# unit9 value at risk

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ACTSC 445: Asset-Liability Management
Department of Statistics and Actuarial Science, University of Waterloo
Unit 9 – Value-at-Risk
References (recommended readings): Chap. 18 of Hull.
Introduction
So far in this course, we have studied asset-liability 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
value-at-risk (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 Lnthe random variable corresponding to the portfolio’s loss over ndays (i.e., V0Vn=Ln,
where Vtis the value of portfolio at time t), then VaRα,n is such that
P(Ln>VaRα,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 100α% conﬁdent that the portfolio will
not lose more than VaRα,n over the next ndays.
Note: VaRα,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 Vbe such that P(VnV) = 1 α. Then VaRα,n =V0V.
VaR in terms of change in portfolio value
Denote ∆V=VnV0. Since 1 α=P(VnV) = P(∆VVV0) = P(∆V≤ −VaR).
VaR in terms of rate of change
Let Rbe such that Vn=V0(1 + R). Deﬁne Rto be such that V=V0(1 + R). Then 1 α=P(Vn
V) = P(RR). Also, VaRα,n=V0V=V0V0(1 + R) = V0R.
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:
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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.
In what follows, we’ll be looking at diﬀerent approaches that model risk factors as being multinormal
random variables.
(1) Multinormal Linear Models
With this approach, the following assumptions are made: the change in the value of the portfolio
is linearly related to the risk factors. We’ll start by looking at cases where this assumption can be
supported by ﬁnancial theory (examples are stock portfolios, bonds, and forward contracts), and then
study cases where this is only an approximation (examples are options).
Let us start with the simple case where we assume that there is only one risk factor.
One-Factor Case
Assume there is only one risk factor Rsuch that RN(µ, σ2) and
VnV0= ∆V=V0(1 + R)V0=RV0.
Hence
VN(V0µ, V 2
0σ2).
We want to compute VaRα,n, which is such that
1α=P(∆V≤ −VaRα,n)
=P(ZVaRα,n V0µ
V0σ).
Deﬁning zαto be such that α=P(Zzα) (so for example, z0.99 = 2.326, z0.975 = 1.96, z0.95 =
1.645, z0.9= 1.282), we get that
VaRα,n =z1αV0σ+V0µ,
and therefore (using the fact that z1α=zα)
VaRα,n =V0σzαV0µ.
Now, usually, this part of the computation is done for daily returns, i.e., nis set to 1. Because of that, µ
is usually taken to be 0. (Why does it make sense? Shouldn’t we also set σto 0 then? If we have a stock
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with annual return µand annual volatility σ, roughly we can think of the daily return/volatility as
being µ/365 and σ/365, respectively. So the daily return becomes negligible, but not the volatility.)
Therefore we have VaRα,n =V0σzα. Then, if we want to compute VaRα,n for n > 1, we can make the
assumption that the return R(n) over nperiods can be written as
R(n) =
n
X
t=1
Rt,
where the Rt’s are i. i. d. N(0, σ2). Hence R(n)N(0, nσ2), and thus
VaRα,n =V0σnzα.
Example:Suppose you have a portfolio that consists of \$10 millions in shares of Microsoft. The goal
is to compute VaRα,n for α= 0.99 and n= 10 days. We assume the stock’s return over one day has a
volatility σof 0.02. Since z0.99 = 2.33, we have
VaR0.99,1= 0.02 ×(10 ×106)×2.33 = \$466,000,
and
VaR0.99,10 =10VaR0.99,1= \$1,473,621.
Similarly, for a portfolio where you hold \$5 millions in shares of AT&T and you assume the daily
volatility of the AT&T stock is 0.01, then
VaR0.99,1= 0.01 ×(5 ×106)×2.33 = \$116,500
VaR0.99,10 =10VaR0.99,1= \$368,405.
Two-Factor Case
Suppose we now have two risk factors, so that we can write
VnV0= ∆V=V0(w1(1 + R1) + w2(1 + R2)) V0,
where w1and w2represent the proportion of the portfolio subjected to risk 1 and risk 2, respectively.
For instance, if the two risk factors are returns on two assets, then wi=niSi/V0for i= 1,2, where ni
is the number of shares of asset iheld at time 0 and Siis the share’s value.
The multinormal assumption means we believe
R1
R2BV N  µ1
µ2,σ2
1ρσ1σ2
ρσ1σ2σ2
2,
where σ1and σ2are the volatility of the return on asset 1 and 2, respectively, and ρis the correlation
between these two returns. We are interested in the return RVof the portfolio, deﬁned as
RV=V
V0
.
The distribution assumption on R1and R2implies that
RVN(µV, σ2
V),
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