OPTIMAL PORTFOLIO WITHDRAWAL DURING RETIREMENT IN THE PRESENCE OF LONGEVITY RISK

• US 20140067722A1
• Filed: 09/04/2012
• Published: 03/06/2014
• Est. Priority Date: 09/04/2012
• Status: Active Grant

• Current Assignee: Cannex Financial Exchanges Limited
• Original Assignee: Huaxiong Huang, Moshe A. Milevsky
• Inventors: MILEVSKY, MOSHE A., HUANG, HUAXIONG
• Granted Patent: US 8,781,937 B2
• Time in Patent Office: Days
• US Class Current: 705/36.T
• 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 recommending an optimal withdrawal amount, for a given period, from a retiree'"'"'s portfolio accounts comprised of relatively risky and relatively safe financial assets used to finance retirement. The user supplies information about the retiree'"'"'s personal characteristics, including age, gender, and health status. Details of the retiree'"'"'s financial situation are also supplied, including the retiree'"'"'s total liquid wealth, the current value of relatively risky and relatively safe assets, and any after-tax pension and other annuity income. Risk (standard deviation of return), return (expected rate of return based on a lognormal or other random distribution), or other measurable differentiating characteristics are retrieved for a portfolio comprised of relatively risky assets. A valuation rate is also retrieved. Based on these inputs, an actuarial discount rate is computed, an optimal wealth depletion time is located, and the optimal withdrawal amount is computed and displayed to the user.

Claims

1. A computer implemented method for calculating, by a computer processor, an optimal withdrawal amount from a retiree'"'"'s portfolio accounts consisting of financial assets, the method comprising the steps of:

Retrieving current values of any relatively risky and relatively safe assets of the retiree;

Retrieving amounts of any after-tax pension and annuity income of the retiree;

Calculating total after-tax amount of pension and annuity funds receivable by the retiree per time unit, which is referred to as Pie;

Calculating financial wealth of the retiree at time 0, which, is referred to as W(0), as the sum of the current value of the relatively risky assets and the current value of the relatively safe assets;

Retrieving a valuation rate, which is referred to as R, of the retiree;

Calculating or retrieving the retiree'"'"'s level of longevity risk aversion, which is referred to as LRA;

Calculating the retiree'"'"'s optimal wealth depletion time, which is referred to as WDT, by solving the equation OWA(WDT)−

Pie=0, where OWA(T) is the retiree'"'"'s estimated optimal withdrawal amount at time T;

Calculating the retiree'"'"'s optimal withdrawal amount at time 0, which is referred to as OWA(0), by performing the steps of;

Calculating B1=EXP(R*WDT);

Calculating N2=(W(0)+Pie/R)*B1−

(Pie/R);

Calculating actuarial present value for time WDT, which is referred to as APVX(WDT);

Calculating OWA(0)=N2/APVX(WDT)/B1; and

Displaying or communicating the value of OWA(0) to the retiree.

2. The method of claim 1, wherein the relatively risky and relatively safe assets have different risk and return characteristics.

63. The method of claim 1, wherein Pie is equal to 0.

64. The method of claim 1, further comprising the step of calculating Alpha being the current value of the retiree'"'"'s relatively risky assets divided by the sum of W(0) and the present value of the retiree'"'"'s after-tax pension and annuity income, wherein LRA is calculated as the Sharpe ratio inversely scaled by Alpha.

65. The method of claim 64, further comprising the steps of:

retrieving an expected rate of return on a portfolio of relatively risky assets, which is referred to as Mu; and

retrieving a standard deviation of returns for a portfolio of relatively risky assets, which is referred to as Sigma,

wherein LRA is calculated as ((Mu−

R)/Sigma)/Alpha if Alpha is not equal to 0 and a relatively large number.

66. The method of claim 65, wherein the relatively large number is 100.

67. The method of claim 65, wherein OWA(T) is calculated by performing the steps of:

calculating an actuarial present value for time T, which is referred to as APVX(T);

calculating A0=Pie/R;

calculating A1=EXP(R*T);

calculating N1=(W(0)+A0)*A1−

A0;

calculating D1=APVX(T)*A1;

retrieving a subjective discount rate, which is referred to as SDR, of the retiree;

calculating K=(R−

SDR)/LRA;

calculating A2=EXP(K*T);

calculating A3=a conditional probability of the retiree surviving for T time units;

calculating A4=A3̂

(1/LRA); and

calculating OWA(T)=N1/D1*A2*A4.

68. The method of claim 67, wherein solving OWA(WDT)−

Pie=0 comprises steps of;

(a) setting a provisional time estimate WDTprte equal to a pre-determined initial estimate of WDT;

(b) calculating the retiree'"'"'s optimal wealth withdrawal amount (OWA(WDTprte)) for the provisional time estimate;

(c) if OWA(WDTprte)−

Pie is approximately 0 then setting WDT=WDTprte, otherwise if OWA(WDTprte)−

Pie is greater than 0 then increasing the value of WDTprte and repeating steps (b) and (c), otherwise if OWA(WDTprte)−

Pie is less than 0 then decreasing the value of WDTprte and repeating steps (b) and (c).

69. The method of claim 67, wherein APVX(T) is calculated as the sum of individual actuarial present values APV(Ti) computed for each value of time Ti ranging from 0 to T time units, where APV(Ti) is calculated as the conditional probability of the retiree surviving for Ti time units divided by (1+R)̂

Ti.

70. The method of claim 69, wherein the conditional probability the retiree surviving for a particular number of time units is calculated using the Gompertz-Makeham Law of Mortality.

71. The method of claim 68, wherein the time unit is one year, and wherein in step (c) of solving the equation OWA(WDT)−

Pie=0, if OWA(WDTprte)−

Pie is greater than 0 then the value of WDTprte is increased by one month and steps (b) and (c) are repeated, and if OWA(WDTprte)−

Pie is less than 0 then the value of WDTprte is decreased by one month and steps (b) and (c) are repeated.

72. The method of claim 71, wherein the pre-determined initial estimate of the WDT is 20 years.

73. The method of claim 68, wherein the increase or decrease in the value of WDTprte is calculated using Newton'"'"'s method.

74. The method of claim 68, wherein the increase or decrease in the value of WDTprte is calculated using a bisection method.

75. A computer implemented method for calculating an optimal withdrawal amount, for a given time period T, OWA(T), from a retiree'"'"'s portfolio accounts comprising financial assets, the method according to claim 1 comprising the steps of:

calculating OWA(0);

retrieving a subjective discount rate, which is referred to as SDR, of the retiree;

calculating K=(R−

SDR)/LRA;

calculating E0=EXP(K*T);

calculating A3=the conditional probability of the retiree surviving for T time units;

calculating OWA(T)=OWA(0)*E0*(A3̂

(1/LRA)); and

displaying or otherwise communicating the value of OWA(T) to the retiree.

76. The method of claim 75, wherein the conditional probability of the retiree surviving for a particular number of time units is calculated using the Gompertz-Makeham Law of Mortality.

3-26. -26. (canceled)

27. A system for recommending an optimal withdrawal amount, for a given period, from portfolio accounts consisting of relatively risky and relatively safe assets of a given user, comprising:

A server or computer having a CPU and memory;

Means for providing to the server or computer the user'"'"'s age, gender and health status;

Means for providing to the server or computer the user'"'"'s subjective discount rate;

Means for providing to the server or computer a set of Gompertz mortality parameters;

Means for providing to the server or computer the current value of the user'"'"'s relatively risky and relatively safe assets;

Means for providing to the server or computer the after-tax pension and annuity income of the retiree;

Means for providing to the server or computer the risk and return characteristics of a portfolio of relatively risky assets;

Means for providing to the server or computer the valuation rate;

Means of computing the user'"'"'s level of longevity risk aversion, which is a function of the current value of the user'"'"'s relatively risky and relatively safe assets, the after-tax pension and annuity income of the user, the risk and return characteristics of a portfolio of relatively risky assets, and the valuation rate;

Means for computing the user'"'"'s probability of survival as a function of the user'"'"'s age, gender, health status and the set of Gompertz mortality parameters;

Means for computing the actuarial discount rate, which is a function of the user'"'"'s probability of survival, subjective discount rate, the valuation rate, and a time variable;

Means for solving the user'"'"'s optimal wealth depletion time, which is a function of the user'"'"'s total financial wealth, the user'"'"'s after-tax pension and annuity income, the valuation rate, the user'"'"'s probability of survival, the user'"'"'s subjective discount rate, the user'"'"'s level of longevity risk aversion, and the actuarial discount rate;

Means for computing the user'"'"'s optimal withdrawal amount, which is a function of the user'"'"'s total financial wealth, the user'"'"'s after-tax pension and annuity income, the valuation rate, the user'"'"'s optimal wealth depletion time, and the actuarial discount rate; and

Means for displaying or otherwise communicating the recommended optimal withdrawal amount to the user.

28. The system of claim 27, wherein the actuarial discount rate is computed over multiple iterations.

29. The system of claim 27, wherein the actuarial discount rate is computed using a Gamma Function.

30. The system of claim 27, wherein the actuarial discount rate is computed using market quotes for immediate and deferred annuities.

31. The system of claim 27, wherein the optimal wealth depletion time is solved over multiple iterations.

32. The system of claim 27, wherein the optimal wealth depletion time is solved using the Bisection method.

33. The system of claim 27, wherein the optimal wealth depletion time is solved using the Newton method.

34. A non-transitory machine-readable medium storing a computer program that, when executed by a processor, causes the processor to performs the following steps:

retrieve current values of any relatively risky and relatively safe assets of the retiree;

retrieve amounts of any after-tax pension and annuity income of the retiree;

calculate total after-tax amount of pension and annuity funds receivable by the retiree per time unit, which is referred to as Pie;

calculate financial wealth of the retiree at the time 0, which is referred to as W(0), as the sum of the current value of the relatively risky assets and the current, value of the relatively safe assets;

retrieve a valuation rate, which is referred to as R, of the retiree;

Calculating or otherwise retrieving the retiree'"'"'s level of longevity risk aversion, which is referred to as LRA;

calculate the retiree'"'"'s optimal wealth depletion time, which is referred to as WDT, by solving the equation OWA(WDT)−

Pie=0, where OWA(T) is the retiree'"'"'s estimated optimal withdrawal amount at time T;

calculate the retiree'"'"'s optimal withdrawal amount at time 0, which is referred to as OWA(0), by performing the steps of;

Calculating B1=EXP(R*WDT);

Calculating N2=(W(0)+Pie/R)*B1−

(Pie/R);

Calculating an actuarial present value for lime WDT, which is referred to as APVX(WDT); and

Calculating OWA(0)=N2/APVX(WDT)/B1; and

Displaying or otherwise communicating the value of OWA(0) to the retiree.

35. The non-transitory machine-readable medium of Claim 34, wherein relatively risky and relatively safe assets have different risk and return characteristics.

91. The non-transitory machine-readable medium of claim 34, wherein Pie is equal to 0.

92. The non-transitory machine-readable medium of claim 34, wherein the computer program, when executed by the processor, further causes the processor to calculate Alpha being the current value of the retiree'"'"'s relatively risky assets divided by the sum of W(0) and the present value of the retiree'"'"'s after-tax pension and annuity income, wherein LRA is calculated as the Sharpe ratio inversely scaled by Alpha.

93. The non-transitory machine-readable medium of claim 92, wherein the computer program, when executed by the processor, further causes the processor to:

retrieve an expected rate of return on a portfolio of relatively risky assets, which is referred to as Mu; and

retrieve a standard deviation of returns for the portfolio of relatively risky assets, which is referred to as Sigma,

wherein the LRA is calculated as ((Mu−

R)/Sigma)/Alpha if Alpha is not equal to 0 and a relatively large number.

94. The medium of claim 93, wherein the relatively large number is 100.

95. The non-transitory machine-readable medium of claim 93, wherein the computer program, when executed by the processor, further causes the processor to calculate the retiree'"'"'s optimal withdrawal amount (OWA(T)) for a particular time (T) by performing steps of:

calculating an actuarial present value for time T, which is referred to as APVX(T);

calculating A0=Pie/R;

calculating A1=EXP(R*T);

calculating N1=(W(0)+A0)*A1−

A0;

calculating D1=APVX(T)*A1;

retrieving a subjective discount rate, which is referred to as SDR, of the retiree;

calculating K=(R−

SDR)/LRA;

calculating A2=EXP(K*T);

calculating A3=a conditional probability of the retiree surviving for T time units;

calculating A4=A3̂

(1/LRA); and

calculating OWA(T)=N1/D1*A2*A4.

96. The medium of claim 95, wherein solving the equation OWA(WDT)−

Pie=0 comprises the steps of;

(a) setting a provisional time estimate (WDTprte) equal to a pre-determined initial estimate of WDT;

(b) calculating the retiree'"'"'s optimal wealth withdrawal amount (OWA(WDTprte)) for the provisional time estimate;

(c) if OWA(WDTprte)−

Pie is approximately 0 then setting WDT=WDTprte, otherwise if OWA(WDTprte)−

Pie is greater than 0 then increasing the value of WDTprte and repeating steps (b) and (c), otherwise if OWA(WDTprte)−

Pie is less than 0 then decreasing the value of WDTprte and repeating steps (b) and (c).

97. The medium of claim 95, wherein APVX(T) is calculated as the sum of individual actuarial present values APV(Ti) computed for each value of time Ti ranging from 0 to T time units, where APV(Ti) is calculated as the conditional probability of the retiree surviving for Ti time units divided by (1+R)̂

Ti.

98. The non-transitory machine-readable medium of claim 97, wherein the conditional probability of the retiree surviving for a particular number of time units is calculated using the Gompertz-Makeham Law of Mortality.

99. The non-transitory machine-readable medium of claim 96, wherein the time unit is one year, and wherein in step (c) of solving the equation OWA(WDT)−

Pie=0, if OWA(WDTprte)−

Pie is greater than 0 then the value of WDTprte is increased by one month and steps (b) and (c) are repeated, and if OWA(WDTprte)−

Pie is less than 0 then the value of WDTprte is decreased by one month and steps (b) and (c) are repeated.

100. The non-transitory machine-readable medium of claim 99, wherein the pre-determined initial estimate of the WDT is 20 years.

101. The non-transitory machine-readable medium of claim 96, wherein the increase or decrease in the value of WDTprte is calculated using Newton'"'"'s method.

102. The non-transitory machine-readable medium of claim 96, wherein the increase or decrease in the value of WDTprte is calculated using a bisection method.

36-62. -62. (canceled)

77. A system for calculating an optimal withdrawal amount from a retiree'"'"'s portfolio accounts comprising relatively risky and relatively safe assets, the system comprising:

a computer processor;

a display screen on which the computer processor can display data;

a non-transitory memory storing computer readable instructions that, when executed

by the computer processor, cause the computer processor to;

retrieve current values of any relatively risky and relatively safe assets of the retiree;

retrieve amounts of any after-tax pension and annuity income of the retiree;

calculate total after-tax amount of pension and annuity funds receivable by the retiree per time unit (Pie);

calculate financial wealth of the retiree at time 0, which is referred to as W(0), as the sum of the current value of the relatively risky assets and the current value of the relatively safe assets;

retrieve a valuation rate, which is referred to as R, of the retiree;

calculate or retrieve the retiree'"'"'s level of longevity risk aversion, which is referred to as LRA;

calculate the retiree'"'"'s optimal wealth depletion time, which is referred to as WDT, by solving the equation OWA(WDT)−

Pie=0, where OWA(T) is the estimated financial capital of the retiree at time T;

calculate the retiree'"'"'s optimal withdrawal amount at time 0, which is referred to as OWA(0), by performing the steps of;

calculating B1=EXP(R*WDT);

calculating N2=(W(0)+Pie/R)*B1−

(Pie/R);

calculating actuarial present value for time WDT, which is referred to as APVX(WDT); and

calculating OWA(0)=N2/APVX(WDT)/B1; and

display or otherwise communicate the value of OWA(0) to the retiree.

78. The system of claim 77, wherein the relatively risky and relatively safe assets have different risk and return characteristics.

79. The system of claim 77, wherein Pie is equal to 0.

80. The system of claim 77, wherein the computer readable instructions further cause the computer processor to:

calculate Alpha being the current value of the retiree'"'"'s relatively risky assets divided by the sum of W(0) and the present value of the retiree'"'"'s after-tax pension and annuity income, wherein the LRA is calculated as the Sharpe ratio inversely scaled by Alpha.

81. The system of claim 80, wherein the computer readable instructions further cause the computer processor to:

retrieve an expected rate of return on a portfolio of relatively risky assets, which is referred to as Mu; and

retrieve a standard deviation of returns for the portfolio of relatively risky assets, which is referred to as Sigma,

wherein the LRA is calculated as ((Mu−

R)/Sigma)/Alpha if Alpha is not equal to 0 and a relatively large number.

82. The system of claim 81, wherein the relatively large number is 100.

83. The system of claim 81, wherein the computer readable instructions further cause the computer processor to calculate the retiree'"'"'s optimal withdrawal amount (OWA(T)) for a particular time (T) by performing steps of:

calculating an actuarial present value for time T, which is referred to as APVX(T);

calculating A0=Pie/R;

calculating A1=EXP(R*T);

calculating N1=(W(0)+A0)*A1−

A0;

calculating D1=APVX(T)*A1;

retrieving a subjective discount rate, which is referred to as SDR, of the retiree;

calculating K=(R−

SDR)/LRA;

calculating A2=EXP(K*T);

calculating A3=a conditional probability of the retiree surviving for T time units;

calculating A4=A3̂

(1/LRA); and

calculating OWA(T)=N1/D1*A2*A4.

84. The system of claim 83, wherein solving the equation OWA(WDT)−

Pie=0 comprises the steps of;

(a) setting a provisional estimated time WDTprte equal to a pre-determined initial estimate of the WDT;

(b) calculating the retiree'"'"'s optimal wealth withdrawal amount (OWA(WDTprte)) for the provisional time estimate;

(c) if OWA(WDTprte)−

Pie is approximately 0 then setting WDT=WDTprte, otherwise if OWA(WDTprte)−

Pie is greater than 0 then increasing the value of WDTprte and repeating steps (b) and (c), otherwise if OWA(WDTprte)−

Pie is less than 0 then decreasing the value of WDTprte and repeating steps (b) and (c).

85. The system of claim 83, wherein the actuarial present value for time Tn, which is referred to as APVX(Tn), is calculated as the sum of individual actuarial present values APV(T) computed for each value of time T ranging from 0 to Tn time units, where APV(T) is calculated as the conditional probability of the retiree surviving for T time units divided by (1+R)̂

T.

86. The system of claim 85, wherein the conditional probability of the retiree surviving for a particular number of time units is calculated using the Gompertz-Makeham Law of Mortality.

87. The system of claim 84, wherein the time unit is one year, and wherein in step (c) of solving the equation OWA(WDT)−

Pie=0, if OWA(WDTprte)−

Pie is greater than 0 then the value of WDTprte is increased by one month and steps (b) and (c) are repeated, and if OWA(WDTprte)−

Pie is less than 0 then the value of WDTprte is decreased by one month and steps (b) and (c) are repeated.

88. The system of claim 87, wherein the pre-determined initial estimate of the WDT is 20 years.

89. The system of claim 84, wherein the increase or decrease in the value of WDTprte is calculated using Newton'"'"'s method.

90. The system of claim 84, wherein the increase or decrease in the value of WDTprte is calculated using a bisection method.