Introduction To Value At Risk
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INTRODUCTION TO
VALUE AT RISK (VaR)
ALAN ANDERSON, Ph.D.
ECI Risk Training
www.ecirisktraining.com
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Value at Risk (VaR) is a statisticaltechnique designed to measure themaximum loss that a portfolio of assetscould suffer over a given time horizonwith a specified level of confidence
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Value at Risk was originally used tomeasure market risk
It has since been extended to othertypes of risk, such as credit risk andoperational risk
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EXAMPLE
Suppose that it is determined that a$100 million portfolio could potentiallylose $20 million (or more) once every20 trading days
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The VaR of this portfolio equals $20million with a 95% level of confidenceover the coming trading day; 19 out of20 trading days (95% of the time),losses are less than $20 million
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At the 95% confidence level, VaR representsthe border of the 5% “left tail” of the normaldistribution, also known as the fifth percentileor .05 quantile of the normal distribution
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This diagram shows that:
95% of the time, the portfolio’svalue remains above $80 million
5% of the time, the portfolio’svalue falls to $80 million or less
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The VaR of this portfolio is therefore
$100 million - $80 million = $20 million
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VaR is based on the assumption that therates of return of the assets held in aportfolio are jointly normally distributed
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VaR has the advantage that the risksof different assets can be combined toproduce a single number that reflectsthe risk of a portfolio
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Further, the probability of a givenloss can be calculated using VaR
VaR can also be used to determinethe impact on risk of changes in aportfolio’s composition
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VaR has the disadvantage that itis computationally intensive andrequires major adjustments fornon-linear assets, such as options
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COMPUTING VaR
Value-at-Risk is based on the work ofHarry Markowitz, who was awardedthe Nobel Prize in Economics in 1990for his pioneering research in the areaof portfolio theory
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Portfolio theory shows howrisk can be reduced by holdinga well-diversified set of assets
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A collection of assets is considered to be well-diversified if the assets are affected differentlyby changes in economic variables, such asinterest rates, exchange rates, etc.
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As a result, a well-diversified portfolio isless likely to experience extreme changesin value; in this way, risk is reduced
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In statistical terms, a well-diversified portfoliocontains assets whose rates of return havevery low or negative correlations with eachother
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EXAMPLE
A portfolio consisting exclusively of oilstocks would not be well-diversified, sincechanges in the price of oil would have ahuge impact on the portfolio’s value
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A portfolio invested in both oil stocksand automotive stocks would be farmore diversified:
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Rising oil prices would hurt the automotivestocks while helping the oil stocks
Falling oil prices would hurt the oil stockswhile helping the automotive stocks
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As a result, the impact of oil priceswings would be offset by changes inthe value of the automotive stocks
On balance, risk would be reduced
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The risk of holding a portfolio containing twoassets, X and Y, is measured by its standarddeviation, as follows:
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2 2 2 22
P X X Y Y X Y X Yw w w w= + +
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where:
P = the standard deviationof the returns to the portfolio
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X = standard deviation ofthe returns to asset X
Y = standard deviation ofthe returns to asset Y
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wX = weight of asset X
wY = weight of asset Y
The weights represent the proportion
of the portfolio invested in each asset;the sum of the weights is one
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NOTE
If short-selling is not possible, then:
0 wX 1
0 wY 1
If short-selling is possible, theweights can be negative
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= “rho”
this represents the correlation
between the returns to assetsX and Y; -1 1
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The lower is the correlationbetween assets, the lower willbe the risk of the portfolio
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The Value at Risk of aportfolio is a function of:
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the dollar value of the portfolio
the portfolio standard deviation
the confidence level
the time horizon
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COMPUTING VaR FOR
A SINGLE ASSET
For a single asset, using dailyreturns data at a confidence levelof c, the VaR is computed as:
0V
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where:
V0 = initial value of the asset
= standard deviation of the asset’s daily returns
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= the number of standard deviationsbelow the mean corresponding tothe (1-c) quantile of the standardnormal distribution
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EXAMPLE
For a 95% confidence level, c = 0.95
(1-c) is the fifth quantile (1-.95 = .05 =5%) of the standard normal distribution
The corresponding value of is 1.645
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The value of corresponding to anyconfidence level can be found with anormal table or with the Excel functionNORMSINV
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EXAMPLE
For a 99% confidence level, the valueof can be determined as follows:
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c = 0.99
(1-c) = 0.01 = 1%
NORMSINV(0.01) = -2.33
= 2.33
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EXAMPLE
Suppose that an investor’s portfolio consistsentirely of $10,000 worth of IBM stock.
Since the portfolio only contains IBM stock,it can be thought of as a single asset
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Assume that the standard deviation of thestock’s returns are 0.0189 (1.89%) per day
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If the investor wants to know hisportfolio’s VaR over the comingtrading day at the 95% confidencelevel, this would be calculated asfollows:
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V0 = (10,000)(1.645)(0.0189)
= $310.905
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This means that over the coming day,there is a 5% chance that the investor’slosses could reach $310.905 or more(i.e., the portfolio’s value could fall to$9,689.095 or less)
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NOTE
VaR can be extended to differenttime horizons by applying the square
root of time rule
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According to this rule, the standarddeviation increases in proportion tothe square root of time:
t periods = t 1 period
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If the investor wants to know hisportfolio’s VaR over the comingmonth at the 95% confidence level,based on the assumption that thereare 22 trading days in a month, thiswould be calculated as follows:
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0 (10,000)(1.645)(0.0189 22)V =
(10,000)(1.645)(0.0189 22) $1,458.27=
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Similarly, if the investor wants to knowwhat his portfolio’s VaR is over the comingyear, assuming that there are 252 tradingdays in a year, the calculations would be:
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0 (10,000)(1.645)(0.0189 252)V =
(10,000)(1.645)(0.0189 252) $4,935.46=
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COMPUTING PORTFOLIO VaR
In order to compute the Value atRisk of a portfolio of two or moreassets, the correlations among theassets must be explicitly considered
The lower these correlations, thelower will be the resulting VaR
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The Value at Risk of a portfolio
is calculated by determining the:
weight (proportion of the totalinvested) of each asset in theportfolio
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standard deviation of each asset’srate of return in the portfolio
correlations among the assets’ ratesof return in the portfolio
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Once a confidence level and a timehorizon have been chosen, theweights, volatilities and correlationscan be combined using Markowitz’sapproach to derive the portfolio’s VaR
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EXAMPLE
Assume that a $100,000 portfoliocontains $60,000 worth of Stock Xand $40,000 worth of Stock Y.
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Given the following data, computethe VaR of this portfolio with a 95%confidence level over the coming:
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day
month
year
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DATA
wX = 0.60 wY = 0.40
X = 0.016284 Y = 0.015380
= -0.19055
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= 0.01144627 = 1.144627%
2(0.6)(0.4)( 0.19055)(0.016284)(0.015380)
2 2 2 2(0.6) (0.016284) (0.4) (0.015380)P= + +
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The portfolio VaR over the coming day is:
= $1,882.91
0 (100,000)(1.645)(0.01144627)P
V =
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The portfolio VaR over the coming month is:
= $8,831.638
0 (100,000)(1.645)(0.01144627 22)P
V =
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The portfolio VaR over the coming year is:
= $29,890.29
0 (100,000)(1.645)(0.01144627 252)P
V =