Question

2. A portfolio consists of $10,000 of stock A and $20,000 of stock B. Stock A has an expected return of 12% and a standard deviation of 14%, while stock B has an expected return of 10% and a standard deviation of 12%.

a. Find the mean and standard deviation of the portfolio return (percentage form) when the correlation between the two stocks’ return is -1, 0, 1? (i got the answers for these but i want to be certain that i got my answered correctly)

b. Assume that the stock returns are jointly normally distributed, what is the probability that the portfolio return is less than 0% when the correlation between the two stock’s return is 0%.

c. How do correlations between the stocks affect the mean and variance of the portfolio return? If we use the standard deviation of portfolio return as a measure of portfolio risk, what is the value of the correlation when the portfolio risk is largest? , what is the value of the correlation when the portfolio risk is smallest?

Answer 1

	Share	Mean	SD
A	10000	12%	14%
B	20000	10%	10%

Total number of shares = 30000

Proportion of A= 10000 / 30000 = = 1 / 3

Proportion of B= 20000 / 30000 = = 2 / 3

a. Find the mean and standard deviation of the portfolio return (percentage form) when the correlation between the two stocks’ return is -1, 0, 1? (i got the answers for these but i want to be certain that i got my answered correctly)

Mean of the portfolio =

= 1/3 * 12%+ 2/3 * 10%

Mean =

Standard deviation of the protfolio =

=

Where correlation

	Corr = -1
			SD
A	0.33	0.11	14%	0.0196
B	0.67	0.44	10%	0.01
	Var	0.0035
	SD	0.0593
	SD =

	Corr = 0
	Proportion		SD
A	0.33	0.11	14%	0.0196
B	0.67	0.44	10%	0.01
	Var	0.0066
	SD	0.0814
	SD =

	Corr = 1
	Proportion		SD
A	0.33	0.11	14%	0.0196
B	0.67	0.44	10%	0.01
	Var	0.0097
	SD	0.0987
	SD =

b. Assume that the stock returns are jointly normally distributed, what is the probability that the portfolio return is less than 0% when the correlation between the two stock’s return is 0%.

Let Rp represent portfolio return variable

The mean and the SD (corr = 0) are calculate above.

We want to find the probability that the return is less than 0%.

P(Rp < 0) = P( Z < )

= P(Z < -1.31)

= 1 - P(Z < 1.31)

= 1 - 0.90503 ...........found using the normal distribution tables

  

c. How do correlations between the stocks affect the mean and variance of the portfolio return? If we use the standard deviation of portfolio return as a measure of portfolio risk, what is the value of the correlation when the portfolio risk is largest? , what is the value of the correlation when the portfolio risk is smallest?

As seen in the formual above, the correlation does not affect the mean return but only the variance of the portfolio. Since thee correlation term is added in the a variance, we can conclude that the higher the correlation higher is the variance (risk). Looking at it in terms of diversity, a diversified portfolio will reduce the risk is the assets are negatively correlated, since decrease ones return will be balanced by the profit in other. If the assets are positvely correlated then the risk is in the same direction and there will be a big loss if there is an unwanted market change. When the assets are not correlated, one's performance will not affect the other. The loss will be there but only for one asset and it will not be able to be balanced due to other.

We can see that when correlation is 1, the protfolio risk is highest and when correlation is -1, the protfolio risk is lowest.