CAPM - Capital Asset Pricing Model

In finance, the capital asset pricing model (CAPM) 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 symbol beta (β), as well as the expected return of the market and the expected return of a theoretical risk-free asset.

Risk and return
In general, investors expect that it is necessary to take a high risk to receive a high return. Conversely, investors are willing to sacrifice return (accept less than their current return) 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 of returns. 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 of returns is also reduced as the number of securities in the portfolio increases.

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

Beta
From a fundamental perspective, Beta (β) is calculated by taking the risk-adjusted return Ri (return over the risk-free rate, also known as the "Risk-adjusted" or "excess" return) over a time period of the individual asset (i) (or portfolio) divided by the risk-adjusted return (Rm) of the market. Simplifying further, β is the slope of the line of the returns over this period.


 * β = Slope of the line = Ri / Rm.

For example: If an equity X has a return of 8%, the expected return of the market m is 10%, and the risk-free rate is 2%:


 * β = 0.75 = Ri / Rm = (8% - 2%) / (10% - 2%)

On a more comprehensive level, the parameter beta (β) is used to describe the relationship between the returns of a security or portfolio (an asset) and the returns of the market as a whole; it combines the correlation of the asset's returns and the market's returns with the relative volatility of those returns:


 * β = cov(rA, rM) / σM2 = ρ(rA, rM) × (σA / σM)
 * where:
 * rA is the set of returns of the asset
 * rM is the set of returns of the market
 * σM2 is the variance of the returns of the market
 * cov(rA, rM) is the covariance between the returns of the asset and the returns of the market
 * ρ(rA, rM) is the correlation between the returns of the asset and the returns of the market
 * σA is the standard deviation of the returns of the asset
 * σM is the standard deviation of the returns of the market

Market risk (β) 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.

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 of returns. Therefore, expected returns are maximized for any given level of expected volatility of returns.
 * 2) All investors have homogeneous beliefs about the risk/reward trade-offs in the market.
 * 3) There is only one risk factor 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 of returns (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 of returns exactly equal to the market's has a beta of one. In other words, the returns are perfectly positively correlated. By definition, its expected return is equal to the market's expected return:
 * E(rA) = E(rM)

3. 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(rA) = rf + βA(E(rM) - rf)
 * where:
 * E(rA) is the expected return of the asset
 * rf is the risk-free rate
 * E(rM) is the expected return of the market portfolio

(Note: the quantity E(rM) - rf, which is the expected excess return of the market above the risk-free rate, is called the market risk premium, often abbreviated MRP.)

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 (rf) 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, (βA), and multiplying by the MRP, (E(rM) - rf).

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 alpha (α). 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 were up 10%, about 7% of the return can be explained by the advance of all stocks and the other 3% is the result of other factors not related to beta. This observation led to the development of 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) × Kd[cost of debt] × (1 - marginal tax rate for the firm)
 * + (% of the firm in equity, at market value) × Ke[cost of equity]

(Note: this formula assumes the firm has only debt and common equity; i.e., no preferred equity.)


 * Ke is estimated using the CAPM:
 * Ke = rf + βe(E(rM) - rf)
 * where:
 * Ke is the cost of equity
 * rf is the risk-free rate of return
 * E(rM) is the expected return of the market
 * E(rM) - rf is the market risk premium and is held to be, normally, between 4% - 5% (real) per year (the last 110 years of UK data, similar to the US)

(Another method of calculating Ke uses the Dividend discount model.)