Question

Problem 1: What is the variance of a portfolio with: w1 =0.2, w2 =0.8, σ₁² =10, σ₂² =20, and σ₁₂ =5.

Problem 2: a) If the stocks 1 and 2 have negative correlation $\rho$ ^₁₂ then their covariance σ¹² is also negative. Yes, no, uncertain. Explain. b) If stocks 1 and 2 are uncorrelated, i.e. $\rho$ ^₁₂=0 then their covariance is zero, Yes, no, uncertain. Explain c) If stocks 1 and 2 have variance σ²=16 each, could their covariance be equal to σ¹²= 20?

Problem 3: You have a portfolio of 4 stocks with equal shares invested in each stock. Variances of individual stocks are the same and equal to σ2 =16. Correlations between each pair of stocks is $\rho$ a) What would be the covariances between each pair of stocks as function of $\rho$ b) Find the variance of equally-portfolio of these 4 stocks (again as a function of $\rho$ c) What happens to the variance of your portfolio when $\rho$ increases. d) What would be the variance of your portfolio when = 1. Would it be larger or smaller than the variance of individual stocks $\sigma^2$ = 16?

problem 5. Assume you have N stocks σ2 =16. Correlations between each pair of stocks is $\rho$ = 0.125 = 1/8. a) What would be pairwise covariances? b) What would be the variance of an equally-weighted portfolio of N stocks c) Using Excel calculate the variance you found in b) for N=1,2,3,…, 30 stocks and plot it d) Calculate the % of total portfolio variance coming from pairwise covariances and also plot it as a function of N for N=1,2,3,…, 30. Comment on your results.

Answer 1

1. Standard deviation of a two stock portfolio is given by

Op = W;*W, *0; * 0;* Pij

and variance of portfolio is just the square of standard deviation

Var(p) = 0.2² * 10² + 0.8² * 20² + 2* 0.2* 0.8 *5

= 261.6

2. The covariance of two stocks = standard deviation of stock 1 * standard deviation of stock 2* Correlation coefficient between stock 1 and stock 2

a) From the above formula , if correlation coefficient is negative, the covariance has to be negative as standard deviations are always positive. So answer is YES

b) If stocks are uncorrelated i.e. correlation coefficient is 0 , covariance is also zero from the above formula. So answer is YES

c) If variance of two stocks is 16 each , their standard deviation is 4 each, So,

covariance = 4 * 4* Correlation coefficient between stock 1 and stock 2

Now, as the correlation coefficient can only take values between -1 and +1 , the covariance can also range from -16 to +16.

Hence under no circumstance, the covariance can be 20. So it is not possible for covariance to be 20