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## 11.6 The Capital Asset Pricing Model (CAPM)

PLEASE NOTE: This book is currently in draft form; material is not final.

### Learning Objectives

1. Explain the difference between market risk and firm-specific risk.
2. Calculate a portfolio’s beta from its components.
3. Calculate an asset’s expected return using CAPM.

Riskier assets should demand higher returns. But not all risk is the same, because of diversification. For example:

Figure 11.2 Standard Deviation of a Stock Portfolio vs. Number of Random US Securities

Once we approach about 40 stocks in a portfolio (assuming they are relatively random and not, for example, all technology stocks) we cease to see significant reductions in standard deviation. There are some mutual funds and ETFs that attempt to own virtually the entire market of available securities! Nonetheless, there is a certain minimum amount of risk that must be borne by those investing in stocks due to factors that affect the entire market, which we call market riskMinimum amount of risk that must be borne by those investing in stocks due to factors that affect the entire market. Also called systematic risk or non-diversifiable risk. (also called systematic risk or non-diversifiable risk). Only by adding other asset types (for example, bonds or real estate) can we gain some additional diversification benefits.

The additional risk that any stock has above its market risk is called firm-specific riskThe additional risk that any stock has above its market risk, that encompasses all factors that would affect the company’s stock price, but not the market in general. Also called unsystematic, diversifiable, or unique risk. (also called unsystematic, diversifiable, or unique risk). This encompasses all factors that would affect the company’s stock price, but not the market in general. For example, the CEO being indicted for fraud or the approval of a new popular pharmaceutical would be firm-specific event. Diversification reduces firm-specific risk but leaves market risk.

Since firm-specific risk can be virtually eliminated by diversification, investors shouldn’t demand a higher return to compensate for it. Market risk, however, can’t be diversified away, so investors should demand higher returns for those securities which have a larger sensitivity to broad market events (like the state of the economy). For example, airlines tend to do poorly when the economy tanks (since in a bad economy, people travel less for both business and leisure), so investors should demand a higher return for airlines. Discount retailers, like Walmart, tend to be relatively less affected by a downturn in the economy, so they have less market risk; as a result, investors should demand a lower return for Walmart than an airline. Specifically, we are looking at a stock’s risk premiumThe expected return in excess of the risk free rate. (the expected return in excess of the risk free rate), as this is the portion of the return compensating investors for bearing the extra market risk associated with holding the stock.

One way to attempt to quantify a company’s market risk is to calculate its historical betaThe ratio of a company’s risk premium versus the market’s risk premium.. Beta represents the ratio of a company’s risk premium versus the market’s risk premium. Typically, beta is found by some method of statistical regression of past company returns in excess of the risk-free rate compared to a broad market index’s returns in excess of the risk-free rate on a daily, weekly, or monthly basis. Of course, what investors really want is a company’s beta going forward, so the historical beta will be a proxy for this value. Most financial sites provide their projection of public companies’ betas and will usually describe any adjustments they make from the historical calculation.

So how is beta used? A stock that tends to move in the same direction as the market (most stocks) will have a positive beta. Stocks with a beta of 1.0 will move in the same proportion as the market: if the market is up 1% over the risk-free rate, these stocks will also tend to be up 1% (statistically speaking). Stocks with betas higher than 1.0 are more sensitive to market moves; betas lower than 1.0 indicate less sensitivity. The beta of the broad market index must be, by definition 1.0 (since it must move in perfect tandem with itself).

Since the betas of all stock are perfectly correlated with market risk (and, hence, each other), the beta of a portfolio of stocks is the average (weighted mean) of the betas of the components.

Equation 11.6 Portfolio Beta (weighted mean)

Portfolio Beta = (weight of security 1 × beta of security 1) + (weight of security 2 × beta of security 2) + … + (weight of security n × beta of security n) = w1β1 + w2β2 + … + wnβn

For example, if we have $20 thousand invested in ABC stock (which has a beta of 1.5) and$80 thousand invested in DEF stock (which has a beta of 0.7), then our portfolio’s beta is: $(2020+80×1.5)+(8020+80×0.7)=(0.2×1.5)+(0.8×0.7)=0.86$

A portfolio holding every stock in the same weights as the market overall should have a risk premium equal to the market’s risk premium (also called the equity risk premium). It should also have a beta of exactly 1.0. Since the relationship of each component to both the portfolio’s return and beta is the same (weighted mean), the two measures must be proportional to each other. Specifically, each component’s contribution to the risk premium of the overall market portfolio must be proportional to the component’s market risk (the beta):

$Risk Premium of Stock ABeta of Stock A=Risk Premium of MarketRPAβA=RPMKTrA−rRFβA=rMKT−rRF$

Continuing with our example stocks: if the market risk premium was 4.0%, then ABC (beta=1.5) should have a risk premium of 1.5×0.04=0.06 or 6%, and DEF (beta=0.7) should have a risk premium of 0.7×0.04=0.028 or 2.8%.

Of course, any given observation might deviate from the relationship predicted (beta is a statistical number), but the key idea is that our expected returns should hold to this relationship. If we rearrange this equation to solve for the expected return of stock A, we get what is called the Capital Asset Pricing Model (CAPM):

Equation 11.8 Capital Asset Pricing Model (CAPM)

$rA−rRFβA=rMKT−rRFrA=βA×(rMKT−rRF)rA=rRF+βA×(rMKT−rRF)$

This is a very important result, as it implies that we can predict a stock’s expected return if we know the risk-free rate, beta of the stock, and the market risk premium.

Thus, if stock GHI has a beta of 1.2, and we know that the risk-free rate is 2% and the expected market returns are 7%, the expected returns for GHI should be 0.02 + 1.2 × (0.07 − 0.02) = 0.08 or 8%.

We can also use the relationship to compare two stocks directly:

Equation 11.9 Proportional Risk Premiums (two stocks)

$Risk Premium of Stock ABeta of Stock A=Risk Premium of Stock ΒBeta of Stock BRPAβA=RPBβBrA−rRFβA=rB−rRFβB$

## Issues with CAPM

While the realationship embodied in CAPM is very important to understand, the theory is not without its limitations. While determining the expected risk-free rate is fairly accurately obtained from observing the bond market (typically using long-term US government bond yields as a proxy), figuring an accurate forward looking beta and equity risk premium is much more difficult. The bulk of estimates of the equity risk premium in the US fall in the 2%–5% range, though arguments for rates well outside this range are not unheard of!

As a more general critique, there are proposals that there are systematic risk factors beyond just market risk. For example, some multi-factor models (such as including a variable for company size) seem to be better predictors of expected returns. Ideally, the market portfolio shouldn’t just include stocks, but the entire field of investments (including real estate, fine art, and even baseball cards). Empirically, CAPM is not as accurate as we wish it to be in predictive power.

### Key Takeaways

• Firm-specific risk can be diversified away, but market risk can’t.
• The risk premium of assets should be proportional to their betas.
• The beta of a portfolio is the average (weighted mean) of the betas of its components.

### Exercises

1. Are the following examples of firm-specific or market risk?

1. An oil company failing to find oil in one of its oil fields.
2. GDP numbers beating expectations.
3. The SEC finding accounting irregularities with a company.
2. How might a drop in the price of oil cause both firm-specific and market risk effects?
3. We have $25 thousand invested in ABC stock (beta=1.5),$50 thousand in DEF stock (beta=0.7), and \$15 thousand in GHI stock (beta=1.2). What is the portfolio’s beta?
4. Long term gov’t bonds are returning 4%. The equity risk premium is 5%. If XYZ stock has a beta of 0.6, what is its expected return?