Abnormal Portfolio Asset Allocation Model: Review

It has been widely studied how investors will allocate their assets to an investment when the return of assets is normally distributed. In this context usually, the problem of portfolio optimization is analyzed using mean-variance. When asset returns are not normally distributed, the mean-variance analysis may not be appropriate for selecting the optimum portfolio. This paper will examine the consequences of abnormalities in the process of allocating investment portfolio assets. Here will be shown how to adjust the mean-variance standard as a basic framework for asset allocation in cases where asset returns are not normally distributed. We will also discuss the application of the optimum strategies for this problem. Based on the results of literature studies, it can be concluded that the expected utility approximation involves averages, variances, skewness, and kurtosis, and can be extended to even higher moments.


Introduction
How investors will allocate capital when income (return) is not normally distributed. Two approaches can be used to study this problem. The first approach is based on the direct maximization of utility expectations, under alternative assumptions of distribution for capital income (assets). The advantage of this approach is that it provides a real evaluation of the utility of expectations, also that the optimal portfolio is the actual resolution of the original problem (Ruppert, 2004). Furthermore that for most applications, numerical integration is difficult to use to maximize the expectation of the established utility. Consequently, most studies focus on a very small amount of income (two or three) to reduce computational burden (Denuit et al., 2005;Sukono et al., 2019). e-ISSN 2722-1156p-ISSN 27722-1164 The second approach is based on the optimization of a problem that does not require numerical integration. Typically, this approximation includes the moment of income-portfolio distribution. The main difficulty with this approach is how to define high moments that influence the utility of hope. Although various solutions have been introduced, the discussion in this paper focuses on Taylor's expansion of the utility function, which naturally produces utility expectations that depend on the high moment linearity of portfolio income (Batuparan, 2001).

Direct Maximum Utility Hope
In the direct maximization of this utility, it will cover optimization issues, case-variance cases, and numerical integration.

Optimization Problem
Suppose an investor allocates portfolio assets by maximizing the utility of hope for his assets in the next period . It is assumed that there are no costs for short-selling. Wealth at the beginning of the period is denoted by t W , assign the changes equal to one. The next period's wealth is given by where is the fractional vector of wealth allocated to capital, with the limitation that the weight of the portfolio, at the time t , is one, that is where e is a vector ) 1 , (n . Therefore, income on the portfolio is provided by (Jondeau et al., 2007;Denuit et al., 2005).
It is assumed that the investor has a utility function U that depends on the level of wealth in the coming period 1  t W . Formally, optimal portfolio weights are obtained by maximizing the degree of utility utility The first-degree condition of the optimization problem is where ) ( j U states the derivative jth of the utility function. Because the weight of the portfolio must amount to one, the first-degree requirement for investor problems reduces to the limit ) where 1  t  is the income vector from asset 1 to 1  n the excess asset n . In some cases, the optimization-level requirement (2) can help get a solution to the optimization problem. But in general, it is more appropriate to directly maximize the utility of hope, write back as stated pdf combined income vectors in time (Beronilla et al., 2007).

Case Variance Mean
In some cases, the problem of double integration reduces average problems -mere variance. Consider for this case where the utility function is chosen exponential utility and where income is assumed to be normally distributed. The exponential utility function (or HOW, short for Constant Absolute Risk is the absolute risk aversion coefficient. After all, the expectation income vector ) 1 , (n , and the covariance matrix ) , ( n n for risk assets are successively expressed with )' ,..., ( Therefore, problem (1) can be rewritten as is the expected gross income of the portfolio, and form, which is the average objective function -variance. On the other hand, some utility functions state directly the average size -the variance for variable income distribution. This is the case of the quadratic utility function: only includes the mean and variance of the distribution. Therefore, if (i) the income distribution for a portfolio is asymmetric, (ii) the investor's utility function is a degree higher than quadratic, and (iii) the average and variance are not completely determined by the distribution, then the third or higher moment and the coefficient mark must be considered (Denuit et al., 2005).

Numerical Integration
Tauchen and Hussey (1991) provide a numerical solution to equations such as (2.3) with quadratic. A quadratic rule for functions The choice of abscissa i u and weight i w , n i ,..., 1  depends only on the pdf f , but not on the functions h that are integrated. For a univariate quadratic, the Gauss rule is a discrete approximation to be f determined by the moment method using the moment beyond the above 1 2  M . The multiple-dimension quadrature is more in demand, because it has to calculate the integral dimension-n point so that the actual integration of all polynomials of five or more degrees (Jondeau et al., 2007).

Settlement Approximation Based on Moment
Solving general problems, which may not be simple, we may focus on the approximation of this problem based on high moments.

Approximate Utility Utilities
Since we have been primarily interested in measuring the effect of high moments on asset allocation, we are now approximating the utility of hope by expanding the Taylor series around wealth of hope. In this context, the utility function can be approximated by the following form, when the expansion is formed above the fourth degree (to alleviate, we ignore the time index) and  is the rest of Taylor's. The expectation utility is simply approximated by ) ( Optimal portfolio weights can be alternatively obtained by maximizing form (26) or by solving equation (7). Investigation of relations (7) shows that calculating this form would be simpler if the variance, skewness, and kurtosis of portfolio income and derivatives are known (Denuit et al., 2005).
We can also pay attention to the rank of utility functions (or CRRA, for Constant Relative Riskversion), defined as where  the size of the investor's relative risk aversion constant (CRRA). In contrast to the HOW utility, CRRA does not converge to an asymptote for an increase in wealth.
In the case of the CRRA utility function, using the form (5), we obtain the following approximation for the utility of expectation The FOC for the CRRA function can be rewritten, after simplification Optimal portfolio weights can be obtained alternatively by maximizing form (2.9) by solving equation (11).

Investor's Preferences Based on Moments
In the case of the CARA and CRRA utility functions, skewness weights and high moments in the expected utility are estimated depending on the risk-avoidance parameters. this is a preference for skewness and risk-avoidance for kurtosis. Therefore, moment jth weights are strongly related to derivatives jth of utility functions (Dowd, 2002;Jorion, 2002).
Usually, risk aversion investors are assumed to have utility functions with the first and second derivatives as follows The first condition means that the tool has a positive marginal utility for wealth, for example, unpleasant concerning wealth. The second condition is that the decline in marginal utility with wealth affects risk avoidance.
To investigate preferences concerning high moments, the following definitions are very useful. Definition 1. An investor is consistent in the direction of preference for the moment that has a utility function for which the derivative has the same sign whatsoever. The investor is loudly consistent in the direction of preference for the moment if the derivative of-has an exact inequality concerning zero.
Then we have the following theorem Theorem 2 (Scott and Horvath, 1980). Investors with the positive marginal utility of wealth for all levels of wealth (condition (2.12)), risk aversion consistent at all levels of wealth (condition (2.13)), and hard consistency of preference moments, will have positive preferences for positive skewness, ie (14) Theorem 3 (Scott and Horvath, 1980). Consistent risk retention (condition (13)), hard consistency of moment preference and positive preference for positive skewness (condition (14) , there will be 15 different elements for the covariance matrix, 35 elements for the co-skewness matrix, and 70 elements for the co-kurtosis matrix. Define co-skewness and co-kurtosis between asset income as

Optimal Portfolio Allocation
Now how to complete the allocation of assets, when high moments are included in the problem of optimization. Athayde and Flores (2004) make a quasi-analytic solution to the problem of efficient portfolios by defining moments as tensors and then solving optimization problems. The solution, using non-linear optimization methods. Equation (2.11) can be rewritten as  , and 3  are nonlinear functions of  . As many n of these equations can be easily solved numerically, using a standard optimization package (Denuit et al., 2005;Mood et al., 1963).

Conclusion
In optimizing portfolio asset allocation where returns are non-normal distributed, it can be done by direct maximization of utility expectations, and approximation of settlement based on moments. The direct maximization approach to expectation utility assumes that investors have a utility function that depends on the wealth of expectations for the coming period, and in certain cases, exponential or quadratic utility functions are often used. Approximate settlement based on the moment, measuring the effect of the high moment on asset allocation is used to estimate the utility of expectations by expanding the Taylor series around the wealth of expectations. The expected utility of this expectation will involve averages, variances, skewness, and kurtosis, and can be extended to even higher moments.