Chapter 20

Value at RiskValue at Riskpart 1

資管所 陳竑廷

AgendaAgenda

20.1 The VaR measure

20.2 Historical simulation

20.3 Model-building approach

20.4 Linear model

20.1 The VaR measure The VaR measure

Value at RiskValue at Risk

• Provide a single number summarizing the total risk in a

portfolio of financial assets.

• We are X percent certain that we will not lose more than

V dollars in the next N days.

ExampleExample

When N = 5 , and X = 97,

VaR is the third percentile of the distribution of

change in the value of the portfolio over the next 5

days.

( 100-X ) %

VaR

Advantages of VaRAdvantages of VaR

• It captures an important aspect of risk in a

single number

• It is easy to understand

• It asks the simple question: “How bad can things

get?”

ParametersParameters

• We are X percent certain that we will not lose

more than V dollars in the next N days.

– X

• The confidence interval

– N

• The time horizon measured in days

Time HorizonTime Horizon

• In practice , set N =1, because there’s not enough data.

• The usual assumption:

VaRday -1VaRday -N N

ExampleExample

• Instead of calculating the 10-day, 99% VaR

directly analysts usually calculate a 1-day 99%

VaR and assume

day VaR1-day VaR-10 10

20.2 Historical Simulation Historical Simulation

• One of the popular way of estimate VaR

• Use past data in a vary direct way

When N = 1 , X = 99•Step1

– Identify the market variables affecting the portfolio

•Step2– Collect data on the movements in these market

variables over the most recent 500 days

•Provide 500 alternative scenarios for what can happen between today and tomorrow

42.2633.20

78.2085.25

1

i

im v

vv

The fifth-worst daily change is the first percentile of the distribution

20.3 The Model-Building Approach The Model-Building Approach

• Daily Volatilities

– In option pricing we measure volatility “per year”

– In VaR calculations we measure volatility “per day”

252year

day

Single AssetSingle Asset

• Portfolio A consisting of $10 million in Microsoft

• Standard deviation of the return is 2% (daily)

• N = 10 , X = 99

– N(-2.33) = 0.01

– 1-day 99% ： 2.33 x ( 10,000,000 x 2% ) = $ 466,000

– 10-day 99% ： $1,473,621 =10 466,000

Two AssetTwo Asset

• Portfolio B consisting of $10 million in Microsoft and $5

million in AT&T

•

1-day 99% ：

10-day 99% ：

227,220000,50000,2003.02000,50000,200 22

YXYXYX 222

129,513$33.2227,220

0.3 000,50 000,200 that Suppose YX

657,622,1$10129,513

20.4 The Linear Model The Linear Model

We assume

• The daily change in the value of a portfolio is linearly

related to the daily returns from market variables

• The returns from the market variables are normally

distributed

deviation standard sportfolio' theis and

variableofy volatilit theis where

21

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P

i

n

iijjiji

jiiiP

n

iii

i

xP

Linear Model and OptionsLinear Model and Options

define

define

• As an approximation

• Similarly when there are many underlying market variables

where i is the delta of the portfolio with respect to the ith

asset

xSSP

i

iii xSP

ExampleExample

• Consider an investment in options on Microsoft and AT&T. Suppose

that SMS = 120 , SA= 30 , MS = 1000 , and = 1000

Thank you.Thank you.