Question

Suppose there are three stocks with the same expected returns of 10% per year and the same risk (standard deviation) of 100%. The correlation between any two of them is 50%.

a. What is the risk of the equal-weighted portfolio of two stocks?

b. What is the risk of the equal-weighted portfolio of three stocks?

c. What is the minimum possible risk of the portfolio of the three stocks?
  
d. If the third stock has a correlation of -50% instead of 50% with the rest, what is the risk of the equal-weighted portfolio of three stocks, and what is the minimum possible risk?

Answer 1

Answer -

Following information is given in the question

	Stock 1	Stock 2	Stock 3
Return	R1 = 10%	R2 = 10%	R3 = 10%
Standard Deviation	SD1 = 100%	SD2 = 100%	SD3 = 100%
Correlation	r1,2 = 0.50	r2,3 = 0.50	r1,3 = 0.50

Answer - a

Portfolio risk of equal-weighted portfolio of two stocks

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2

Where -

W1 & W2 is 0.50 each and remaining all is given above

On putting these figures in the formula, we get -

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2

(Portfolio Risk)2 = (100)2 * (0.5)2 + (100)2 * (0.5)2 + 2 * 100 * 0.5 * 100 * 0.5 * 0.5

(Portfolio Risk)2 = 2500 + 2500 + 2500

(Portfolio Risk)2 = 7500

Portfolio Risk =

Portfolio Risk = 86.60 or 86.60%

Answer will be same if we pick any of the two stock out of three for calculation of portfolio risk.

Answer - b

Portfolio risk of equal-weighted portfolio of three stocks

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + (SD3)2 * (W3)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2 + 2 * SD2 * W2 * SD3 * W3 * r2,3 + 2 * SD1 * W1 * SD3 * W3 * r1,3​​ ​​​​​

Where -

W1,  W2 & W3 is 0.3333 each and remaining all is given above in the beginning

On putting these figures in the formula, we get -

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + (SD3)2 * (W3)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2 + 2 * SD2 * W2 * SD3 * W3 * r2,3 + 2 * SD1 * W1 * SD3 * W3 * r1,3​​ ​​​​​

(Portfolio Risk)2 = (100)2 * (0.3333)2 + (100)2 * (0.3333)2 + (100)2 * (0.3333)2 + 2 * 100 * 0.3333 * 100 * 0.3333 * 0.5 + 2 * 100 * 0.3333 * 100 * 0.3333 * 0.5 + 2 * 100 * 0.3333 * 100 * 0.3333 * 0.5

​​ ​​​(Portfolio Risk)2 = 1111.11 + 1111.11 + 1111.11 + 1111.11 + 1111.11 + 1111.11

(Portfolio Risk)2 = 6666.67

​​ Portfolio Risk =

Portfolio Risk = 81.65 or 81.65%

Answer - c

In the instant question the risk and return in the three stocks are equal so as the correlation between any two of them, and from the portfolio risk calculated in Answer-b above we can observe that the risk would be reduced to 81.65%, if equal weights are put in each of three stocks in the portfolio, from 100% if no portfolio was made. Hence 81.65% is the minimum possible risk of the portfolio of three stocks if equal weights are put in each of three stocks in the portfolio.

Answer - d

If the third stock has a correlation of -50% instead of 50% with the rest, then following are the correlations between each of two stocks:

	Stock 1	Stock 2	Stock 3
Return	R1 = 10%	R2 = 10%	R3 = 10%
Standard Deviation	SD1 = 100%	SD2 = 100%	SD3 = 100%
Correlation	r1,2 = 0.50	r2,3 = -0.50	r1,3 = 0.50

Now, let us calculate he risk of the equal-weighted portfolio of three stocks:

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + (SD3)2 * (W3)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2 + 2 * SD2 * W2 * SD3 * W3 * r2,3 + 2 * SD1 * W1 * SD3 * W3 * r1,3​​ ​​​​​

Where -

W1,  W2 & W3 is 0.3333 each and remaining all is given above

On putting these figures in the formula, we get -

(Portfolio Risk)2 = (SD1)2 * (W1)2 + (SD2)2 * (W2)2 + (SD3)2 * (W3)2 + 2 * SD1 * W1 * SD2 * W2 * r1,2 + 2 * SD2 * W2 * SD3 * W3 * r2,3 + 2 * SD1 * W1 * SD3 * W3 * r1,3​​ ​​​​​

(Portfolio Risk)2 = (100)2 * (0.3333)2 + (100)2 * (0.3333)2 + (100)2 * (0.3333)2 + 2 * 100 * 0.3333 * 100 * 0.3333 * 0.5 + 2 * 100 * 0.3333 * 100 * 0.3333 * (-0.5) + 2 * 100 * 0.3333 * 100 * 0.3333 * (-0.5)

​​ ​​​(Portfolio Risk)2 = 1111.11 + 1111.11 + 1111.11 + 1111.11 - 1111.11 - 1111.11

(Portfolio Risk)2 = 2222.22

​​ Portfolio Risk =

Portfolio Risk = 47.14 or 47.14%

In the instant question the risk and return in the three stocks are equal, and from the portfolio risk calculated in above we can observe that the risk would be reduced to 47.14%, if equal weights are put in each of three stocks in the portfolio, from 100% if no portfolio was made. Hence 47.14% is the minimum possible risk of the portfolio of three stocks if equal weights are put in each of three stocks in the portfolio.