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INTRODUCTION TO
VALUE AT RISK (VaR)

  ALAN ANDERSON, Ph.D.
     ECI Risk Training
   www.ecirisktraining.com
Value at Risk (VaR) is a statistical
technique designed to measure the
maximum loss that a portfolio of assets
could suffe...
Value at Risk was originally used to
measure market risk

It has since been extended to other
types of risk, such as credi...
EXAMPLE

Suppose that it is determined that a
$100 million portfolio could potentially
lose $20 million (or more) once eve...
The VaR of this portfolio equals $20
million with a 95% level of confidence
over the coming trading day; 19 out of
20 trad...
At the 95% confidence level, VaR represents
the border of the 5% “left tail” of the normal
distribution, also known as the...
(c) ECI Risk Training
www.ecirisktraining.com   7
This diagram shows that:

 95% of the time, the portfolio’s
 value remains above $80 million

 5% of the time, the portfol...
The VaR of this portfolio is therefore

 $100 million - $80 million = $20 million




                  (c) ECI Risk Train...
VaR is based on the assumption that the
rates of return of the assets held in a
portfolio are jointly normally distributed...
VaR has the advantage that the risks
of different assets can be combined to
produce a single number that reflects
the risk...
Further, the probability of a given
loss can be calculated using VaR

VaR can also be used to determine
the impact on risk...
VaR has the disadvantage that it
is computationally intensive and
requires major adjustments for
non-linear assets, such a...
COMPUTING VaR

Value-at-Risk is based on the work of
Harry Markowitz, who was awarded
the Nobel Prize in Economics in 1990...
Portfolio theory shows how
risk can be reduced by holding
a well-diversified set of assets




              (c) ECI Risk ...
A collection of assets is considered to be well-
diversified if the assets are affected differently
by changes in economic...
As a result, a well-diversified portfolio is
less likely to experience extreme changes
in value; in this way, risk is redu...
In statistical terms, a well-diversified portfolio
contains assets whose rates of return have
very low or negative correla...
EXAMPLE

A portfolio consisting exclusively of oil
stocks would not be well-diversified, since
changes in the price of oil...
A portfolio invested in both oil stocks
and automotive stocks would be far
more diversified:




                (c) ECI R...
Rising oil prices would hurt the automotive
stocks while helping the oil stocks

Falling oil prices would hurt the oil sto...
As a result, the impact of oil price
swings would be offset by changes in
the value of the automotive stocks

On balance, ...
The risk of holding a portfolio containing two
assets, X and Y, is measured by its standard
deviation, as follows:




   ...
P   = w
      2
      X
          2
          X   +w   2
                   Y
                         2
                 ...
where:


     P  = the standard deviation
    of the returns to the portfolio



               (c) ECI Risk Training
    ...
X   =   standard deviation of
        the returns to asset X

Y   =   standard deviation of
        the returns to asset Y...
wX =    weight of asset X
wY =    weight of asset Y

The weights represent the proportion
of the portfolio invested in eac...
NOTE

If short-selling is not possible, then:

         0   wX        1
         0   wY        1

If short-selling is poss...
= “rho”

this represents the correlation
between the returns to assets
X and Y; -1       1


            (c) ECI Risk Trai...
The lower is the correlation
between assets, the lower will
be the risk of the portfolio




             (c) ECI Risk Tra...
The Value at Risk of a
portfolio is a function of:




           (c) ECI Risk Training
          www.ecirisktraining.com ...
the dollar value of the portfolio
the portfolio standard deviation
the confidence level
the time horizon



            (c...
COMPUTING VaR FOR
A SINGLE ASSET

For a single asset, using daily
returns data at a confidence level
of c, the VaR is comp...
where:

    V0 = initial value of the asset

         = standard deviation of the
           asset’s daily returns


     ...
= the number of standard deviations
 below the mean corresponding to
 the (1-c) quantile of the standard
 normal distribut...
EXAMPLE
For a 95% confidence level, c = 0.95

(1-c) is the fifth quantile (1-.95 = .05 =
5%) of the standard normal distri...
(c) ECI Risk Training
www.ecirisktraining.com   37
The value of corresponding to any
confidence level can be found with a
normal table or with the Excel function
NORMSINV


...
EXAMPLE

For a 99% confidence level, the value
of can be determined as follows:




               (c) ECI Risk Training
 ...
c = 0.99
(1-c) = 0.01 = 1%
NORMSINV(0.01) = -2.33
  = 2.33


      (c) ECI Risk Training
     www.ecirisktraining.com   40
(c) ECI Risk Training
www.ecirisktraining.com   41
EXAMPLE

Suppose that an investor’s portfolio consists
entirely of $10,000 worth of IBM stock.

Since the portfolio only c...
Assume that the standard deviation of the
stock’s returns are 0.0189 (1.89%) per day




                 (c) ECI Risk Tra...
If the investor wants to know his
portfolio’s VaR over the coming
trading day at the 95% confidence
level, this would be c...
V0   = (10,000)(1.645)(0.0189)

         = $310.905




            (c) ECI Risk Training
           www.ecirisktraining.c...
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., t...
NOTE
VaR can be extended to different
time horizons by applying the square
root of time rule




               (c) ECI Ri...
According to this rule, the standard
deviation increases in proportion to
the square root of time:


    t periods   = t  ...
If the investor wants to know his
portfolio’s VaR over the coming
month at the 95% confidence level,
based on the assumpti...
V0   = (10, 000)(1.645)(0.0189 22)


(10, 000)(1.645)(0.0189 22) = $1, 458.27



                (c) ECI Risk Training
   ...
Similarly, if the investor wants to know
what his portfolio’s VaR is over the coming
year, assuming that there are 252 tra...
V0    = (10, 000)(1.645)(0.0189 252)


(10, 000)(1.645)(0.0189 252) = $4,935.46


                 (c) ECI Risk Training
 ...
COMPUTING PORTFOLIO VaR

 In order to compute the Value at
 Risk of a portfolio of two or more
 assets, the correlations a...
The Value at Risk of a portfolio
is calculated by determining the:

weight (proportion of the total
invested) of each asse...
standard deviation of each asset’s
rate of return in the portfolio

correlations among the assets’ rates
of return in the ...
Once a confidence level and a time
horizon have been chosen, the
weights, volatilities and correlations
can be combined us...
EXAMPLE
Assume that a $100,000 portfolio
contains $60,000 worth of Stock X
and $40,000 worth of Stock Y.




             ...
Given the following data, compute
the VaR of this portfolio with a 95%
confidence level over the coming:




             ...
day
  month
  year


 (c) ECI Risk Training
www.ecirisktraining.com   59
DATA
wX = 0.60     wY = 0.40
 X = 0.016284  Y = 0.015380
  = -0.19055




           (c) ECI Risk Training
          www.e...
P   = (0.6) (0.016284) + (0.4) (0.015380) +
            2               2                2   2




2(0.6)(0.4)( 0.19055)(0...
The portfolio VaR over the coming day is:


V0   P   = (100,000)(1.645)(0.01144627)

             = $1,882.91

           ...
The portfolio VaR over the coming month is:


V0   P   = (100, 000)(1.645)(0.01144627 22)

               = $8,831.638

  ...
The portfolio VaR over the coming year is:


V0   P   = (100, 000)(1.645)(0.01144627 252)


                = $29,890.29

...
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Introduction To Value At Risk

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Transcript of "Introduction To Value At Risk"

  1. 1. INTRODUCTION TO VALUE AT RISK (VaR) ALAN ANDERSON, Ph.D. ECI Risk Training www.ecirisktraining.com
  2. 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 (c) ECI Risk Training www.ecirisktraining.com 2
  3. 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 (c) ECI Risk Training www.ecirisktraining.com 3
  4. 4. EXAMPLE Suppose that it is determined that a $100 million portfolio could potentially lose $20 million (or more) once every 20 trading days (c) ECI Risk Training www.ecirisktraining.com 4
  5. 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 (c) ECI Risk Training www.ecirisktraining.com 5
  6. 6. 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 (c) ECI Risk Training www.ecirisktraining.com 6
  7. 7. (c) ECI Risk Training www.ecirisktraining.com 7
  8. 8. 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 (c) ECI Risk Training www.ecirisktraining.com 8
  9. 9. The VaR of this portfolio is therefore $100 million - $80 million = $20 million (c) ECI Risk Training www.ecirisktraining.com 9
  10. 10. VaR is based on the assumption that the rates of return of the assets held in a portfolio are jointly normally distributed (c) ECI Risk Training www.ecirisktraining.com 10
  11. 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 (c) ECI Risk Training www.ecirisktraining.com 11
  12. 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 portfolio’s composition (c) ECI Risk Training www.ecirisktraining.com 12
  13. 13. VaR has the disadvantage that it is computationally intensive and requires major adjustments for non-linear assets, such as options (c) ECI Risk Training www.ecirisktraining.com 13
  14. 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 (c) ECI Risk Training www.ecirisktraining.com 14
  15. 15. Portfolio theory shows how risk can be reduced by holding a well-diversified set of assets (c) ECI Risk Training www.ecirisktraining.com 15
  16. 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. (c) ECI Risk Training www.ecirisktraining.com 16
  17. 17. As a result, a well-diversified portfolio is less likely to experience extreme changes in value; in this way, risk is reduced (c) ECI Risk Training www.ecirisktraining.com 17
  18. 18. In statistical terms, a well-diversified portfolio contains assets whose rates of return have very low or negative correlations with each other (c) ECI Risk Training www.ecirisktraining.com 18
  19. 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 portfolio’s value (c) ECI Risk Training www.ecirisktraining.com 19
  20. 20. A portfolio invested in both oil stocks and automotive stocks would be far more diversified: (c) ECI Risk Training www.ecirisktraining.com 20
  21. 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 (c) ECI Risk Training www.ecirisktraining.com 21
  22. 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 (c) ECI Risk Training www.ecirisktraining.com 22
  23. 23. The risk of holding a portfolio containing two assets, X and Y, is measured by its standard deviation, as follows: (c) ECI Risk Training www.ecirisktraining.com 23
  24. 24. P = w 2 X 2 X +w 2 Y 2 Y + 2wX wY X Y (c) ECI Risk Training www.ecirisktraining.com 24
  25. 25. where: P = the standard deviation of the returns to the portfolio (c) ECI Risk Training www.ecirisktraining.com 25
  26. 26. X = standard deviation of the returns to asset X Y = standard deviation of the returns to asset Y (c) ECI Risk Training www.ecirisktraining.com 26
  27. 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 (c) ECI Risk Training www.ecirisktraining.com 27
  28. 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 (c) ECI Risk Training www.ecirisktraining.com 28
  29. 29. = “rho” this represents the correlation between the returns to assets X and Y; -1 1 (c) ECI Risk Training www.ecirisktraining.com 29
  30. 30. The lower is the correlation between assets, the lower will be the risk of the portfolio (c) ECI Risk Training www.ecirisktraining.com 30
  31. 31. The Value at Risk of a portfolio is a function of: (c) ECI Risk Training www.ecirisktraining.com 31
  32. 32. the dollar value of the portfolio the portfolio standard deviation the confidence level the time horizon (c) ECI Risk Training www.ecirisktraining.com 32
  33. 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 (c) ECI Risk Training www.ecirisktraining.com 33
  34. 34. where: V0 = initial value of the asset = standard deviation of the asset’s daily returns (c) ECI Risk Training www.ecirisktraining.com 34
  35. 35. = the number of standard deviations below the mean corresponding to the (1-c) quantile of the standard normal distribution (c) ECI Risk Training www.ecirisktraining.com 35
  36. 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 (c) ECI Risk Training www.ecirisktraining.com 36
  37. 37. (c) ECI Risk Training www.ecirisktraining.com 37
  38. 38. The value of corresponding to any confidence level can be found with a normal table or with the Excel function NORMSINV (c) ECI Risk Training www.ecirisktraining.com 38
  39. 39. EXAMPLE For a 99% confidence level, the value of can be determined as follows: (c) ECI Risk Training www.ecirisktraining.com 39
  40. 40. c = 0.99 (1-c) = 0.01 = 1% NORMSINV(0.01) = -2.33 = 2.33 (c) ECI Risk Training www.ecirisktraining.com 40
  41. 41. (c) ECI Risk Training www.ecirisktraining.com 41
  42. 42. 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 (c) ECI Risk Training www.ecirisktraining.com 42
  43. 43. Assume that the standard deviation of the stock’s returns are 0.0189 (1.89%) per day (c) ECI Risk Training www.ecirisktraining.com 43
  44. 44. 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: (c) ECI Risk Training www.ecirisktraining.com 44
  45. 45. V0 = (10,000)(1.645)(0.0189) = $310.905 (c) ECI Risk Training www.ecirisktraining.com 45
  46. 46. 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) (c) ECI Risk Training www.ecirisktraining.com 46
  47. 47. NOTE VaR can be extended to different time horizons by applying the square root of time rule (c) ECI Risk Training www.ecirisktraining.com 47
  48. 48. According to this rule, the standard deviation increases in proportion to the square root of time: t periods = t 1 period (c) ECI Risk Training www.ecirisktraining.com 48
  49. 49. 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: (c) ECI Risk Training www.ecirisktraining.com 49
  50. 50. V0 = (10, 000)(1.645)(0.0189 22) (10, 000)(1.645)(0.0189 22) = $1, 458.27 (c) ECI Risk Training www.ecirisktraining.com 50
  51. 51. 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: (c) ECI Risk Training www.ecirisktraining.com 51
  52. 52. V0 = (10, 000)(1.645)(0.0189 252) (10, 000)(1.645)(0.0189 252) = $4,935.46 (c) ECI Risk Training www.ecirisktraining.com 52
  53. 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 (c) ECI Risk Training www.ecirisktraining.com 53
  54. 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 (c) ECI Risk Training www.ecirisktraining.com 54
  55. 55. standard deviation of each asset’s rate of return in the portfolio correlations among the assets’ rates of return in the portfolio (c) ECI Risk Training www.ecirisktraining.com 55
  56. 56. 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 (c) ECI Risk Training www.ecirisktraining.com 56
  57. 57. EXAMPLE Assume that a $100,000 portfolio contains $60,000 worth of Stock X and $40,000 worth of Stock Y. (c) ECI Risk Training www.ecirisktraining.com 57
  58. 58. Given the following data, compute the VaR of this portfolio with a 95% confidence level over the coming: (c) ECI Risk Training www.ecirisktraining.com 58
  59. 59. day month year (c) ECI Risk Training www.ecirisktraining.com 59
  60. 60. DATA wX = 0.60 wY = 0.40 X = 0.016284 Y = 0.015380 = -0.19055 (c) ECI Risk Training www.ecirisktraining.com 60
  61. 61. P = (0.6) (0.016284) + (0.4) (0.015380) + 2 2 2 2 2(0.6)(0.4)( 0.19055)(0.016284)(0.015380) = 0.01144627 = 1.144627% (c) ECI Risk Training www.ecirisktraining.com 61
  62. 62. The portfolio VaR over the coming day is: V0 P = (100,000)(1.645)(0.01144627) = $1,882.91 (c) ECI Risk Training www.ecirisktraining.com 62
  63. 63. The portfolio VaR over the coming month is: V0 P = (100, 000)(1.645)(0.01144627 22) = $8,831.638 (c) ECI Risk Training www.ecirisktraining.com 63
  64. 64. The portfolio VaR over the coming year is: V0 P = (100, 000)(1.645)(0.01144627 252) = $29,890.29 (c) ECI Risk Training www.ecirisktraining.com 64
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