7/28/2019 Value at Risk Final Ppt
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Anu Bumra (14054)
Pratibha Virdi
Ramandeep
7/28/2019 Value at Risk Final Ppt
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Value at Risk is an estimate of the worstpossible loss an investment could realizeover a given time horizon, under normal
market conditions (defined by a givenlevel of confidence).
To estimate Value at Risk a confidencelevel must be specified.
7/28/2019 Value at Risk Final Ppt
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• “We are X percent certain that we will notlose more than V dollars in time T .”
• Function of confidence level X and time T
7/28/2019 Value at Risk Final Ppt
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Normal market conditions – the returns that accountfor 95% of the distribution of possible outcomes.
Abnormal market conditions – the returns that accountfor the other 5% of the possible outcomes.
7/28/2019 Value at Risk Final Ppt
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If a 95% confidence level is used toestimate Value at Risk for a monthlyhorizon;
losses greater than the Value at Riskestimate are expected to occur one intwenty months (5%).
7/28/2019 Value at Risk Final Ppt
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Step 1: Transform simple monthly stock returns intocontinuously compounded stock returns.Note: Technically, log stock returns are “more likely” to be normally
distributed.
Step 2: Choose a level of confidence.
90%, 95%, 99%, etc. Banks are required to report Value at Risk estimated
with a 99% level of confidence to determine regulatorycapital requirements.
Step 3: Compute Value at Risk from sample estimates of
and . For example, the largest likely loss in the household
industry over the next month under normal marketconditions with a 95% level of confidence is: $18,000.
Note: It is possible to realize a loss greater than $18,000.
7/28/2019 Value at Risk Final Ppt
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It captures an important aspect of risk in a single number.
It is easy to understand.
Firm wide summary and handles all the futures , optionsand other complications
Relatively model free. Deviations from normal distributions.
VaR translates portfolio volatility into a dollar value.
VaR is useful for monitoring and controlling risk within the
portfolio. As a tool, VaR is very useful for comparing a portfolio with
the market portfolio (S&P500).
It asks the simple question : “How bad can things get?”
7/28/2019 Value at Risk Final Ppt
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VaR does not measure "event" (e.g.,market crash) risk. That is why portfoliostress tests are recommended tosupplement VaR.
VaR does not readily capture liquiditydifferences among instruments.
VaR doesn't readily capture model risks,
which is why model reserves are alsonecessary.
7/28/2019 Value at Risk Final Ppt
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Instead of calculating the 10-day, 99% VaRdirectly ,analysts usually calculate a 1-day 99% VaR and assume
This is exactly true when portfolio
changes on successive days come fromindependent identically distributednormal distributions.
day VaR1-day VaR-10 10
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Historical simulation
Model building approach /Variance- Covariance approach
Linear approachQuadratic model
Monte Carlo simulation
7/28/2019 Value at Risk Final Ppt
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Create a database of the daily movementsin all market variables.
The first simulation trial assumes that
the percentage changes in all marketvariables are as on the first day
The second simulation trial assumes thatthe percentage changes in all market
variables are as on the second day and soon
7/28/2019 Value at Risk Final Ppt
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Suppose we use m days of historical data
Let vi be the value of a variable on day i
There are m-1 simulation trials
The ith trial assumes that the value of themarket variable tomorrow (i.e., on daym+1) is
1i
i
m
v
vv
7/28/2019 Value at Risk Final Ppt
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The main alternative to historicalsimulation is to make assumptions aboutthe probability distributions of return on
the market variables and calculate theprobability distribution of the change inthe value of the portfolio analytically
This is known as the model building
approach or the variance-covarianceapproach
7/28/2019 Value at Risk Final Ppt
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We have a position worth $10 million inMicrosoft shares
The volatility of Microsoft is 2% per day
(about 32% per year)We use N =10 and X =99
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The standard deviation of the change in theportfolio in 1 day is $200,000
The standard deviation of the change in 10
days is
200 000 10 456, $632,
7/28/2019 Value at Risk Final Ppt
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We assume that the expected change in thevalue of the portfolio is zero (This is OK forshort time periods)
We assume that the change in the value ofthe portfolio is normally distributed
Since N (–2.33)=0.01, the VaR is
2 33 632 456 473 621. , $1, ,
7/28/2019 Value at Risk Final Ppt
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We assume
The daily change in the value of a portfolio islinearly related to the daily returns from
market variables The returns from the market variables are
normally distributed
7/28/2019 Value at Risk Final Ppt
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Portfolio of stocks
Portfolio of bonds
Forward contract on foreign currency
Interest-rate swap
7/28/2019 Value at Risk Final Ppt
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Consider a portfolio of options dependent ona single stock price, S. Define
and S P
S
S x
7/28/2019 Value at Risk Final Ppt
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As an approximation
Similarly when there are many underlyingmarket variables
where i is the delta of the portfolio withrespect to the ith asset
xS S P
i
iiixS P
7/28/2019 Value at Risk Final Ppt
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Consider an investment in options onMicrosoft and AT&T. Suppose the stockprices are 120 and 30 respectively and thedeltas of the portfolio with respect to thetwo stock prices are 1,000 and 20,000respectively
As an approximation
where x1 and x2 are the percentagechanges in the two stock prices
21 000,2030000,1120 x x P
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For a portfolio dependent on a single stockprice it is approximately true that
this becomes2)(
21 S S P
22 )(21 xS xS P
7/28/2019 Value at Risk Final Ppt
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With many market variables we get anexpression of the form
where
This is not as easy to work with as thelinear model
n
i
n
i jiij jiiii x xS S xS P 1 1 2
1
ji
ij
i
iS S
P
S
P
2
7/28/2019 Value at Risk Final Ppt
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To calculate VaR using M.C. simulation we
Value portfolio today
Sample once from the multivariate
distributions of the xi Use the x
ito determine market variables at
end of one day
Revalue the portfolio at the end of day
7/28/2019 Value at Risk Final Ppt
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Calculate P
Repeat many times to build up a probabilitydistribution for P
VaR is the appropriate fractile of thedistribution times square root of N
For example, with 1,000 trial the 1percentile is the 10th worst case.
7/28/2019 Value at Risk Final Ppt
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Model building approach assumes normaldistributions for market variables. It tends togive poor results for low delta portfolios
Historical simulation lets historical datadetermine distributions, but iscomputationally slower
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