Optimal asset allocation during retirement in the presence of fixed and variable immediate life annuities (payout annuities)

• US 20030233301A1
• Filed: 06/18/2002
• Published: 12/18/2003
• Est. Priority Date: 06/18/2002
• Status: Active Grant

• Current Assignee: Ibbotson Associates, Inc. (Morningstar Incorporated)
• Original Assignee: Ibbotson Associates, Inc. (Morningstar Incorporated)
• Inventors: Chen, Peng, Milevsky, Moshe A.
• Granted Patent: US 7,120,601 B2
• Time in Patent Office: Days
• US Class Current: 705/36.00R
• CPC Class Codes: G06Q 40/06 Asset management; Financial planning or analysis
• CPC Class Codes: G06Q 40/08 Insurance

Abstract

A method, system and medium for optimally allocating investment assets for a given investor within and between annuitized assets and non-annuitized assets retrieves an investor'"'"'s utility of consumption, utility of bequest, objective and subjective probabilities of survival and expected rates of return from each of a plurality of annuity and nonannuity assets having varying degrees of risk and return. Based on these inputs, an objective utility function is maximized by adjusting the asset allocation weights. The optimal asset allocation weights may be used to allocate the assets of the investor'"'"'s portfolio among predetermined investment vehicles or as an analytical tool by portfolio managers.

Claims

1. A method for allocating assets of an investor portfolio among annuity and nonannuity assets, comprising the steps of:

retrieving at least one probability of survival of the investor;

selecting a utility of consumption and a utility of bequest of the investor;

retrieving for each of a plurality of nonannuity assets, an expected rate of return, the nonannuity assets having expected rates of return which are different from each other;

retrieving, for each of a plurality of annuity assets, an expected rate of return, the annuity assets having expected rates of return which are different from each other;

maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors each corresponding to a nonannuity asset or an annuity asset; and

allocating wealth in the portfolio to the nonannuity assets and the annuity assets according to receptive ones of the investment weighting factors.

2. The method of claim 1, wherein pairs of the annuity and nonannuity assets have the same expected rate of return.

3. The method of claim 1, wherein at least one of the annuity assets and the nonannuity assets itself comprises a portfolio of investment instruments.

4. The method of claim 1, wherein the annuity and nonannuity assets are paired in degrees of risk.

5. The method of claim 1, wherein the wealth of the portfolio is allocated among two annuity assets and two nonannuity assets.

6. The method of claim 1, wherein said step of maximizing includes the further step of calculating each of the investment weighting factors for the annuity assets as a function of the probability of survival.

7. The method of claim 1, wherein said step of maximizing includes the further steps of

calculating each of the investment weighting factors as components of the utility of the live state;

and

calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.

8. The method of claim 1, and further comprising the steps of

retrieving a subjective probability of survival for the investor;

and

performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.

9. The method of claim 1, and further comprising the steps of

retrieving an objective probability of survival for the investor;

and

calculating the utility of the live state as a function of the objective probability of survival of the investor.

10. The method of claim 9, wherein the investor'"'"'s objective probability of survival is also chosen as a function of the investor'"'"'s age, gender and ethnic origin.

11. The method of claim 1, wherein said step of maximizing is performed in multiple iterations.

12. The method of claim 1, wherein said step of maximizing the objective utility function comprises using the Method of Feasible Directions.

13. The method of claim 1, wherein said step of maximizing the objective utility function comprises using a Golden Mean grid search.

14. The method of claim 1, wherein said step of maximizing the objective utility function comprises using Simpson'"'"'s Composite Rule.

15. The method of claim 1, wherein said step of maximizing the objective function comprises using a Fibonacci search algorithm.

16. The method of claim 1, wherein the investor is a joint investor and the probability of survival is a joint probability of survival.

17. The method of claim 1, wherein the investor is a sole investor and the probability of survival is a sole probability of survival.

18. The method of claim 1, wherein the weight for utility of consumption≧

0, the weight for utility of bequest≧

0 and the sum of the weights for utilities of consumption and bequest is one.

19. The method according to claim 1, in which the following factors are considered in maximizing the objective utility function:

(i) the investor'"'"'s risk tolerance; and

(ii) the investor'"'"'s age.

20. The method according to claim 1, wherein the objective utility function is maximized using a Two-Phase Simplex Method in which Phase I determines an initial feasible point of the linear program if it exists, and Phase II determines the optimal point of the linear program.

21. A system for allocating portfolio assets of an investor among annuity and nonannuity vehicles, comprising:

a memory for storing at least one probability of survival of the investor, at least one of a utility of consumption attributed to the investor and a utility of bequest attributed to the investor, data identifying a plurality of nonannuity assets having different rates of return and degrees of risk, data identifying a plurality of annuity assets having different rates of return and degrees of risk;

a processor coupled to the memory to retrieve the stored probability of survival, the stored utility of bequest or utility of consumption and said data identifying the annuity and nonannuity assets, the processor maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors, each factor being a weight of a respective nonannuity or annuity asset; and

a portfolio asset allocator coupled to an output of the processor for allocating assets in the portfolio responsive to the adjusted values of the investment weighting factors which result in a maximum of the objective utility function.

22. The system of claim 21, wherein pairs of annuity and nonannuity assets have the same degree of risk.

23. The system of claim 21, wherein the assets to which the portfolio can be allocated are limited to first and second nonannuity assets and first and second annuity assets.

24. The system of claim 21, wherein the processor, in maximizing the utility function, calculates the annuity investment weighting factors as a function of the probability of survival.

25. The system of claim 21, wherein the processor calculates the annuity and nonannuity investment weighting factors as components of the utility of consumption, and calculates the nonannuity investment weighting factors as components of the utility of bequest.

26. The system of claim 21, wherein the memory stores a subjective probability of survival, the processor, in maximizing the objective utility function, retrieving the subjective probability of survival and apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.

27. The system of claim 21, wherein the memory stores an objective probability of survival of the investor, the processor retrieving an objective probability of survival for the investor and calculating the utility of the live state as a function of the objective probability of survival of the investor.

28. The system of claim 21, wherein the processor maximizes the objective utility function by performing multiple iterations of calculation steps.

29. The system of claim 21, wherein the memory stores a risk aversion factor for the investor, the processor calculating the investment weighting factors as a function of the risk aversion factor.

30. The system of claim 21, wherein the processor, in maximizing the objective utility function, uses The Method of Feasible Directions.

31. The system of claim 21, wherein the processor, in maximizing the objective utility function, uses a Golden Mean grid search.

32. The system of claim 21, wherein the processor, in maximizing the objective utility function, uses Simpson'"'"'s Composite Rule.

33. A system for optimally allocating investment assets for a given user within and between annuitized assets and non-annuitized assets, comprising:

a computer having a CPU and memory;

means for providing to the computer at least one of the user'"'"'s utility of consumption and utility of bequest;

means for providing to the computer the user'"'"'s objective probability of survival based on the user'"'"'s age;

means for providing to the computer the user'"'"'s subjective probability of survival based on the user'"'"'s physical well-being;

means for providing to the computer an expected rate of return from each of a plurality of assets, ones of the assets being riskier than others of the assets; and

means for maximizing an objective utility function which is calculated as a function of the asset expected rates of return, a utility of a live state of the user and a utility of a dead state of the user and a plurality of asset allocation weights respectively corresponding to the different assets.

34. The system of claim 33 wherein the objective utility function is maximized using the Method of Feasible Directions.

35. The system of claim 33 wherein the objective utility function is maximized using a Golden Mean grid search.

36. The system of claim 33, wherein the objective utility function is maximized using Simpson'"'"'s Composite Rule.

37. The system of claim 33, wherein the following decision factors are taken into consideration in maximizing the objective utility function:

(i) the investor'"'"'s risk tolerance, and

(ii) the investor'"'"'s age.

38. The system according to claim 33 wherein the objective utility function is maximized using a Two-Phase Simplex Method in which Phase I determines an initial feasible point of the linear program if it exists, and Phase II determines the optimal point of the linear program.

39. A machine-readable medium on which has been prerecorded a computer program which, when executed by a processor, performs the following steps:

retrieving at least one probability of survival of an investor;

selecting a utility of consumption and a utility of bequest of the investor;

retrieving, for each of a plurality of annuity assets and nonannuity assets, an expected rate of return, each nonannuity asset and each annuity asset having an expected rate of return and a degree of risk which is different from others of the assets;

maximizing an objective utility function as the sum of a utility of a live. state and a utility of a dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors corresponding to respective ones of the annuity and nonannuity assets; and

allocating wealth in a portfolio of the investor comprising the nonannuity assets and the annuity assets according to the investment weighting factors for which the objective utility function is maximized.

40. The medium of claim 39, wherein pairs of annuity and nonannuity assets have the same expected rate of return.

41. The medium of claim 39, wherein at least one of the assets itself comprises a portfolio of investment instruments.

42. The medium of claim 39, wherein pairs of annuity and nonannuity assets have the same degree of risk.

43. The medium of claim 39, wherein the investment weighting factors corresponding to the annuity assets are calculated as a function of the probability of survival.

44. The medium of claim 39, wherein said step of maximizing includes the further steps of

calculating the investment weighting factors of the annuity assets and the nonannuity assets as components of the utility of the live state;

and

calculating the investment weighting factors of the nonannuity assets as components of the utility of the dead state.

45. The medium of claim 39, wherein the processor performs the further steps of

retrieving a subjective probability of survival for the investor;

and

performing said step of maximizing by apportioning the objective utility function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.

46. The medium of claim 39, wherein the processor performs the further steps of

retrieving an objective probability of survival for the investor;

and

calculating the utility of the live state as a function of the objective probability of survival of the investor.

47. The medium of claim 39, wherein the processor performs said step of maximizing in multiple iterations.

48. The medium of claim 39 wherein the processor maximizes the objective utility function by using the Method of Feasible Directions.

49. The medium of claim 39, wherein the processor maximizes the objective utility function by using a Golden Mean grid search.

50. The medium of claim 39, wherein the processor maximizes the objective utility function by using Simpson'"'"'s Composite Rule.

51. The medium of claim 39, wherein the processor maximizes the objective utility function by using a Fibonacci search algorithm.

52. The medium of claim 39, wherein the investor is a joint investor and the retrieved probability of survival is a joint probability of survival.

53. The medium of claim 39, wherein the investor is a sole investor and the retrieved probability of survival is a sole probability of survival.

54. The medium of claim 39 wherein the objective utility function is a function of the investor'"'"'s risk tolerance and the investor'"'"'s age.

55. The medium of claim 39 wherein the objective function is maximized using the Two-Phase Simplex Method, a first phase of the two-phase simplex method determining an initial feasible point of the linear program if it exists, the second phase of the two-phase simplex method determining the optimal point of the linear program.

56. A method for allocating assets of an investor to a plurality of annuity and ponannuity investment vehicles, comprising the steps of:

retrieving personal characteristics of the investor, the characteristics including age, a risk aversion measure and at least one of a utility of consumption and a utility of bequest;

retrieving data about the financial wealth of the investor, including, for each investment vehicle presently employed by the investor, a measure of risk of the investment vehicle, a rate of expected return for the investment vehicle, and an amount of wealth in the investment vehicle, the sum of the wealth in the investment vehicles being the total financial wealth of the investor;

classifying the investment vehicles of the investor into allocatable investment vehicles whose assets may be reallocated, and nonallocatable investment vehicles whose assets may not be reallocated;

providing a plurality of asset classes having varying degrees of risk and expected return, the asset classes including nonannuity asset classes and annuity asset classes;

attributing the present investment vehicles of the investor to the asset classes as a function of the risk and expected return of the asset class and the investment vehicle;

maximizing an objective utility function as the sum of a utility of a live state and a utility of a dead state given the rates of expected return of the asset classes by adjusting values of a plurality of investment weighting factors each corresponding to an asset class; and

rebalancing the total financial wealth of the investor among the asset classes as a function of the investment weighting factors by adjusting the amount of assets held in each of the allocatable investment vehicles.

57. The method of claim 56, wherein pairs of the annuity and nonannuity asset classes have the same expected rate of return.

58. The method of claim 56, wherein at least one of the annuity asset classes and the nonannuity asset classes itself comprises a portfolio of investment instruments.

59. The method of claim 56, wherein the annuity and nonannuity asset classes are paired in degrees of risk.

60. The method of claim 56, wherein said step of maximizing includes the further step of calculating each of the investment weighting factors for the annuity assets as a function of the probability of survival.

61. The method of claim 56, wherein said step of maximizing includes the further steps of

calculating each of the investment weighting factors as components of the utility of the live state;

and

calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.

62. The method of claim 56, and further comprising the steps of

retrieving a subjective probability of survival for the investor;

and

performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.

63. The method of claim 56, and further comprising the steps of

retrieving an objective probability of survival for the investor;

and

calculating the utility of the live state as a function of the objective probability of survival of the investor.

64. The method of claim 63, wherein the user'"'"'s objective probability of survival is also chosen as a function of the user'"'"'s gender and ethnic origin.

65. The method of claim 56, wherein said step of maximizing is performed in multiple iterations.

66. The method of claim 56, wherein said step of maximizing the objective utility function comprises using the Method of Feasible Directions.

67. The method of claim 56, wherein said step of maximizing the objective utility function comprises using a Golden Mean grid search.

68. The method of claim 56, wherein said step of maximizing the objective utility function comprises using Simpson'"'"'s Composite Rule.

69. The method of claim 56, wherein said step of maximizing the objective function comprises using a Fibonacci search algorithm.

70. The method of claim 56, wherein the investor is a joint investor and the probability of survival is a joint probability of survival.

71. The method of claim 56, wherein the investor is a sole investor and the probability of survival is a sole probability of survival.

72. The method of claim 56, wherein the utility of consumption≧

0, the utility of bequest≧

0 and the sum of the utilities of consumption and bequest is one.

73. The method according to claim 56, in which the following factors are considered in maximizing the objective utility function:

(i) the investor'"'"'s risk tolerance; and

(ii) the investor'"'"'s age.

74. The method according to claim 56, wherein the objective utility function is maximized using a Two-Phase Simplex Method in which Phase I determines an initial feasible point of the linear program if it exists, and Phase II determines the optimal point of the linear program.

75. A method of comparing a current allocation of assets in an investment portfolio by a portfolio manager with an optimum allocation of the wealth of the portfolio among a plurality of annuity and nonannuity assets, comprising the steps of:

retrieving, for the investor for which the portfolio has been established, at least one probability of survival;

retrieving for the investor a utility of consumption and a utility of bequest;

retrieving, for each of the annuity and nonannuity assets, an expected rate of return, ones of the annuity assets having rates of return which are different from others of the annuity assets, ones of the nonannuity assets having rates of return which are different from others of the nonannuity assets;

relating a utility of a live state of the investor to a utility of a dead state of the investor as a function of the utility of consumption and the utility of bequest;

maximizing an objective utility function as the sum of the utility of the live state and the utility of the dead state given the retrieved rates of return by adjusting the values of a plurality of investment weighting factors each corresponding to one of the annuity assets or one of the nonannuity assets;

making a proposed allocation of the wealth of the portfolio among the nonannuity assets and the annuity assets according to the relative values of the investment weighting factors;

comparing the proposed allocation to the current allocation; and

modifying the current allocation in view of the proposed allocation.

76. The method of claim 75, wherein pairs of the annuity and nonannuity assets have the same expected rate of return.

77. The method of claim 75, wherein at least one of the annuity assets and the nonannuity assets itself comprises a portfolio of investment instruments.

78. The method of claim 75, wherein the annuity and nonannuity assets are paired in degrees of risk.

79. The method of claim 75, wherein the wealth of the portfolio is allocated among two annuity assets and two nonannuity assets.

80. The method of claim 75, wherein said step of maximizing includes the further step of calculating each of the investment weighting factors for the annuity assets as a function of the probability of survival.

81. The method of claim 75, wherein said step of maximizing includes the further steps of

calculating each of the investment weighting factors as components of the utility of the live state;

and

calculating each of the investment weighting factors of the nonannuity assets as components of the utility of the dead state.

82. The method of claim 75, and further comprising the steps of

retrieving a subjective probability of survival for the investor;

and

performing said step of maximizing by apportioning the objective function between the utility of the live state and the utility of the dead state according to the value of the subjective probability of survival.

83. The method of claim 75, and further comprising the steps of

retrieving an objective probability of survival for the investor;

and

calculating the utility of the live state as a function of the objective probability of survival of the investor.

84. The method of claim 83, wherein the user'"'"'s objective probability of survival is also chosen as a function of the user'"'"'s gender and ethnic origin.

85. The method of claim 75, wherein said step of maximizing is performed in multiple iterations.

86. The method of claim 75, wherein said step of maximizing the objective utility function comprises using the Method of Feasible Directions.

87. The method of claim 75, wherein said step of maximizing the objective utility function comprises using a Golden Mean grid search.

88. The method of claim 75, wherein said step of maximizing the objective utility function comprises using Simpson'"'"'s Composite Rule.

89. The method of claim 75, wherein said step of maximizing the objective function comprises using a Fibonacci search algorithm.

90. The method of claim 75, wherein the investor is a joint investor and the probability of survival is a joint probability of survival.

91. The method of claim 75, wherein the investor is a sole investor and the probability of survival is a sole probability of survival.

92. The method of claim 75, wherein the utility of consumption≧

0, the utility of bequest≧

0 and the sum of the utilities of consumption and bequest is one.

93. The method according to claim 75, in which the following factors are considered in maximizing the objective utility function:

(i) the investor'"'"'s risk tolerance; and

(ii) the investor'"'"'s age.

94. The method according to claim 75, wherein the objective utility function is maximized using a Two-Phase Simplex Method in which Phase I determines an initial feasible point of the linear program if it exists, and Phase II determines the optimal point of the linear program.