A Stochastic Model of Dynamic Consumption and Portfolio Decisions

This paper sets out a basic framework for solving a stochastic portfolio problem using dynamic programming (DP). Dynamic portfolio decisions are concerned with simultaneous decisions on savings and asset allocation whereby asset returns, such as on equity and bonds, are stochastic as in Campbell and Viceira (Strategic asset allocation, portfolio choice for long-term investors, 2002). In contrast to CV (2002) we do not use a local approximation method to solve the stochastic model but rather use a global solution procedure such as DP. Whereas CV (2002) solve their model by assuming a constant consumption-wealth ratio and equity premium, we can allow both to be time varying. Different variances of equity and bond returns are explored in their impact on saving and asset allocation decisions and on the value function. The stochastic dynamic portfolio decision method proposed here allows for online decisions as data on asset returns are available in real time. The method is set up in a way such that it also helps to make fund decisions online for various types of investment opportunities.

1. Further recent dynamic portfolio models can be found in Cochrane (2006), Di Giorgi and Mayer (2007), Gruene et al. (2007), Munk et al. (2004) and Wachter (2002).

2. This is only a notation for:

   $$\begin{aligned} X(t)=X(t_{0})+\int _{t_{0}}^{t}a(t,X(t))dt+\int _{t_{0}}^{t}b(t,X(t))dW(t) \end{aligned}$$

3. Because it is numerical easier to maximize on a finite set.

4. Referring to Gruene (2008), pp. 102–103, modified to model (17).

5. Compare Algorithm in step 2.

6. See Mueller (2009).

7. For calculation: see Eq. (17).

\({\mathbb {R}}\) :

Real number

x :

State

u :

Control

z :

Stochastic influence

\(\beta \) :

Discount factor

l :

Running costs in J

\(J_{\infty }\) :

Objective function of stochastic control problem on infinite time horizon

\(V_{\infty }\) :

Optimal value function on infinite time horizon

h :

Increment

\({\mathcal {W}}\) :

Function space

\(\widetilde{V}_{\infty }\) :

Numerical approximation of \(V_{\infty }\)

\(\pi \) :

Projection operator

T(W):

Operator for right side of principle of optimality

\(\Gamma \) :

Grid

X(t):

State at the time of t

u(t):

Control at the time of t

z(t):

Stochastic influence at the time of t

\({\mathcal {N}}(\mu ,\sigma ^{2})\) :

Normal distribution with expected value \(\mu \) and standard deviation \(\sigma \), denoted with \((\mu ,\sigma ^{2})\)-normal-distributed

\(\widetilde{W}(\cdot ,\omega )\) :

Approximated path of Wiener Process

The paper is inspired by joint work with Lars Gruene, University of Bayreuth, Germany. We would like to thank Lars Gruene for extensive communications

1. Maik Mueller

   Present address: University of Bayreuth, Bayreuth, Germany

Authors and Affiliations

1. New School University, New York, USA

   Willi Semmler

2. ZEW Mannheim and CEM Bielefeld University, Mannheim, Germany

   Willi Semmler

Corresponding author

Correspondence to Willi Semmler.

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