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INTRODUCTION TO
VALUE AT RISK (VaR)
ALAN ANDERSON, Ph.D.
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Value at Risk (VaR) is a statistical
technique designed to measure the
maximum loss that a portfolio of assets
could suffer over a given time horizon
with a specified level of confidence
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Value at Risk was originally used to
measure market risk
It has since been extended to other
types of risk, such as credit risk and
operational risk
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EXAMPLE
Suppose that it is determined that a
$100 million portfolio could potentially
lose $20 million (or more) once every
20 trading days
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The VaR of this portfolio equals $20
million with a 95% level of confidence
over the coming trading day; 19 out of
20 trading days (95% of the time),
losses are less than $20 million
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At the 95% confidence level, VaR represents
the border of the 5% “left tail” of the normal
distribution, also known as the fifth percentile
or .05 quantile of the normal distribution
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This diagram shows that:
95% of the time, the portfolio’s
value remains above $80 million
5% of the time, the portfolio’s
value 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 the
rates of return of the assets held in a
portfolio are jointly normally distributed
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VaR has the advantage that the risks
of different assets can be combined to
produce a single number that reflects
the risk of a portfolio
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Further, the probability of a given
loss can be calculated using VaR
VaR can also be used to determine
the impact on risk of changes in a
portfolio’s composition
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VaR has the disadvantage that it
is computationally intensive and
requires major adjustments for
non-linear assets, such as options
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COMPUTING VaR
Value-at-Risk is based on the work of
Harry Markowitz, who was awarded
the Nobel Prize in Economics in 1990
for his pioneering research in the area
of portfolio theory
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Portfolio theory shows how
risk can be reduced by holding
a well-diversified set of assets
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A collection of assets is considered to be well-
diversified if the assets are affected differently
by changes in economic variables, such as
interest rates, exchange rates, etc.
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As a result, a well-diversified portfolio is
less likely to experience extreme changes
in value; in this way, risk is reduced
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In statistical terms, a well-diversified portfolio
contains assets whose rates of return have
very low or negative correlations with each
other
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EXAMPLE
A portfolio consisting exclusively of oil
stocks would not be well-diversified, since
changes in the price of oil would have a
huge impact on the portfolio’s value
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A portfolio invested in both oil stocks
and automotive stocks would be far
more diversified:
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Rising oil prices would hurt the automotive
stocks while helping the oil stocks
Falling oil prices would hurt the oil stocks
while helping the automotive stocks
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As a result, the impact of oil price
swings would be offset by changes in
the value of the automotive stocks
On balance, risk would be reduced
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The risk of holding a portfolio containing two
assets, X and Y, is measured by its standard
deviation, as follows:
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P = w
2
X
2
X +w 2
Y
2
Y + 2wX wY X Y
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where:
P = the standard deviation
of the returns to the portfolio
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X = standard deviation of
the returns to asset X
Y = standard deviation of
the 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, the
weights can be negative
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= “rho”
this represents the correlation
between the returns to assets
X and Y; -1 1
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The lower is the correlation
between assets, the lower will
be the risk of the portfolio
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The Value at Risk of a
portfolio 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 daily
returns data at a confidence level
of c, the VaR is computed as:
V0
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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 deviations
below the mean corresponding to
the (1-c) quantile of the standard
normal 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 any
confidence level can be found with a
normal table or with the Excel function
NORMSINV
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EXAMPLE
For a 99% confidence level, the value
of 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 consists
entirely 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 the
stock’s returns are 0.0189 (1.89%) per day
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If the investor wants to know his
portfolio’s VaR over the coming
trading day at the 95% confidence
level, this would be calculated as
follows:
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This means that over the coming day,
there is a 5% chance that the investor’s
losses 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 different
time horizons by applying the square
root of time rule
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According to this rule, the standard
deviation increases in proportion to
the square root of time:
t periods = t 1 period
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If the investor wants to know his
portfolio’s VaR over the coming
month at the 95% confidence level,
based on the assumption that there
are 22 trading days in a month, this
would be calculated as follows:
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Similarly, if the investor wants to know
what his portfolio’s VaR is over the coming
year, assuming that there are 252 trading
days in a year, the calculations would be:
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COMPUTING PORTFOLIO VaR
In order to compute the Value at
Risk of a portfolio of two or more
assets, the correlations among the
assets must be explicitly considered
The lower these correlations, the
lower 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 total
invested) of each asset in the
portfolio
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standard deviation of each asset’s
rate of return in the portfolio
correlations among the assets’ rates
of return in the portfolio
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Once a confidence level and a time
horizon have been chosen, the
weights, volatilities and correlations
can be combined using Markowitz’s
approach to derive the portfolio’s VaR
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EXAMPLE
Assume that a $100,000 portfolio
contains $60,000 worth of Stock X
and $40,000 worth of Stock Y.
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Given the following data, compute
the 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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