How suboptimal are linear sharing rules?

Abstract

The objective of this paper is to analyze criteria for portfolio choice when two investors are forced to invest in a common portfolio and share the proceeds by a linear sharing rule. A similar situation with many investors is typical for defined contribution pension schemes. The restriction implies two sources of suboptimal investment decisions as seen from each of the two investors individually. One is the suboptimal choice of portfolio, the other is the forced linear sharing rule. We measure the combined consequence for each investor by their respective loss in wealth equivalent. We show that significant losses can arise when investors are diverse in their risk attitude. We also show that an investor with a low degree of risk aversion, like the logarithmic or the square root investor, often applied in portfolio choice models, can either inflict or be subject to severe losses when being forced to participate in such a common investment pool.

Appendices

Appendix 1: Proof of Theorem 3

For any given constant proportional portfolio policy with portfolio weight $\pi$ allocated to the risky asset, the terminal wealth is, cf. Eq. (5):

$$\begin{aligned} W_T=W_0e^{\left( r+\pi \eta \sigma -\frac{1}{2}\pi ^2\sigma ^2\right) T+\pi \sigma Z_T}. \end{aligned}$$

(24)

Routine calculations with expected values of log-normal distributions lead to:

$$\begin{aligned} W_T^{1-\gamma }&=W_0^{1-\gamma }e^{(1-\gamma )\left( r+\pi \eta \sigma -\frac{1}{2}\pi ^2\sigma ^2\right) T+(1-\gamma )\pi \sigma Z_T}\end{aligned}$$

(25)

$$\begin{aligned} \mathbb {E}\left[ W_T^{1-\gamma }\right]&=W_0^{1-\gamma } e^{(1-\gamma )\left( r+\pi \eta \sigma -\frac{1}{2}\pi ^2\sigma ^2+\frac{1}{2}(1-\gamma )\pi ^2\sigma ^2 \right) T} \end{aligned}$$

(26)

$$\begin{aligned}&=W_0^{1-\gamma } e^{(1-\gamma )\left( r+\pi \eta \sigma -\frac{\gamma }{2}\pi ^2\sigma ^2\right) T}. \end{aligned}$$

(27)

The optimal portfolio weight $\pi$ for any given value of $\gamma$ is found by maximizing the expression $\pi \eta \sigma -\frac{\gamma }{2}\pi ^2\sigma ^2$, which leads to the classical result that $\pi =\frac{\eta }{\gamma \sigma }$.

If a suboptimal portfolio weight is imposed, arising from an “erroneous” choice of $\bar{\gamma }$ instead of the “true” $\gamma$, we can insert $\bar{\pi }=\frac{\eta }{\bar{\gamma }\sigma }$ in Eq. (27) and get the following closed form expression for the expected utility:

$$\begin{aligned} \mathbb {E}\left[ W_T^{1-\gamma }\right] =W_0^{1-\gamma } e^{(1-\gamma )\left( r+\frac{\eta ^2}{ \bar{\gamma }}-\frac{\gamma }{2}\frac{\eta ^2}{\bar{\gamma }^2}\right) T}. \end{aligned}$$

(28)

Compare this with an optimally invested inital wealth $\widehat{W}_0$:

$$\begin{aligned} \mathbb {E}\left[ \widehat{W}_T^{1-\gamma }\right]&=\widehat{W}_0^{1-\gamma } e^{(1-\gamma )\left( r+\frac{\eta ^2}{\gamma } -\frac{\gamma }{2}\frac{\eta ^2}{\gamma ^2}\right) T}. \end{aligned}$$

(29)

The wealth equivalent is the magnitude of $\widehat{W}_0$ that equates the expected utility in Eq. (28) with the expected utility in Eq. (29). That is:

$$\begin{aligned} \frac{\widehat{W}_0}{W_0}&=e^{\left( \frac{\eta ^2}{ \bar{\gamma }} -\frac{\gamma }{2}\frac{\eta ^2}{\bar{\gamma }^2}\right) T- \left( \frac{\eta ^2}{\gamma } -\frac{\gamma }{2}\frac{\eta ^2}{\gamma ^2}\right) T} =e^{\frac{-\eta ^2 T}{2 \gamma } \left( \frac{-2\gamma }{\bar{\gamma }}+\frac{\gamma ^2}{\bar{\gamma }^2}+1 \right) }= e^{-\frac{1}{2 \gamma }\left( 1-\frac{\gamma }{\bar{\gamma }}\right) ^2\eta ^2 T}. \end{aligned}$$

(30)

Appendix 2: The derivations behind Fig. 7

The Pareto optimal allocation is clearly affected by the choice of $\mu$ as shown in Fig. 3. The technical derivations behind Fig. 7 are given here. The starting point is the first order condition (18), which is subjected to a series of differentiations. We denote the conglomerate utility function of the “representative investor” by $\mathcal{U}$.

$$\begin{aligned} \lambda M_T&=\mu \xi ^{1-\gamma _1}{\mathcal{W}}^{-\gamma _1}+(1-\mu )(1-\xi )^{1-\gamma _2}{\mathcal{W}}^{-\gamma _2}~\left( =\mathcal{U}'({\mathcal{W}})\right) \end{aligned}$$

(31)

$$\begin{aligned} \frac{\partial \lambda }{\partial \mu }M_T&= \left[ \mu \xi ^{1-\gamma _1}(-\gamma _1){\mathcal{W}}^{-\gamma _1-1}+ (1-\mu ) (1-\xi )^{1-\gamma _2}(-\gamma _2){\mathcal{W}}^{-\gamma _2-1}\right] \frac{\partial {\mathcal{W}}}{\partial \mu }\nonumber \\&\quad +\xi ^{1-\gamma _1}{\mathcal{W}}^{-\gamma _1}-(1-\xi )^{1-\gamma _2}{\mathcal{W}}^{-\gamma _2}\quad \ \left( =\frac{\partial \mathcal{U}'({\mathcal{W}})}{\partial \mu }\right) \end{aligned}$$

(32)

$$\begin{aligned} \lambda \frac{\partial M_T}{\partial {\mathcal{W}}}&= \left[ \mu \xi ^{1-\gamma _1}(-\gamma _1){\mathcal{W}}^{-\gamma _1-1}+ (1-\mu ) (1-\xi )^{1-\gamma _2}(-\gamma _2){\mathcal{W}}^{-\gamma _2-1}\right] \nonumber \\&\quad ~\left( =\mathcal{U}''({\mathcal{W}})\right) . \end{aligned}$$

(33)

Upon dividing (33) by (31) we arrive at the following expression for the relative risk aversion of the “representative investor”

$$\begin{aligned} -\frac{\partial M_T}{\partial {\mathcal{W}}}\frac{{\mathcal{W}}}{M_T}= -\frac{\mathcal{U}''({\mathcal{W}}){\mathcal{W}}}{\mathcal{U}'({\mathcal{W}})} =\varPhi _1 \gamma _1+\varPhi _2 \gamma _2~(\equiv \mathcal{R}({\mathcal{W}})), \end{aligned}$$

(34)

where

$$\begin{aligned} \varPhi _1=\frac{\mu \xi ^{1-\gamma _1}{\mathcal{W}}^{-\gamma _1}}{\mu \xi ^{1-\gamma _1}{\mathcal{W}}^{-\gamma _1}+ (1-\mu )(1-\xi )^{1-\gamma _2}{\mathcal{W}}^{-\gamma _2}},~~~~~\varPhi _2=1-\varPhi _1. \end{aligned}$$

(35)

Relation (34) shows that the representative investor has a relative risk aversion $\mathcal{R({\mathcal{W}})}$ equal to an arithmetically weighted average of the two $\gamma$-values with weights given by their contribution to the central planner’s marginal utility of wealth.⁷ Furthermore, by dividing (32) by (31) and making use of the relations in (35) defining the weights $\varPhi _1$ and $\varPhi _2=1-\varPhi _1$ we get the following relation:

$$\begin{aligned} \frac{\partial \lambda }{\partial \mu }\frac{1}{\lambda }&= -\mathcal{R({\mathcal{W}})}\frac{\partial {\mathcal{W}}}{\partial \mu }\frac{1}{{\mathcal{W}}}+ \frac{\varPhi _1}{\mu }-\frac{\varPhi _2}{1-\mu }\quad \Leftrightarrow \nonumber \\ \frac{\partial \lambda }{\partial \mu }\frac{\mu }{\lambda }&= -\mathcal{R({\mathcal{W}})}\frac{\partial {\mathcal{W}}}{\partial \mu }\frac{1}{{\mathcal{W}}}+ \frac{\varPhi _1}{1-\mu }-\frac{\mu }{1-\mu }. \end{aligned}$$

(36)

The variable $\lambda$ is the Lagrangian multiplier for the budget constraint; it depends on the weight $\mu$, but is clearly independent of any particular realization ${\mathcal{W}}$:

$$\begin{aligned} \mathcal{R({\mathcal{W}})}\frac{\partial {\mathcal{W}}}{\partial \mu }\frac{\mu }{{\mathcal{W}}} + \underbrace{\frac{\partial \lambda }{\partial \mu }\frac{\mu }{\lambda }+\frac{\mu }{1-\mu }}_{}&\equiv \frac{\varPhi _1}{1-\mu }.\nonumber \\ {\text {constant across}} {\mathcal{W}} \end{aligned}$$

(37)

Let $\varPsi ({\mathcal{W}})$ denote the elasticity of ${\mathcal{W}}$ with respect to the weighting parameter $\mu$:

$$\begin{aligned} \varPsi ({\mathcal{W}})\equiv \frac{\partial {\mathcal{W}}}{\partial \mu } \frac{\mu }{{\mathcal{W}}}, \end{aligned}$$

(38)

and define k as:

$$\begin{aligned} k=\frac{\partial \lambda }{\partial \mu }\frac{\mu }{\lambda } +\frac{\mu }{1-\mu }. \end{aligned}$$

(39)

Then relation (37) turns into:

$$\begin{aligned} \mathcal{R({\mathcal{W}})}\varPsi ({\mathcal{W}})=\frac{\varPhi _1}{1-\mu }-k. \end{aligned}$$

(40)

Assume without loss of generality that $\gamma _1<\gamma _2$. Then it follows from (35) that $\varPhi _1\nearrow 1$ as ${\mathcal{W}}\nearrow \infty$, i.e. the least risk averse individual dominates the right hand tail. In the other end $\varPhi _1\searrow 0$ as ${\mathcal{W}}\searrow 0$.

It can be ruled out that k can be negative. In that case $\varPsi ({\mathcal{W}})$ must always be positive, which will violate the budget constraint. Hence we can conclude that

$$\begin{aligned} \varPsi ({\mathcal{W}}) \rightarrow {\left\{ \begin{array}{ll} -\frac{k}{\gamma _2}&{}\quad (<0~\text{ for }~{\mathcal{W}}\rightarrow 0)\\ \frac{\frac{1}{1-\mu }-k}{\gamma _1}&{}\quad (>0~\text{ for }~{\mathcal{W}}\rightarrow \infty ). \end{array}\right. } \end{aligned}$$

(41)