Portfolio Volatility for Two Independent, Equal-Sized Stock Positions

Summary

The note explains how to calculate portfolio volatility when adding an independent stock position of the same size to an existing holding. Because the two returns are assumed independent, their variances add for the combined dollar positions. The example uses the stated volatilities for stocks A and B and takes the square root of the summed variances to obtain the portfolio standard deviation.

The key caveat is the meaning of “one stock” in the exercise. The answer treats the investment as buying one position in A and then an additional position of equal size in B, so it does not normalize the combined holdings to a fixed portfolio value. If the question instead intended weights within a fixed budget, those weights would be needed and portfolio volatility would be calculated differently. The prompt itself is acknowledged as ambiguous.

Key ideas

• For independent returns, the covariance term is zero and variances of dollar positions add.
• The example assumes the two stock positions have equal size and adds money for the second holding.
• Portfolio standard deviation is the square root of the combined variance.
• A fixed-budget portfolio requires explicit weights, which the exercise does not specify.

Tags

# A portfolio with two risky assets - Simple exercise


# A portfolio with two risky assets - Simple exercise












I was trying to solve the following exercise:

"Stocks A have $\mu_A=8\%$, $\sigma_A=2,5\%$ and stocks B have $\mu_B=6\%$, $\sigma_B=1,2\%$. Let us suppose that expexted returns are independent. What is the standard deviation of a portfolio made up of one stock A and one stock B?"

Since expected returns are independent, I immediately thought of the formula $$\sigma_P^2=w\sigma_A^2+(1-w)\sigma_B^2,$$ where $w$ represents the weight of stock A in the portfolio.

So I was wondering whether in the text there is a missing value, such as the weight of one stock in the portfolio. I don't think I can assume stocks are equally weighted. Anyway the final result is $2.77\%$. Thanks in advance.

## Answer by nbbo2 (score 0, accepted)

https://quant.stackexchange.com/a/33479

If you buy a position in stock A, the variance is $0.025^2$. If now you buy an additional position of the same size in stock B, the variance is $0.025^2+0.012^2=0.000769$. Independent variances can be added. However I have to say this is not well explained in the problem statement, which is very unclear. We are adding a new position with additional money, not splitting a given amount of money between the 2 stocks. $\sqrt{0.00769}=0.02773$

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