Tutorial 2 1. Write the t-distribution for p = 2 without using matrix. 2. For SP500 index of 2014 (symbol gspc), please (a) t a t-distribution to the daily log-return. (b) calculate sample Skewness and test the symmetry with signicant level 0.01 (c) calculate sample Kurtosis and test whether it is 3 with signicance level 0.01 (d) t t-distribution and calculate the value-at-risk 3. For daily log returns of prices of MMM, MSFT, APPLE in 2014 (symbols, MMM, MSFT, AAPL), please (a) calculate their correlation coecient matrix (b) test whether the coecnents can be accepted as 0 with signicant level 0.01 4. Suppose R1 and R2 are returns of 2 assets and with expected return and variance 2 2 respectively r1 and r2 and variance-covariance 1 , 2 and 12. Find w 1 such that the portfolio wR1 + (1 w)R2 has the smallest risk. (Use the Dierential calculus) 5. Suppose R1, ...., Rp are returns of p assets and with expected return and variance 2 2 respectively r1, ..., rp and 1 , ..., p . If they are independent, what is the optimal weight, w1, ..., wp, with p w1 0, ..., wp 0, wi = 1 i=1 such that the portfolio (no short-selling) w1R1 + w2R2 + ... + wpRp has the smallest risk (Use the Dierential calculus) 6. For daily log returns of prices of MMM, MSFT, AT&T in 2014 (symbols, MMM, MSFT, T), suppose short-selling is not allowed. (a) calculate their average return and variance-covariance matrix (b) Find the ecient portfolio with targeted return 0.0005 (c) Find the minimum risk portfolio 2 (d) Find the tangency portfolio with risk-free return 0.0001 (e) With risk-free return 0.0001, if someone wants to reduce risk = 0.05 but remain the same Sharpe ratio, what is his investment. 7. Suppose that there are two risky assets, A and B, with expected returns equal to 2.3% and 4.5%, respectively. Suppose that the standard deviations of the returns are 6%and 11% and that the returns on the assets have a correlation of 0.17, (a) What portfolio of A and B achieves a 3% rate of expected return? (b) What portfolios of A and B achieve a 5.5% standard deviation of return? Among these, which has the largest expected return? 8. Suppose two stocks' returns has mean 0.01, 2 = 2 3 0.02, 0.03 and covariance matrix 2 3 5 4 4 7 Find the minimum risk portfolio based on the formula given in Lecture Notes 3(A) page 36 (which allows short-selling) 3 Find the minimum risk portfolio based on package 'fPortfolio' assuming short-selling is not allowed. Do you nd any dierence between the above calculations Draw the ecient frontier 9. suppose there are two assets with the same expected return r0, and the same risk 0. If the portfolio has not sort selling, RP = wR1 + (1 w)R2 with 0 w 1. Prove that (1) The return is always r0 (2) with the target return r0, the optimal portfolio has w = 0.5 (3) if the risk free rate rf is smaller than r0, what is the tangency portfolio 4