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### Chapter 6

### The End

Further Developments in the Theory of Optimal Consumption and Portfolio Selection

Rui Zhang

zhangruimail@etang.com

Outline:

- The limitation of dynamic programming technique
- Introduction of Cox-Huang methodology
- The relationship between above two
- Optimal portfolio rules when nonnegativity constraint on consumption is binding
- Generalized preferences and their impact on optimal portfolio demands

Assumption of of dynamic programming technique

- a single consumption good
- investor preferences that are additive and independent with no intertemporal complementarity of consumption

Limitation of dynamic programming technique

- Even if an optimal solution exist,dynamic programming can only be used to find it if J is continuously differentiable.
- As a nonlinear partial differential equation,the Bellman equation is difficult to solve either in closed form of by numerical methods
- There are no easily applied general conditions that ensure existence of a solution

Breakthrough of Cox-Huang approach

- Do not require differentiability of the derived-utility function
- The optimal consumption and portfolio policies can be determined by solving one algebraic transcendental equation and a linear partial differential equation of the classic parabolic type
- It is a major computational breakthrough because there is substantial literature on numerical methods for solving this equation when no closed-form solution can be found

The growth-optimum portfolio strategy

W(t)—value of a portfolio that reinvests all earnings

W(T)/W(t)—cumulative total return

ACCR(t,T)—average continuously compounded return

Consider a portfolio policy chosen so as to maximize

,We get the “growth-optimal” portfolio

Optimal portfolio fractions with logarithmic preferences

Note: depend only on the current values of the investment opportunity set ,independently of whether or not these values will change in the future. are also independent of both the level of the portfolio’s value and the planning horizon T

Let X(t) denote the value of the growth-optimum portfolio, then

where

Note: X(t) is jointly Markov in X and p.And the growth-optimum portfolio is instantaneously mean-variance efficient.

If the elements of are constant over time, then, the risky-asset returns are jointly log-normally distributed, and X(t) is, by itself, a Markov process.Here, (6.2) can be explicitly integrated so that, for and for all t

- An alternative expression with constant

Conclusion

Because X(t) is a mean-variance efficient portfolio, it follows from corollary 5.2 that , when the investment opportunity set is constant, all investors’ optimal portfolios can be generated by a simple combination of the growth-optimum portfolio and the riskless asset.

Alternative expression

objection

Constraints

, for all

,

Solution

The Cox-Huang solution of the intertemporal consumption-investment problem- The lifetime consumption and portfolio-selection problem

Theorem 6.1

Under quite mild regularity conditions,there exists a solution to (6.7) if and only if

- there exists a solution to (6.8)
- for and

Because the joint probability distribution for X(t) and P(t) is not affected by the investor’s choices for and , (6.8) has the structure of a static optimization problem which can be solved by Lagrange-Kuhn-Tucker methods

If the investor’s preferences are such that he becomes satiated at some finite level of consumption, ,then .However,for , and

- If there exists a finite number such that ,then and

for

If, for any t, ,where

then the investor will be satiated in wealth.

- For the balance of analysis, we assume that the investor’s initial wealth is such that he is not satiated. It assures us that strict equality applies in constraint (6.8a) for the optimal program and is strictly positive

If, for any t,there exists a number such that ,then and

- If there exists a number such that , then and
- Therefore, these marginal utility conditions applying for all are sufficient to ensure that the unconstrained solution to (6.8) and (6.8a) is a feasible solution

To complete the solution, only is to be determined, and under the assumption of no satiation, is the solution to the transcendental algebraic equation given by

Substituting for ,we have a complete solution for the time path of optimal consumption and the bequest of wealth.

Cox-Huang vs. Dynamic programming

- Cox-Huang requires only the solution of a single transcendental algebraic equation to derive a complete description of the optimal intertemporal consumption bequest allocation.
- Using Cox-Huang,we cannot determine the dynamic portfolio strategy. However, given G and H,we can derive the optimal portfolio strategy

Define

suppose the investor reexamines his decisions at some time, ,then his optimal problem will be to

constraint

Solution

- W(t) must be such that ,

and are feasible choices

- G and H are the optimal rules that the investor would select as the solutions to (6.18)

If F is twice continuously differentiable, using Ito’s lemma ,we can derive the dynamics of the investor’s optimally allocated wealth.

note: is the instantaneous expected rate of growth of the investor’s optimally allocated wealth

Theorem 6.2

If there exists an optimal solution to (6.8),

for , then the optimal portfolio strategy that achieves this allocation is given by

with the balance of the investor’s wealth in the riskless asset.

Proof

- As theorem 6.1 indicates, the portfolio strategies required to implement the common allocation in (6.7) and (6.8) are identical. Using dynamics technique

Because ,and therefore we can get

substituting

Summarization of Cox-Huang technique

- determine the joint probability density function for X(t) and P(t) which involves at most the solution of a linear partial differential equation;
- determine the optimal intertemporal consumption-bequest allocations which requires solution of the transcendental algebraic equation;
- determine F[X(t),P(t),t] which requires mere quadrature;
- determine the optimal portfolio strategy for each t from the formula.

Theorem 6.3

If F is twice continuously differentiable, then F is a solution to the linear partial differential equation

subject to the boundary condition that

While

Substituting and arranging, we have that

Relationship between Cox-Huang and Dynamics method

- The Cox-Huang methodology is especially well suited for solving problems in which the nonnegativity constraints on consumption and wealth are binding
- Although the Cox-Huang methodology does not dominate the dynamic programming approach , it is a powerful new tool

Optimal portfolio rules with nonnegativity constraint on consumption

- We use dynamic programming to analyze the portfolio behavior of long-lived investors for whom the nonnegativity constraint on consumption is binding.
- However, we apply the Cox-Huang technique to determine the optimal consumption and portfolio rules for those HARA preference functions for which the unconstrained solutions are invalid.

Assumption:

For simplification, we keep the assumption of an infinite-time-horizon investor with exponential time preference

Conditions

- and for
- is sufficiently large to satisfy the transversality condition
- for all and some fixed number M

(a) and (b) ensure the existence of an optimal policy. (c) implies that, for each t, there exists a level of wealth such that for

Equation of optimality

- constraint:
- boundary conditions:

We can divide the problem into two linked but unconstrained optimization problems

- for
- for
- Because J should be twice continuously differentiable for Thus, and must satisfy

Equation′ of optimality

- constraint:

Deriving the first-order condition

and substituting in equation′ to arrive at

Solution

where

Both solutions satisfy and . However, because is a positive, strictly increasing and concave function, and only

is consistent with , hence, the optimal solution J for is given by

In the similar analysis, we can derive

Theorem 6.4

If asset returns are jointly log-normally distributed and if an investor’s preferences satisfy exponential preference, then for every t such that , the optimal portfolio strategy is a constant-proportion levered combination of the growth-optimum portfolio and the riskless security, with fraction

allocated to the growth-optimum portfolio and fraction in the riskless security

Under such hypothesized conditions, we have that, for

By inspection, when the nonnegativity constraint on consumption is binding, depend only on the investment opportunity set and the investor’s rate of time preference ,while not on the functional form of V(C), the investor’s current wealth, explicit time, or

Corollary 1

At each t, all investors with the same constant rate

of time preference, and for whom ,will

- Hold the identical optimal portfolio fractions
- Choose their portfolios “as if” they had in common the isoelastic utility function and for any T > t

Proof

- Because depends only on the investment opportunity set and . From theorem 6.4, if ,then . Thus, all investors with the same , and for whom ,will choose at time t the same portfolio allocation.
- As noted before, an investor with preferences such that

and , , for some , will hold

optimal portfolio fraction in the risky asset,

independently of W(t), t, or T. Here, and

.Hence, for , such an investor’s optimal portfolio allocation is

Corollary 2

If an investor’s optimal consumption satisfies for , then the investor’s wealth at time s, , can be expressed as

where

Proof

From theorem6.4, ,for all

Hence, the dynamics of the investor’s wealth

can be written as

Using It ’s lemma,we can integrate this

stochastic differential equation to obtain the

equation

Corollary 3

If the investor’s initial wealth W(0) is positive, and if X(0) > 0, then, for all

- W(t) = 0 only if X(t) = 0
- Prob{W(t) > 0|W(0)>0} = 1

Proof

- By hypothesis, , and . For all t such that , . Because the investor’s optimally invested wealth has a continuous sample path, if for some , then there exists a time such that and where e . Hence, the hypothesized conditions of corollary 2 are satisfied. By inspection of the formula for W(t) , only if
- From (6.3), . Hence,from (a), (b) proves

Implication

- In the special case of infinite-lived investors and a constant investment opportunity set,the probability of bankruptcy is zero. So, the nonnegativity constraint on wealth does’t affects
- However, for finite-lived investors facing relatively general stochastic investment opportunity sets, it doesn’t obtain.

Define the random variable time interval

conditional on , by

then at time t, the indirect utility function can be expressed as

clock

Imagine a clock that keeps time according to the time

scale , where

- Initially ,
- As time passes, the clock moves more slowly than ordinary-clock time, and the larger is , the slower is the clock
- For ,

So, we can express the first-passage time interval until optimal consumption is positive in terms of -clock time as

The first-order condition

- The second-order condition

for

By inspection, (c) is identical with equation′ with .

Hence, is given by (6.38) with .Because

, we have that and .

Note that for ,

Implication

- When the nonnegativity constraint on consumption is binding, is identical with the strategy that minimizes the expected time until optimal consumption becomes positive,with time measured with a clock.
- Investors with the same rate of time preference have the same clock
- All such investors follow the same portfolio strategy

Proof ＊( -clock)

- Suppose each investor uses a clock with his own rate of preference to keep track of time
- a Wiener process measured in -clock and ordinary-clock times satisfies the relations

, the optimal portfolio strategy formulated in -clock time can be expressed as the solution to

the greater degree of concavity induced by this truncation of the distribution for reflects an urgency to get wealth up to the

level before it does not matter

boundary condition

because the nonnegativity constraints on consumption and wealth do not affect, it can be solved as an unconstrained optimization problem.

To ensure that C=0 falls within the domain and that V’(0) is finite, we consider only those HARA functions with

- Define

and suppose

unconstrained optimal consumption and portfolio policies for

- note: for and for

optimal consumption and portfolio policies with nonnegativity constraints

- implication: does not either depend on P(t) or explilcit time. And is the critical value such that for and for

Derive the relationship between &

- From section 6.2, the investor’s optimally invested wealth will satisfy ,where F satisfies

boundary conditions

only if

for

As , , but as the optimal consumption policy approaches from the unconstrained optimal policy. Therefore, with

Define , noting that G depends only on y, we have that ,independent of t. By substitution, we can rewrite (6.54) as an ordinary differential equation by

- boundary conditions
- general homogeneous solution

Where

- note: A and B are arbitrary constants

Define

Then , and ,

and it follows that

- In the region , ,and for the appropriate selection of A and B. Because

which implies , replace A by , we have that

- Note: for existence of a nontrivial optimal policy, the parameters , , and must satisfy (6.52a), it follows that , so

In the region in which ,we have that , for ,the solution for f is

Conclusion

- where

Generalized preferences and their impact

- In the preceding analyses, we assume that utility depends only on age and the time path of consumption and wealth, and the marginal utility of consumption at time t is uncorrelated with that at any other time
- In this section, we explore the utility which is additive in time but depends on other variables in addition to current consumption,wealth, and age,which is more general

Assumption

- To isolate the impact of generalized preferences on behavior, we assume throughout the analysis here that , and are constants over time, i, j=1,…,m
- Let denote a vector of generic state variables at time t

Objection

with the dynamics for S(t) written as the vector It process

Define

along the dynamic programming methods, we have that the optimal consumption-investment program can be written as

The first-order condition

in the 2nd derivation, the linearity characteristic of the optimal portfolio demands still obtains. Hence, these demands can be solved explicitly by matrix inversion

Denotation:

(note:A and are alterable,while and are the same for all investors)

- If preferences are not state dependent, then

,and we can get .Hence, to analyze the impact of state-dependent preferences on optimal portfolio demands,we examine the “differential-demand” function

Define a systematic state variable . for some j,

- idiosyncratic state variable can affect the investor’s risk aversion and consumption behavior, while only systematic state variables differentially affect the structure of his portfolio demands.

Further intuition

- Suppose for each state variable, there exists a security whose instantaneous return is perfectly correlated with the unanticipated change in the state variable, then

Implication

- Relative to an investor with state-independent preferences but the same current level of absolute risk aversion, the investor with state-dependent preferences will hold more of asset I if and less if
- Such strategy is not only to attain a preferred risk-return tradeoff in wealth, but also to “hedge” against unanticipated and unfavorable changes in the other state variables that enter into their preferences

Note :

- Optimal portfolio demand structures like (6.68)-(6.71) are also induced by a stochastic investment-opportunity set, even when investors’ direct utility functions are not state dependent.
- Two-fund separation theorem fails here, but later, a generalized (m+2)-fund theorem will be proved.

Specific applications

- Suppose M different consumption goods, and that the dynamics of the prices can be expressed by

where F and G are suitably restricted to ensure that

- Then investor’s lifetime preference function is

Breeden(1979) and Fischer(1975) show,the solution can be decomposed into two parts.

- At each t, solve for the utility-maximizing consumption of individual goods

subject to

the first-order condition

Define , thus U is the standard indirect utility function, and (6.72) can be written as

- Note1: there exists the sensible possibility that utility of bequests will depend on both the amount bequeathed and the prices of consumption goods.
- Note2: as long as we permit state-dependent preferences, intertemporal consumption-investment behavior can be analyzed as if there is a single consumption good

Money as consumption good

- Suppose that the utility derived from money depends only on the total consumption expenditure at each point in time
- The optimal individual demand for money is modified into
- The nominal accumulation equation for wealth

As (6.68) and (6.69) still apply, we have that

- Conclusion: the inclusion of money for commodity transactions in this way does not change the basic structure of optimal portfolio demands

Grossman and Laroque(1990) develop a different model which still uses a single good in addition that the good is durable and illiquid

- It assumes that the level of consumption services can be changed only by selling the existing durable and purchasing a new one
- The optimal behavior is to maintain a constant level of consumption until wealth either rises or falls to optimally determined thresholds. If either threshold is attained, the investor changes his level of consumption services to a new target value
- Note: in this model, transactions costs in consumption do not affect the basic structure of the portfolio demand functions in the continuous-time model

Intertemporal complementarity of consumption

- Let S(t) denote an exponentially weighted average of past consumption defined by
- The investor’s lifetime program is written as

Instantaneous utility at time t depends on both current consumption and the time path of past consumption, where the more recent past consumption is given greater weight than the more distant past.

- Note: (6.80) and (6.65) are different in two ways: the instantaneous change in the state variable is nonstochastic and the dynamics of the state variable are controllable by the investor.

The optimal consumption-investment program is

the first-order conditions

Implication

- The introduction of nonzero intertemporal complementarity of consumption leads to failure of the envelope condition that the marginal utility of current consumption equals the marginal utility of current wealth
- Once more, the structure of the optimal portfolio demand functions is unaffected by this generalization of preferences

Mayer(1970) posits a multiplicative separable form for lifetime preferences. The Meyer-type lifetime. The Meyer-type lifetime utility function can be written as

note: for , it reduces to the standard additive preference orderings

- The Bellman equation for optimal decisions with preferences given above

The Bellman principle of optimality:

for a program to be optimal between t and T, it must be optimal between t+h and T whatever rules are followed between t and t+h. So we can rewrite the Bellman equation as

If the optimal consumption is right-continuous, then by the Mean Value Theorem there exists a , , such that . By Taylor’s theorem

substituting into (6.86) and making some transformation, we can get

The first-order conditions

- The intertemporal complementarity of consumption for

causes the envelope condition to fail so that the marginal utility of current consumption is not equal to the marginal utility of current wealth

- Investor’s instantaneous absolute risk aversion function

For the same instantaneous absolute risk aversion of consumption and the same marginal propensity to consume, the multiplicative utility investor will have a larger absolute risk aversion for investments than his additive counterpart

- For the same instantaneous absolute risk aversion of both consumption and wealth, the multiplicative utility investor will have smaller marginal propensity to consume
- The structure of optimal portfolio, particularly the relative holdings of risky assets are the same whether state-independent preferences are additive or multiplicative

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