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DIPARTIMENTO DI ECONOMIA, MANAGEMENT E METODI QUANTITATIVI
Via Conservatorio 7 20122 Milano
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PORTFOLIO MANAGEMENT USING PROSPECT THEORY: COMPARING GENETIC ALGORITHMS AND PARTICLE SWARM
OPTIMIZATION
SEYEDEHZAHRA NEMATOLLAHI GIANCARLO MANZI
Working Paper 3/2018
MARCH 2018
FRANCESCO GUALA
Working Paper n. 2011-18
SETTEMBRE 2011
ARE PREFERENCES FOR REAL?
CHOICE THEORY, FOLK PSYCHOLOGY,
AND THE HARD CASE FOR COMMONSENSIBLE REALISM
FRANCESCO GUALA
Working Paper n. 2011-18
SETTEMBRE 2011
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Portfolio Management Using Prospect Theory: Comparing Genetic Algorithms and Particle Swarm Optimization
SEYEDEHZAHRA NEMATOLLAHI1, GIANCARLO MANZI2 Department of Economics, Management and Quantitative Methods
Università degli Studi di Milano
Abstract In this work, we compare the performance of two metaheuristic optimization algorithms, namely the Genetic Algorithms (GA) and the Particle Swarm Optimization (PSO), in finding an optimized investing portfolio. This comparison is based on two performance criteria: the consistency and quality of the solution and the speed of convergence of these two algorithms. These metaheuristic algorithms will be developed further to specify the weights of assets in an optimal portfolio, which is a portfolio with a maximum level of return (or a minimum level of risk) using a portfolio optimization model. We chose the prospect theory portfolio optimization as our background model. The prospect theory model is the main behavioral alternative to the expected utility theory and is still a relatively new subject in the financial literature. A mean‐variance portfolio optimization has also considered as a benchmark to our behavioral model. The performance of these two models has been evaluated in practice using several criteria such as the CPU time and the ratio between the portfolio mean returns and the standard deviation (as a measure of portfolio’s risk). Finally, an out‐of‐sample test as well as simulated bullish and bearish markets were performed to analyze the efficiency of these models based on historical data.
1 Email: [email protected] 2 Email: [email protected]
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1. Introduction Portfolio selection can be described as a procedure of estimation, analysis and collection of risky assets with different weights to provide an investor with an acceptable trade‐off between portfolio return and risk. There are many different theories, which try to formulate this portfolio optimization problem. Among them, the modern portfolio theory and the behavioral portfolio theory are two widely known theories to explain the investors’ behavior in financial markets. The modern portfolio theory was introduced by Nobel winner Harry Markowitz in 1952 as a parametric optimization model for portfolio selection [1]. Markowitz’s theory, known as the mean‐variance model, considers the portfolio selection problem as a procedure for the estimation, analysis and choice of which financial assets a risk‐averse investor should hold for a specific period with the aim of maximizing the expected return (as a measure of portfolio return) given a constant variance (as a measure of portfolio risk) [2]. In this model, the number of different securities and their correlation are factors worthy of consideration in constructing a diversified portfolio [3]. Markowitz analysis shows that the risk of a position is as important as its realized returns, and should therefore be included in an investor’s analysis [4]. Contrary to the assumptions in Markowitz’s theory, the work of Daniel Kahneman and Amos Tversky [5] shows that the behavior of an investor in the real market completely violates the mean‐variance theory’s assumption about the investor to be a risk‐averse maximizer of expected returns. They have suggested the ‘prospect theory’ as a behavioral alternative to the mean‐variance model. Their suggested model tries to describe a decision‐making framework to choose between several risky alternatives, considering the effects of behavioral biases on an investor’s decision [6]. The prospect theory model includes a value function, which is described on the domain of gains and losses instead of the final wealth (in case of the expected utility theory), and is weighted by a non‐linear probability in various states of the world, called scenarios. This theory assumes that investors are loss‐averse in general, and that they are risk‐averse when comparing two gains, but risk seeking when comparing two losses [7]. In 2000, the behavioral portfolio theory (BPT) was developed by [8] based on the security, potential and aspiration theory (SP/A) [9] and prospect theory as two psychological theories to describe the process of choice under uncertainty. Today, both modern portfolio theory (MPT) and behavioral portfolio theory (BPT) are being used to explain the investors’ behavior in financial markets. We can think of modern portfolio theory as a model describing the financial market mechanism in an ideal world, while behavioral portfolio theory tries to characterize this mechanism in the real world. In this paper, we apply a prospect theory model to construct an optimal portfolio based on the desire of a hypothetical investor with certain loss and risk aversion levels to be discussed in later sections. We also use a mean‐variance portfolio optimization model as a benchmark to check the performance of our behavior‐based portfolio selection model.
Since our prospect theory model is non‐convex, it cannot be efficiently solved by exact optimization algorithms [10]. Therefore, many sources in the literature suggest metaheuristic algorithms to obtain an “optimal” solution for this portfolio optimization model. For some examples, see [11] and [12]. In this study, we implement the genetic algorithm (GA) and particle swarm optimization algorithm (PSO) as two examples of the increasingly popular nature‐inspired metaheuristic algorithms to solve portfolio optimization models. In the GA, we represent the practical solutions of our problem using chromosomes. Then for each generation, a “fitness function” (the objective function of the optimization) is assigned to the population of these chromosomes to evaluate them. The chromosomes with more fitness value are given higher opportunities to “recombine” and form a new generation as “offspring” sharing some features taken from each “parent.” The least fit members of the population are less likely to get selected for recombination and so “die out.” Moreover, a mutation operator will change some features of this offspring to ensure the diversity of the next generation [13]. Like GAs, PSO is a nature‐inspired optimization method based on the movement behavior of bird and fish flocks. Considering the feasible solutions of our problem as a bird swarm and the food resource as the best solution in the whole search area, the birds moving from one place to another is equal to the development of this solution using the information‐sharing property of a swarm. Indeed, bird swarm individuals are transmitting information, especially the good information, at any time while searching from one place to another for food. The PSO algorithm is very simple, easily computable and needs only a few parameters. These properties have contributed to the algorithm becoming a widely‐used method to solve many complex optimization problems [14].
2. Problem formulation 2.1. Mean‐variance model
The mean‐variance model states that the most desirable portfolios have the minimum expected risk at any given expected rate of return or, conversely, the maximum expected rate of return at any given level of expected risk. In this paper, we analyze the latter case. This classic portfolio optimization theory is based on four fundamental behavioral assumptions: 1. Each asset is presented by a probability distribution of
return measured over a specified time interval (its holding period).
2. The investors’ estimates of risk are proportional to the standard deviation of returns .
3. The investors’ decisions are only based on the expected return and risk statistics. Considering the distribution of asset returns to be normal, this means that the investors’ utility is a function of the standard deviation of returns and the expected value of return .
4. For any given level of risk, the rational investors prefer higher returns to lower returns. Conversely, for any given level of rate of return, the rational investors prefer less risk to more risk. In other words, all rational investors are risk‐averse and rate of return‐maximizers [15].
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Given these assumptions, we describe the mean‐variance model considering a market of different assets which are traded at the initial time 0 with prices
0 , 0 ,⋯ , 0 0. We know that at the final time , the asset prices , , ⋯ , are non‐negative
random variables on a probability space Ω, , Ρ . For constructing our Markowitz‐based portfolio, we calculate the return of each asset as:
≔ (1)
for 1,⋯ , .
Then, we can obtain the expected rate of return and risk for a given asset using the historical mean and variance of the return as an estimate:
(2)
(3)
1,⋯ , .
As mentioned earlier, the correlation coefficient is one of the key factors in Markowitz’ portfolio selection. This component helps us to know whether a positive deviation in one asset will lead to positive or negative deviations in the other assets or whether they are independent [16]. We can calculate the correlation coefficient between assets and using the next equation:
(4)
where the covariance of assets and is calculated as: , (5)
A perfect positive linear relationship between and will have a value of +1, while a value of –1 indicates a perfect negative relationship between these assets’ rates of return [17]. Given these assets’ financial relationships, now we can simply calculate the performance of each hypothetical portfolio. Technically speaking, the risk and return of a portfolio can be treated as a convolution of the individual assets’ returns and covariance when these assets can be described by the distributions of their returns. So, suppose that an investor is interested in investing in risky assets. We can present her choice as an 1 vector array
, … , of asset weights where each represents the share of asset in this portfolio wherefore the sum of these weights must be equal to 1:
1 (6)
Then, the expected return of our portfolio with the weight of allocated assets presented as , … , is:
(7)
And the variance of this portfolio will be:
(8)
Accordingly, for a portfolio with a low or negative correlation among asset returns, the standard deviation of the portfolio’s rates of return and hence the risk of the portfolio itself will be reduced. In fact, equation (8) shows us the effect of diversification in which combining different asset categories in a portfolio may reduce the overall portfolio risk without harming potential returns.
As stated already, the basic idea of Markowitz is that the rational investor takes decisions based on a tradeoff between the expected return (i.e. portfolio mean) and risk (i.e. portfolio variance). Indeed, the investor solves the portfolio optimization problem by setting a lower bound for the portfolio return, and then he/she tries to determine a portfolio vector with the lowest possible risk. In the classical mean‐variance model, the task of minimizing the portfolio risk given a lower bound for the expected return can be illustrated as follows:
(9)
Subject to: 0 for 1,⋯ , (10)
1 (11)
(12)
We can also add transaction costs and other particular constraints, such as a short sell allowance in the market or weights bounds on the investor’s portfolio to our portfolio optimization problem (if short selling is allowed, then the weights of assets in the portfolio can be negative). Therefore, for finding the optimal mean‐variance portfolio, we have a convex quadratic programming problem to solve. Here, the first and second constraints indicate that the weights of assets in our portfolio are non‐negative (no short sale) and sum up to 1. The third constraint shows that the expected return is no less than a boundary value when our objective function is the total variance of the portfolio. Solving this problem for different values of the expected return boundary , we obtain a set of feasible portfolios. We use the term efficient portfolios for those feasible portfolios with maximum return among all portfolios with the same variance (risk). The set of all these efficient portfolios constructs the mean‐variance efficient frontier or simply efficient frontier, which is a two‐dimensional curve. Each point of the efficient frontier corresponds to the standard deviation and the expected return of an efficient portfolio [18]. So, given this efficient frontier, the investor could choose his/her preferred portfolio, based on his/her personal risk return preferences. 2.2. Prospect Theory Model First, we define a prospect as a function from a finite set of states of nature to a set of probable monetary outcomes which assigns a monetary outcome to each state ∈ . Considering as the probability of each state of nature , we can show a lottery as the corresponding pairs of all monetary outcomes and their probability of occurrence as , ; … ; , where the sum of all probabilities is equal to
one.
⋯ 1 (13)
Here, 1,2, … , 1, denotes the finite set of states of nature while , is representing a scenario (prospect) in which the probability of having an amount of outcome is equal to . Assuming a reference point , we can classify the monetary outcomes as gains (positive outcomes which are greater than ) or losses (negative outcomes which are lower than ) [19].
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Then we can write the subjective utility function for this lottery as:
: , (14)
where the prospect theory utility function is a non‐linear function wherein ∶ → is the value function defined over changes in wealth instead of final wealth (in case of mean variance model), and ∶ 0,1 → 0,1 is the probability weighting function which transfers real probabilities into subjective probabilities [20]. The probability weighting function has some properties:
x It is a continuous and monotonous increasing function.
x For small values of 0 , we have , and for large values of 1 , we have . This captures the fact that individuals tend to overweight the small probabilities and underweight the large probabilities. Moreover, we have 0 0 and 1 1.
It is important to note that the probability weighting function represents not just the probability of events but also the impact of these events on the desirability of prospects and so is a distortion of the given probability. Therefore, 0.5 is not necessarily equal to 0.5. The value function has also three key features: x It is a continuous and monotonous increasing function
defined over the domain of gains and losses, considering some neutral reference point .
x It is strictly concave in the gain domain and convex in the loss domain.
x The value function is steeper for losses than for gains. (Its slope at is greater than its slope at ) [20].
It should also be mentioned that the concavity of the value function in the domain of gain justifies the risk‐averse behavior of investors while the convexity of the value function in the domain of loss justifies their risk‐seeking behavior. Besides, the higher slope of the value function in the domain of losses demonstrates the loss aversion attitude of investors in the prospect theory model. Now, we need to find and which are appropriate for our optimization problem. Our choice of value function is the value function proposed [21] as a piecewise power function for , ∈ 0,1 and 1 and a reference point , as:
,, (15)
Here, and are risk aversion coefficients with respect to gains and losses while is the loss aversion coefficient which underlines the observed fact that investors respond to losses more strongly than to gains. Kahneman and Tversky suggest the loss aversion coefficients
0.88 and the loss aversion coefficient 2.25 , based on their laboratory results. For the probability weighting function , different forms have been suggested as well. The original choice of Kahneman and Tversky is as follows:
≔1
(16)
where captures the perception of probabilities by investors and 0.27 1. A lower indicates a stronger distortion in the probabilities they perceived.
We can also consider different weighting functions for gains and losses denoted by and . However, for the sake of simplicity in this study, we considered in our constructed prospect theory model. The form of value function and probability weighting function for a standard prospect theory model is shown in Figure . In the following, we fit the prospect theory model to our asset allocation problem. Assuming a portfolio of risky assets where the proportional weight of asset is denoted by for 1,… , and the sum of all weights is equal to 1, i.e. ∑ 1. Moreover, we consider the weights to be non‐negative 0, ∀ which means that short selling is not allowed in the market.
Figure 1. Value function and weighting function of the prospect theory model
Now, the return of a portfolio , … . can be defined as a weighted average of all assets in different states of nature. So, we have the following equation:
(17)
where is the proportional weight of asset in the portfolio and is the return of asset in scenario . (1,… , ). The standard prospect theory asset allocation problem consists of maximizing the objective function of this portfolio with a desired level of return above . This optimization problem is formulated as follows:
(18)
subject to the constraints:
1 (19)
0 , ∀ (20)
(21)
To solve this optimization problem, we commonly confront two types of computational complexity: First, our objective function is not differentiable for some ; for them, is equal to zero. Second, this problem is categorized as a non‐smooth optimization problem. The available algorithms for non‐smooth problems require the objective function to be convex. This is not the case for our prospect theory‐based asset allocation problem which is a non‐linear and non‐convex optimization model with the objective function being non‐linear and non‐convex. Despite this, in case we have many assets in our portfolio, the asset allocation problem will become too complex to be solved by classical numerical methods [21]. Therefore, many researchers and financial experts use heuristic methods to solve this type of portfolio optimization
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problems. These heuristics optimize the problem by using a procedure of producing and evaluating some new points and finally yield the best solution found during this process as the optimal point. In fact, heuristic methods combine the advantages of classical algorithms, yet they are more flexible and less restricted to constraints [16]. In this study, we have used genetic algorithms (GA) and particle swarm optimization (PSO) as the two most common metaheuristic methods for the asset allocation problems with prospect theory objective function.
3. Solution approaches 3.1 Genetic algorithms (GA) Genetic algorithms are based on concepts derived from both genetics and evolution theory. Nowadays, Holland’s Theory is known as an operative tool in the domain of search and optimization problems and is widely used by scientist and engineers [22]. Genetic algorithms can be taught as a broad collection of stochastic optimization algorithms that let the fittest survive and the weak die [23]. A typical continuous GA works as follows. 1. Initialize the population of chromosomes. 2. Evaluate each chromosome using an evaluation
function . 3. Repeat the following steps until a stopping criterion is
satisfied. � Select parents from the population. � Perform a mating operation on parents, creating a
new population of offspring. � Perform a mutation operation on a percentage of
the offspring. � Evaluate the chromosomes of a new population.
A genetic algorithm, like any other optimization algorithm, begins by specifying the model variables known as genes. A potential solution (a chromosome) to our optimization problem can be represented as a set of these variables. For example, a chromosome with variables (in the case of an N‐dimensional optimization problem) can be written as an array with 1 elements:
, , , … , (22)
where is the value of the variable. In this way, each chromosome can be defined either as a binary or a real code string [24]. A real GA has several advantages. For instance, it requires less storage space and is particularly faster than a binary GA. In addition, it is more convenient to solve continuous optimization problems. Therefore, we decided to construct our GA algorithms based on real encoding. In a GA with real encoding, a chromosome is a vector of floating point numbers and the size of each chromosome is the same as the length of this vector [25]. The GA procedure starts with defining a group of random chromosomes known as the initial population . The total number of variables in each generation can be presented by a matrix. Then we evaluate each chromosome of this initial population to find the chromosomes which are fit enough to survive and more likely to participate in producing offspring in the next generation. Thus, we employ an objective function to obtain the fitness of each chromosome:
, , , … , (23)
The aim of our optimization problem is to find the global optimum for this objective function. After evaluating each chromosome in a standard genetic algorithm, the next generation of chromosomes (probable solutions) will be produced using three operations: selection, mating (crossover) and mutation. In the selection stage, we choose the fittest chromosomes in the initial population to survive and possibly reproduce the next generation. These selection processes are mostly stochastic, and their aim is to preserve the diversity of the population and prevent an early termination of the process (in case of trapping in some sub‐optimal solutions) [26]. The selection operation can be undertaken by several methods including rank‐based selection, roulette wheel selection and the tournament selection. For this study, we implemented the tournament selection as one of the most common selecting methods. In the tournament selection method, several chromosomes are randomly chosen from the population. These chromosomes are compared with each other, and the best fitting one is chosen to be the parent of the next offspring. This procedure will continue until we find the adequate number of parents to mate [27]. After choosing the set of parents using a selection method, it is time to produce the next generation of chromosomes. In a genetic algorithm, two operators are used to produce the next generation: the mating operator (sometimes called crossover) and the mutation operator. In the mating operation, two parents are chosen and combined to make the offspring. There are many mating operators, but the most common procedure is to randomly select two parents and then exchange some portion of their characteristics between them to create two new offspring [28]. Many different approaches have been tried for mating in continuous GAs. We will review and apply the whole arithmetic mating to our portfolio optimization model. The whole arithmetic mating is probably the most commonly used operator in which the offspring are produced by taking the weighted mean of two parents. With two parents, A and B, each offspring will be generated as:
1 ∗ 1 ∗ (24)
2 ∗ 1 ∗ (25)
Here, is a random value between [0,1]. The mutation operation is the final step of the GA procedure. This operator allows the GA process to explore new potential solutions to our problem while keeping the GA from an excessively fast convergence through finding a local maximum. This operator randomly modifies both offspring resulting from each mating operation to form the new generation [29]. There are many different forms of mutation for the different kinds of representation. In this study, we have decided to implement the uniform mutation. In the uniform mutation, ‐th element of the chromosome , , , … , , … , are randomly selected. Then, a normally distributed random number is added to . The new chromosome is a vector , , , … , , … , where . This process of selection, mating and mutation operations iterates until an acceptable optimal solution is found by the algorithm. The GA algorithm’s speed of convergence is
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affected by different methods of selection, mating and mutation employed in our optimization problem. So, deciding on the appropriate operators is a major issue in a genetic algorithm. Indeed, each of these methods leads to a different set of parents so the composition of the next generation will be different for each selection method [30]. 3.2 Particle swarm optimization (PSO) The Particle Swarm Optimization (PSO) algorithm is a population‐based stochastic optimization technique introduced by [31]. This algorithm is based on a population of search points called particles. These particles move stochastically in the search space, and the best position ever attained by these particles, also known as its experience, is retained in a specific memory. This experience is then communicated to a part of or the whole particle population, leading its movement towards the best areas detected so far. This communication design is determined by a fixed or adaptive social network and plays a key role in the convergence properties of the particle swarm optimization algorithm [32]. As mentioned earlier, the PSO model consists of a population of individuals where each member of this population represents a potential solution in a search space. This population is called a swarm, and its members are called particles; We can represent the swarm as an ‐dimensional set:
, , … , (26)
where represents the current positions of the particles and each particle is defined as:
, , ⋯ , ∈ , 1,⋯ , (27)
Here ⊂ is the feasible space of the problem, and the objective function of our optimization problem : → ⊂ is as follows:
∈ (28)
First, we initialize the particles’ positions randomly, using a uniform distribution on the search space A. Then, each particle is iteratively moved through the search space A, with some velocity denoted as:
, , ⋯ , , 1,⋯ , (29)
This velocity vector determines the next direction and distance of the particles, allowing them to be investigated throughout the search space area. The velocity is updated based on information obtained in previous steps of the algorithm. This information is stored in a memory, where each particle can save the best position it has ever visited during its search. For this purpose, we introduce the PSO memory set as:
, , ⋯ , (30)
which includes the best positions ever visited by each particle:
, , ⋯ , , 1,2,⋯ , (31)
These positions are demonstrated by: max (32)
where represents the iteration numerator and is our objective function. So, for denoting our iteration numerator, each particle in the swarm consists of three parts: (1) its position in the search space , (2) its velocity and (3) its memory , where the best positions are saved. Then, the PSO algorithm will approximate the best position ever visited by whole particles using an information
exchange mechanism. This mechanism allows particles to mutually communicate their experience to find the best solution within our search space. So, at a given iteration , the best position of the entire population is obtainable using the formula:
argmax (33)
In the early version of PSO presented by [31], the velocity is updated by using an equation as follows:
1
(34)
1, … and 1,… ,
where is the number of iterations. and denote random variables uniformly distributed within 0,1 ; and and are weighted acceleration factors adjusting the impact of the local and global information. Then, the position of particle in the next iteration will be calculated as:
1 1 (35) 1,⋯ , 1,⋯ ,
The particles are evaluated at each iteration after which the best position will be updated. Hence the new best position of at iteration 1 can be formulated as follows:
Each iteration of PSO will be completed by finding and updating the general best position [32]. The basic concept of PSO lies in accelerating each particle toward its and positions, with a random weighted acceleration at each iteration. The particle swarm optimization (PSO) procedure can be summarized with this pseudo‐code: � Objective function , , … , � Initialize locations and velocity of particles � Find from , … , 0 � While (criterion) � For loop over all particles � Generate new velocity 1 � Calculate new locations 1 � Evaluate objective functions at new locations 1 � Find the current best for each particle � Update 1 � End While � Output the results Although this initial version of PSO can be perfectly applied to a simple optimization problem, researchers have confronted some deficiencies in using this algorithm for harder problems with a vast search space and several local maxima. The swarm explosion effect is one of these deficiencies caused by an increase in the value of a particle’s velocities, leading to divergence in the PSO algorithm. In fact, this explosion indicates the lack of a proper mechanism for constricting velocities in initial PSO. Therefore, [33] suggested a more generalized PSO by appending constriction coefficients to the initial version of the algorithm. Their model is defined by the following equations:
11 , 1,
(36)
7
1
(37)
1 1 (38)
for 1,2,⋯ , 1,2,⋯ , .
where the constriction coefficient controls the convergence of the particle and prevents the model from the explosion effect and divergence [34]. This coefficient is calculated using the follow equation:
22 4
(39)
Here, and 4. In Clerc and Kennedy’s constriction method, we usually set 0.729 and 2.05. In this study, we have applied a PSO algorithm with adjusted velocity and position limits to handle the particles’ positions and velocities which are placed outside our search space. The feasible search space for our asset allocation problem is an 0,1 interval for each particle (that is, it corresponds to the weight of each asset in our portfolio). Thus, to maintain the particles within the interval of feasible solutions, we have implemented two constraint‐handling mechanisms, namely the reflect method and the deterministic back method, on the position and velocity of particles in each iteration of algorithm. As a position limit, we have applied the reflect method in which the boundary of search space operates like a mirror and if the updated velocity refers to a non‐feasible point (outside our search space), it will be reflected at the boundaries of our feasible interval. Moreover, we have used a deterministic back method to restrict the particles’ velocity. In this method, the velocity of particles outside the search space will be multiplied with for some 0. In our particle swarm optimization algorithm, we set 1 [35].
4. Experimental set‐up 4.1. Data In this paper, we try to solve the portfolio optimization problem using the publicly available data of 30 assets in the Nasdaq stock market as a major market index for 260 time periods each (weekly data), taken from the Yahoo Finance! website. The data is presented in the form of matrices of asset adjusted close prices. We transformed the original data set into matrices of asset returns using the logarithmic rate of return. Then, to test the performance of our models in different problem conditions, several subsets of the main data set with 10, 15 and 20 assets were randomly picked. We also conducted an out‐of‐sample test to evaluate the performance of the proposed models. To do so, the first 90 data points were considered as the in‐sample data and the remaining periods (170 data points) were used as the historical out‐of‐sample test inputs. Further, we applied the data set to create two simulated markets (namely a bearish market and a bullish market) as out‐of‐sample tests to analyze the efficiency of these models. In [36], a bearish market is defined as a period in which the prices of securities or commodities fall by 20 percent or more. During such a period, the investment interest is generally limited, the investors’ concern about the state of the economy will rise, and the investors are more interested
in selling their investment than to increase their risk by holding them. On the other hand, a bullish market denotes a period in which prices are rising, securities are traded in high volumes, the investment interest is high, and investors view the economy as strong and getting stronger. To simulate these two markets’ behaviors, bearish and bullish, we used the built‐in functions accessible in the Statistics Toolbox in MATLAB. To construct a sample with the characteristics of a bullish market, we picked out a set of assets returns from 4‐Jan‐2013 to 2‐Jan‐2014 which indicates a progressive economic market. Then the simulated data of the bullish market were obtained using the MATLAB function , that returns k observations sampled uniformly at random, with replacements filled in from our data set. The bearish market sample was based on a set of assets returns from 4‐Jan‐2008 to 31‐Dec‐2014, which encompasses the 2008 financial crisis as the worst economic situation since the great depression of 1929. We used the MATLAB function ; ; that returns a matrix of random numbers chosen from the multivariate distribution, where is the matrix of historical data, is the degrees of freedom and is either a scalar or is a vector with elements (we choose 4). For , the number of columns in , the output has rows and columns. The student t distribution was selected because of the fat tails characteristic of asset return distributions which allocate more probability weights to low returns imitating the real market situation [37]. 4.2. Experimental design During this study, we run each GA and PSO algorithm 5 times to eliminate the probability of a lucky initialization. To test the performance of these algorithms, a prospect theory model is applied. The best solution of each run with the same parameters and objective function will be stored. The error of each algorithm as a metric of the solution quality can be obtained by calculating the average deviation of the best solution of each run of an algorithm from the best solution of all runs of this algorithm. However, since the optimal solution of the prospect theory problem is typically unknown, and we have no benchmark for comparative analysis, the errors of our optimization algorithms from the optimal solution using above metrics are not the actual errors [38]. Moreover, since the consistency of an algorithm is defined as the ability of an algorithm to always find a good (near‐optimal) solution, we can take the standard deviation of algorithm errors as a consistency measure of our metaheuristics. To measure the speed of our optimization algorithms, we use the average CPU time and number of function evaluation (NFE) which is required by each algorithm to find an acceptable solution. Here, NFE reports the total number of objective function evaluations. We will also analyze the sensitivity of these two metaheuristic algorithms to different values of the population size and the number of iterations. After testing several cases we found out that in our non‐linear problems with continuous design variables, PSO outperforms GA; so, we performed the next part of the study using PSO algorithm.
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In this step, we implement the selected metaheuristic algorithm in our behavioral asset allocation model to find an optimal solution which maximizes the utility of an assumptive investor. We also apply this heuristic algorithm to solve a mean‐variance portfolio optimization model as a benchmark to analyze the performance of our behavior‐based model. While the convex mean‐variance model will be solved easily using the portfolio object in the Financial Toolbox of MATLAB, the non‐convexity of the prospect theory model makes its solution approach a mathematical challenge. To reduce the complexity of the behavior‐based problem, we define the prospect theory’s weighting function as , and employ original Tversky’ value function. The risk aversion and loss aversion coefficients are also chosen as 2.25, 0.88 which are exactly the parameter values suggested by [19]. In addition, we calculate the desired level of return for the selected portfolio considering the mean return of assets in the market using this equation: ∗ 0.25 (40)
Here, and are the maximum and minimum mean of asset returns for every specific set of assets [39]. In Table 1, the desired level of return along with the chosen amount of reference point for markets with 10, 15 and 20 assets are reported. The value of the reference point in our model is proportional to the 3‐month Treasury Bill Rate for November 2012.
Table 1. Desired returns and reference points of current models
10 0.0031 0.0007
15 0.0032 0.0007
20 0.0034 0.0007
4.2.2. Parameters of the heuristic algorithms Both genetic algorithm and particle swarm optimization have parameters that must be tuned before applying the algorithms to our portfolio optimization models. Since the efficiency of these algorithms is strongly affected by chosen parameters, it is important to find an adequate set of parameters for each metaheuristic algorithm to increase their performance. As we mentioned earlier, the performance of an algorithm can be measured using several criteria. To choose a proper set of parameters, we will evaluate the efficiency of each algorithm using computational time, the average value of the best solution of the algorithm and the error term . To analyze the efficiency of heuristic algorithms, we use a prospect theory utility as our objective function to be maximized. Hereby, we evaluated the performance of each optimization algorithms to solve a prospect theory based asset allocation model without boundary on the number of assets or portion of asset weights to be held in the optimal portfolio. The parameter sets leading each algorithm to find a better solution for the prospect theory optimization problem, with lower mean errors and standard deviation of errors , will be passed on to the next part of our study. In fact, we are interested in achieving a balance between computational time and the consistency of solutions.
We know that the parameters of heuristic algorithms are highly dependent on the objective function characteristics. There are a lot of cases in the literature which have studied sets of parameters suitable for both PSO and GA heuristic algorithms. See for example [40‐43]. The number of particles/chromosomes (population size) and the number of iterations are two common parameters which greatly affect the efficiency of our intended heuristic algorithms. To find the most appropriate population size and number of iterations for both GA and PSO in this study, we have performed a test with different values for each of these two parameters using the prospect theory model. Then the test was repeated 5 times for each set of parameters, and the performance of our model was evaluated. We test several population sizes, namely 40, 50, 60, for both algorithms to find a suitable value for this parameter considering the CPU time, consistency and the quality of the solution. Then, to choose the adequate number of iterations, we repeat the tests for 150, 200, 250 and evaluate again the CPU time, consistency and the quality of the solution of both the GA and PSO algorithms. In the genetic algorithms, apart from the size of the population and the maximum number of iterations, we must choose other parameters, including the mutation rate, the mating rate and the selection method. The mating rate is the probability that a mating procedure will be performed on two selected chromosomes while the mutation rate is likewise the probability that a mutation procedure will happen on the offspring of two chromosomes. The appropriate rate of the mutation operation is generally indeterminable. It depends on the problem characteristics, the population size, encoding methods and other factors. In fact, the mutation rate used in genetic algorithms as observed in nature is very small [44]. Implementing an arithmetic mating as well as a uniform mutation to our genetic algorithm, the mating and mutation rate was chosen to be 0.8 (80%) and 0.1 (10%) respectively. Moreover, we use the tournament selection as our selection method since its results are more efficient compared to other selection methods, especially in the cardinality constraint model. For the particle swarm optimization algorithm, we must as well make a decision about the values of the personal learning coefficient and the swarm learning coefficient which are applied in the constriction coefficients of the PSO model. We choose the constriction factors based on the work of [45] which suggests 1.49445 . All algorithms are implemented in MATLAB® version 9.1 (R2016b). The system runs under MS Windows 7 64‐bit, and in our experiments, we used an Intel® Core i7‐3517U with 1.90 GHz and 2.40 GHz processor and 4.00 GB RAM.
5. Experimental Results 5.1. The influence of the population size and the number of iterations The analyses and selection procedures to choose an adequate set of parameters for the PSO and GA algorithms for our asset allocation problem is presented in Table 2. Thus, based on these preliminary analysis and other recommendations from literature, we have decided to
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continue our study with a population size equal to 50 (50) and a number of iteration (It=200) for both PSO and GA. It is important to note that the value of the prospect theory objective function is almost the same for selected algorithms. It supports our choice of implementing the heuristic approach to solve the prospect theory model. 5.2. Comparison of the performance of the algorithms 5.2.1. Consistency, solution quality and Speed of convergence In the first experimental run, we check the solution quality and consistency as well as the convergence speed of our algorithms for solving the behavioral portfolio optimization problem. Here, we use a prospect theory model for a market containing 20 assets. Each PSO and GA algorithm is executed 5 times with 50 particles/chromosomes for a maximum of 200 iterations. Table 3 presents the results of this experiment considering ε as a criterion for the quality of the solution; the PSO algorithm had a higher quality solution (less amount of error) than its GA counterpart. The PSO optimal solutions also are more consistent due to the lower
value of σ, indicating the lower volatility of the algorithm’s best solutions. However, the difference between their consistency is very small. For testing the speed of convergence, an average CPU time, where the number of fitness evaluations (NFE) and the minimum number of iterations needed for each algorithm to converge within 0.1% of the best solution of each algorithm are examined. These results show that the particle swarm optimization algorithm needed less iterations and hence CPU time to find an acceptable solution within 0.1% of the optimal solution ever found by an algorithm. Also, the average number of fitness evaluations to solve the prospect theory optimization problem is much lower for PSO than for GA. To make the result of the consistency test more visible, we illustrate the optimal solutions of the PSO and GA algorithms for 12 different runs in Figure 2, left side. As the figure shows, the results of the genetic algorithm have more fluctuation and thus less consistency in the best solution of all runs.
Table 2. The influence of the population size and number of iterations on the performance of the PSO and GA
Particle Swarm Optimization Genetic Algorithms
Na Parameter Parameter Value NFE CPU Time PT(x) ε σ NFE CPU Time PT(x) ε σ
20
P
40 8040 571.57 0.020 0.003 0.002 10440 798.30 0.02036 0.0038 0.00232
50 1005 730.34 0.021 0.001 0.001 13050 994.12 0.02084 0.0022 0.00129
60 1206 906.06 0.021 0.002 0.002 15660 1195.62 0.02056 0.0040 0.00224
It
150 7550 559.40 0.019 0.002 0.001 9800 759.85 0.01956 0.0027 0.00141
200 1005 730.34 0.021 0.001 0.001 13050 994.12 0.02084 0.0022 0.00129
250 1255 925.01 0.021 0.001 0.001 16300 1247.98 0.02116 0.0017 0.00113
Table 3. Comparing the consistency, solution quality and speed of convergence of PSO and GA Na Algorithm PT(x) ε σ Nit NFE CPU Time
20 PSO 0.0216 0.0017 0.0012 344.6 17280 1278.33
GA 0.0208 0.0022 0.00129 354.4 23086 1633.12
Figure2. Comparing the volatility of results and Trend of Convergence for GA and PSO
The process of finding the optimal solution during iterations undertaken by both algorithms is also illustrated in Figure 2, right side. It shows that the convergence of particle swarm optimization occurs faster and in a lower number of iterations compared to the genetic algorithm. PSO has also found a better solution in a reasonable number of iterations. One can see that the PSO fitness curve changes step by step, while GA presents a smoother and more continuous progression.
5.4. Comparison of the portfolio optimization models’ performance During the experiment, we noticed that the value of the genetic algorithm’s best solution is somewhat lower than the best solution achieved by the particle swarm optimization algorithm. At the same time, the CPU time needed for the convergence of PSO to an acceptable optimum is much less than GA, thus reducing the overall time required for further tests. Therefore, we have decided
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to implement the PSO algorithm for our prospect theory and mean‐variance portfolio optimization models. 5.4.1. Portfolio optimization models 5.4.1.1. In‐sample data In this section, the performance of the prospect theory and the mean‐variance models using 3 random subsets of in‐sample data (with 10, 15 and 20 assets) are evaluated. The summary of this evaluation is displayed in Table 4. This table contains the portfolio mean return , the portfolio standard deviation as the proxy of the risk, the mean return to standard deviation ratio , the number of assets in the optimal portfolio as well as the CPU time for each model. Clearly, the CPU time needed for the convergence to an optimal solution is significantly high in the case of the prospect theory model compared to the mean‐variance model with the same number of assets in the market which was predictable considering the mathematical complexity of behavior‐based models. The CPU time also increases in
markets with more assets for both the mean‐variance and the prospect theory based models. Since the prospect theory model finds portfolios with better returns and accordingly higher risks, it is difficult to compare the average return and risk of the optimal portfolio selected by our studied models. So [39] suggests using the ratio as a unified measure of performance which contains the mean return as well as the risk of the selected portfolio. A model with a higher ratio is more efficient.
However, in our case, it is hard to compare the ratio for the prospect theory and mean‐variance portfolio using the in‐sample data. For instance, for a market with 10 and 20 assets, PT model outperforms the MV model which is against the result achieved in market with 15 assets. Despite this, the return of portfolio is often extremely higher for the prospect theory model than the mean‐variance model. This can be caused by the existence of a reference point which encourages the model to always choose assets with higher returns and less focus on the related risk of the portfolio.
Table 4. Comparison of PT and MV basic models (in‐sample data)
Model CPU Time
10 MV 16.93 9 0.2172 0.0043 0.0198
PT 368.70 10 0.2245 0.0047 0.0209
15
MV 17.03 10 0.3006 0.0052 0.0173
PT 701.18 15 0.2472 0.0052 0.0211
20 MV 21.37 13 0.2000 0.0034 0.0170
PT 701.79 20 0.2024 0.0043 0.0212
It can also be seen that the number of assets in the mean‐variance model portfolio is always lower than the entirety of available assets in the market, but the prospect theory model has a greater tendency to use all existing assets to construct its optimal portfolio. In fact, the portfolio selected by our prospect theory model is more diversified than the mean‐variance portfolio. 5.4.1.2. Out‐of‐sample data In this section, we conduct a performance analysis on the MV and PT models using the out‐of‐sample data. First, we consider the historical data for the remaining 170 data points as our out‐of‐sample data which we use to test the efficiency of the models’ portfolio. Then, we use this historical data sample to calculate the mean return , the risk and the return to risk ratio of the portfolio selected by the MV and PT models in the previous section. For further study, we extend our out‐of‐sample tests to analyze the models’ performance for a simulated bullish and bearish market, respectively. As one can see in Table 5 for historical data, despite the higher amount of portfolio risk, the ratio is always better for a behavior‐based portfolio than MV’s. Moreover, as in the case of in‐sample data, the prospect theory portfolio gains a far greater average return whereas MV’s portfolio carries less portfolio risk.
In Figure 3 the cumulative returns for the MV and PT models’ portfolios during in‐sample and out‐of‐sample periods are demonstrated. Evidently, the ascending slopes of the curves for the in‐sample test is followed by the historical out‐of‐sample test in both the MV and PT portfolios. However, as we have anticipated from the results in Table 5, the PT curve is always above the MV curve for out‐of‐sample data, indicating the higher average return of behavior‐based portfolios.
Figure 3. Comparison of basic PT and MV models using in‐sample
data vs historical out‐of‐sample data (the results of a market with 20 assets)
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Table 5. Comparison of PT and MV models (out‐of‐sample test)
historical data simulated bullish market simulated bearish market
Model
10 MV 0.0833 0.0020 0.0240 0.1587 0.0013 0.0083 ‐0.1141 ‐0.0861 0.7548
PT 0.1312 0.0030 0.0230 0.1395 0.0012 0.0086 ‐0.1305 ‐0.0909 0.6969
15 MV 0.0784 0.0016 0.0204 0.1724 0.0015 0.0087 ‐0.1516 ‐0.1365 0.9005
PT 0.1199 0.0027 0.0225 0.1719 0.0015 0.0088 ‐0.1161 ‐0.0845 0.7280
20 MV 0.0777 0.0016 0.0206 0.1609 0.0014 0.0087 ‐0.0409 ‐0.0401 0.9815
PT 0.1486 0.0034 0.0230 0.1893 0.0017 0.0090 ‐0.0392 ‐0.0302 0.7713
Figure 4. Comparison of PT and MV models using out‐of‐sample data (simulating a bullish market and a bearish market)
For simulated bullish market, our behavior‐based model does not perform well in markets with a lower number of assets both in terms of portfolio mean return and risk. But for a market of 20 assets, the PT model’s portfolio shows a higher average return and ratio than MV’s portfolio. At the same time, the MV model is more beneficial considering the risk parameter for all three markets. The cumulative returns for the prospect theory and mean‐variance model portfolios for a bullish market are also shown in Figure 4, left side. It seems that both the PT and MV curves show the same trend during this simulated bullish market. Finally, we checked the performance of the basic mean‐variance and prospect theory models’ optimal portfolios for a market with declining securities prices, also known as a bearish market. As can be seen, for simulated bearish market, the risk of the prospect theory’s optimal portfolio is surprisingly lower than the MV’s portfolio for all three markets, even though its returns are low in markets with 10 assets. Moreover, this portfolio shows a better performance for both 15 and 20 assets markets in terms of the ratio. It seems that the behavior‐based portfolio provides a lower risk in critical market situations in comparison with the traditional mean‐variance approach. It is a surprising result since, unlike the mean‐variance model, we didn’t set any boundary for our level of portfolio risk in the prospect theory model. In Figure 4 right side, we also illustrate the cumulative returns of our basic models’ portfolio for a simulated bearish market. It is obvious that both the PT and MV curves have a
decreasing trend with similar fluctuations while the cumulative returns of both portfolios fall to a negative point. 6.Conclusion In this study, two metaheuristic approaches, namely Particle Swarm Optimization (PSO) and Genetic Algorithm (GA), were applied to find an optimal solution for a prospect theory portfolio optimization model. In addition, a mean‐variance portfolio optimization model was used as a benchmark to compare the performance of our behavior‐based model. The objective function to be maximized was the utility function of each model calculated by using in‐sample historical data. To effectively use the available data, several subsets of the main dataset were randomly chosen. As a part of our study, the performance of these two metaheuristic algorithms in finding an optimal solution for a prospect theory model was compared as well. This comparison was based on the consistency and quality of the solution as well as the speed of convergence for each algorithm. The results showed that both heuristic algorithms could find an optimal solution in an acceptable amount of time. However, for the prospect theory model, PSO outperforms GA both in terms of consistency and speed of convergence to an optimal solution. We also performed several tests to specify the adequate population size and number of iterations for each heuristic algorithm to converge to the optimal solution of our models in a reasonable amount of time and with equally reasonable consistency.
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Further, we used the particle swarm optimization to analyze the performance of both prospect theory and mean‐variance models. Then, we tested the optimal portfolios selected by the prospect theory and mean‐variance models, using a historical out‐of‐sample data set as well as simulated data obtained from hypothetical bearish and bullish markets. This comparison seems to point in favor of the solutions obtained with the prospect theory model, especially for datasets with a greater number of assets. The optimal portfolio of this behavioral model is also more diversified than the optimal portfolio of our classical model. However, the portfolio of the mean‐variance model usually has a lower level of risk in both the basic and the constrained asset allocation problems. As we mentioned earlier, the return from the prospect theory model’s optimal portfolio is higher than that of our mean‐variance model. That is probably due to the existence of a reference point which induces the model to find a portfolio with a higher return without considering the increasing degree of risk. However, in a simulated bearish and bullish market, the performance of our prospect theory model is rather worse than that of the mean‐variance model. Finally, it is also important to know that these heuristic algorithms are not guaranteed to always find the best solution for an optimization problem. Yet, they can find an acceptable solution to a problem. References 1. Markowitz, H., 1952. Portfolio selection. The journal of finance, 7(1), pp. 77‐91. 2. Elton, E.J. and Gruber, M.J., 1997. Modern portfolio theory, 1950 to date. Journal of
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