How to calculate standard deviation of a portfolio?

Question

So i have this information:

Suppose that 60% of your portfolio is invested in Johnson & Johnson (JNJ) and the remainder is invested in Ford. You expect that over the coming year JNJ will give a return of 8% and Ford, 18.8%.

Then I know that expected return on the portfolio is simply a weighted average of the expected returns on the individual stocks:

Expected portfolio return = (.60 × 8) + (.40 × 18.8) = 12.3%

However, then the book says:

We said earlier that if the two stocks were perfectly correlated, the standard deviation of the portfolio would lie 40% of the way between the standard deviations of the two stocks.

There have been no other calculations on the book and only after this statement should i start the calculations in order to understand the portfolio risk.

Can somebody please tell me where this 40% is coming from ? Thanks!

Answer 1

You could say the return on Johnson & Johnson is a random variable $R_J$ with expected value $\mu^{\,}_J$ and variance $E[(R_P-\mu^{\,}_P)^2] = \sigma^2_J$, and on Ford is $R_F$ with expected value $\mu^{\,}_F$ and variance $E[(R_F-\mu^{\,}_F)^2] = \sigma^2_F$

$R_J$ and $R_F$ perfectly correlated means $\frac{\mathbb E[(R_J-\mu^{\,}_J)(R_F-\mu^{\,}_F)]}{\sigma^{\,}_J\,\sigma^{\,}_F}=1$ so $\mathbb E[(R_J-\mu^{\,}_J)(R_F-\mu^{\,}_F)] = \sigma^{\,}_J\sigma^{\,}_F$

Now consider your portfolio $60\%$ Johnson & Johnson and $40\%$ Ford. It has return $R_P= 0.6 R_J +0.4 R_F$ with expected value $\mu^{\,}_P= 0.6 \mu^{\,}_J +0.4 \mu^{\,}_F$ by linearity of expectation, so the actual return and expected return are $40\%$ of the way from the corresponding figures for Ford to the figures for Johnson & Johnson

Now consider the variance of the return on the portfolio $$\sigma^2_P = \mathbb E[(R_P-\mu^{\,}_P)^2] \\= \mathbb E[(0.6 (R_J-\mu^{\,}_J) +0.4 (R_F-\mu^{\,}_F))^2] \\ = 0.6^2 \sigma^2_J +2\times 0.6 \times 0.4 \sigma^{\,}_J\,\sigma^{\,}_F + 0.4^2 \sigma^2_F \\= (0.6 \sigma^{\,}_J + 0.4 \sigma^{\,}_F)^2 $$ so taking square roots gives $\sigma^{\,}_P = 0.6 \sigma^{\,}_J + 0.4 \sigma^{\,}_F$, i.e. the standard deviation of the return on the portfolio is again $40\%$ of the way from the corresponding figure for Ford to the figure for Johnson & Johnson