1. Chp.4 Lifetime Portfolio Selection Under Uncertainty
Hai Lin
Department of Finance, Xiamen University,361005

2. 1.Introduction
Examine the combined problem of optimal portfolio selection and consumption rules for individual in a continuous time model.
The rates of return are generated by Wiener Brownian-motion process.
Particular case:
Two asset model with constant relative risk aversion or isoelastic marginal utility.
Constant absolute risk aversion.

3. 2.Dynamics of the Model: The Budget Equation
W(t): the total wealth at time t;
Xi(t): the price of ith asset at time t, i=1,2,…,m;
C(t): the consumption per unit time at time t;
wi(t):the proportion of total wealth invested in the ith asset at time t, i=1,2,…,m.

4. The budget equation
At time t0, the investment between t0 and t(t0+h) is :
The value of this investment at time t is:

5. The process of g(t)
Suppose g(t) is the geometric Brownian motion. In discrete time,
:the expected return of asset i;
: the volatility of asset i;

6. Momentum

7. Continuous time

8. 3. The two asset model :the proportion invested in the risky asset;
:the proportion invested in the sure asset.
: the return on risky asset.

9. Two asset model(2)

10. The objective problem

11. The dynamic programming form
Define
Then the objective function can be written:

12. The dynamic programming(2)
If ,then by the Mean Value Theorem and Taylor Rule,

13. The dynamic programming(3)
Take the conditional expectation on both sides and use the previous results, divide the equation by h and take the limit as

14. The solution
Define

15. First order condition

16. Second order condition
If is concave in W,

17. Summary
The maximum problem can be rewritten as:

18. 4.A special case: constant relative risk aversion
The above mentioned nonlinear partial equation coupled with two algebraic equations is difficult to solve in general.
But for the utility function with constant relative risk aversion, the equations can be solved explicitly.

19. Optimality conditions

20. Optimality conditions(2)

21. Bequest value function
The boundary condition can cause major changes in the solution.
means no bequest.
A slightly more general form which can be used as without altering the resulting solution substantively is

22. The trial solution
Suppose

23. The trial solution(2)

24. Sufficient condition for the solution
be real (feasibility);
To ensure the above conditions,

25. The optimal consumption and portfolio selection rules

26. The Bequest valuation function
The economic motive is that the true function for no bequest
Then when
This does not mean the infinite rate of consumption, but because the wealth is driven to 0.

27. Dynamic properties of consumption
Then the instantaneous marginal propensity to wealth is an increasing function of time.

28. Dynamic properties of consumption
Define

29. Dynamic behavior of wealth
Remember that
Then

30. Dynamic behavior of consumption(2)
This implies that, for all finite-horizon optimal paths, the expected rate of growth of wealth is diminishing function of time.
: the investor save more than expected return.
: the investor consume more than expected return.
Then, if

31. 6. Infinite time horizon Consider the infinite time horizon case,
Suppose
It is independent of time, can be rewritten as J(W).
Remark: conditional expectation or unconditional expectation?

32. The ordinary differential equation
Then the partial differential equation can be changed into a ordinary differential equation by J(W).

33. The ordinary equation(2)
Then,
First order conditions are:

34. The additional conditions
Similar to case of finite time horizon, to ensure the solution to be maximum,
The boundary condition is satisfied.
Using ito theorem, we can get

35. remark
Note that:
The second item on the right side is very similar to a return or yield.
Then it is a generalization of the usual consumption required in deterministic optimal consumption growth models when the production function is linear.

36. The consumption and portfolio selection under infinite time horizon
Summary: in the case of infinite time horizon, the partial differential equation is reduced to an ordinary differential equation.

37. 7. Economic interpretation
Samuelson(1969) proved by discrete time series, for isoelastic marginal utility, the portfolio-selection decision is independent of the consumption decision.
For special case of Bernoulli logarithmic utility, the consumption is independent of financial parameters and is only dependent upon level of wealth.
Two assumption:
Constant relative risk aversion which implies that one’s attitude toward financial risk is independent of one’s wealth level
The stochastic process which generate the price changes.
Under the two assumptions, the only feedbacks of the system, the price change and resulting level of wealth have zero relevance for the optimal portfolio decision and is hence constant.

38. The relative risk aversion
The optimal proportion in risky asset can be rewritten in terms of relative risk aversion,
Then the mean and variance of optimal composite portfolio are

39. Phelps-Ramsey problem
Then after determining the optimal proportion, we can think of the original problem as a simple Phelps-Ramsey problem which we seek an optimal consumption rule given that the income is generated by the uncertain yield of an asset.

40. Comparative analysis

41. Comparative analysis(2)
Consider the case
Remark: the substitution effect is minus and the income effect is plus.

42. Comparative analysis(3)
One can see that,
The individuals with low risk aversion,
The substitution effect dominates the income effect and the investor chooses to invest more.
For high risk aversion,
The income effect dominates the substitution effect.
For log utility, the income effect and substitution effect offset each other.

43. The other case
Consider

44. Elasticity analysis The elasticity of consumption to the mean is
The elasticity of consumption to the variance is

45. Elasticity analysis(2)
When

46. Some cases
For relatively high variance, high risk averter will be more sensitive to the variance change than to the mean.
For relatively low variance, low risk averter will be sensitive to the mean.
The sensitivity is depending on the size of k since the investors are all risk averters. For large k, risk is the dominant factor, the risk has more effect. If k is small, it is not the dominant factor, the yield has more effect.

47. 8.Extension to many assets
The two asset model can be extended to m asset model without any difficulty. Assume the mth asset to be certain asset, and the proportion in ith asset is wi(t).

48. Solution
Under the infinite time horizon, the ordinary differential equation becomes
The optimal decision rules are:

49. 9.Constant absolute risk aversion
The other special case of utility function which can be solved explicitly is the constant absolute risk aversion.

50. The optimal problem
After some mathematics, the optimal system can be written by

51. solution
Take a trial solution:
Then, we can get:

52. Implications
The differences between constant relative risk aversion and constant absolute risk aversion are:
The consumption is no longer a constant proportion of wealth although it is still linear in wealth.
The proportion invested in the risky asset is no longer constant, although the total dollar value invested in risky asset is constant.
As a person becomes wealthier, the proportion invested in risky falls. If the wealth becomes very large, the investor will invest all his wealth in certain asset.

53. 10. Other extensions The model can be extended to the other cases.
Simple Wiener model can be generalized to multi Wiener model.
A more general production function, Mirrless(1965).
Requirements:
The stochastic process must be Markovian;
The first two moments of distribution must be proportional to delta t and higher moments on o(delt).
Remark: although this model can be generalized in large amount, the computational solution is quite difficult since it involves a partial differential equation.