When are two portfolios better than one? A prospect theory approach

Abstract

We investigate whether the display of portfolio performance as coming from one large portfolio or two smaller subportfolios matters to individuals and whether prospect theory can explain this preference. To this end, we run a large survey experiment of 3267 individuals in 5 European countries presenting an identical overall return as coming from one portfolio or two smaller subportfolios to individuals. We also elicited the coefficients of the prospect theory value function through price list lotteries. In losses, following prospect theory and mental accounting predictions, we observe emprically a preference for the display of returns as coming from 2 subportfolios, one displaying a small gain and the other a large loss, over a unique portfolio displaying the aggregated resulting loss. As expected, for an overall gain, individuals favor one portfolio over two subportfolios, one displaying a small loss and the other a larger gain. The shape of the value function does not explain individuals’ preferences. The reference point seems to be the main factor involved in explaining individuals’ choice of two subportfolio presentations of the outcome.

Appendix

1.1 Proof of Proposition 1

Proof

Following Jarnebrant et al. (2009), for any outcome of portfolio $r<0$: If the regret-proof policy is optimal for a gain $r_g$ and a loss $r_l$ where $r=r_g+r_l$, it is optimal for smaller gains and losses $(0<r_g<|r|)$ and $(r_l<r<0)$ when the value function is concave in gains and convex in losses and when loss aversion is not extreme.

Similar to Jarnebrant et al. (2009), we define $x_0$ and $x_1$ as the gain achieved from using a regret-proof policy and $y_0$ and $y_1$ as the loss reduction achieved from not using the regret-proof policy. Under segregated portfolios gains and losses correspond to $r_{g0}$ and $r_{l0}$ and $r_{g1}$ and $r_{l1}$ respectively, where $0<r_{g1}<r_{g0}<|r|$, $r_{l0}<r_{l1}<r<0$ and $r=r_{g0}+r_{l0}=r_{g1}+r_{l1}$.

$$\begin{aligned}&x_0=v(r_{g0}) , x_1=v(r_{g1})\\&y_0=v(r)-v(r_{l0}) , y_1=v(r)-v(r_{l1}) \end{aligned}$$

Proposition 1(a): The monotonicity and concavity of the value function in the gain domain and convexity in the loss domain ($v^{''}<0$ for x>0 and $v^{''}>0$ for x<0 ), imply that

$$\begin{aligned}&\frac{x_1}{r_{g1}}> \frac{x_0}{r_{g0}} \Leftrightarrow x_1 > x_0\frac{r_{g1}}{r_{g0}} \\&\frac{y_1}{r_{g_1}}< \frac{y_0}{r_{g0}} \Leftrightarrow y_1 < y_0\frac{r_{g1}}{r_{g0}} \end{aligned}$$

For conditions where the regret-proof policy is optimal, we have

$$\begin{aligned}y_0<x_0 \Rightarrow y_1< y_0\frac{r_{g1}}{r_{g0}}< x_0\frac{r_{g1}}{r_{g0}}< x_1 \Rightarrow y_1<x_1\end{aligned}$$

We see that if it is optimal to use the regret-proof policy with $r_{g0}$ and $r_{l0}$, it is also optimal for any smaller value of $r_{g1} < r_{g0}$ and $r_{l0}< r_{l1}$.

This ensures that there exists a region around r for which the regret proof-policy is optimal if the value function is concave in gains and convex in losses.

Conversely, if the value function is convex in gains and concave in losses, the following inequalities hold:

$$\begin{aligned}&\frac{x_1}{r_{g1}}< \frac{x_0}{r_{g0}} \Leftrightarrow x_1 < x_0\frac{r_{g1}}{r_{g0}} \\&\frac{y_1}{r_{g1}}> \frac{y_0}{r_{g0}} \Leftrightarrow y_1 > y_0\frac{r_{g1}}{r_{g0}} \end{aligned}$$

which implies that the loss reduction resulting by not using the regret policy is greater than the gain resulting from using the regret policy, namely

$$\begin{aligned}x_0< y_0 \Rightarrow x_1< x_0\frac{r_{g1}}{r_{g0}}< y_0\frac{r_{g1}}{r_{g0}}< y_1 \Rightarrow x_1<y_1\end{aligned}$$

This shows that, in such a case (convexity in gains and concavity in losses), there exists no $r_g^*>0$ and $r_l^*<0$ around r for which the regret proof policy is optimal.

Proposition 1(b):

When $r_g^*>0$ exists then $v(r_g^*)+v(r_l^*)=v(r) \Leftrightarrow v(r_g^*)+v(r_l^*) - v(r) = 0$. Following Jarnebrant et al. (2009), we define $r_g^*$ by $F(r_g^*, r, \lambda , \alpha ) = 0$ where $F(r_g, r, \lambda , \alpha ) = r_g^\alpha - \lambda (r_g-r)^\alpha +\lambda (-r)^\alpha$.

Using envelope theorem $\frac{\partial F}{\partial \lambda } = -(r_g-r)^\alpha + (-r)^\alpha$. For $\alpha >0$, we always have $\frac{\partial F}{\partial \lambda }<0$. Thus, lower loss aversion levels leads to preference for segregation of outcomes. $\square$

1.2 Proof of Proposition 2

Proof

Proposition 2 (a): For any outcome of portfolio $r>0$: If the regret-proof policy is optimal for a gain $r_g$ and a loss $r_l$ where $r=r_g+r_l$, it is optimal for smaller gains and losses $(r<r_g)$ and $(-r<r_l<0)$ when the value function is convex in gains and concave in losses and when loss aversion is not extreme.

Similar to the proof of Proposition 1, let us define $x_0$ and $x_1$ as a gain achieved from using a regret-proof policy and $y_0$ and $y_1$ as the loss reduction achieved from not using the regret-proof policy. Under segregated portfolios gains and losses correspond to $r_{g0}$ and $r_{l0}$ and $r_{g1}$ and $r_{l1}$ respectively, where $r<r_{g1}<r_{g0}$, $-r<r_{l0}<r_{l1}<r<0$ and $r=r_{g0}+r_{l0}=r_{g1}+r_{l1}$.

$$\begin{aligned}&x_0=v(r_{g0})-v(r) , x_1=v(r_{g1})-v(r)\\&y_0=v(r_{l0}) , y_1=v(r_{l1})\end{aligned}$$

The monotonicity and convexity of the value function in the gain domain and concavity in the loss domain imply that

$$\begin{aligned}&\frac{x_1}{r_{l1}}> \frac{x_0}{r_{l0}} \Leftrightarrow x_1 > x_0\frac{r_{l1}}{r_{l0}} \\&\frac{y_1}{r_{l1}}< \frac{y_0}{r_{l0}} \Leftrightarrow y_1 < y_0\frac{r_{l1}}{r_{l0}} \end{aligned}$$

Under the condition where the regret-proof policy is optimal, we have

$$\begin{aligned}y_0<x_0 \Rightarrow y_1< y_0\frac{r_{l1}}{r_{l0}}< x_0\frac{r_{l1}}{r_{l0}}< x_1 \Rightarrow y_1<x_1\end{aligned}$$

We see that if it is optimal to use the regret-proof policy with $r_{l0}$ and $r_{l0}$, it is also optimal for any smaller value of $r_{g1} < r_{g0}$ and $r_{l0}< r_{l1}$.

This ensures that there exists a region around r for which the regret proof-policy is optimal if the value function is convex in gains and concave in losses.

Proposition 2 (b):

Similar to the proof given by Proposition 1 (b), when $r_g^*>0$ exists the preference for segregation is equal to preference for integration: $v(r_g^*)+v(r_l^*)=v(r)$. Then $r_g^*$ is defined by $F(r_g^*, r, \lambda , \alpha ) = 0$, where $F(r_g, r, \lambda , \alpha ) = r_g^\alpha - r^\alpha - \lambda (r_g-r)^\alpha$. Using envelope theorem $\frac{\partial F}{\partial \lambda } = -(r_g-r)^\alpha < 0$. This shows that the preference for segregation is decreasing in loss aversion $\lambda$. $\square$

1.3 Example of the impact of the shape of the value function on preferences

In this example, we assume that the overall return, r, is negative, the reference point is set at zero and $\lambda$ is constant (at 3 levels of $\lambda =4, 2, 1$). $r_g$ and $r_l = r-r_g$ are the outcomes of the two subportfolios. $r_g>0$ shows that the subportfolios present the mixed outcomes of r, and $r_g<0$ denotes that both subportfolios are negative. Figure 5 shows how the presentation of two subportfolios (segregation) can create more value for the investor. The overall performance for this example is $-2.5$. The portfolio is equally divided into two subportfolios with $r_g$ and $r-r_g$. As shown in Fig. 5, the preference for segregation over integration depends on the loss aversion and risk aversion coefficient (curvature of the value function). For mixed outcomes ($r_g>0$), we can see that the regret-proof policy is more effective for individuals who are less loss averse and who have a more concave value function in gains. When the outcomes are both negative ($r_g<0$), the segregation of the portfolio is preferred for the value function that is concave in losses.

Fig. 5

Demonstration of preference for segregation with value function. The y-axis represents the difference between the value derived from the segregation and integration of outcomes. The area below 0 reflects the preference for integration, and the area above 0 reflects the preference for segregation (regret-proof policy presentation). The $\alpha$ parameter ranges from 0.05 to 1.5 in these figures

1.4 Robustness check

See Appendix Tables 11 and 12.

Table 11 Panel probit estimates for segregation preference

Table 12 Probit estimates for segregation preference

1.5 Survey question

See Appendix Fig. 6.

Fig. 6

Example of the reference point question for the positive treatment