Investment Management

Table of Contents

Question 1 - Mean-variance portfolio optimization        

(a) Market risk premium and standard deviation        

(b) Capital allocation        

(c) Portfolio strategy        

(d) Expected utility        

Question 2- Fama-French 3 factors        

(a) Expected values and standard deviations        

(b) Profitability of portfolios        

(c) Expected utility        

Question 3- Testing the CAPM        

(a) Expected returns and CAMP betas        

(b) Slop coefficient and p- value of industrial portfolio        

(c) Slop coefficient and p- value of industrial portfolio for Market, SMB and HML        

Question 1 - Mean-variance portfolio optimization

(a) Market risk premium and standard deviation

In this scenario, we can assess an expected return by sum up the rate of return in excess of the risk-free rate. Base on the monthly returns provided from January 2000 to December 2015, we estimate the equilibrium value of the market risk premium as approximately 0.3169% per annum, which represent there is only 0.3069% rate of return that investors require to accept the uncertain outcomes associated with investment, relative to the return provided by a risk-free asset. Assuming that variance of market risk premium is approximately same weighty as variance of market portfolio, the variance and the standard deviation of market portfolio are approximately 0.2035% and 4.5110% respectively, which represent a stable portfolio investing by client can be supported with a lower standard deviation. Small standard deviations indicate lower degrees of risk.

(b) Capital allocation

In this circumstance, risk aversion with A is given 4.0 extremely high, the investor will prefer to have almost all their portfolio tied up in risk-free asset. According to the utility function, then one equation to determine their optimal allocation for the risky portfolio will be:

E (rm – rf )

A*Var (rm)

And (1-W*) is the subsequent weight in the risk-free asset. Given market risk premium and variance of market return are approximately 0.3169% and 0.2035%, respectively. Then, the optimal weight for portfolio calculated approximately is 0.3893, in what we can make an advise for investors to invest 38.93% of their wealth in the equity index and the remainder 61.06% of wealthy in risk-free bond to achieve capital optimal allocation.

(c) Portfolio strategy

To perform the portfolio’s value with optimal strategy relative to the other two strategies, one is market rate strategy which have 100% of initial wealth invested in the equity index while another one is risk-free strategy which held 100% of initial wealth invested in the risk-free asset. Figure 1.1 provides a more detailed perspective of the relationship between these three strategies. We notice that the market return strategy is extremely volatile and risk- free return is not, by comparing optimal return. Hence, we do not advise client to invest all initial wealth in the equity index because it is very risky.

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FIGURE 1.1 Optimal strategy returns.

(d) Expected utility

Note that an expected return on market portfolio be used to estimate the expected utility, where U= E(rm)- 0.5*A*Var(rm). The expected utility calculated with optimal portfolio approximately is 0.1577%. If the client had invested 100% of initial wealth in the equity index, the expected utility approximately is 0.0059%. If the client had invested 100% of initial wealth in the risk-free bond, the expected utility is equal to expected free rate E(rf) because the variance is zero, which approximately is 0.0960%. As shown in table1.1:

TABLE 1.1 Expected Utilities

Table 1.1 suggests investing in the optimal portfolio that offers the highest expected utility. To prefer more wealth in investing, we expect the increase in utility. In conclusion, investors should choose to invest in this optimal portfolio to achieve a maximum profit.

Question 2- Fama-French 3 factors

(a) Expected values and standard deviations

The expected values and standard deviations for SMB, HML, and Mkt-RF, calculated by using Excel, are shown in the table 2.1. The expected values of SMB and HML are very similar to that of Mkt-RF, and the standard deviations of SMB and HML are smaller than that of Mkt-RF.