ACTSC 445: AssetLiability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 9 – ValueatRisk
References
(recommended readings): Chap. 18 of Hull.
Introduction
So far in this course, we have studied assetliability issues related to risks associated with movements
in interest rates. In this unit, we take a broader point of view and study risks that arise from a variety
of factors:
not only movements in interest rates, but also in stock prices, currency rates, etc.
To
quantify this more global risk or
total risk
, banks and insurance companies often use a measure called
valueatrisk
(VaR). Although this measure is not perfect, its widespread use makes it relevant to study.
Definition:
VaR is a statistical measure of a portfolio’s risk that estimates the maximum loss that
may be experienced by the portfolio over a given period of time and with a given level of confidence.
More precisely, for a period of time of
n
(typically measured in days) and a confidence level of
α
, if we
denote by
L
n
the random variable corresponding to the portfolio’s loss over
n
days (i.e.,
V
0

V
n
=
L
n
,
where
V
t
is the value of portfolio at time
t
), then VaR
α,n
is such that
P
(
L
n
>
VaR
α,n
) = 1

α.
We can think of VaR as summarizing in a single number the global exposure of the portfolio to market
risks and adverse moves in financial variables (or risk factors).
In plain words, we can think of VaR as being such that we are 100
α
% confident that the portfolio will
not lose more than VaR
α,n
over the next
n
days.
Note:
VaR
α,n
is nonnegative and measured in $.
As an alternative to the notion of loss in the definition of VaR, we can also use other related random
variables:
VaR in terms of portfolio value
Let
V
*
be such that
P
(
V
n
≤
V
*
) = 1

α
. Then VaR
α,n
=
V
0

V
*
.
VaR in terms of change in portfolio value
Denote Δ
V
=
V
n

V
0
. Since 1

α
=
P
(
V
n
≤
V
*
) =
P
(Δ
V
≤
V
*

V
0
) =
P
(Δ
V
≤ 
VaR).
VaR in terms of rate of change
Let
R
be such that
V
n
=
V
0
(1 +
R
). Define
R
*
to be such that
V
*
=
V
0
(1 +
R
*
). Then 1

α
=
P
(
V
n
≤
V
*
) =
P
(
R
≤
R
*
). Also, VaR
α,
n
=
V
0

V
*
=
V
0

V
0
(1 +
R
*
) =

V
0
R
*
.
To compute VaR, we’ll see two family of approaches: analytical approximations and simulation.
In
both cases, there are two main tasks that need to be done before computing VaR:
1
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1. Identify the risk factors (typically, market prices and rates). Usually, we try to first decompose
the (possibly complex) instruments in the portfolio into more basic instruments. Then, we try
to restrict the number of risk factors so that they can be quantified more easily. We must also
make assumptions on how these factors affect the portfolio’s value.
2. Must make assumptions on the distribution of these factors.
Getting an analytical formula to compute VaR
Here, we try to come up with a model that identifies some risk factors and then tells us how the portfolio
of interest is related to these factors. In addition, we need to have a model for how these risk factors
behave. If the chosen model is sufficiently simple, then we can usually compute VaR analytically.
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 Fall '09
 ChristianeLemieux
 Normal Distribution, Standard Deviation, Modern portfolio theory, Mathematical finance, Monte Carlo methods in finance, Value at risk

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