Portfolio optimization under safety first expected utility with nonlinear probability distortion

Highlights

• Firstly, we study a portfolio selection problem under safety first expected utility model with distortion (SFEUD model) where the underlying probability scale is transformed by a nonlinear distortion function.

• Secondly, the general form of optimal solution to the problem is obtained, and the necessary and sufficient condition for existing such an optimal solution is also proved.

• Thirdly, an analytical solution to the optimal strategies under four different monotonic cases is further demonstrated.

Abstract

In this paper, we study a portfolio selection problem under safety first expected utility model with distortion (SFEUD model) where the underlying probability scale is transformed by a nonlinear distortion function. By employing the quantile formulation and the relaxation method, we obtain the general form of optimal solution to the problem, and the necessary and sufficient condition for existing such an optimal solution is also proved. We further demonstrate an analytical solution to the optimal strategies under four different monotonic cases. What’s more, the optimal terminal wealth process is replicated into a portfolio of options and deposits under specific circumstances. In addition, we empirically verify the model implication and figure out that investors’ realization of disaster risk can explain the price change of call options to some degree.

Introduction

Both safety first principle and expected utility theory are widely used in portfolio decision-making, and they are suitable for investors with different investment principles. The theory of expected utility tends to be adopted by individuals who hope to get greater asset utility in the future, as their target is often to maximize the expected utility of the terminal assets. Other than the former, investors under the safety first principle intend to avoid extreme risks and generally their goal is to minimize the probability of a disaster. The level of the disaster varies from person to person, and each individual can offer a specific level of disaster based on his own situation. Safety first theory was proposed by Roy [16], who defined a disaster level D in his article: if the terminal wealth of an investor is lower than D, it will be regarded as a disaster. In fact, since the original problem of safety first is difficult to solve, Roy [16] obtained the upper bound of the disaster probability by using Chebyshev inequality, and dealt with a surrogate problem by minimizing the upper bound in a single period. Based on his work, safety first principle has been expanded in many aspects of academia, see for example, Chiu and Li [4], Chiu et al. [5], Kataoka [11], Li et al. [13], Li and Ng [14], Telser [17], and Gao et al. [7].

However, the safety first principle has limited effects in the real investment market. To be specific, portfolio decisions based on safety first principle sometimes can be counterintuitive so that it is unacceptable to investors. For instance, in two portfolios, suppose D = 0. If the investor chooses portfolio 1, he will definitely get $0.1; if the investor chooses portfolio 2, he will have a 0.1% chance of getting $0.1 while 99.9% probability of getting $10,000. In this case, rational investors would favor portfolio 2, while other investors would choose the portfolio 1 rather than 2 according to safety first principle, this is because avoiding extreme risk is not the only determining factor in the investment decision-making. In order to overcome the defects of safety first principle, Levy and Levy [12] proposed the safety first and expected utility model (SFEU model). They believe that individual investment decisions depend on the weighted average of safety first principle and expected utility. Chiu et al. [6] provided an analytical solution to the SFEU model and replicated an investment portfolio consisting of a call option, a put option, a digital put option and cash deposit.

Nevertheless, it is beyond the ability of the SFEU model to explain several puzzles, which EUT model also fails to do, such as the Allais paradox problem. That’s because all of the above models and theories assume that investors in markets are rational and risk averse. However, this is often not the case in reality. In fact, when people enjoy profits, they become risk-averse, but when they suffer losses, they become risk-preference. In order to deal with the shortcomings of the classical portfolio selection model, several theories were proposed, such as dual theory by Yarri [21], rank-dependent utility (RDU) theory by Quiggin [15], cumulative prospect theory (CPT) by  Tversky and Kahneman [22], and other important behavioral finance theories. The core of these theories to improve the traditional portfolio selection model is to add a probability weighting function (distortion function), and this paper is inspired by the RDU theory.

In RDU theory, the measure of wealth at the end of the period is

$$V^{RDU}\left(X\right)={\int }_0^{+\infty }u\left(x\right)d[1-w(1-F_X\left(x\right))]={\int }_0^{+\infty }u\left(x\right)w^{\prime }(1-F_X\left(x\right))d{F}_X\left(x\right),$$

and the investor’s goal is to maximize such value. Here, $w^{\prime }(1-F_X\left(X\right))$ is a weight on the outcome $x$ of random variable $X,$ and such weight depends on the cumulative distribution function of the random variable $X$. In fact, in the original literature of Quiggin, the inversed-S shaped probability weighting function $w(·)$ was set, namely, it is concave first and then convex. In addition, when $x$ is large enough or small enough, $w^{\prime }(1-F_X\left(x\right))>1$. Such setting addresses a common phenomenon in reality that people tend to overestimate low-probability events and underestimate high-probability events. However, on the one hand, due to the addition of probability weighting function, the concavity of the utility function is destroyed in the objective function, making the traditional optimization method, Lagrange multiplier method, unusable. On the other hand, due to the existence of nonlinear probability weighting function, which destroys several good properties such as probability additive and linear expectation, the problems is time-inconsistent, so the classical dynamic programming principle and probabilistic approaches can not be used any more, which brings great difficulties.

Jin and Zhou [10] studied the expected utility model of continuous time under cumulative prospect theory, and they used the quantile function technique to overcome the difficulties caused by the existence of probability weighting function. The main approach is to change the decision variable in the model from the wealth process to the form of its quantile function. The new objective function obtained is the concave form of the new decision variable. At this point, we can use the Lagrange multiplier method to solve the problem. However, their work also has a major flaw, that is, Jin and Zhou assumes a monotonic relationship between the pricing kernel and the probability weighting function, but such an assumption is too harsh and does not conform to the reality. Despite this, their work is still of great significance and opens the space for further research. He and Zhou [8] studied five models under the condition of law-invariance, and relaxed the monotonic assumption of Jin and Zhou to the case of piecewise monotone. Bi et al. [3] further relaxed the assumption to four monotonic variations and studied the mean semi-variance problem in continuous time under RDU framework. Bi et al. [2], on the basis of Bi et al. [3], continued to study the mean-variance problem with nonlinear probability weighting functions existing. Nonetheless, all the above studies need to assume the monotonic relationship between the probability weighting function and the pricing kernel function. With the deepening of the research, although the assumptions are keeping being relaxed, there are still a huge gap between the reality. Xia and Zhou [18] made a major breakthrough. The use of variational method made it no longer necessary to make any monotonicity assumption to solve the problem. But, their proof process is very complicated and skillful, and there is no additional discussion on the feasibility of the solution and other properties. Xu [19] started from another perspective with the so-called relaxation method and reached the same conclusion briefly, and related the problem to a classic expected utility problem to solve the feasibility, well-posedness of the solution. For further related works under probability distortion framework, see Bernard et al. [1], Jin et al. [9] and Xu et al. [20].

In the scope of our knowledge, the safety first expected utility with probability distortion model (SFEUD model) has been studied in little literature. This paper will be the first to study the portfolio selection problem under SFEUD model.

We decompose the SFEUD portfolio selection problem into two sub-problems. The first sub-problem is to find out an optimal attainable wealth of $X^{*},$ which is the optimal value among all the terminal wealth $x\left(T\right)$ generated by admissible portfolios. The second sub-problem is to find out an optimal strategy ${\pi }^{*}$ that replicates $X^{*}$.

First, we change the decision variable from the wealth process to its quantile function, and the objective function turns into the locally concave function form of the decision variable. Second, we use Lagrange multiplier method, relaxation method and concave envelope to solve the first sub-problem. Then, we study the explicit form of the solution to the first sub-problem through another path, namely, the method based on Bi et al. [2], which assumed four monotonic relationships between the distortion function and the pricing kernel function. Fourthly, the optimal strategy is deduced to replicate the optimal terminal wealth. Finally, we use a special probability weighting function and a special utility function, i.e., the CRRA utility function, to investigate the optimal wealth and their theoretical implications.

The structure of this paper is as follows: In Section 2, we formulate the SFEUD portfolio selection problem. In Section 3, we solve the first sub-problem. In Section 4, the optimal investment strategy are replicated under specific circumstances and numerical analysis will be given to explain the implication of the SFEUD model.