On a neglected aspect of portfolio choice: the role of the invested capital

Abstract

In return-based portfolio choice models à la Markowitz, the amount of capital invested does not play a role, while in expected utility models it does. The aim of this paper is to bring out the connection between the amount of capital invested and the portfolio formation. In a general one-period framework, which also comprises illiquid assets held by the investor and the period income, we present results in finding the optimal portfolio weights and study the influence of the invested capital, the illiquid assets and the income on the portfolio formation. Special emphasis is put on HARA utility functions and the dependence of the amount invested risklessly on the invested capital. As a side result, we find that Markowitz portfolio choice is not fully compatible with expected utility reasoning.

Appendix

Proof of Theorem 1

We assume that the optimal weighting vectors are always identical and show that we get a contradiction.

For this purpose, we tie together some well-known facts from decision theory. First, we consider situations in which the additive term in (1), (v ₀ − v)(1 + R ₀) + I is non-stochastic. We make use of the fact (Pfanzagl 1959; Schneeweiß 1967, p. 87) that an additive constant is irrelevant for preferences and indifferences between random variables if and only if CARA prevails.

Now if we restrict to CARA utility functions we have

$$ \hbox{CE}_\alpha\left(vR(x)\right) = v \hbox{CE}_{v\alpha}\left(R(x)\right), $$

(16)

where CE_α symbolizes the certainty equivalent under absolute risk aversion (ARA) α > 0, i.e.,

$$ \hbox{CE}_\alpha\left(Z\right) = -\frac{1}{\alpha} \ln\hbox{E}({\exp(-\alpha Z)}). $$

(17)

The equation (16) is valid for arbitrary v > 0 and random variables R(x) without restriction. Note that different ARA values are involved on both sides of (16). In terms of certainty equivalents, our initial assumption reads as follows

$$ \hbox{argmax}_{x}\hbox{CE}_\alpha\left(R(x)\right) = \hbox{argmax}_{x}v\hbox{CE}_{\alpha v}\left(R(x)\right)=\hbox{argmax}_{x}\hbox{CE}_{\alpha v}\left(R(x)\right) , $$

(18)

no matter which amount of invested v will be chosen. The left hand side of (18) does not depend on v while the right hand side does so heavily. To result in the desired contradiction, it is sufficient to consider very low and very high values of v.

In the limiting case $v\rightarrow 0,$ the ARA αv also tends to zero and the behavior approaches risk neutrality. Therefore, CE approaches

$$ \hbox{E}({R(x)})=1+r_f+\sum_{i=1}^n x_i\left[\hbox{E}({R_i})-r_f\right] . $$

The optimal portfolio weights thus approach the values $+\infty$ (for the asset with maximum expected return) and $-\infty$ (for all the other assets).

In the limiting case $v\rightarrow \infty$ we have an ever increasing risk aversion. The certainty equivalent tends to the worst possible outcome of R(x). The optimizer is defined by the maxmin criterion (see, for instance, Bamberg and Spremann 1981). It is easy to see that in our case the optimizer is x ^* = 0, leading to the certainty equivalent 1 + r _f .

With these extreme cases, the contradiction is established and the claim is proven.$\square$

1.1 Proof of Theorem 2

Consider the expected utility of V ₁:

$$ \begin{aligned} &\hbox{E}({u(V_1)})=-\hbox{E}\left({e^{-\alpha\left[(v_0-v)(1+R_0)+I+v\left(1+r_f+\sum\limits_{i=1}^n x_i(R_i-r_f)\right)\right]}}\right) = -e^{-\alpha v(1+r_f)}\hbox{E}({e^{-\alpha(v_0-v)(1+R_0)}})\hbox{E}\left({e^{-\alpha I}}\right)\hbox{E}\left({e^{-\alpha v\sum\limits_{i=1}^n x_i(R_i-r_f)}}\right). \end{aligned} $$

The first three factors are greater than zero and not dependent on x and, thus, constants in the optimization problem. Therefore, the optimal weights are not dependent on R ₀ and I. $\square$