Chapter 6

1. Chapter 6 Further Developments in the Theory of Optimal Consumption and Portfolio Selection Rui Zhang zhangruimail@etang.com

2. 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

3. Assumption of of dynamic programming technique • a single consumption good • investor preferences that are additive and independent with no intertemporal complementarity of consumption

4. 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

5. 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

6. 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

7. 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

8. 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.

9. 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

10. 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.

11. Dynamic programming approach Alternative expression objection Constraints , for all , Solution The Cox-Huang solution of the intertemporal consumption-investment problem • The lifetime consumption and portfolio-selection problem

12. 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

13. 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

14. Deriving the first-order conditions

15. 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

16. 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

17. 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

18. Solution

19. 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

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

21. 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

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

23. 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)

24. 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

25. 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.

26. 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

27. Because ,and therefore we can get substituting

28. Because is not perfectly correlated with , ,this condition can only be satisfied if k=1,…,m

29. 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.

30. 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

31. Proof • In theorem 6.2 • In theorem 6.1 Substituting ,we have that

32. While Substituting and arranging, we have that

33. 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

34. 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.

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

36. 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

37. Equation of optimality • constraint: • boundary conditions:

38. 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

39. Equation′ of optimality • constraint:

40. Deriving the first-order condition and substituting in equation′ to arrive at

41. Solution where

42. 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

43. 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

44. 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

45. 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

46. 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

47. Corollary 2 If an investor’s optimal consumption satisfies for , then the investor’s wealth at time s, , can be expressed as where

48. 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

49. 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

50. 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