Market Risk VaR: Historical Simulation Approach
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Market Risk VaR: Historical Simulation Approach N. Gershun. Historical Simulation. 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

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Market risk var historical simulation approach n gershun

Market Risk VaR: Historical Simulation Approach

N. Gershun


Historical simulation

Historical Simulation

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


Historical simulation continued

Historical Simulation continued

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


Market risk var historical simulation approach n gershun

Historical Simulation continued

  • Rank the n-1 resulting values

  • VaR is the required percentile rank


Example of historical simulation

Example of Historical Simulation

  • 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


Example of historical simulation cont

Example of Historical Simulation cont.

  • Apply all changes to the current value of assets in the portfolio

  • T-bond value = 102 x % change

    T-bill value = 97 x % change


Example of historical simulation cont1

Example of Historical Simulation cont.

  • 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


Notes on historical simulation

Notes on Historical Simulation

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

  • Correlations between the market factors are implicit in this method because we are using historical information

  • In our example, short bonds and long bonds would typically move in the same direction


  • Accuracy

    Accuracy

    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


    Example

    Example

    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


    Example cont

    Example (cont.)

    • 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%


    Extension 1

    Extension 1


    Extension 2

    Extension 2

    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

    Extreme Value Theory

    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.


    Extreme value theory1

    Extreme Value Theory

    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)


    Extreme value theory2

    Extreme Value Theory


    Extreme value theory3

    Extreme Value Theory

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

    17


    Generalized pareto distribution gpd

    Generalized Pareto Distribution(GPD)

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

    The cumulative distribution is

    The probability density function


    Market risk var historical simulation approach n gershun

    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


    Generalized pareto distribution cont

    Generalized Pareto Distribution(cont).

    • G.P.D. is appropriate distribution for independent observations of excesses over defined thresholds

    • GPD can be used to predict extreme portfolio losses


    Maximum likelihood estimator

    Maximum Likelihood Estimator

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

    We choose  and  to maximize

    21


    Tail probabilities

    Tail Probabilities

    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


    Estimating var using extreme value theory

    Estimating VaR Using Extreme Value Theory

    The estimate of VaR at the confidence level q

    is obtained by solving


    Estimating expected shortfall using extreme value theory

    Estimating Expected Shortfall Using Extreme Value Theory

    The estimate of ES, provided that the losses exceed the

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


    Example1

    Example

    • 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


    Example2

    Example

    • The VaR with the 99% confidence limit is


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