✏️ CAPM

$$\begin{aligned} &(E(r_S) - r_F) = β[E(r_M) - r_F] \\ &E(r_S) = r_F + β[E(r_M) - r_F] \end{aligned}$$

✏️ Suppose that the rate of return on T-bills (generally considered to be risk free) is 5%. Suppose, also, that the Expected return on the market is $E(r_M) = 12\%$. Suppose that you are analyzing a stock and predict that the expected return of this stock is 17%. What would it’s beta have to be for it to have an expected return of 17%.

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We can solve this question using “Plug and Chug:”

Plug and chug: (help)

1. Equation:
   

   We know:
   $rF = 5\%$
   $E(r_M) = 12\%$
   $E(r_S) = 17\%$
   $β = ?$
   An equation that connects all of these is:
   $E(r_S) = r_F + β[E(r_M) - r_F]$

2. Plug:🔌
   $17\% = 5\% + β[12\% - 5\%]$
3. Solve: 🚂
   

   $17\% = 5\% + β[7\%]$
   $12\% = β[7\%]$
   $\frac{12\%}{7\%}= β = \frac{12}{7}=1.7143$

4. Reflect: 🧠
   β is 1.7143. Given that this stock is very risky (ie high beta), it should have a high return. ✅

You can apply the CAPM formula not just to specific securities, but also to portfolios of securities:

✏️ Suppose β=.9 for the portfolio that I am managing. The expected return on the market is 13%, and the risk free rate is 2%, then what is the expected return on my portfolio? (based on the CAPM).

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We would expect a number less than 13%, because $β < .9$
$E(r_S) = r_F + β[E(r_M) - r_F] = 2\% + .9[13\% - 2\%] = 11.9\%$

✏️ Suppose $r_F = 5\%$
$E(r_M) - r_F = 7\%$
Fill in the rest of the table…

Stock	Beta	$E(r_S)$
Amazon	2.20	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 2.2 \times (7\%) = 20.4\%$
IBM	1.59	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 1.59 \times (7\%) = 16.13\%$
Disney	1.26
Microsoft	1.13
Boeing	1.09
Starbucks	.69
ExxonMobil	.65

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Stock	Beta	$E(r_S)$
Amazon	2.20	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 2.2 \times (7\%) = 20.4\%$
IBM	1.59	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 1.59 \times (7\%) = 16.13\%$
Disney	1.26	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 1.26 \times (7\%) = 13.8\%$
Microsoft	1.13	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 1.13 \times (7\%) = 12.9\%$
Boeing	1.09	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + 1.09 \times (7\%) = 12.6\%$
Starbucks	.69	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + .69 \times (7\%) = 9.8\%$
ExxonMobil	.65	$E(r_S) = r_F + β[E(r_M) - r_F] = 5\% + .65 \times (7\%) = 9.6\%$

✏️ Suppose that the rate of return required by the market (ie E(r_S)) for a stock with β=1.3 is 15% and suppose that $r_F = 2\%$. What is the Expected Return on the Market Portfolio? See the “jargon reference”:
Jargon reference:
“Expected Return of the Market Portfolio” = $E(r_M)$

• Whenever you see $r_M$, you should think of the return that you would earn if you owned every single security in the market. This is known as the “market portfolio.” $E(r_M)$ is just the return on this market portfolio

“Return required by the market for a specific stock” = $E(r_S)$

• The CAPM formula, $E(r_S) = r_F + β[E(r_M) - r_F]$, tells us the return that investors will require if they are going to buy the shares of a given security. In other words, $E(r_S)$ always refers to the return required by the market.

The word “market” appears in both phrases, but it means different things.

• In “market portfolio” it means that the portfolio contains all securities in the stock and bond markets.
• In “required by the market” it means “what investors will require.”

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This problem is a little different because we are told $E(r_S)$. We need to use algebra to calculate $E(r_M)$. We’ll just “Plug and Chug:”

Plug and chug: (help)

1. Equation:
   $E(r_S) = r_F + β[E(r_M) - r_F]$
2. Plug:🔌
   $15\% = 2\% + 1.3 \times [E(r_M) - 2\%]$
3. Solve: 🚂
   

   $13\% = 1.3 \times [E(r_M) - 2\%]$
   $\frac{13\%}{1.3} = E(r_M) - 2\% = 10\%$
   $E(r_M) = 12\%$

4. Reflect: 🧠
   The β is greater than 1, so we expect the return on the stock to be greater than the return on the market. It is.