25 Nov 2011

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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., V0−Vn=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 V∗be such that P(Vn≤V∗) = 1 −α. Then VaRα,n =V0−V∗.

VaR in terms of change in portfolio value

Denote ∆V=Vn−V0. Since 1 −α=P(Vn≤V∗) = P(∆V≤V∗−V0) = P(∆V≤ −VaR).

VaR in terms of rate of change

Let Rbe such that Vn=V0(1 + R). Deﬁne R∗to be such that V∗=V0(1 + R∗). Then 1 −α=P(Vn≤

V∗) = P(R≤R∗). Also, VaRα,n=V0−V∗=V0−V0(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 R∼N(µ, σ2) and

Vn−V0= ∆V=V0(1 + R)−V0=RV0.

Hence

∆V∼N(V0µ, V 2

0σ2).

We want to compute VaRα,n, which is such that

1−α=P(∆V≤ −VaRα,n)

=P(Z≤−VaRα,n −V0µ

V0σ).

Deﬁning zαto be such that α=P(Z≤zα) (so for example, z0.99 = 2.326, z0.975 = 1.96, z0.95 =

1.645, z0.9= 1.282), we get that

−VaRα,n =z1−αV0σ+V0µ,

and therefore (using the fact that z1−α=−zα)

VaRα,n =V0σzα−V0µ.

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 =V0σ√nzα.

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

Vn−V0= ∆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

R2∼BV 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

RV∼N(µV, σ2

V),

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