value at risk chapter 20 value at risk part 1 資管所 陳竑廷
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Chapter 20
Value at RiskValue at Riskpart 1
資管所 陳竑廷
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AgendaAgenda
20.1 The VaR measure
20.2 Historical simulation
20.3 Model-building approach
20.4 Linear model
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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.
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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
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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?”
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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
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Time HorizonTime Horizon
• In practice , set N =1, because there’s not enough data.
• The usual assumption:
VaRday -1VaRday -N N
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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
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20.2 Historical Simulation Historical Simulation
• One of the popular way of estimate VaR
• Use past data in a vary direct way
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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
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42.2633.20
78.2085.25
1
i
im v
vv
The fifth-worst daily change is the first percentile of the distribution
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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
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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
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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
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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
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deviation standard sportfolio' theis and
variableofy volatilit theis where
21
222
1
P
i
n
iijjiji
jiiiP
n
iii
i
xP
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Linear Model and OptionsLinear Model and Options
define
define
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• 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
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ExampleExample
• Consider an investment in options on Microsoft and AT&T. Suppose
that SMS = 120 , SA= 30 , MS = 1000 , and = 1000
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Thank you.Thank you.