# 金融代写|投资组合代写Investment Portfolio代考|FINC3017

## 金融代写|投资组合代写Investment Portfolio代考|PORTFOLIO CONSTRAINTS COMMONLY USED IN PRACTICE

Institutional features and investment policy decisions often lead to more complicated constraints and portfolio management objectives than those present in the original formulation of the mean-variance problem. For example, many mutual funds are managed relative to a particular benchmark or asset universe (e.g., S\&P 500, Russell 1000) so that their tracking error relative to the benchmark is kept small. ${ }^3$ A portfolio manager might also be restricted on how concentrated the investment portfolio can be in a particular industry or sector. These restrictions, and many more, can be modeled by adding constraints to the original formulation.

In this section, we describe constraints that are often used in combination with the mean-variance problem in practical applications. Specifically, we distinguish between linear, quadratic, nonlinear, and combinatorial/ integer constraints.

Throughout this section, we denote the current portfolio weights by $\mathbf{w}_0$ and the targeted portfolio weights by w, so that the amount to be traded is $\mathbf{x}=\mathbf{w}-\mathbf{w}_0$.

High portfolio turnover can result in large transaction costs that make portfolio rebalancing inefficient. One possibility is to limit the amount of turnover allowed when performing portfolio optimization. The most common turnover constraints limit turnover on each individual asset
$$\left|x_i\right| \leq U_i$$
or on the whole portfolio
$$\sum_{i \in I}\left|x_i\right| \leq U_{\text {portfolio }}$$
where $I$ denotes the available investment universe. Turnover constraints are often imposed relative to the average daily volume (ADV) of a stock. For example, we might want to restrict turnover to be no more than $5 \%$ of average daily volume. Modifications of these constraints, such as limiting turnover in a specific industry or sector, are also frequently applied.

## 金融代写|投资组合代写Investment Portfolio代考|Risk Factor Constraints

In practice, it is very common for portfolio managers to use factor models to control for different risk exposures to risk factors such as market, size, and style. ${ }^4$ Let us assume that security returns have a factor structure with $K$ risk factors, that is
$$R_i=\alpha_i+\sum_{k=1}^K \beta_{i k} F_k+\varepsilon_i$$
where $F_k, k=1, \ldots, K$ are the $K$ factors common to all the securities, $\beta_{i k}$ is the sensitivity of the $i$-th security to the $k$-th factor, and $\varepsilon_i$ is the nonsystematic return for the $i$-th security.

To limit a portfolio’s exposure to the k-th risk factor, we can impose the constraint
$$\sum_{i=1}^N \beta_{i k} w_i \leq U_k$$
where $U_k$ denotes maximum exposure allowed. To construct a portfolio that is neutral to the $k$-th risk factor (for example, market neutral) we would use the constraint
$$\sum_{i=1}^N \beta_{i k} w_i=0$$

## 金融代写|投资组合代写Investment Portfolio代考|Risk Factor Constraints

$$R_i=\alpha_i+\sum_{k=1}^K \beta_{i k} F_k+\varepsilon_i$$

$$\sum_{i=1}^N \beta_{i k} w_i \leq U_k$$

$$\sum_{i=1}^N \beta_{i k} w_i=0$$

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