The Multi-Period Investment Model

Grauer & Hakansson's [1986] basic discrete-time MPI model is a pure reinvestment model that ignores consumption, and describes an investor with a power utility function. An investor's utility of wealth w, with n periods to the horizon, is defined as:

where γ is a measure of the investor's relative risk-aversion.

If returns are assumed to be independent and non-stationary from period to period, then (3.1) holds for a very broad class of terminal utility functions (Hakansson [1974]). Convergence implies that use of the following stationary, myopic, decision rule is optimal for each period:

max E [Un (γ, ω)]

The advantages of this specification are twofold. First, it is tractable, providing an explicit function to maximise. Secondly, γ can be used to describe the degree of an investor's risk-aversion.

Let µit be the anticipated total return on asset category i in period t, and ∑ijt be the variance-covariance between the returns on asset categories i and j in period t.

Assuming that returns are normally distributed and that there are no transaction costs, the objective function15 then becomes:

Constraint (3.2) above rules out short sales; constraint (3.3) is the budget constraint.

Thus by varying γ, the risk attitude of an investor can be altered. As γ approaches 1, the investor becomes increasingly risk neutral, with the variance of the portfolio becoming less important as the investor aims to maximise his expected returns. Alternatively, as γ tends to minus infinity, the emphasis placed on expected returns diminishes, with the investor increasingly concerned with the variance of his portfolio. At the limit he would choose the minimum-variance portfolio.

Annual revision strategies are examined, with an investor choosing amongst the four basic assets. Table 3.2 below shows the means and standard deviations of the realised returns of the asset categories. The model used ten years of historical data to forecast portfolio weights for the next period, with investment first taking place in 1977. The performance of portfolios with and without real estate, held by investors with varying attitudes toward risk, was examined, with consideration being given to the effect of `unsmoothing' real estate returns upon its inclusion within a multi-asset portfolio. Due to the results obtained from the above study, Grauer & Hakansson's [1986] model was then extended, to investigate the effect of placing a constraint on the amount invested in real estate.

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15Note that a concave function has to be maximised, subject to a set of constraints: a Kuhn & Tucker [1951] optimisation problem. A gradient ascent algorithm which is often termed the Multiplier Method (Uzawa algorithm), was applied to the dual functional with augmented Lagrangian; written for Mathematica by Jean-Christophe Culioli and available via anonymous ftp from Mathsource at ftp.wri.com.