<p>You could say the return on Johnson &amp; Johnson is a random variable <span class="math-container">$R_J$</span> with expected value <span class="math-container">$\mu^{\,}_J$</span> and variance <span class="math-container">$E[(R_P-\mu^{\,}_P)^2] = \sigma^2_J$</span>, and on Ford is <span class="math-container">$R_F$</span> with expected value  <span class="math-container">$\mu^{\,}_F$</span> and variance <span class="math-container">$E[(R_F-\mu^{\,}_F)^2] = \sigma^2_F$</span></p>

<p><span class="math-container">$R_J$</span> and <span class="math-container">$R_F$</span> perfectly correlated means <span class="math-container">$\frac{\mathbb E[(R_J-\mu^{\,}_J)(R_F-\mu^{\,}_F)]}{\sigma^{\,}_J\,\sigma^{\,}_F}=1$</span> so <span class="math-container">$\mathbb E[(R_J-\mu^{\,}_J)(R_F-\mu^{\,}_F)] = \sigma^{\,}_J\sigma^{\,}_F$</span></p>

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

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

CC BY-SA 4.0
2019-05-26T00:25:25.913Z
2019-05-26T00:25:25.913Z
Henry
<p>So i have this information:</p>

<p>Suppose that 60% of your portfolio is invested in Johnson &amp; 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%. </p>

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

<p>Expected portfolio return = (.60 × 8) + (.40 × 18.8) = 12.3%</p>

<p>However, then the book says:</p>

<p>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.</p>

<p>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.</p>

<p>Can somebody please tell me where this 40% is coming from ? Thanks!</p>

CC BY-SA 4.0
2019-05-25T21:20:41.047Z
What is "the book"?
Giskard
<stock-market><bonds><portfolio-theory>
How to calculate standard deviation of a portfolio?
Sara Salvante
# How to calculate standard deviation of a portfolio?

(Asked by: username_0 on 2019-05-25)

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!

**Tags:** stock-market, bonds, portfolio-theory
> **Comments:**
> **username_1**: What is "the book"?  

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## 1 answers

### Answer by username_2 on 2019-05-26. Score: 1 | Accepted answer

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


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