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!

  • $\begingroup$ What is "the book"? $\endgroup$ – Giskard May 25 '19 at 21:20

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

| improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.