Capital Asset Pricing Model
Overview
The Capital Asset Pricing Model, developed by Sharpe, Lintner and Mossin, is a model which seeks to tie the price
of an asset to the risk characteristics of that asset.
(
Luenberger pg 173)
It starts by quantifying risk as portfolio standard deviation
(see
variance as risk).
However, some risks are diversifiable. That is, they disappear in a portfolio of a large number
of diverse assets. Other risks cannont be diversified away.
(total market risk for example)
In general this means that the risks that an asset faces can be
split between diversifiable risk and
non diversifiable risk.
{% asset \; risk = diversifiable \; risk + non \; diversifiable \; risk %}
The innovation of the CAPM was the assumption that the market only rewards investors for taking non-diversifiable risk.
As such, the expected returns should be tied only to non-diversifiable risk.
Defintions
The Capital Asset Pricing Model postulates the follwoing equation for the return of a given
asset.
{% r_i = \alpha_i + r_{risk free} + \beta_i \times r_m + e_i %}
{% r_i %} is the rate of return of a given asset, {% r_m %}
is the return on the market, {% r_{risk free} %} is the risk free rate
(return on Treasuries) and {% e %} is the random component.
The residual returns are defined as
{% \theta_i = \alpha_i + e_i %}
That is, the residual return is the return that the asset acieves above (or below) that expected given its beta to the market.
Assertions
The CAPM makes the following assertions:
- {% \mathbb{E}(e_i) = 0 %} - the expected value of the random term is zero
- {% \alpha_i = 0 %} - there is no alpha
Together, these assumptions can be stated as {% \mathbb{E}(\theta_i) = 0 %}, or the expected residual return is zero.
As such, the following equations follows from the
linearity of the expectation.
{% \mathbb{E}(r_i) = r_{risk \, free} + \beta_i \times (r_m - r_{risk \, free}) %}
or
{% \mathbb{E}(r_i - r_{risk \, free}) = \beta_i \times (r_m - r_{risk \, free}) %}
Graphical Representation of the CAPM. The x-axis is the market return, the asset return is on the y-axis. Expected asset return
is linearly related to the market return.
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CAPM in Portfolio Management
CAPM, if accepted, dramatically simplifies portfolio management.
It essentially asserts that the market only rewards takers of
systematic risk, not idiosyncratic risk. If that is true,
then active management consists simply of choosing a total risk target
through managing portfolio beta and then diversifying away
all idiosyncratic risk.
CAPM as Risk Tool
As a risk tool, the CAPM finds its expression as a single index
factor model. For information about single index models, please
see:
single index model
Testing and Measuring CAPM
Measuring CAPM : discusses ways to measure beta and/or test
the theory.
