领航UTS IM-专题 2 Portfolio management & Markowitzportfolio theoryGiven constraint (1) Sum of weight =1 (2) expectedreturn à minimize varianceLimitations of Markowitz portfolio theory1) To solve the problem, we need lots of input parameters,assume n assets: expected return; standard deviation and covariance): 2n+n (n-1)/2parameters. 需要估计过多parameters，在信息处理上技术难度比较大，而且存在误差2) Have lots of estimation, difficult to measure meanof different assets. People hold different expectations with the parameters.MVS（minimum variance set)o 每个在MVS上的点代表着一种资产组合(a combination of assets = a portfolio) o 如何得到MVS？§ 给定回报率(R)，使的风险最小化(standard deviation) · à知道如何推导（对于仅有two asset的情况下）§ 给定风险(standard deviation)，使的回报率最大化(R) o All points onthe MVS are attainable, but not all points on MVS are efficient o 分清以下几种情况，和相对应的diversification效果§ All risky asset§ With a risk-free asset? § With short selling? § Without short selling?《图一》 · Efficient frontier§ If consist all risky asset?If include a risk-freeasset? § G-MVP (风险最小，回报率最大—> 注意该点不是最优点(not optimal combination) 为什么？最优点是由什么决定的？) o Efficient frontier 只是MVS的一部分 (only the upper part of MVS, starting fromMVP)o 不在efficient frontier上但可以达到(attainable)的投资组合，均是inefficient的。为什么？ · Find optimal combination o 必须知道two things§ 1. Constraint: efficient frontier à MVS§ 2. Utility function à indifference curveo 当indifference curve和efficient frontier相切时获得的点即为最优点o 如何获得？§ Optimal risky portfolio§ Optimal investor portfolio 《图二》 · Effect of diversificationo Reduce risk by combining more asserts into aportfolio. This is because not all asset prices move in the same direction atthe same time.o Diversification is affected by§ Correlations between assets: lower correlation means greater diversification benefits(because less tendency for stock returns to move together)· Correlation = +1 à no diversification· Correlation = -1 à greatest diversification is achieved Number of assets included in the portfolio à more assets mean reater diversification
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