Value at Risk (VaR) is a statistical technique used to measure potential portfolio losses over a specified time period and confidence level. It was originally used to measure market risk but has been extended to other risk types like credit and operational risk. VaR calculates the maximum dollar amount a portfolio could lose with a given level of confidence, usually 95%. Lower correlations between assets in a portfolio reduce overall risk. VaR is computed using weights, volatilities, and correlations of assets in a portfolio along with the confidence level and time horizon.
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Introduction To Value At Risk
1. INTRODUCTION TO
VALUE AT RISK (VaR)
ALAN ANDERSON, Ph.D.
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2. 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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3. 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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4. 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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5. 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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6. At the 95% confidence level, VaR represents
the border of the 5% โleft tailโ of the normal
distribution, also known as the fifth percentile
or .05 quantile of the normal distribution
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8. This diagram shows that:
95% of the time, the portfolioโs
value remains above $80 million
5% of the time, the portfolioโs
value falls to $80 million or less
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9. The VaR of this portfolio is therefore
$100 million - $80 million = $20 million
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10. 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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11. 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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12. 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
portfolioโs composition
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13. VaR has the disadvantage that it
is computationally intensive and
requires major adjustments for
non-linear assets, such as options
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14. 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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15. Portfolio theory shows how
risk can be reduced by holding
a well-diversified set of assets
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16. 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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17. 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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18. 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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19. 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 portfolioโs value
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20. A portfolio invested in both oil stocks
and automotive stocks would be far
more diversified:
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21. 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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22. 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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23. The risk of holding a portfolio containing two
assets, X and Y, is measured by its standard
deviation, as follows:
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24. P = w
2
X
2
X +w 2
Y
2
Y + 2wX wY X Y
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25. where:
P = the standard deviation
of the returns to the portfolio
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26. X = standard deviation of
the returns to asset X
Y = standard deviation of
the returns to asset Y
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27. 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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28. 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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29. = โrhoโ
this represents the correlation
between the returns to assets
X and Y; -1 1
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30. The lower is the correlation
between assets, the lower will
be the risk of the portfolio
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31. The Value at Risk of a
portfolio is a function of:
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32. the dollar value of the portfolio
the portfolio standard deviation
the confidence level
the time horizon
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33. 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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34. where:
V0 = initial value of the asset
= standard deviation of the
assetโs daily returns
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35. = the number of standard deviations
below the mean corresponding to
the (1-c) quantile of the standard
normal distribution
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36. 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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37. (c) ECI Risk Training
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38. 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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39. EXAMPLE
For a 99% confidence level, the value
of can be determined as follows:
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40. c = 0.99
(1-c) = 0.01 = 1%
NORMSINV(0.01) = -2.33
= 2.33
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41. (c) ECI Risk Training
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42. EXAMPLE
Suppose that an investorโs 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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43. Assume that the standard deviation of the
stockโs returns are 0.0189 (1.89%) per day
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44. If the investor wants to know his
portfolioโs VaR over the coming
trading day at the 95% confidence
level, this would be calculated as
follows:
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46. This means that over the coming day,
there is a 5% chance that the investorโs
losses could reach $310.905 or more
(i.e., the portfolioโs value could fall to
$9,689.095 or less)
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47. NOTE
VaR can be extended to different
time horizons by applying the square
root of time rule
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48. 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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49. If the investor wants to know his
portfolioโs 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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51. Similarly, if the investor wants to know
what his portfolioโs VaR is over the coming
year, assuming that there are 252 trading
days in a year, the calculations would be:
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53. 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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54. 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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55. standard deviation of each assetโs
rate of return in the portfolio
correlations among the assetsโ rates
of return in the portfolio
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56. Once a confidence level and a time
horizon have been chosen, the
weights, volatilities and correlations
can be combined using Markowitzโs
approach to derive the portfolioโs VaR
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57. 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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58. Given the following data, compute
the VaR of this portfolio with a 95%
confidence level over the coming:
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59. day
month
year
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60. DATA
wX = 0.60 wY = 0.40
X = 0.016284 Y = 0.015380
= -0.19055
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