

Use a volatility updating scheme and adjust the percentage change observed on day i for a market variable for the differences between volatility on day i and current volatility

Value of market variable under ith scenario becomes

Where n+1 is the current estimate of the volatility of the market variable and i is the volatility estimated at the end of day i-1

Extreme value theory can be used to investigate the properties of the right tail of the empirical distribution of a variable x. (If we are interested in the left tail we consider the variable –x.)

We then use Gnedenko’s result which shows that the tails of a wide class of distributions share common properties.

Suppose F(*) is a the cumulative distribution function of the losses on a portfolio.

We first choose a level u in the right tail of the distribution of losses on the portfolio

The probability that the particular loss lies between u and u +y (y>0) is

F(u+y) – F(u)

The probability that the loss is greater than u is:

1-F(u)

Gnedenko’s result shows that for a wide class of distributions, Fu(y) coverges a Generalized Pareto Distribution

17

GDP has two parameters (the shape parameter) and (the scale parameter)

The cumulative distribution is

The probability density function

fx(x)

=+0.5

0

=-0.5

/

Generalized Pareto Distribution

- = 0 if the underlying variable is normal
- increases as tails of the distribution become heavier
- For most financial data >0 and is between 0.1 and 0.4

- G.P.D. is appropriate distribution for independent observations of excesses over defined thresholds
- GPD can be used to predict extreme portfolio losses

The observations, i, are sorted in descending order. Suppose that there are nu observations greater than u

We choose and to maximize

21

Our estimator for the cumulative probability that the variable is greater than x is

Extreme Value Theory therefore explains why the power law holds so widely

The estimate of VaR at the confidence level q

is obtained by solving

The estimate of ES, provided that the losses exceed the

VaR, at the confidence level q, is given by:

- Consider an example in the beginning of the lecture. Suppose that u= 4 and nu = 20. That is there are 20 scenarios out of total of 100 where the loss is greater than 4.
- Suppose that the maximum likelihood estimation results in = 34 and = 0.39
- The VaR with the 99% confidence limit is

- The VaR with the 99% confidence limit is