1 a) The graph of the probability distribution of X is shown below:

1 b) As the sum of the probabilities of all expectations equals to 1 (0.108+0.562+0.33+0=1), X is a random variable.

1 c&d)The expected value, variance and standard deviation of X are calculated as follows:

Interest rate, xi (%) Probability, Pr Pr * xi(%) Pr * (xi-E(X))^2
1.75 0.000 0.0000 0.00000
2 0.330 0.6600 0.01248
2.25 0.562 1.2645 0.00173
2.5 0.108 0.2700 0.01008 Expected value of X, E(X) 2.1945 Variance of X 0.02429 Std dev of X (var) 0.15587

2 a) The formula for calculating sample covariance (sxy) and correlation (rxy)of X and Y are as follows:

Based on the above formulae, the sample covariance (Sxy) and correlation (rxy) of X and Y are calculated as follows:-

HSBC monthly CLP monthly return, xi (%) (xi-E(X))^2 return, yi (%) (yi-E(Y))^2 (xi-E(X))(yi-E(Y))
Mar -9.4632 39.6726 -0.7076 11.4956 21.3556
Apr 3.0562 38.6981 1.1867 2.2387 -9.3077
May -2.7877 0.1420 3.6863 1.0068 0.3782
Jun -5.9793 7.9226 3.6814 0.9970 -2.8104
Jul -0.6489 6.3286 5.5678 8.3225 7.2574 Expected return E(X), E(Y) -3.16458 2.6829 Cov x, y (Sxy) 4.2183 n-1 = 5-1 = 4 Variance of X, Y 23.1910 6.0151
Std derivation Sx Sy 4.8157 2.4526 Correlation (rxy) 0.3572

2 b) The correlation between HSBC and CLP is 0.3572 as calculated in 2(a) above. It means that the performances of the two shares are positively correlated though the positive correlation is weak.

One simple measure of financial risk is variance. The variance of a portfolio of stocks can be calculated using below formula.

var(aX + bY ) = a2var(X) + b2var(Y ) + 2abcov(X; Y )

If all money invested in HSBC, the variance of the portfolio is 23.191, i.e., var of X, see 2(a)

If all money invested in CLP, the variance of the portfolio is 6.0151, i.e., var of Y, see 2(a)

If 50% invested in HSBC and 50% invested in CLP, the variance of the portfolio calculated from the above formula is 9.4107.

Hence, investing 50% of one’s money in HSBC and CLP shares respectively with a portfolio variance of 9.4107 can effectively reduce his risk as compared to investing all his money in HSBC, which has a greater variance of 23.191. However, investing 50% in each of HSBC and CLP shares cannot reduce his risk as compared to investing all his money in CLP, which has a lower variance of 6.0151.

If he invests small portion of his money in HSBC and CLP and holding the remaining money in cash, the variance of his portfolio can be effectively reduced as compared to putting all his money in HSBC or CLP based on the above formula. However, if he invests the remaining money into other securities, the variance of his full portfolio may vary depending on his asset allocation. For example, if he invests the remaining money into one highly risky asset, he may not be able to reduce his risk. On the other hand, if he invests his remaining money into a number of securities, say, 20 securities, he may be able to reduce his risk by the effect of diversification.