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Chp.4 Lifetime Portfolio Selection Under Uncertainty

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Chp.4 Lifetime Portfolio Selection Under Uncertainty

Hai Lin

Department of Finance, Xiamen University,361005

- 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.

- 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.

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

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

- :the proportion invested in the risky asset;
- :the proportion invested in the sure asset.
- : the return on risky asset.

- Define
- Then the objective function can be written:

- If ,then by the Mean Value Theorem and Taylor Rule,

- Take the conditional expectation on both sides and use the previous results, divide the equation by h and take the limit as

- Define

- If is concave in W,

- The maximum problem can be rewritten as:

- 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.

- 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

- Suppose

- be real (feasibility);
- To ensure the above conditions,

- 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.

- Then the instantaneous marginal propensity to wealth is an increasing function of time.

- Define

- Remember that
- Then

- 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

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

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

- Then,
- First order conditions are:

- 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

- 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.

- Summary: in the case of infinite time horizon, the partial differential equation is reduced to an ordinary differential equation.

- 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.

- 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

- 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.

- Consider the case
- Remark: the substitution effect is minus and the income effect is plus.

- 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.

- Consider

- The elasticity of consumption to the mean is
- The elasticity of consumption to the variance is

- When

- 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.

- 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).

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

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

- After some mathematics, the optimal system can be written by

- Take a trial solution:
- Then, we can get:

- 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.

- 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.