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Figure 2.1: VaRs from HS with 250 and 1,000 Return Days Jul 1, 2008 - Dec 31, 2010

# Figure 2.1: VaRs from HS with 250 and 1,000 Return Days Jul 1, 2008 - Dec 31, 2010

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Weighted Historical Simulation

• WHS relieves the tension in the choice of m

• It assigns relatively more weight to the most recent

observations and relatively less weight to the returns

further in the past

• It is implemented as follows –

• Sample of m past hypothetical returns, {RPF,t+1-τ}mτ=1

is assigned probability weights declining exponentially

through the past as follows

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

10

Weighted Historical Simulation

– Today’s observation is assigned the weight η1 =

(1- η) / (1- ηm)

– ητ goes to zero as t gets large, and that the weights

ητ for τ = 1,2,...,m sum to 1

– Typical value for η is between 0.95 and 0.99

• The observations along with their assigned weights

are sorted in ascending order.

• The 100p% VaR is calculated by accumulating the

weights of the ascending returns until 100p% is

reached.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

11

Pros and Cons of WHS

Pros

•Once η is chosen, WHS does not require estimation

and becomes easy to implement

•It’s weighting function builds dynamics into the

WHS technique

•The weighting function also makes the choice of m

somewhat less crucial.

•WHS responds quickly to large losses

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

12

Pros and Cons of WHS

Cons

•No guidance is given on how to choose η

•Effect on the weighting scheme of positive versus

negative past returns

•If we are short the market, a market crash has no

impact on our VaR. WHS does not respond to large

gains

•the multiday VaR requires a large amount of past

daily return data, which is not easy to obtain.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

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• It can pick up the increase in market variance

from the crash regardless of whether the crash

meant a gain or a loss

• In this model, returns are squared and losses and

gains are treated as having the same impact on

tomorrow’s variance and therefore on the portfolio

risk.

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

14

Figure 2.2 A:

Historical Simulation VaR and Daily Losses from

Long S&P500 Position, October 1987

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

15

Figure 2.2 B:

Historical Simulation VaR and Daily Losses from

Short S&P500 Position, October 1987

Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen

16

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Figure 2.1: VaRs from HS with 250 and 1,000 Return Days Jul 1, 2008 - Dec 31, 2010

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