CAPM - Capital Asset Pricing Model

From Bogleheads

In finance, the capital asset pricing model (CAPM) ^[1] is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk. The model takes into account the asset's sensitivity to non-diversifiable risk (also known as systemic risk or market risk), often represented by the quantity beta (β) in the financial industry, as well as the expected return of the market and the expected return of a theoretical risk-free asset.

Contents 1 Risk and Return 1.1 Beta 2 CAPM 2.1 Assumptions 2.2 Model Description 3 Shortcomings of the CAPM Model 4 Other uses of CAPM 5 See Also 6 References 7 How to Cite

Risk and Return

In general, investors expect that it is necessary to take a high risk in order to receive a high return. Conversely, investors are willing to sacrifice return (accept less than the present value) in order to reduce risk.

An asset exhibits two types of risk: systematic and unsystematic

Unsystematic risk only affects an individual security or portfolio and does not affect the market as a whole. Unsystematic risk is treated as "random noise" in a portfolio. Consider, for example, volatility. Through the use of diversification (adding many securities), the random noise component will eventually have a mean of zero (the definition of random noise). Standard deviation is also reduced as the number of securities in the portfolio increases.

If there are enough assets in a portfolio such that diversification can not affect the performance, the portfolio volatility matches that of the overall market. Risk that can not be mitigated through diversification is known as systematic risk. The portion of its volatility which is considered systematic is measured by the degree to which its returns vary relative to those of the overall market.

Beta

A parameter called beta (β) is used to describe how well a security or portfolio correlates to the return of the market as a whole.

β = cov(r_A,r_M) / σ²_M

• where r_A is the return of the asset
• r_M is the return of the market
• σ²_M is the variance of the return of the market
• cov(r_A,r_M) is the covariance between the return of the market and the return of the asset.

Market risk (beta) is calculated using historical returns for both the asset and the market, with the market portfolio being represented by a broad index such as the S&P 500 or the Russell 2000.

CAPM

Assumptions

The Capital Asset Pricing Model (CAPM) attempts to quantify the relationship between the beta of an asset and its corresponding expected return. Several assumptions are made:

1. Investors care only about expected returns and volatility. Therefore, expected returns are maximized for any given level of expected volatility.
2. All investors have homogeneous beliefs about the risk/reward trade-offs in the market.
3. There is only one risk factor is common to a broad-based market portfolio, called systematic market risk. Investors are assumed to hold diversified portfolios. As a result, the CAPM model states that if an asset's beta is known, the corresponding expected return can be predicted.

Model Description

There are three areas of interest:

1. β = 0: An asset that has no volatility (no risk) does not have returns that vary with the market and therefore has a beta of zero and an expected return equal to the risk-free rate.

2. β = 1: An asset that moves with a volatility exactly equal to the market has a beta of one. In other words, it is perfectly correlated. By definition, it's return rate is equal to the market, E(r_A) = E(r_m)

2. β > 1: An asset that experiences greater swings in periodic returns than the market, which, by definition, has a beta greater than one. This asset is expected to earn returns superior to those of the market as compensation for this extra risk.

Making a lot of generalizations leads to the CAPM model:

E(r_A) = r(_f) + β_A(E(r_m) - r_f)

• where r(_f) is the risk-free rate and
• E(r_m) is the expected excess return of the market portfolio beyond the risk-free rate, often called the equity risk premium.

The general idea of CAPM is that investors should be compensated in two ways: time value of money and risk.

The time value of money is represented by the risk-free (r(_f)) rate in the formula and compensates the investors for placing money in any investment over a period of time.

The other part of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is done by taking an estimate of risk, (β_A), and multiplying by the difference of the (returns of the asset) to the (returns of the market) over a period of time, (E(r_m)) - r_f).

An asset is expected to earn the risk-free rate plus a reward for bearing risk as measured by that asset’s beta. The chart below demonstrates this predicted relationship between beta and expected return – this line is called the Security Market Line.

For example, a stock with a beta of 1.5 would be expected to have an excess return of 15% in a time period where the overall market beat the risk-free asset by 10%.

The CAPM model is used for pricing an individual security or a portfolio. For individual securities, the security market line (SML) and its relation to expected return and systematic risk (beta) shows how the market must price individual securities in relation to their security risk class.

As the CAPM predicts expected returns of assets or portfolios relative to risk and market return, the CAPM can also be used to evaluate the performance of active fund managers. The difference is “excess return”, which is often referred to as “α” (or, alpha). If α is greater than zero, the portfolio lies above the Security Market Line.

Shortcomings of the CAPM Model

Several shortcomings of the CAPM model exist. Incorrectly predicting results compared to realized returns and the affect of other risk factors have put this model under criticism. The assumption of a single risk factor limits the usefulness of this model.

Eugene F. Fama and Kenneth R. French found that on average, a portfolio’s beta explains about 70% of its actual returns. For example, if a portfolio was up 10%, about 70% of the return can be explained by the advance of all stocks and the other 30% is due to other factors not related to beta. The result is a model known as the Fama and French Three-Factor Model.

Other uses of CAPM

CAPM is also used to cost equities in applications other than investing. For example, in the Weighted Average Cost of Capital (the rate that a company is expected to pay to finance its assets), CAPM is used to calculate the cost of equity:

Weighted Average Cost of Capital (for a firm) = (% of the firm in debt, at market value) X K_d(cost of debt) X (1 - marginal tax rate for the firm) +

(% of the firm in equity, at market value) * K_e(cost of equity)

K_e is estimated using the CAPM:

K_e = r(_f) + β_A(E(r_m) - r_f)

• where K_e is the cost of equity
• r(_f) is the risk-free rate of return and
• E(r_m) is the equity risk premium and is held to be, normally between 4-5% real per annum (the last 110 years of UK data, similar to the US)

(There is also a second method of calculating K_e which uses the Dividend Discount Model.)

See Also

• Understanding Risk and Return, the CAPM, and the Fama-French Three-Factor Model (also available here)
  
• Capital Asset Pricing Model on Wikipedia
  
• Fama-French Three Factor Model (from Portfolio Solutions)
  
• CAPM on Investopedia
  
• Definition of CAPM on Google
  
• Definition of Beta on Wikipedia
  
• Weighted Average Cost of Capital on Wikipedia
  

Source for Graph (Google Docs Presentation)

References

1. ↑ Press Release, 16 October, 1990. William F. Sharpe won the The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1990 for his pioneering work on the CAPM.

How to Cite

You can link to this page from the Bogleheads Forum by copying and pasting the following text into your forum posting:

Please see [url=http://www.bogleheads.org/wiki/CAPM_-_Capital_Asset_Pricing_Model]CAPM - Capital Asset Pricing Model[/url] on the [url=http://www.bogleheads.org/wiki/Main_Page]Bogleheads Wiki[/url].

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