Method and device for calculating value at risk

• US 20030139993A1
• Filed: 09/28/2002
• Published: 07/24/2003
• Est. Priority Date: 03/28/2000
• Status: Abandoned Application

• Current Assignee: Andrey Feuerverger
• Original Assignee: Andrey Feuerverger
• Inventors: Feuerverger, Andrey
• Application Number:
  US10/240,333
• Publication Number:
  US 20030139993A1
• Time in Patent Office: Days
• US Class Current: 705/36
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis
• CPC Class Codes: G06Q 40/08 Insurance

Abstract

The invention is a method and system for determining VaR. The invention does not require Monte Carlo sampling. Alternatively, if Monte Carlo sampling is used, it requires only a reduced number of such trials. The invention is based on reducing the pricing function of the overall portfolio to a delta-gamma approximaiton, which in effect is a quadratic form in the risk factors; the distribution of the risk factors is, in turn, assumed to be a known multivariate normal distribution; the distribution of this quadratic form in normal variables is then determined by means of first evaluating the moment generating function (Laplace transform) of this distribution, and then applying highlt accurate methods of saddlepoint approximation to this moment generating function to determine the distribution and its quantiles.

Claims

1. A method of determining the risk in possessing a portfolio having a portfolio price and a portfolio return, the portfolio including holdings each having a holding return, the holdings having been mapped to risk factors for which the parameters of a multivariate normal statistical distribution have been determined, the method including:

expressing each holding return as a quadratic form in the returns of the risk factors;

aggregating the quadratic forms in the holdings to obtain a quadratic form approximation for the portfolio;

determining a cumulant generating function of the quadratic form in the portfolio return and the first and second derivatives of the cumulant generating function;

inputting the cumulant generating function and the derivatives into a saddlepoint approximation of first order or higher order from which the statistical distribution function of the portfolio return is provided, and

providing a Value at Risk quantity from a tail area of the statistical distribution function of the portfolio return.

2. The method of claim 1, wherein the wherein the holdings comprise financial instruments.

3. The method of claim 1 or 2, wherein the quadratic form is a function which is a sum of a first part and a second part, the first part including a linear term in the risk factor returns;

the second part including a quadratic term in the risk factor returns.

4. The method of any of claims 1 to 3, wherein the cumulant generating function is obtained from a transform including a characteristic function or a moment generating function of the statistical distribution of the quadratic form.

5. The method of any of claims 1 to 4, comprising determining a cumulant generating function of the quadratic form in the portfolio return and its first, second and/or higher derivatives.

6. The method of any of claims 1 to 5, wherein the cumulant generating function is determined from a Laplace transform, a Fourier transform, a Mellin transform, or a probability generating function.

7. The method of any of claims 1 to 6, wherein the saddlepoint approximation includes a Lugannani and Rice saddlepoint approximation, a Barndorff-Nielsen saddlepoint approximation, a Rice saddlepoint approximation, a Daniels saddlepoint approximation, or a higher order saddlepoint approximation.

8. The method of any of claims 1 to 7, wherein the portfolio return is expressed as a sum of two functions, the first term of which is a linear term, a quadratic term or a sum thereof, and the second term being a residual term.

9. The method of any of claims 1 to 8, further comprising Monte Carlo trials to determine the Value at Risk.

10. The method of any of claims 1 to 9, wherein the quadratic form is determined from a pricing formula for derivative securities.

11. The method of claim 10, wherein the pricing formula comprises a Black and Scholes formula, a Cox-Ingersol-Ross formula, a Heath-Morton-Jarrow formula, a binomial pricing formula or a Hull-White formula.

12. The method of any of claims 1 to 9, wherein the quadratic form is determined analytically or numerically with the gradient and/or Hessian of a function or of a computing program which determines the return or the price of the portfolio.

13. The method of any of claims 1 to 12, wherein the method is performed with a computer.

14. A value at risk provided in accordance with any of claims 1 to 13.

15. A system for determining the risk in possessing a portfolio having a portfolio return and a portfolio price, the portfolio including holdings each having a holding return the holdings having been mapped to risk factors (i) for which the multivariate normal distribution has been determined or (ii) for which the parameters of a discrete or continuous mixture of multivariate normal distributions has been determined, the method including:

a) means for expressing each holding return as a quadratic form in the returns of the risk factors;

b) means for aggregating the quadratic forms in the holdings to obtain a quadratic form approximation for the overall portfolio;

c) means for determining a cumulant generating function of the quadratic form in the portfolio return and the first and second derivatives of the cumulant generating function; and

d) means for inputting the cumulant generating function and the derivatives into a saddlepoint approximation of first order or higher order from which the statistical distribution function of the portfolio return is provided,

wherein a Value at Risk quantity can be provided from a tail area of the statistical distribution function of the portfolio return.

16. The system of claim 15, wherein the wherein the holdings comprise financial instruments.

17. The system of claim 15 or 16, wherein the quadratic form is a function which is a sum of a first part and a second part, the first part including a linear term in the risk factor returns;

the second part including a quadratic term in the risk factor returns.

18. The system of any of claims 15 to 17, wherein the cumulant generating function is obtained from a transform including a characteristic function or a moment generating function of the statistical distribution of the quadratic form.

19. The system of any of claims 15 to 18, comprising means for determining a cumulant generating function of the quadratic form in the portfolio return and the first, second and/or higher derivatives.

20. The system of any of claims 15 to 19, wherein the cumulant generating function is determined from a Laplace transform, a Fourier transform, a Mellin transform, or a probability generating function.

21. The system of any of claims 15 to 20, wherein the saddlepoint approximation includes a Lugannani and Rice saddlepoint approximation, a Barndorff-Nielsen saddlepoint approximation, a Rice saddlepoint approximation, a Daniels saddlepoint approximation, or a higher order saddlepoint approximation.

22. The system of any of claims 15 to 21, wherein the portfolio return is expressed as a sum of two functions, the first term of which is a linear term, a quadratic term or a sum thereof, and the second term being a residual term.

23. The system of any of claims 15 to 22, further comprising Monte Carlo trials to determine the Value at Risk.

24. The system of any of claims 15 to 23, wherein the quadratic form is determined from a pricing formula for derivative securities.

25. The system of claim 24, wherein the pricing formula comprises a Black and Scholes formula, a Cox-Ingersol-Ross formula, a Heath-Morton-Jarrow formula, a binomial pricing formula or a Hull-White formula.

26. The system of any of claims 15 to 23, wherein the quadratic form is determined analytically or numerically with the gradient and/or Hessian of a function or of a computing program which determines the return or price of the portfolio.

27. The system of any of claims 15 to 26, wherein the method is performed with a computer.

28. A value at risk provided in accordance with any of claims 15 to 27.

29. A method of determining the risk in possessing a portfolio having a portfolio return and a portfolio price, the portfolio including holdings each having a holding return, the holdings having been mapped to risk factors for which the parameters of a discrete or continuous mixture of multivariate normal distributions has been determined, the method including:

expressing each holding return as a quadratic form in the returns of the risk factors;

aggregating the quadratic forms in the holdings to obtain a quadratic form approximation for the portfolio;

determining a cumulant generating function of the quadratic form in the portfolio return and the first and second derivatives of the cumulant generating function;

inputting the cumulant generating function and the derivatives into a saddlepoint approximation of first order or higher order from which the statistical distribution function of the portfolio return is provided, and

providing a Value at Risk quantity from a tail area of the statistical distribution function of the portfolio return.

30. The method of claim 29, wherein the mixture of multivariate normal distributions includes a convolution and/or a kernel density estimator.

31. The method of claim 29 or 30, wherein the wherein the holdings comprise financial instruments.

32. The method of any of claims 29 to 31, wherein the quadratic form is a function which is a sum of a first part and a second part, the first part including a linear term in the risk factor returns;

the second part including a quadratic term in the risk factor returns.

33. The method of any of claims 29 to 32, wherein the cumulant generating function is obtained from a transform including a characteristic function or a moment generating function of the statistical distribution of the quadratic form.

34. The method of any of claims 29 to 33, comprising determining a cumulant generating function of the quadratic form in the portfolio return and the first, second and/or higher derivatives.

35. The method of any of claims 29 to 34, wherein the cumulant generating function is determined from a Laplace transform, a Fourier transform, a Mellin transform, or a probability generating function.

36. The method of any of claims 29 to 35, wherein the saddlepoint approximation includes a Lugannani and Rice saddlepoint approximation, a Barndorff-Nielsen saddlepoint approximation, a Rice saddlepoint approximation or a Daniels saddlepoint approximation, or a higher order saddlepoint approximation.

37. The method of any of claims 29 to 36, wherein the portfolio return is expressed as a sum of two functions, the first term of which is a linear term, a quadratic term or a sum thereof, and the second term being a residual term.

38. The method of any of claims 29 to 37, wherein the quadratic form is determined from a pricing formula for derivative securities.

39. The method of claim 38, wherein the formula comprises a Black and Scholes formula a Cox-Ingersol-Ross formula, a Heath-Morton-Jarrow formula, a binomial pricing formula or a Hull-White formula.

40. The method of any of claims 29 to 37, wherein the quadratic form is determined analytically or numerically with the gradient and/or Hessian of a function or of a computing program which determines the return of the portfolio return.

41. The method of any of claims 29 to 40, further comprising Monte Carlo trials to determine the Value at Risk.

42. The methods of any of claims 29 to 41, wherein the method is performed with a computer.

43. A value at risk provided in accordance with any of claims 29 to 42.

44. A method of determining the risk in possessing a portfolio having a portfolio return, the portfolio including holdings each having a holding return, the holdings having been mapped to risk factors for which the parameters of a multivariate normal statistical distribution have been determined, the method including:

expressing each holding return as an expanded polynomial of the third or higher order in the returns of the risk factors;

aggregating the multivariate polynomials for the holdings to obtain a multivariate form approximation for the portfolio return;

determining a predetermined number of the first cumulants of the expanded polynomial;

determining a cumulant generating function of the expanded polynomial in the portfolio return using the first cumulants;

determining the first and second derivatives of the cumulant generating function;

inputting the cumulant generating function and first and second derivatives into a saddlepoint approximation of first order or higher order from which the statistical distribution function of the portfolio return is provided, and

providing a Value at Risk quantity from a tail area of the statistical distribution function of the portfolio return.

45. The method of claim 44, wherein the holdings comprise financial instruments.

46. The method of any of claims 44 and 45, wherein at least four of the first cumulants are determined.

47. The method of any of claims 44 to 46, wherein the pre-determined number of the first cumulants are determined by applying a method which comprises the Leonov-Shiryaev formula for multivariate cumulants of products of random variables.

48. The method of any of claims 44 to 46, wherein the pre-determined number of the first cumulants are determined from an empirical distribution of the collection of historical data of the returns of the risk factors during a pre-determined time period.

49. The method of any of claims 44 to 46, wherein the pre-determined number of the first cumulants are determined from the convolution of a kernel function with an empirical distribution of the collection of historical data of the returns of the risk factors during a pre-determined time period.

50. The method of any of claims 44 to 49, wherein the cumulant generating function of the expanded polynomial is approximated by constructing a truncated power series using the first cumulants as coefficients.

51. The method of any of claims 44 to 50, wherein the cumulant generating function of the expanded polynomial is approximated by a method comprising:

approximating each holding return as a quadratic form in the returns of the risk factors;

aggregating the quadratic forms in the holdings to obtain a quadratic form approximation for the portfolio;

determining a cumulant generating function of the quadratic form in the portfolio and a pre-determined number of the first derivatives of the cumulant generating function of the quadratic form;

determining a pre-determined number of the first coefficients of the Taylor series expansion of the cumulant generating function of the quadratic form using the derivatives of the cumulant generating function of the quadratic form of order one to the number of cumulants.

approximating the cumulant generating function of the expanded polynomial as the sum of the cumulant generating function of the quadratic form and a polynomial with coefficients equal to the differences between the cumulants as determined from the quadratic form and as determined from the coefficients of the Taylor series expansion.

52. The method of claim 51, wherein the quadratic form is a function which is a sum of a first part and a second part, the first part including a linear term in the risk factor returns;

the second part including a quadratic term in the risk factor returns.

53. The method of any of claims 51 or 52, wherein the cumulant generating function of the quadratic form is obtained from a transform including a characteristic function or a moment generating function of the statistical distribution of the quadratic form.

54. The method of any of claims 51 to 53, comprising determining a cumulant generating function of the quadratic form in the portfolio return and its first, second and/or higher derivatives.

55. The method of any of claims 51 to 54, wherein the cumulant generating function of the quadratic form is determined from a Laplace transform, a Fourier transform, a Mellin transform, or a probability generating function.

56. The method of any of claims 51 to 55, wherein the portfolio return is expressed as a sum of two functions, the first term of which is a linear term, a quadratic term or a sum thereof, and the second term being a residual term.

57. The method of any of claims 51 to 56, wherein the saddlepoint approximation includes a Lugannani and Rice saddlepoint approximation, a Barndorff-Nielsen saddlepoint approximation, a Rice saddlepoint approximation, a Daniels saddlepoint approximation, or a higher order saddlepoint approximation.

58. The method of any of claims 51 to 57, wherein the quadratic form is determined from a pricing formula for derivative securities.

59. The method of claim 58, wherein the pricing formula comprises a Black and Scholes formula, a Cox-Ingersol-Ross formula, a Heath-Morton-Jarrow formula, a binomial pricing formula or a Hull-White formula.

60. The method of any of claims 51 to 57, wherein the quadratic form is determined analytically or numerically with the gradient and/or Hessian of a function or of a computing program which determines the return of the portfolio.

61. The method of any of claims 44 to 60, further comprising Monte Carlo trials to determine the Value at Risk.

62. The method of any of claims 44 to 61, wherein the coefficients of the multivariate polynomial, being the Taylor expansion, of the portfolio return are obtained by summing the coefficients of multivariate polynomials of holding returns.

63. The method of any of claims 44 to 61, wherein the coefficients of the multivariate polynomial, being the Taylor expansion, of the portfolio return are obtained by averaging the coefficients of multivariate polynomials of holding returns.

64. The method of any of claims 44 to 63, wherein the coefficients of multivariate polynomials of holding returns are obtained by computing the holding return and its derivatives.

65. The method of any of claims 44 to 64, wherein the method is implemented by a computer.

66. A value at risk provided in accordance with any of claims 44 to 65.

67. A system of determining the risk in possessing a portfolio having a portfolio return, the portfolio including holdings each having a holding return, the holdings having been mapped to risk factors for which the parameters of a multivariate normal statistical distribution have been determined, the system including:

means for expressing each holding return as an expanded polynomial of the third or higher order in the returns of the risk factors;

means for aggregating the multivariate polynomials for the holdings to obtain a multivariate form approximation for the portfolio return;

means for determining a pre-determined number of the first cumulants of the expanded polynomial;

means for determining a cumulant generating function of the expanded polynomial in the portfolio return using the first cumulants;

means for determining the first and second derivatives of the cumulant generating function;

means for inputting the cumulant generating function and first and second derivatives into a saddlepoint approximation of first order or higher order from which the statistical distribution function of the portfolio return is provided, and

means for providing a Value at Risk quantity from a tail area of the statistical distribution function of the portfolio return.

68. The system of claim 67, wherein the holdings comprise financial instruments.

69. The system of any of claims 67 and 68, wherein at least four of the first cumulants are determined.

70. The system of any of claims 67 to 69, comprising means for determining the predetermined number of the first cumulants by applying the Leonov-Shiryaev formula for multivariate cumulants of products of random variables.

71. The system of any of claims 67 to 69, comprising means for determining the predetermined number of the first cumulants from an empirical distribution of the collection of historical data of the returns of the risk factors during a predetermined time period.

72. The system of any of claims 67 to 69, comprising means for determining the predetermined number of the first cumulants from the convolution of a kernel function with an empirical distribution of the collection of historical data of the returns of the risk factors during a pre-determined time period.

73. The system of any of claims 67 to 72, comprising means for determining the cumulant generating function of the expanded polynomial by an approximation by constructing a truncated power series using the first cumulants as coefficients.

74. The system of any of claims 67 to 72, comprising means for determining the cumulant generating function of the expanded polynomial by an approximation by:

approximating each holding return as a quadratic form in the returns of the risk factors;

aggregating the quadratic forms in the holdings to obtain a quadratic form approximation for the portfolio;

determining a cumulant generating function of the quadratic form in the portfolio and a pre-determined number of the first derivatives of the cumulant generating function of the quadratic form;

determining a pre-determined number of the first coefficients of the Taylor series expansion of the cumulant generating function of the quadratic form using the derivatives of the cumulant generating function of the quadratic form of order one to the number of cumulants.

approximating the cumulant generating function of the expanded polynomial as the sum of the cumulant generating function of the quadratic form and a polynomial with coefficients equal to the differences between the cumulants as determined from the quadratic form and as determined from the coefficients of the Taylor series expansion.

75. The system of claim 74, wherein the quadratic form is a function which is a sum of a first part and a second part, the first part including a linear term in the risk factor returns;

the second part including a quadratic tern in the risk factor returns.

76. The system of any of claims 74 to 75, comprising means for determining the cumulant generating function of the quadratic form from a transform including a characteristic function or a moment generating function of the statistical distribution of the quadratic form.

77. The system of any of claims 74 to 75, comprising means for determining a cumulant generating function of the quadratic form in the portfolio return and its first, second and/or higher derivatives.

78. The system of any of claims 74 to 77, comprising means for determining the cumulant generating function of the quadratic form from a Laplace transform, a Fourier transform, a Mellin transform, or a probability generating function.

79. The system of any of claims 74 to 78, wherein the portfolio return is expressed as a sum of two functions, the first term of which is a linear term, a quadratic term or a sum thereof, and the second term being a residual term.

80. The system of any of claims 74 to 78, wherein the saddlepoint approximation includes a Lugannani and Rice saddlepoint approximation, a Barndorff-Nielsen saddlepoint approximation, a Rice saddlepoint approximation, a Daniels saddlepoint approximation, or a higher order saddlepoint approximation.

81. The system of any of claims 74 to 80, wherein the quadratic form is determined from a pricing formula for derivative securities.

82. The system of any of claims 81, wherein the pricing formula comprises a Black and Scholes formula, a Cox-Ingersol-Ross formula, a Heath-Morton-Jarrow formula, a binomial pricing formula or a Hull-White formula.

83. The system of any of claims 74 to 80, wherein the quadratic form is determined analytically or numerically with the gradient and/or Hessian of a function or of a computing program which determines the return of the portfolio.

84. The system of any of claims 67 to 83, wherein the coefficients of the multivariate polynomial, being the Taylor expansion, of the portfolio return are obtained by summing the coefficients of multivariate polynomials of holding returns.

85. The system of any of claims 67 to 83, wherein the coefficients of the multivariate polynomial, being the Taylor expansion, of the portfolio return are obtained by averaging the coefficients of multivariate polynomials of holding returns.

86. The system of any of claims 67 to 85, wherein the coefficients of multivariate polynomials of holding returns are obtained by computing the holding return and its derivatives.

87. The system of any of claims 67 to 86, further comprising Monte Carlo trials to determine the Value at Risk.

88. A value at risk provided in accordance with any of claims 67 to 87.