## Portfolio Optimization, Risk Aversion, Constraints 组合投资最优化限制条件

Portfolio Optimization

Since an investor can trade risk for return along the efficient frontier, the motivation of optimizing a portfolio originates from the idea that a greater expected return for the same level of risk can be achieved if one can rearrange the portfolio.

Equity portfolios should always be constructed in a manner that risk, style, quality, safety, tax, social responsibility, concentration (sector or geographic), and turnover are all at an optimal level. The optimization is the process of changing the set of assets held to meet various criteria. It calculates optimal capital weightings for a basket of investments that gives the highest return for the lowest possible risk.

Like any models, key features of a portfolio optimization framework include:

• Flexibility of inputs, with embedded assumptions
• Ability to specify minimum and maximum constraints
• Options to maintain a certain level of return to ensure that the return will not be traded off at the expense of risk
• Ability to modify the correlation matrix, which is the key to optimization process; typically, quadratic, nonlinear, mixed integer programming are used in this phase
• Ability to modify portfolio dynamics both prior to and during the process
• Ability to count for downside risk
• Graphical display where the Monte Carlo simulation, including probability analysis on the specified target return level, can be observed

It came to my mind that one inevitable outcome is that if the portfolio is optimized too frequently, transaction costs may quickly offset or even deteriorate the value of the portfolio.

Risk Aversion

Portfolio optimization assumes that an investor may have some risk aversion and stock prices may exhibit significant differences among historical value, forecast values, and what the portfolio has experienced. Risk aversion represents the amount of return required by an investor for an extra unit of risk – the tradeoff between risk and return. Risk aversion identifies which portfolio on the efficient frontier a portfolio manager selects as optimal.

In the case of passive optimization, the absence of expected returns turns the optimization process above into a pure minimum risk optimization process, thus the risk aversion parameter is irrelevant.

Approaches of Determining Risk Aversion Parameter

1. Default Setting – Simply derive a default value, an average excess return, a variance number from the broad equity market, and set a decent starting point. Then a portfolio manager can modify by adopting a combination for a particular portfolio.
2. Risk Target – This optimization process maximizes the expected return for a given level of risk, also involves an objective function, and the constraints. However, risk target is more difficult to solve computationally.
3. Efficient Frontier – Such choice of risk aversion parameter requires the selection of one specific optimal portfolio on the efficient frontier. With increasing risk aversion, the optimal mean-variance portfolio moves along the active return to active variance efficient frontier through a linear combination of the maximum Sharpe Ratio and the minimum risk portfolios to the origin.
4. Alternatively, a portfolio manager can run an efficient frontier optimization and select the risk and return combination that best suits his expectation, and then extract the implied risk aversion parameter associated with his or her choice and return to the original optimization, where a full set of constraints are specified.

Constraints of Optimization

On the asset level, convex constraints include beta and turnover limit; non-convex constraints are restrictions such as maximum number of names or combination. Constraints may force a portfolio to take unwanted risk but yield no return, which are typically represented by closed convex sets and support function. A constrained optimal portfolio is the sum of positions that are aligned with the manager’s alpha and positions that are orthogonal to these alphas.