Questions:
How do you compute the standard deviation of a portfolio with two stocks?

James has a portfolio of two companies, Kensho and Bloomberg. He has invested $60,000 in Kensho and $40,000 in Bloomberg. Based on past performance Kensho has had a standard deviation of 19% in its returns whereas the standard deviation in the returns of Bloomberg is 15%. Moreover, the correlation between these two stocks is 0.30. James is trying to figure out what is the standard deviation of his portfolio so that he can decide whether he should change the composition of his portfolio or not. Based on his risk tolerance, he prefers to hold a portfolio with a standard deviation of at most 14%. Should James alter his portfolio?

Reference answers:
# Formula

The standard deviation of a portfolio can be found by calculating the variance of a portfolio and squaring it. The formula of variance of a portfolio is as follows:

Portfolio variance σp² = w1²σ1² + w2²σ2² + 2w1w2 Covariance(R1,R2)

- σp² = the variance of the portfolio 
- w1 = the weight of the first stock in the portfolio
- σ1 = the standard deviation of the first stock
- w2 = the weight of the second stock in the portfolio
- σ2  = the standard deviation of the second stock in the portfolio 
- Covariance(R1, R2) = This is the covariance between the two stocks

# Calculations

Formula: Portfolio variance σp² = w1²σ1² + w2²σ2² + 2w1w2 Covariance(R1,R2)

1. w1 = the weight of the first stock (Kensho) = ($40,000) / ($40,000 + $60,000) = 40% = 0.4

2. σ1 = the standard deviation of the first stock = 19% = 0.19

3. w2 = the weight of the second stock (Bloomberg) = ($60,000) / ($40,000 + $60,000) = 60% = 0.6

4. σ2  = the weight of the second stock = 15% = 0.15

5. Covariance(R1, R2) = This is the covariance between two stocks which can be found using the formula of correlation. 

The correlation between two stocks is given as follows:
Correlation (R1, R2) = Covariance (R1, R2) / (σ1 * σ2)

From this, the Covariance (R1, R2) = Correlation (R1, R2) * (σ1 * σ2)

Covariance (R1, R2) = 0.30 * (0.19 * 0.15) = 0.00855

These values can be inserted in the formula now 

Formula: Portfolio variance σp² = w1²σ1² + w2²σ2² + 2w1w2 Covariance(R1,R2)

Portfolio variance σp² = w1²σ1² + w2²σ2² + 2w1w2 Covariance(R1,R2)

Portfolio variance σp² = (0.4)²(0.19)² + (0.6)²(0.15)² + 2(0.4)(0.6)(0.00855)

Portfolio variance σp² = 0.005776 + 0.0081 + 0.004104

Portfolio variance σp² = 0.01798

Standard deviation σp can be found by taking the square root of the variance

Standard deviation σp = ( 0.01798)^½  = 0.1341 = 13.41%

The standard deviation of James’ current portfolio is 13.41% which is below his preference of 14%. Therefore, James should not change the composition of his current portfolio