
Maximum loss optimization: search over the losses that occur for intermediate as
well as extreme values of the risk variables
Chapter 9
Market
Risk
VaR:
Historical
Simulation
Approach
1.
Methodology
Suppose that we calculate VaR for a portfolio using a oneday time horizon, a 99%
confidence level, and 500 days of data.

Identify the market variables affecting the portfolio. They typically are exchange
rates, equity prices, interest rates, and so on.

Collect data on the movements in these variables over the most recent 500 days.

The value of the market variable tomorrow under i
th
scenario is
1
i
n
i
v
v
v
−
(n –
today, say n=500, i = 0, 1, … 499)

Get change in portfolio value under each scenario according to the calculated
market variables.

VaR = 5
th
worst number of the change in portfolio value.
2.
The confidence interval:

Kendall and Stuart calculate a confidence interval for the quantile of a distribution
when it is estimated from sample data.

The standard errors of the estimate is:
1
(1
( )
q
q
f x
n
)
−
, where f(x) is the
probability density function of the loss evaluated at x.
3.
Weighting of observations
: the basic historical simulation approach assumes that
each day in the past is given equal weight. Boudoukh et alpha. Suggest that more
recent observations should be given more weight – the weight given to the change
between day ni and ni+1 is
1
(1
)
1
i
n
λ
λ
λ
−
−
−
. VaR is calculated by ranking the
observations from the worst outcome to the best. Starting at the worst outcome,
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