# 金融代写|投资组合代写Investment Portfolio代考|FINC3001

## 金融代写|投资组合代写Investment Portfolio代考|The Mathematics of Portfolio Selection with Higher Moments

Dealing with the third and higher portfolio moments quickly becomes cumbersome algebraically and can also be computationally inefficient unless caution is used. It is convenient to have similar formulas for the skew and kurtosis as for the portfolio mean and standard deviation
\begin{aligned} r_p &=\mathbf{w}^{\prime} \boldsymbol{\mu} \ \sigma_p^2 &=\mathbf{w}^{\prime} \mathbf{\Sigma} \mathbf{w} \end{aligned}
where $\boldsymbol{\mu}$ and $\boldsymbol{\Sigma}$ are the vector of expected returns and the covariance matrix of returns of the assets. In full generality, each moment of a random vector can be mathematically represented as a tensor. In the case of the second moment, the second moment tensor is the familiar $N \times N$ covariance matrix, whereas the third moment tensor, the so-called skew tensor, can intuitively be seen as a three-dimensional cube with height, width, and depth of $N$. The fourth moment tensor, the kurtosis tensor, can similarly be visualized as a four-dimensional cube.

When dealing with higher moments in the portfolio choice problem, it is convenient to “slice” the higher moment tensors and create one big matrix out of the slices. For example, the skew tensor (a three-dimensional cube) with $N^3$ elements and the kurtosis tensor (a fourth-dimensional cube) with $N^4$ elements, can each be represented by an $N \times N^2$ and an $N \times N^3$ matrix, respectively. Formally, we denote the $N \times N^2$ and $N \times N^3$ skew and kurtosis matrices by ${ }^{52}$
\begin{aligned} &\mathbf{M}3=\left(s{i j k}\right)=E\left[(\mathbf{R}-\boldsymbol{\mu})(\mathbf{R}-\boldsymbol{\mu})^{\prime} \otimes(\mathbf{R}-\boldsymbol{\mu})^{\prime}\right] \ &\mathbf{M}4=\left(\kappa{i j k l}\right)=E\left[(\mathbf{R}-\boldsymbol{\mu})(\mathbf{R}-\boldsymbol{\mu})^{\prime} \otimes(\mathbf{R}-\boldsymbol{\mu})^{\prime} \otimes(\mathbf{R}-\boldsymbol{\mu})^{\prime}\right] \end{aligned}
where each element is defined by the formulas
\begin{aligned} s_{i j k} &=E\left[\left(R_i-\mu_i\right)\left(R_j-\mu_j\right)\left(R_k-\mu_k\right)\right], i, j, k=1, \ldots, N \ \kappa_{i j k} &=E\left[\left(R_i-\mu_i\right)\left(R_j-\mu_j\right)\left(R_k-\mu_k\right)\left(R_l-\mu_l\right)\right], i, j, k, l=1, \ldots, N \end{aligned}

## 金融代写|投资组合代写Investment Portfolio代考|POLYNOMIAL GOAL PROGRAMMING FOR PORTFOLIO

In this section we discuss an approach to the portfolio optimization problem with higher moments that is referred to as the polynomial goal programming (PGP) approach. ${ }^{56}$ We suggested in the previous section that investors have a preference for positive odd moments, but strive to minimize their exposure to even moments. For example, an investor may attempt to, on the one hand, maximize expected portfolio return and skewness, while on the other, minimize portfolio variance and kurtosis. Mathematically, we can express this by the multiobjective optimization problem:
\begin{aligned} \max {\mathbf{w}} O_1(\mathbf{w}) &=\mathbf{w}^{\prime} \boldsymbol{\mu} \ \min {\mathbf{w}} O_2(\mathbf{w}) &=\mathbf{w}^{\prime} \mathbf{\Sigma} \mathbf{w} \ \max {\mathbf{w}} O_3(\mathbf{w}) &=\mathbf{w}^{\prime} \mathbf{M}_3(\mathbf{w} \otimes \mathbf{w}) \ \min {\mathbf{w}} O_4(\mathbf{w}) &=\mathbf{w}^{\prime} \mathbf{M}_4(\mathbf{w} \otimes \mathbf{w} \otimes \mathbf{w}) \end{aligned}
subject to desired constraints. The notation used in this formulation was introduced in the previous section. This type of problem, which addresses the trade-off between competing objectives, is referred to as a goal programming (GP) problem. The basic idea behind goal programming is to break the overall problem into smaller solvable elements and then iteratively attempt to find solutions that preserve, as closely as possible, the individual goals.

Because the choice of the relative percentage invested in each asset is the main concern in the portfolio allocation decision, the portfolio weights can be rescaled and restricted to the unit variance space $\left{\mathbf{w} \mid \mathbf{w}^{\prime} \mathbf{\Sigma} \mathbf{w}=1\right}$. This observation allows us to formulate the multiobjective optimization problem as follows:

\begin{aligned} \max {\mathbf{w}} O_1(\mathbf{w}) &=\mathbf{w}^{\prime} \boldsymbol{\mu} \ \max {\mathbf{w}} O_3(\mathbf{w}) &=\mathbf{w}^{\prime} \mathbf{M}3(\mathbf{w} \otimes \mathbf{w}) \ \min {\mathbf{w}} O_4(\mathbf{w}) &=\mathbf{w}^{\prime} \mathbf{M}4(\mathbf{w} \otimes \mathbf{w} \otimes \mathbf{w}) \end{aligned} subject to $$\begin{gathered} \mathbf{l}^{\prime} \mathbf{w}=1, \mathbf{‘}^{\prime}=[1,1, \ldots, 1] \ \mathbf{w}^{\prime} \mathbf{\Sigma}{\mathbf{w}}=1 \end{gathered}$$

## 金融代写|投资组合代写Investment Portfolio代考|The Mathematics of Portfolio Selection with Higher Moments

$$r_p=\mathbf{w}^{\prime} \boldsymbol{\mu} \sigma_p^2 \quad=\mathbf{w}^{\prime} \boldsymbol{\Sigma} \mathbf{w}$$

$$\mathbf{M} 3=(s i j k)=E\left[(\mathbf{R}-\boldsymbol{\mu})(\mathbf{R}-\boldsymbol{\mu})^{\prime} \otimes(\mathbf{R}-\boldsymbol{\mu})^{\prime}\right] \quad \mathbf{M} 4=(\kappa i j k l)=E\left[(\mathbf{R}-\boldsymbol{\mu})(\mathbf{R}-\boldsymbol{\mu})^{\prime} \otimes(\mathbf{R}-\boldsymbol{\mu})\right.$$

$$s_{i j k}=E\left[\left(R_i-\mu_i\right)\left(R_j-\mu_j\right)\left(R_k-\mu_k\right)\right], i, j, k=1, \ldots, N \kappa_{i j k} \quad=E\left[\left(R_i-\mu_i\right)\left(R_j-\mu_j\right)\left(R_k-\mu_k\right)\right.$$

## 金融代写|投资组合代写Investment Portfolio代考|POLYNOMIAL GOAL PROGRAMMING FOR PORTFOLIO

$$\max \mathbf{w} O_1(\mathbf{w})=\mathbf{w}^{\prime} \boldsymbol{\mu} \max \mathbf{w} O_3(\mathbf{w}) \quad=\mathbf{w}^{\prime} \mathbf{M} 3(\mathbf{w} \otimes \mathbf{w}) \min \mathbf{w} O_4(\mathbf{w})=\mathbf{w}^{\prime} \mathbf{M} 4(\mathbf{w} \otimes \mathbf{w} \otimes \mathbf{w})$$

$$\mathbf{1}^{\prime} \mathbf{w}=1,^{6^{\prime}}=[1,1, \ldots, 1] \mathbf{w}^{\prime} \mathbf{\Sigma} \mathbf{w}=1$$

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