STA 4020: Statistical Modelling in Financial Markets Term 1 2022 Assignment 7
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STA 4020: Statistical Modelling in Financial Markets
CUHK(SZ) Term 1 2022
Assignment 7
(Due Dec 2)
1. Download the monthly returns of Apple, AT&T, Goldman Sachs from Yahoo in the period 2011/1–2020/12. Download the one-month LIBOR rate from website in the same period and use it as the monthly risk-free rate. Assume the risk aversion 1/入 = 4.
(a) (20pt) Use the asset returns in the period 2011/1–2015/12 to find the
MLE, unbiased estimate, James-Stein estimate, and Jorion estimate of the optimal portfolio. Use the asset returns in the period 2016/1–2020/12 to do out-of-sample test: compare the expected return, standard deviation of the return, Sharpe ratio of the estimated optimal portfolio, and the objective value of the mean-variance problem.
(b) (25pt) Starting from the end of 2015, at the end of every month, use the
historical asset returns (from 2011/1 to the end of that month) to find the MLE, unbiased estimate, James-Stein estimate, and Jorion estimate of the optimal portfolio and apply the resulting portfolio in the following month. Calculate the realized monthly returns using these four portfolios from 2016/1 to 2020/12 and perform out-of-sample test, i.e., compare the expected return, standard deviation of the return, Sharpe ratio, and the objective value of the mean-variance problem.
(c) (10pt) Compare the out-of-sample Sharpe ratios of the unbiased estimate, James-Stein estimate, Jorion estimate, in the settings of (a) and (b).
2. (20pt) Derive the posterior distributions of u and C using informative prior and 
prdct and Cˆprdct on lecture notes.
3. (25pt) Consider a market with one risk-free asset and three global assets: US stocks, Euro Stocks, and Japanese Stocks. The market capitalizations of these three assets are 35, 10, and 5 billion, respectively. The covariance matrix of the returns of these three assets are estimated as follows
US Euro Japan
US Euro Japan
7.344% 2.015% 3.309%
2.015% 4.410% 1.202%
3.309% 1.202% 3.497%
The market risk premium is estimated to be 6%. Apply the Black-Litterman model to compute the optimal allocation to US, Euro, and Japanese stocks. Assume his risk aversion degree 1/入 = 4. Assume A = 0r1, 1, and 10, and
P = 
、. q = 
、. Ω = 
、r
4. Suppose you are a wealth manager and believe that monthly stock returns in the market follow the Fama-French three factor model. You are looking into a particular portfolio strategy that is beta neutral, but has loadings 0.5 and 0.3 for SMB and HML, respectively. It has been estimated using historical data that the expected return of the market portfolio is 0.7%, the risk-free return is 0.2%, the expected payoffs for the zero-cost strategies associated with SMB and HML are 0.4% and 0.3%, respectively, and the expected return of the portfolio strategy is 0.4%. Is this portfolio strategy a good one from your perspective?
5. Consider the possibility of investing in a share of a certain oil well that will produce a payoff that is random. The expected payoff is $1,000 and the standard deviation of return is a relatively high 40%. The beta of the asset is - = 0r6. The risk-free rate is Tf = 10%, and the expected return of the market portfolio is 0.17. What is the value of this share of the oil venture, based on CAPM?
2022-12-02