You have to answer to the questions on an excel sheet (calculus and comments); Post your EXCEL sheet on BB OR sent it by e mail to this address: XXXXXXXXXX before 08:00 PM Returns predictability and...

You have to answer to the questions on an excel sheet (calculus and comments);


Post your EXCEL sheet on BB OR sent it by e mail to this address: [email protected] before 08:00 PM



Returns predictability and Asset Allocation


You have to invest in a portfolio “p” with only two assets: the S&P 500 index and the risk free asset. Your investment horizon is one month.
You are a so called “mean-variance” investor. In that case, you have to find the optimal weights w (invested in the S&P) and (1-w) (invested in the risk free asset) such that they maximize your expected utility function :


where E(Rp) denote the expected return of the portfolio, Var (Rp) the variance of the portfolio and A is your risk aversion coefficient (in your case A=7)
The data for the S&P 500 are on the excel sheet (“data”). The one month risk free rate is 5,34% (per year).

Q1
First, you assume that the market is efficient, then the expected return and the variance of the risky asset should be estimated with the historical average and historical variance.


Where is the historical average return and the residuals.


Required :




- Calculate analytically (find the formula) and numerically the weights in your optimal portfolio





- Trace the efficient frontier with these two assets.



Q2
To improve your portfolio, you think that you should test if returns can be predictable in the short run with past data. To test this assumption, you have first to run the following OLS regression:




Required :



-

Estimate your coefficients (and the variance of residuals), then check if they are different from zero with a T test. What can you say about market efficiency?




- Based on the previous results of the regression, find analytically the weights in your optimal portfolio when returns are predictable



Q3
Assume now that your investment horizon is 3 months. You have heard about a phenomenon called “the time diversification” but you don’t really know what it is about.


Required:



-
Calculate analytically (find the formula) and numerically the optimal weight of your portfolio for a 3 month investment horizon, if the returns are predictable or not … Explain what is “time diversification”



Q4
You have also heard that the variance is perhaps not the best way to estimate the risk of an asset in the financial world. You then decide to test the impact of the risk measure on the optimal portfolio weights ; your expected utility function is now of this form:


where E(Rp) denote the expected return of the portfolio, SemiVar (Rp) the semi variance of the portfolio and A is your risk aversion coefficient (in your case A=7)
The semi variance can be defined as an average of the squared deviations of values that areless
than the mean.
The semi variance of an asset is calculated in that way :


Where T is the number of observations in your sample and is the historical average return.


Required :

• 
  
  Calculate the weights in your optimal portfolio with such an utility function
  
  


• 
  
  Can you explain why the weights are different from those of Question 1 ?