**Insurance Abstract**
A system and method for allocating an investor's wealth to at least
one risky asset and life insurance includes retrieving a profile
of the investor. The financial capital available to the investor
and a human capital value for the investor are determined. An objective
function value for the investor is determined and maximized. An
amount of the investor's wealth is allocated to the at least one
risky asset and to life insurance.
**Insurance Claims**
1. A method for allocating an investor's wealth to at least one
risky asset and life insurance comprising: (a) retrieving a profile
of the investor; (b) determining financial capital available to
the investor; (c) determining a human capital value for the investor;
(d) determining an objective function value for the investor; (e)
maximizing the investor's objective function value; and (f) allocating
an amount of the investor's wealth to the at least one risky asset
and to life insurance.
2. The method according to claim 1, wherein retrieving the investor's
profile further comprises: (a1) determining a current age of the
investor; (a2) determining a gender of the investor; (a3) determining
a retirement age of the investor.
3. The method according to claim 1, wherein the at least one risky
asset comprises a stock.
4. The method according to claim 1, wherein determining the investor's
financial capital further comprises: (b1) calculating a value of
the at least one risky asset; and (b2) calculating a plurality of
budget constraints on the investor's capital.
5. The method according to claim 1, wherein determining the investor's
human capital further comprises: (c1) calculating a labor income
of the investor; and (c2) calculating a present value of the investor's
future income.
6. The method according to claim 1, wherein determining the plurality
of objective function values further comprises: (d1) determining
a utility of bequest for the investor's; (d2) retrieving at least
one survival probability of the investor; and (d3) determining a
risk tolerance for the investor.
7. The method according to claim 6, wherein the utility of bequest
of the investor is greater than zero.
8. The method according to claim 6, wherein the utility of bequest
of the investor is zero.
9. The method according to claim 6, wherein the at least one probability
of survival is selected from the group consisting of an objective
probability of survival for the investor and a subjective probability
of survival for the investor.
10. The method according to claim 6, wherein the risk tolerance
of the investor is greater than zero.
11. The method according to claim 1, wherein steps (a)-(d) are
repeated N times.
12. The method according to claim 11, wherein N is 20000.
13. A system for allocating an investor's wealth to at least one
risky asset and life insurance comprising: a memory for storing
a profile of the investor, financial capital available to the investor,
a human capital value for the investor, an objective function value
for the investor, a maximized objective function value for the investor,
and an allocated an amount of the investor's wealth to at least
one risky asset and to life insurance; and a processor coupled to
the memory to retrieve the stored profile of the investor; the stored
financial capital available to the investor; the stored human capital
value for the investor; the stored objective function value for
the investor; the stored maximized objective function value for
the investor; and the stored allocated an amount of the investor's
wealth to at least one risky asset and to life insurance.
14. The system according to claim 13, where the processor, in retrieving
the investor's profile, determines the current age of the investor;
determines the gender of the investor; and determines the retirement
age of the investor.
15. The system according to claim 13, where the processor, in determining
the investor's financial capital, calculates a value of the at least
one risky asset; and calculating a plurality of budget constraints
on the investor's capital.
16. The system according to claim 13, where the processor, in determining
the investor's human capital, calculates a labor income of the investor;
and calculates a present value of the investor's future income.
17. The system according to claim 13, where the processor, in determining
the plurality of objective function values, determines a utility
of bequest for the investor's; retrieves at least one survival probability
of the investor; and determines a risk tolerance for the investor.
18. The system according to claim 13, wherein the system is connected
to at least one peripheral device from the group consisting of a
display or monitor, a keyboard, a mouse, a printer and/or copier.
19. The system according to claim 13, wherein the system further
comprises a communication device from the group consisting of a
modem and a network card.
20. The system according to claim 13, wherein the system further
comprises a web server acting as a host for a website on which can
be displayed a questionnaire or other request for information, which
is accessible to an investor and an investor's computer either remotely
or non-remotely.
21. A machine-readable medium on which has been prerecorded a computer
program, which when executed by a processor, performs method comprising:
retrieves a profile of an investor; determines financial capital
available to the investor; determines a human capital value for
the investor; determines an objective function value for the investor;
maximizes the investor's objective function value; and allocates
an amount of the investor's wealth to the at least one risky asset
and to life insurance.
22. The medium according to claim 21, wherein the retrieving the
investor's profile further comprises: determining a current age
of the investor; determining a gender of the investor; determining
a retirement age of the investor.
23. The medium according to claim 21, wherein determining the investor's
financial capital further comprises: calculating a value of the
at least one risky asset; and calculating a plurality of budget
constraints on the investor's capital.
24. The medium according to claim 21, wherein determining the investor's
human capital further comprises: calculating a labor income of the
investor; and calculating a present value of the investor's future
income.
25. The medium according to claim 21, wherein determining the plurality
of objective function values further comprises: determining a utility
of bequest for the investor's; retrieving at least one survival
probability of the investor; and determining a risk tolerance for
the investor.
**Insurance Description**
FIELD OF THE INVENTION
[0001] The invention general relates to financial planning, more
particularly, to a system, method and medium of financial analysis
utilizing parameters including asset allocation, human capital and
life insurance demand.
BACKGROUND
[0002] An important decision that confronts investors is the allocation
of their wealth either towards the purchase of life insurance or
towards the acquisition of assets to provide savings and financial
growth for their retirement years. This allocation of wealth is
often governed by an investor's human capital since human capital
affects both asset allocation and the demand for life insurance.
However, these two important financial decisions have consistently
been analyzed separately in theory and practice. Financial planners
and advisors have recently begun to recognize that human capital
must be taken into account when building financial portfolios for
individual investors.
[0003] Life insurance has long been used to hedge against mortality
risk (i.e., the loss of human capital in the unfortunate event of
premature death), a unique aspect of an investor's human capital.
Life insurance is the business of human capital securitization--addressing
the uncertainties and inadequacies of an individual's human capital
(Ostaszewski, K., "Is Life Insurance a Human Capital Derivatives
Business?" Journal of Insurance Issues, 26, 1, 1-14 (2003)).
On the other hand, empirical studies on life insurance adequacy
have shown that under insurance is prevalent. Gokhale and Kotlikoff
argue that questionable financial advice, inertia, and the unpleasantness
of thinking about one's death are the likely causes (Gokhale, J.
and Kotlikoff, L. J.; "The Adequacy of Life Insurance, Reasearch
Dialogue," TIAA CREF INSTITUTE, Issue no. 72 (July 2002), www.tiaa-crefinstitute.org).
[0004] Typically, the greater the value of human capital, the more
life insurance the family demands. In fact, popular investment and
financial planning advice regarding how much life insurance one
should acquire is seldom framed in terms of the level of risk associated
with one's human capital. Conversely, asset allocation decisions
are only recently being framed in terms of the risk characteristics
of human capital, and rarely is it integrated with life insurance
decisions.
[0005] Economic theory predicts that investors make asset allocation
and life insurance purchase decisions to maximize their lifetime
utilities of wealth and consumption. An investor's total wealth
typical includes two parts: (1) readily tradable financial assets,
and (2) human capital. Although human capital is not readily tradable,
it is often an investor's single largest asset. As illustrated in
FIG. 1, younger investors typically have far more human capital
than financial capital because younger investors have a greater
number of years to work and have had few years to save and accumulate
financial wealth. Conversely, older investors tend to have more
financial capital than human capital, since they have fewer years
ahead to work, but have accumulated financial capital. This changing
mix of financial capital and human capital impacts investors' decisions
regarding financial asset allocation.
[0006] In the late 1960s, economists established models that implied
individuals should optimally maintain constant portfolio weights
throughout their life cycle (Samuelson, P., "Life Time Portfolio
Selection by Dynamic Stochastic Programming," Review of Economics
and Statistics, Vol. 51, 239-246 (1969); Merton, R., "Life
Time Portfolio Selection Under Uncertainty: The Continuous Time
Case," Review of Economics and Statistics, Vol. 51, 247-257
(1969)). Those models incorrectly assumed investors had no labor
income (i.e., human capital). However in contrast to such models,
when labor income is included in the portfolio choice model, individuals
will optimally change their allocation of financial assets in a
pattern related to their position in their life cycle. In other
words, optimal asset allocation depends on the risk-return characteristics
and flexibility of the labor income (such as how much or how long
the investor works). Thus, the investor has the ability to adjust
the financial portfolio to compensate for the non-tradable human
capital risk exposures (e.g., Merton, R., "Optimum Consumption
and Portfolio Rules in a Continuous-Time Model,"Journal of
Economic Theory, 3(4), 373-413, (December 1971); Bodie, Z., Merton,
R., and Samuelson, W., "Labor supply flexibility and portfolio
choice in a life cycle model," Journal of Economic Dynamics
and Control, Vol. 16, 427-449 (1992); Heaton, J., and Lucas, D.
"Market Frictions, Savings Behavior, and Portfolio Choice,"
Macroeconomic-Dynamics, 1(1): 76-101 (1997); Jaganathan, R., and
Kocherlacota, N., "Why Should Older People Invest Less in Stocks
Than Younger People?" Federal Reserve Bank of Minneapolis Quarterly
Review, 20(3):11-23, Summer (1996); and Campbell, J., and Viceira,
L., "Strategic Asset Allocation--Portfolio Choice for Long-term
Investors," Oxford University Press (2002)). Several key theoretical
implications with respect to risk-return characteristics and the
flexibility of the labor income include: 1) young investors will
invest more in risky assets (e.g. stocks) than older investors;
2) investors with safe labor income and thus, safe human capital
will invest more of their financial portfolio into stocks; 3) investors
with labor income highly correlated with stock markets will invest
their financial assets into less risky assets; and 4) the ability
to adjust labor supply (i.e., higher flexibility) also increases
an investor's allocation toward risky assets. However, empirical
studies show that most investors do not efficiently diversify their
financial portfolio considering the risk of their human capital.
In fact, many investors use primitive methods to determine the asset
allocation and many of them invest very heavily into the stock of
the company for which they work. Benartzi, S., "Excessive Extrapolation
and the Allocation of 401(k) Accounts to Company Stock," Journal
of Finance, 56, 1747-64 (2001) and Benartzi, S., and Thaler, R.,
Naive Diversification Strategies in Defined Contribution Saving
Plans," American Economic Review, 91, 79-98 (2001)
[0007] Additionally, the lifetime consumption and portfolio decision
models in the art need to be expanded to take into account lifetime
uncertainty (or mortality risk). Life insurance and life annuities
may be used to insure against lifetime uncertainty, while also deriving
conditions under which consumers would fully insure against lifetime
uncertainty. (Yaari, M. E., "Uncertain Lifetime, Life Insurance,
and the Theory of the Consumer," The Review of Economic Studies,
Vol. 32, No. 2, 137-150 (1965)).
[0008] Theoretical studies have shown a link between the demand
for life insurance and the uncertainty of human capital. For most
households, labor income uncertainty dominates financial capital
income uncertainty. Solutions have been developed where the optimal
amount of insurance a household should purchase is based on human
capital uncertainty. (Campbell, R. A, "The Demand for Life
Insurance: An Application of the Economics of Uncertainty,"
The Journal of Finance, Vol. 35, No. 5, 1155-1172 (1980)). For example,
model life insurance demand in a portfolio context can be determined
using mean-variance analysis, deriving the optimal insurance demand
and the optimal allocation between risky and risk-free assets where
the optimal amount of insurance depends on two components: the expected
value of human capital and the risk-return characteristics of the
insurance contract (Buser, S., and Smith, M., "Life insurance
in a portfolio context," Insurance: Mathematics and Economics
2, 147-57 (1983)).
[0009] Thus, there is a need in the industry for a method that
links the asset allocation decision with the life insurance purchase
decision into one framework by incorporating human capital, which
takes into consideration the impact of the investor's bequest motive,
objective and/or subjective survival probability, the volatility
of the investor's income in correlation to the financial market
BRIEF DESCRIPTION OF THE DRAWINGS
[0010] For the purpose of illustrating the invention, explanatory
Figures are provided; it being understood, that this invention is
not limited to the information represented by the Figures.
[0011] FIG. 1 depicts the relationship between an investor's expected
financial capital and human capital over the investor's life cycle.
[0012] FIG. 2 is a flow chart depicting the financial capital calculation
in an embodiment of the present invention.
[0013] FIG. 3 is a flow chart depicting the human capital calculation
in an embodiment of the present invention.
[0014] FIG. 4 is a schematic illustrating a computer system suitable
for carrying out the present invention.
[0015] FIG. 5 depicts the relationship between an investor's human
capital, insurance demand and financial asset allocation over the
investor's life cycle.
[0016] FIG. 6 depicts the relationship between an investor's insurance
demand and asset allocation across the investor's strength of bequest.
[0017] FIG. 7 depicts the relationship between an investor's insurance
demand and asset allocation at different risk aversion levels.
[0018] FIG. 8 depicts the relationship between an investor's insurance
demand and asset allocation at different levels of financial wealth.
[0019] FIG. 9 depicts the relationship between an investor's insurance
demand and asset allocation at different correlation levels.
DETAILED DESCRIPTION
[0020] It will be appreciated that the following description is
intended to refer to specific embodiments of the invention and is
not intended to define or limit the invention, other than in the
appended claims. A variety of modifications to the embodiments described
will be apparent to those skilled in the art from the disclosure
provided herein. Thus, the invention may be embodied in other specific
forms without departing from the spirit or essential attributes
thereof.
[0021] When a value or parameter is given as either a range, preferred
range, or a list of upper preferable values and lower preferable
values, this is to be understood as specifically disclosing all
ranges formed from any pair of any upper range limit or preferred
value and any lower range limit or preferred value, regardless of
whether ranges are separately disclosed. Where a range of numerical
values is recited herein, unless otherwise stated, the range is
intended to include the endpoints thereof, and all integers and
fractions within the range. It is not intended that the scope of
the invention be limited to the specific values recited when defining
a range.
[0022] The term "human capital", as used herein, is meant
to refer to the actuarial present financial economic value or future
labor income of all future wages, which is a scalar quantity and
is dependent upon a number of both subjective or market equilibrium
factors.
[0023] The term "retirement", as used herein, is meant
to indicate the termination of the human capital income flow and
the beginning of the pension phase.
[0024] The invention relates to a method, system or medium for
linking an investor's human capital with determining the investor's
allocation of wealth to the acquisition of assets, life insurance,
or both, where the invention accounts for the impact of the investor's
bequest motive and objective and/or subjective survival probability
as well as the volatility of the investor's income in correlation
to the financial market.
[0025] In merging asset allocation and human capital with the optimal
demand for life insurance, understanding the actuarial factors that
impact the pricing of a life insurance contract is important. A
number of life insurance product variations are available (i.e.,
term life, whole life, or universal life), and the system and method
of the invention can be adapted to work with any of them. The most
fundamental type, the one-year renewable term policy, is used as
an example below.
[0026] With a one-year renewable term the policy premium is typically
paid at the beginning of the year, or on the individual's birthday,
and protects the human capital of the insured for the duration of
the year. Thus, should the insured person die within the year covered
by the purchased policy, the insurance company pays the face value
to the beneficiaries, soon after the death or prior to the end of
the year. The next year the contract is guaranteed to start anew
with new premium payments made and protection received.
[0027] The policy premium is typically an increasing function of
the desired face value, where the policy premium calculation is
represented by the general equation: P=[q/(1+r)].theta. (1)
[0028] The premium P is calculated by multiplying the desired face
value of the insurance policy .theta. by the probability of death
q, and then discounting by the interest rate factor (1+r). The theory
behind equation (1) is the well-known "law of large numbers,"
which guarantees that probabilities become percentages when individuals
are aggregated. The implicit assumption of equation (1) is that
although death can occur at any time during the year (or term),
the premium payments are made at the beginning of the year and the
face values are paid at the end of the year. From an insurance company's
perspective, all of the premiums received from the group of N individuals
with the same age (i.e., probability of death q) and face value
.theta., are co-mingled and invested in an insurance reserve earning
a rate of interest r so that at the end of the year, PN(1+r) is
partitioned amongst the qN beneficiaries. No savings component or
investment component is embedded within the policy premium defined
by equation (1). Rather, at the end of the year the survivors lose
any claim to the pool of accumulated premiums, since all funds go
directly to the beneficiaries.
[0029] As an individual ages and the probability of death q.sub.x
increases (denoted by x), the same exact face amount of (face value)
life insurance .theta. will cost more and will induce a higher premium
P.sub.x, as per the formula. In practice, the actual premium is
loaded by an additional factor denoted by (1+.lamda.) to account
for commissions, transaction costs, and profit margins (.lamda.
denotes the fees and expenses, i.e., actuarial and insurance loading,
imposed and charged on a typical life insurance policy) and so the
actual amount paid by the insured is closer to P(1+.lamda.), but
the underlying pricing relationship driven by the law of large numbers
remains the same.
[0030] Typically, when purchasing life insurance, individuals conduct
a budgeting analysis to determine his or her life insurance demands
(i.e., the amount the surviving family and beneficiaries need to
replace the lost wages in present value terms). The life insurance
demand would be taken as the required face value in equation (1),
which would then lead to a premium. Alternatively, one can think
of a budget for life insurance purchases, and the policy purchase
premium would be determined by equation (1).
[0031] The model of the invention will "solve" for the
optimal age-varying amount of life insurance denoted by .theta.,
which then induces an age-varying policy payment P.sub.x, which
maximizes the welfare of the family by taking into account its risk
preferences and attitudes toward bequest.
[0032] As previously set forth, in the invention an investor can
allocate their financial wealth between life insurance and assets.
With respect to the allocation of financial wealth to assets it
is assumed that there are two asset classes, where an investor can
allocate financial wealth to either a risk-free asset (i.e., bonds,
which is representatively utilized herein) and a risky asset (i.e.,
stocks, which is representatively utilized herein). This is consistent
with the two-fund separation theorem that is consistent with traditional
portfolio theory. Of course, this can always be expanded to multiple
asset classes. The investor's objective is to maximize the overall
utility of their wealth, which includes utility from the alive state
and the dead state. It is also assumed that the investor makes asset
allocation and insurance purchase decisions at the start of each
period and that labor income is also received at the beginning of
each period.
[0033] An embodiment of the invention relates to a method for allocating
an investor's wealth to at least one asset (e.g., either at least
one risky asset and/or at least one risk-free asset) and life insurance
comprising: [0034] (a) retrieving a profile of the investor; [0035]
(b) determining financial capital available to the investor; [0036]
(c) determining a human capital value for the investor; [0037] (d)
determining an objective function value for the investor; [0038]
(e) maximizing the investor's objective function value; and [0039]
(f) allocating an amount of the investor's wealth to at least one
risky asset and to fund a life insurance policy.
[0040] Retrieving an investor's profile includes gathering information
pertinent to determining the investor's available financial capital
and human capital value. The investor's profile typically includes,
but is not limited to, determining their current age, gender, and
either the investor's age or projected age for retirement. The investor's
profile may be generated by retrieving the particular data from
the investor's employer, one or more record keepers having the requisite
information or from the investor himself or herself.
[0041] With reference to FIG. 2, determining an investor's available
financial capital comprises calculating the value of at least one
risky asset in their investment portfolio, illustrated at 12; and
calculating the one or more budget constraints or restrictions on
the investor's capital, illustrated at 14.
[0042] The value of a risky asset is represented by the general
equation: S t + 1 = S t .times. exp .times. { .mu. S - 1 2 .times.
.sigma. S 2 + .sigma. S .times. Z S , t + 1 ) ( 2 ) such that this
value follows a discrete version of a Geometric Brownian Motion,
where S.sub.t denotes the at least one risky asset, .mu..sub.S denotes
the expected return of the risky asset, .sigma..sub.S denotes the
standard deviation of the risky asset and Z.sub.S,t is an independent
random variable and Z.sub.S,t.about.N(0,1).
[0043] The budget constraints or restrictions of an investor's
investment portfolio need to be taken into account. These constraints
act as limitations of the financial wealth that an investor may
commit to the purchase of life insurance, the purchase of at least
one asset, of the purchase of both. As a result, the amount of available
financial capital is to be discounted by the budget constraints.
The budget constraints are represented by the general equation:
W t + 1 = [ W t + h t - ( 1 + .lamda. ) .times. q t .times. .theta.
x .times. e - r f - C t ] .times. [ .alpha. x .times. e .mu. S -
1 2 .times. .sigma. S 2 + .sigma. S 2 .times. Z S , t + 1 + ( 1
- .alpha. x ) .times. .times. e r f ] ( 3 ) W.sub.t denotes financial
wealth at time t; h.sub.t denotes the annual labor income; .lamda.
denotes the fees and expenses (i.e., actuarial and insurance loading)
imposed and charged on a typical life insurance policy; q.sub.t
denotes the objective probabilities of death at the end of the year
x+1 conditional on being alive at age x, determined by a given population
(i.e., mortality table); .theta..sub.x denotes the amount of life
insurance; the expected return on the risk-free asset is denoted
bye e.sup.r.sup.f or e.sup.-r.sup.f; C.sub.t denotes the consumption
at year t; .alpha..sub.x denotes the allocation to risky assets;
.mu..sub.S denotes the expected return of the risky asset; .sigma..sub.s
denotes the standard deviation of the return of the risky assets;
and Z.sub.S,t is an independent random variable and Z.sub.S,t.about.N(O,
1).
[0044] Human capital, though not traded and highly illiquid, should
be treated as part of the endowed wealth that must be protected,
diversified and hedged. The correlation between human capital and
financial capital (i.e., whether you are closer to a bond or a stock)
has a noticeable and immediate impact on the demand for life insurance
as well as the usual portfolio considerations. A determination of
how much life insurance is needed and where financial capital should
be invested cannot be solved in isolation. For instance, a person
whose income heavily relies on commissions should consider his human
capital closer to a stock since the income is highly correlated
with the market, which results in great uncertainty in his human
capital. Consequently, he should purchase less insurance and invest
more financial wealth in bonds. Conversely, a tenured university
professor, for example, who considers his/her human capital closer
to a bond, purchases more insurance, and invests more financial
wealth in stocks.
[0045] In between the extremes of classifying an investor as purely
a stock or bond, human capital is a diversified portfolio of stocks
and bonds, plus any idiosyncratic risks. For example, if a person's
human capital is 40% long-term bonds, 30% financial services, and
30% utilities, the unpredictable shocks to future wages have a given
correlation structure with the named sub-indices. Accordingly, a
tenured university professor could be considered to be a 100% real-return
(inflation linked) bond, since shocks to wages, if there are any,
would not be linked to any financial sub-index. However, there are
difficulties involved in calibrating these variables and some of
the parameters relied upon by Davis and Willen are utilized herein
for the case numerical examples (Davis, Stephen J. and Willen, Paul,
"Occupation-Level Income Shocks and Asset Returns: Their Covariance
and Implications for Portfolio Choice", University of Chicago
Graduate School of Business, Working Paper (2000)).
[0046] As illustrated in FIG. 3, determination of an investor's
human capital comprises calculating their annual labor income, illustrated
at 16, and calculating the present value of the investor's future
income, illustrated at 18.
[0047] Calculating an investor's annual labor income is represented
by the general equation: h.sub.t+1=h.sub.t exp{.mu..sub.h+.sigma..sub.hZ.sub.h,t+1}
(4) where labor income h.sub.t is greater than zero, .mu..sub.h
and .sigma..sub.h are the annual growth rate and the annual standard
deviation, respectively, of the income process. Z.sub.h,t is an
independent random variable and Z.sub.h,t.about.N(0,1).
[0048] Thus, based on equation (4), for a person at age x, their
income at age x+t is represented by the general equation: h x +
t = h x ( k = 1 l .times. exp .times. { .mu. h + .sigma. h .times.
Z h , k } ) ( 5 ) where h.sub.t>0, and .mu..sub.h and .sigma..sub.h
are the annual growth rate and the annual standard deviation of
the income process. Z.sub.h,k is an independent random variable
and Z.sub.h,k.about.N(0,1).
[0049] Thus, calculating the present value of future income from
age t+1 until death is represented by the general formula: H x +
t = j = t + 1 Y - x .times. [ h x + j .times. exp .times. { - (
j - t ) .times. ( r f + .eta. h + .zeta. h ) } ] ( 6 )
[0050] where .eta..sub.h is the risk premium or discount rate for
the income process and captures the market risks associated with
the income. .zeta..sub.h is the discount factor of the human capital
calculation which accounts for the illiquidity risk associated with
the investor's occupation (a 4 percent discount rate per year is
typical). Y is the investor's retirement age, either real or projected
and the return on the risk free asset is r.sub.f. Furthermore, we
regard the expected value of H.sub.t (i.e., E[H.sub.x+t]), as the
human capital one has at age x+t+1.
[0051] The risk premium for the income process (.eta..sub.h) is
calculated using the general equation: .eta. h = .rho. .function.
[ .mu. S - ( e r f - 1 ) ] .times. .sigma. h .sigma. S ( 7 ) where
.rho. represents the correlation between the labor income innovation
and the return of the risky asset; .mu..sub.s is the expected return
of the financial markets (the risky asset); r.sub.f is the expected
return on the risk-free asset; .sigma..sub.h is the annual standard
deviation of the income growth rate; and .sigma..sub.S is the standard
deviation of the return on the risky asset. It is assumed that the
correlation between the labor income innovation and the return of
risky asset is .rho. and: Z.sub.h=.rho.Z.sub.S+ {square root over
(1-.rho..sup.2)}Z (8) where, Z is a standard Brownian motion independent
of Z.sub.S. That is, Corr(Z.sub.S,Z.sub.h)=.rho.. (9)
[0052] A determination of the functional value comprises determining
the utility of bequest and risk aversion of the investor and retrieving
the at least one survival probability of the investor. The asset
allocation decision affects well-being in both the alive consumption
state (U.sub.alive) and the dead bequest state (U.sub.dead) while
the life insurance decision mostly affects the bequest state.
[0053] Bequest preference is arguably the most important factor
other than human capital when evaluating the life insurance demand
(a well-designed questionnaire could help elicit the individual's
attitude towards bequest, even though a precise estimate may be
hard to obtain). Investors who weight bequest more (higher D) are
likely to purchase more life insurance). D denotes the relative
strength of the utility of bequest, where individuals with no utility
of bequest will have a value of D=0.
[0054] The invention also considers an investor's subjective survival
probability (1- q). Investors with low subjective survival probability
will tend to buy more life insurance. This adverse selection problem
is well-documented in the insurance literature. The actuarial mortality
tables can be taken as a starting point. Life insurance is already
priced to take into account the adverse selection.
[0055] The function value is represented by the general equation:
(1-D).times.(1- q.sub.x).times.U.sub.alive[W.sub.x+1+H.sub.x+1]+D.times.(
q.sub.x).times.U.sub.dead[W.sub.x+1+.theta..sub.x] (10) such that
U alive .function. ( x ) = U dead .function. ( x ) = x 1 - .gamma.
1 - .gamma. ( 11 ) where for x>0 and .gamma..noteq.1, and U.sub.alive(x)=U.sub.dead(x)=ln(x)
(12) for x>0 and .gamma.=1; where x, W.sub.x+1, H.sub.x+1 and
.theta..sub.x have been described above and .gamma. denotes the
risk aversion parameter.
[0056] Thus arriving at the objective functional values is done
through simulation, where the values of the risky asset is simulated
using equation (2). Then Z.sub.h is simulated through equation (8),
which takes into account the correlation between the income innovation
and the return of the financial market. Human capital (H.sub.x+1)
is calculated using equations (5) and (6). If the wealth level at
age x+1 is less than zero, the wealth is set as equal to zero, to
indicate that an investor does not have any remaining wealth. This
process is carried out N times.
[0057] Thus, based on the above, the objective functional value
can then be calculated, which is represented by the general equation:
f N .function. ( .theta. x , .alpha. x ) = 1 N .times. n = 1 N .times.
{ ( 1 - D ) .times. ( 1 - q _ x 1 ) .times. U alive .function. [
W x + 1 , n + H x + 1 , n ] + D .times. ( q _ x 1 ) .times. U dead
.function. [ W x + 1 , n + .theta. x ] } ( 11 ) where the equation
accounts for the impact of the investor's bequest motive and objective
survival probability and/or subjective survival probability as well
as the volatility of the investor's income in correlation to the
financial market.
[0058] Thus, the investor can determine the optimal amount of life
insurance demand (a.k.a. the face value of the life insurance or
the death benefit) (.theta..sub.x) together with the allocation
of wealth (.alpha..sub.x) to risky assets in order to maximize the
year end utility of the total wealth, which is the human capital
plus the financial wealth, weighted by the alive and dead states.
Therefore, optimization can be express using the equation: max {
.theta. x , .alpha. x } .times. E .times. { ( 1 - D ) .times. (
1 - q _ x ) .times. U alive .function. [ W x + 1 + H x + 1 ] + D
.times. ( q _ x ) .times. U dead .function. [ W x + 1 + .theta.
x ] } ( 12 ) subject to the budget constraints of Equation (3) such
that and .theta. 0 .ltoreq. .theta. x .ltoreq. ( W x + h x - C x
) .times. .times. e r f ( 1 + .lamda. ) .times. .times. q x , (
13 ) and 0 .ltoreq. .alpha. x .ltoreq. 1. ( 14 )
[0059] Equation (13) requires the cost (or price) of the term insurance
policy to be less than the amount of current financial wealth the
client has, and there is a minimum insurance amount (.theta..sub.0>0)
an investor is required to purchase in order to have a minimum protection
from the loss of human capital.
[0060] The invention further relates to a system for carrying out
the determination of the optimal allocation of an investor's wealth.
A representative suitable system is indicated in general at 20 in
FIG. 4 and comprises those computers well known in the art, for
example a personal computer 22. Typically, the computer system includes
a memory 24 (for example either random access memory including DRAM,
SDRAM, or other known types of memory) for storing data such as
the investor's profile and the plurality of formulas of the invention.
The computer also includes a bus 26 and a microprocessor 28 loaded
with an operating system and executable instructions for one or
more special applications capable of carrying of the invention.
The computer also includes electronic read only memory 32 for storing
those programs known in the art that are non-volatile and persist
after the computer is shut down.
[0061] Alternatively, one or more of the computer programs capable
of carrying out the invention may be "hard-wired" into
the read only memory instead of being loaded as software instructions
into the random access memory. The read only memory can comprise
electrically programmable read only memory, electrically erasable
and programmable read only memory of either flash or nonflash varieties
or other sorts of read only memory such as programmable fuse or
antifuse arrays.
[0062] The computer program for carrying out the invention will
be stored or pre-recorded on a machine-readable medium or mass storage
device 34 (FIG. 4), such as an optical disk or magnetic hard drive
or other known device. The data and formula associated with the
invention will typically exist as a data base on the mass storage
device but could also reside on a separate database server and be
accessed remotely through a network.
[0063] The computer system may also be connected to peripheral
devices used to communicate with an operator such as, for example
a display or monitor 36, a keyboard 38, a mouse 40, a printer and/or
copier 42. Additionally, the computer system may also include communication
devices 44 such as a modem or a network card to communicate with
other computers and equipment, where such communication pathways
are preferably secure pathways to protect the confidential nature
of certain data.
[0064] The computer system may also include a web server acting
as a host for a website on which can be displayed a questionnaire
or other request for information, which is accessible to an investor
and investor's computer either remotely or non-remotely.
EXAMPLES
[0065] An analysis for five representative case studies is performed
with respect to the optimal asset allocation and the optimal life
insurance coverage. The problem is solved via simulation. There
are several key results: 1) investors need to make asset allocation
decisions and life insurance decisions jointly; 2) the magnitude
of human capital, its volatility, and its correlation with other
assets have a significant impact on the two decisions over the life
cycle; 3) bequest preferences and the subjective survival probability
have a significant impact on insurance demand, but little influence
on optimal asset allocation; and 4) conservative investors should
invest more in risk-free assets and buy more life insurance.
[0066] In the Examples, it is assumed there are two asset classes
in which the investor can invest his/her financial capital. Table
I provides the capital market assumptions used in all five cases.
It is also assumed that the investor is male, wherein his preference
toward bequest is one-fourth of his preference toward consumption
in the live state, 1-D=0.8 and D=0.2. He is agnostic about his relative
health status (i.e., his subjective survival probability is equal
to the objective actuarial survival probability). His income is
expected to grow with inflation, and the volatility of the growth
rate is 5 percent (the salary growth rate and the volatility are
chosen mainly to show the implications of the model and are not
necessarily representative). His real annual income is $50,000,
and he saves 10 percent each year. He expects to receive a pension
of $10,000 each year (in today's dollars) when he retires at age
65. His current financial wealth is $50,000. The investor is assumed
to follow the constant relative risk aversion (CRRA) utility with
a risk aversion coefficient (.gamma.). Finally, the financial portfolio
is assumed to be rebalanced and the term life insurance contract
is renewed annually (he mortality and insurance loading is assumed
to be 12.50%). These assumptions remain the same for all cases.
Other parameters such as initial wealth will be specified in each
case. TABLE-US-00001 TABLE 1 Capital Market Return Assumptions Compounded
Annual Return Risk (Standard Deviation) Risk-Free (Bonds) 5% --
Risky (Stocks) 9% 20% Inflation 3% --
Example 1
Human Capital, Financial Asset Allocation, and Life Insurance Demand
Over Lifetime
[0067] In this case, it is assumed that the investor has a moderate
risk aversion (relative risk aversion of 4). Also, the correlation
between the investor's income and the market return (risky asset)
is assumed to be 0.20 (Davis and Willen (2000) estimated the correlation
between labor income and equity market returns using the Current
Occupation Survey. They find that correlation between equity returns
and labor income typically lies in the interval from -0.10 to 0.20).
For a given age, the amount of insurance the investor should purchase
can be determined by his consumption/bequest preference, risk tolerance,
and financial wealth. His expected financial wealth, human capital,
and the derived optimal insurance demand over the investor's life
(from age 25 to 65) are presented in FIG. 5.
[0068] Several results are worth noting. First, human capital gradually
decreases as the investor gets older and the remaining number of
working years gets smaller. Second, the amount of financial capital
increases as the investor ages; this is the result of growth of
existing financial wealth and additional savings the investor makes
each year. The allocation to risky asset decreases as the investor
ages. This result is due to the dynamic between human capital and
financial wealth over time. When an investor is young, the investor's
total wealth is dominated by human capital. Since human capital
in this case is less risky than the financial risky asset, young
investors will invest more financial wealth into risky assets to
offset the impact of human capital on the overall asset allocation.
As the investor gets older, the allocation to risky assets is reduced
as human capital gets smaller. Finally, the insurance demand decreases
as the investor ages, as the primary driver of the insurance demand
is the human capital. The decrease in the human capital reduces
the insurance demand. The following cases will vary the investor's
preference of bequest, risk preference, and existing financial wealth
to illustrate the impact of these variables on the investor's optimal
asset allocation and life insurance purchases.
Example 2
Strength of Bequest Motive
[0069] This case shows the impact of bequest motives on the optimal
decisions on asset allocation and insurance demand. In the case,
it is assumed the investor is at age 45 and has an accumulated financial
wealth of $500,000. The investor has a moderate risk aversion coefficient
of 4. The optimal allocations to the risk-free asset and the optimal
insurance demands across various bequest levels are presented in
FIG. 6.
[0070] It can be seen that the insurance demand increases as the
bequest motive gets stronger, i.e., the D gets larger. This results
because an investor with a stronger bequest motive is more concerned
about his/her heirs and has the incentive to purchase a larger amount
of insurance to hedge the loss of human capital. On the other hand,
there is almost no change in the proportional allocation to risk-free
asset at different strengths of bequest motive. This indicates that
the asset allocation is primarily determined by risk tolerance,
returns on risk-free and risky assets, and human capital. This case
shows that bequest motive has a strong impact on insurance demand,
but little impact on optimal asset allocation. In this model, subjective
survival probability has similar impact on the optimal insurance
need and asset allocation as the bequest motive (D). When subjective
survival probability is high, the investor will buy less insurance.
Example 3
Risk Tolerance
[0071] This case shows the impact of the different degrees of risk
aversion on the optimal decisions on asset allocation and insurance
demand. In this case, it is again assumed the investor is at age
45 and has accumulated a financial wealth of $500,000. The investor
has a moderate bequest level, i.e., D=0.2. The optimal allocations
to risk-free asset and the optimal insurance demands across various
risk aversion levels are presented in FIG. 7.
[0072] The allocation to the risk-free asset increases with the
investor's risk aversion level. Actually, the optimal portfolio
is 100 percent in stocks for risk aversion levels less than 2.5.
The optimal amount of life insurance has a very similar pattern.
The optimal insurance demand increases with risk aversion. For a
moderate investor (a CRRA risk aversion coefficient 4), the optimal
insurance demand is about $290,000, which is roughly six times the
current income of $50,000 (This result is very close to the typical
recommendation by financial planners; i.e., purchase a term life
insurance policy that has a face value four to seven times one's
current income. (See, for example, Todd (2004)). Therefore, conservative
investors should invest more in risk-free assets and buy more life
insurance, compared to aggressive investors.
Example 4
Financial Wealth
[0073] This case shows the impact of the different amounts of current
financial wealth on the optimal asset allocation and insurance demand.
The investor's age is held at 45 and the risk preference and the
bequest motive at the moderate levels (a CRRA risk aversion coefficient
4 and bequest level 0.2). The optimal asset allocations to risk-free
asset and the optimal insurance demands for various financial wealth
levels are presented in FIG. 8.
[0074] First, it can be seen that the optimal allocation to the
risk-free asset increases with the initial wealth. This may seem
inconsistent with the CRRA utility function, since the CRRA utility
function implies the optimal asset allocation does not change with
the amount of the investor's wealth. However, it needs to be noted
that the wealth includes both financial wealth and human capital.
In fact, this is an example of the impact of human capital on the
optimal asset allocation. An increase in financial wealth not only
increases the total wealth, but also reduces the percentage of total
wealth represented by human capital. In this case, human capital
is less risky than the risky asset, where the income has a real
growth rate of 0% and a standard deviation of 5%, yet the expected
real return on stock is 8% and the standard deviation is 20%.).
When the initial wealth is low, the human capital dominates the
total wealth and the allocation. As a result, to achieve the target
asset allocation of a moderate investor, say an allocation of 60
percent risk-free asset and 40 percent risky asset, the closest
allocation is to invest 100 percent financial wealth in the risky
asset, since the human capital is illiquid. With the increase in
the initial wealth, the asset allocation is gradually adjusted to
approach the target asset allocation a moderate risk-averse investor
desires.
[0075] Second, the optimal insurance demand decreases with financial
wealth. This result can be intuitively explained through the substitution
effects between financial wealth and life insurance. In other words,
with a large amount of wealth in hand, one has less demand for insurance,
since the loss of human capital has a much lower impact on the well-being
of one's heirs. The optimal amount of life insurance decreases from
over $400,000, when the investor has little financial wealth, to
almost zero, when the investor has $1.5 million in financial assets.
[0076] Thus, for a typical investor whose human capital is less
risky compared to the stock market, the optimal asset allocation
is more conservative and the life insurance demand is smaller for
investors with more financial assets.
Example 5
Correlation Between Wage Growth Rate and Stock Returns
[0077] In this case, the impact of the correlation between the
shocks to wage income and the risky asset returns is examined. In
particular, we want to evaluate the life insurance and asset allocation
decision for investors with highly correlated income and human capital.
This can happen when the investor's income is closely linked to
his employer's company stock performance, or where the investor's
compensation is highly influenced by the financial market (e.g.,
the investor works in the financial industry).
[0078] Again, the investor's age is held at 45 and the risk preference
and the bequest motive at the moderate level. The optimal asset
allocations to the risk-free asset and the optimal insurance demands
for various financial wealth levels are presented in FIG. 9.
[0079] The optimal allocation becomes more conservative (i.e.,
more allocation to risk-free asset) as the income and stock market
return become more correlated. One way to look at this is that a
higher correlation between the human capital and the stock market
results in less diversification, thus a higher risk of the total
portfolio (human capital plus financial capital). To reduce this
risk, an investor will invest more financial wealth in the risk-free
asset. The optimal insurance demand decreases as the correlation
increases. Life insurance is purchased to protect human capital
for the family and loved ones. As the correlation between the risky
(stock) asset and the income flow increases, the ex ante value of
the human capital to the surviving family becomes lower. Therefore,
this lower human capital valuation induces a lower demand for insurance.
Also, less money spent on life insurance also indirectly increases
the amount of financial wealth the investor can invest. This also
allows the investor to invest more in risk-free assets to reduce
the risk associated with the total wealth.
[0080] In summary, the optimal asset allocation becomes more conservative
and the amount of life insurance becomes less, as wage income and
the stock market returns become more correlated. |