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# Introduction To Value At Risk

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Introduction to Value at Risk (VaR)

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### 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