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

• 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

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

• Let vi be 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

1i

in v

vv

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

Historical Simulation continued

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 infoDate T-Bond Value % Change T-Bill Value % Change

12/31/10 102 - 97 -

12/30/10 100 2.00% 98 -1.02%

12/29/10 97 3.09% 98 0.00%

: : : : :

: : : : :

9/12/10 103 -2.91% 96 2.08%

9/11/10 103 0.00% 97 -1.03%

Example of Historical Simulation cont.

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

• T-bond value = 102 x % changeT-bill value = 97 x % change

T-Bond Modeled T-Bill Modeled Portfolio

Date % Change Value % Change Value Value

12/31/10 2.00% 104.04 -1.02% 96.01 200.05

12/30/10 3.09% 105.15 0.00% 97.00 202.15

: : : : : :

: : : : : :

9/12/10 -2.91% 99.03 2.08% 99.02 198.05

9/11/10 0.00% 102.00 -1.03% 96.00 198.00

Example of Historical Simulation cont.

• Rank the resulting 100 portfolio values

• The 5th lowest portfolio value is the VaRRank

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

• 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

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

n

qq

xf

)1(

)(

1

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

67.1500

99.001.0

0027.0

1

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 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

1

111 /)(

i

iniiin v

vvvv

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 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 Theory

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)

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

• The cumulative distribution is

• The probability density function

/ξ1

yβ

ξ11F(y)cdf

1ξ1

β

ξy1

β

1f(y)pdf

0.0

0.5

1.0

0 1 2 3 4

fx(x)

/

=+0.5

=-0.5

0

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).

• 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

un

i

i uv

1

1/1)(

11

ln

21

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

• We choose and to maximize

Tail Probabilities

1

where

)Prob(

lawpower the toscorrespond that thissee we Setting

1

/1

-

/1

n

nK

Kxxv

u

ux

n

n

u

u

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

1)1(

isIt

11/1

qn

nuVaR

uVaR

n

nq

u

u

The estimate of VaR at the confidence level qis obtained by solving

Estimating Expected Shortfall Using Extreme Value Theory

ξ1

ξuβVaRES

The estimate of ES, provided that the losses exceed theVaR, at the confidence level q, is given by:

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

Example

• The VaR with the 99% confidence limit is

25.197

1)99.01(20

100

39.0

344

39.0

VaR