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'''[[Capital asset pricing model]]''' ('''CAPM''') is a model in [[Financial economics|financial economics]] that describes the relationship between the expected return of an asset and its [[Systematic risk|systematic risk]]. It is commonly used in [[Asset pricing|asset pricing]], [[Portfolio management|portfolio management]], [[Corporate finance|corporate finance]], and the estimation of the [[Cost of equity|cost of equity]].

The model states that the expected return on an asset is equal to the [[Risk-free rate|risk-free rate]] plus a risk premium determined by the asset's [[Beta (finance)|beta]] relative to the expected market risk premium.

==CAPM equation==

The CAPM expected-return equation is:

<math>
E(R_i)
=
R_f
+
\beta_i
\left[
E(R_m)-R_f
\right]
</math>

where:

* <math>E(R_i)</math> = expected return of asset <math>i</math>;
* <math>R_f</math> = risk-free rate;
* <math>\beta_i</math> = beta of asset <math>i</math>;
* <math>E(R_m)</math> = expected return of the market portfolio;
* <math>E(R_m)-R_f</math> = expected market risk premium.

==Market risk premium==

The '''market risk premium''' is the additional expected return investors require for investing in the market portfolio rather than a risk-free asset.

It is calculated as:

<math>
\text{Market Risk Premium}
=
E(R_m)-R_f
</math>

The CAPM therefore expresses the expected return of an asset as:

<math>
\text{Expected Return}
=
\text{Risk-Free Rate}
+
\text{Systematic Risk Premium}
</math>

The systematic risk premium is:

<math>
\text{Systematic Risk Premium}
=
\beta_i
\left[
E(R_m)-R_f
\right]
</math>

==Beta==

'''Beta''' measures the sensitivity of an asset's return to movements in the market portfolio.

A simplified beta calculation is:

<math>
\beta_i
=
\frac{\operatorname{Cov}(R_i,R_m)}
{\operatorname{Var}(R_m)}
</math>

where:

* <math>R_i</math> = return of asset <math>i</math>;
* <math>R_m</math> = market return;
* <math>\operatorname{Cov}(R_i,R_m)</math> = covariance between the asset and market returns;
* <math>\operatorname{Var}(R_m)</math> = variance of market returns.

A beta of 1 indicates that an asset has the same systematic risk as the market portfolio.

A beta greater than 1 indicates greater sensitivity to market movements, while a beta below 1 indicates lower sensitivity.

==Risk decomposition==

CAPM distinguishes between systematic and unsystematic risk.

'''Systematic risk''' is risk associated with broad market factors and cannot be eliminated completely through diversification.

'''Unsystematic risk''' is company-specific or asset-specific risk that can potentially be reduced through diversification.

CAPM assumes that investors are compensated for systematic risk rather than for diversifiable unsystematic risk.

A simplified representation is:

<math>
\text{Total Risk}
=
\text{Systematic Risk}
+
\text{Unsystematic Risk}
</math>

==Security market line==

The '''security market line''' (SML) represents the relationship predicted by CAPM between an asset's expected return and its beta.

The equation of the SML is:

<math>
E(R_i)
=
R_f
+
\beta_i
\left[
E(R_m)-R_f
\right]
</math>

In a graphical representation:

* the vertical intercept is the risk-free rate;
* the horizontal axis represents beta;
* the slope represents the market risk premium.

The SML therefore describes the expected return required for different levels of systematic risk.

==Example==

Suppose:

* the risk-free rate is 4%;
* the expected market return is 10%;
* the asset's beta is 1.2.

The expected return under CAPM is:

<math>
E(R_i)
=
4\%
+
1.2(10\%-4\%)
</math>

Therefore:

<math>
E(R_i)
=
4\%
+
7.2\%
=
11.2\%
</math>

The CAPM-implied expected return is therefore 11.2%.

==Cost of equity==

CAPM is commonly used to estimate a company's [[Cost of equity|cost of equity]].

The CAPM cost-of-equity equation is:

<math>
K_e
=
R_f
+
\beta_e
\left[
E(R_m)-R_f
\right]
</math>

where:

* <math>K_e</math> = cost of equity;
* <math>R_f</math> = risk-free rate;
* <math>\beta_e</math> = equity beta;
* <math>E(R_m)-R_f</math> = expected market risk premium.

The estimated cost of equity can be used as an input in financial valuation and [[Weighted average cost of capital|weighted average cost of capital]] calculations.

==Portfolio application==

CAPM can also be applied to a diversified portfolio.

The portfolio beta can be calculated as a weighted average of the betas of its constituent assets:

<math>
\beta_p
=
\sum_{i=1}^{n}
w_i\beta_i
</math>

where:

* <math>\beta_p</math> = portfolio beta;
* <math>w_i</math> = portfolio weight of asset <math>i</math>;
* <math>\beta_i</math> = beta of asset <math>i</math>.

The CAPM expected return for the portfolio is then:

<math>
E(R_p)
=
R_f
+
\beta_p
\left[
E(R_m)-R_f
\right]
</math>

==CAPM and diversification==

CAPM assumes that investors can diversify away asset-specific risk.

Because unsystematic risk can be reduced through diversification, the model does not assign an additional expected return premium for bearing such risk.

The expected return is therefore determined by systematic risk as measured by beta.

==CAPM and Jensen's alpha==

[[Jensen's alpha]] measures the difference between an asset's or portfolio's actual return and its CAPM-implied return.

A simplified expression is:

<math>
\alpha_p
=
R_p
-
\left[
R_f
+
\beta_p
(R_m-R_f)
\right]
</math>

where:

* <math>\alpha_p</math> = Jensen's alpha;
* <math>R_p</math> = portfolio return;
* <math>R_f</math> = risk-free rate;
* <math>\beta_p</math> = portfolio beta;
* <math>R_m</math> = market return.

A positive alpha indicates a return above the CAPM-implied return, while a negative alpha indicates a return below it.

==Assumptions==

The traditional CAPM is based on several simplifying assumptions, including:

* investors are rational and seek to maximize expected utility;
* investors are concerned with expected return and risk;
* investors can borrow and lend at the risk-free rate under the traditional formulation;
* investors have homogeneous expectations about asset returns and risks;
* markets are frictionless;
* investors can hold diversified portfolios;
* assets are infinitely divisible;
* investors have access to the same relevant information;
* there are no transaction costs or taxes in the basic model.

These assumptions simplify the model and may not fully reflect actual financial markets.

==Limitations==

CAPM has several limitations.

These include:

* expected returns are difficult to estimate;
* beta can change over time;
* the true market portfolio cannot be directly observed;
* investors may not be able to borrow or lend at the risk-free rate;
* taxes and transaction costs exist in actual markets;
* markets may not be fully efficient;
* returns may not conform to the assumptions of the model;
* additional risk factors may influence expected returns.

Empirical research has identified factors beyond market beta that can help explain differences in asset returns.

==Importance==

CAPM is one of the foundational models of modern [[Portfolio theory|portfolio theory]] and [[Asset pricing|asset pricing]]. It provides a framework for relating expected investment returns to systematic risk and is widely used in estimating the cost of equity, evaluating portfolio performance, and analyzing investment risk.

==See also==

* [[Modern portfolio theory]]
* [[Security market line]]
* [[Beta (finance)]]
* [[Risk-free rate]]
* [[Risk premium]]
* [[Market risk]]
* [[Systematic risk]]
* [[Unsystematic risk]]
* [[Jensen's alpha]]
* [[Sharpe ratio]]
* [[Cost of equity]]
* [[Weighted average cost of capital]]
* [[Portfolio management]]
* [[Asset pricing]]

[[Category:Capital asset pricing model]]
[[Category:Financial economics]]
[[Category:Asset pricing]]
[[Category:Investment]]
[[Category:Portfolio theory]]
[[Category:Corporate finance]]
[[Category:Financial risk]]