Derivation of the CAPM
We know from Markowtiz’ framework concerning two-fund separation that each investor will have a utility-maximizing portfolio that is a combination of the risk free asset and the tangency portfolio. If all investors see the same capital allocation line, they will all have the same linear efficient set called the Capital Market Line (CML). This forms a linear relationship between expected return of the portfolio and the standard deviation. If market equilibrium is to exist we know that the prices of all assets must adjust such that all assets are held by investors, there can be no excess demand. We get the market portfolio, M. Hence, in equilibrium the market portfolio will consist of all marketable assets held in proportion to their value weights.

If we invest a % in a risky asset, i, and (1-a) % in the market portfolio, we get the following mean and standard deviation:

Change in the mean and standard deviation with respect to the percentage of the portfolio, a, invested in asset i is a follows:

However we notice that by the definition of the market portfolio asset i is already hold in the market portfolio according to its market value weight. Therefore the percentage a in the equations is excess demand for i, which in equilibrium must be zero. We elaborate the new information in our equations:

The slope of the risk-return trade-off evaluated at point M in the graph, in market equilibrium, is:

This slope will also be equal to the slope of the CML (known as the Sharpe Ratio) in the point M:

If we rearrange and solve for :
, where:
This is the capital asset pricing model, graphically called the security market line.

...CAPMCAPM provides a framework for measuring the systematic risk of an individual security and relate it to the systematic risk of a well-diversified portfolio. The risk of individual securities is measured by β (beta). Thus, the equation for security market line (SML) is:
E(Rj) = Rf + [E(Rm) – Rf] βj
(Equation 1)
Where E(Rj) is the expected return on security j, Rf the risk-free rate of interest, Rm the expected return on the market portfolio and βj the...

...
The Capital Asset Pricing Model commonly known as CAPM defines the relationship between risk and the return for individual securities. CAPM was first published by William Sharpe in 1964. CAPM extended “Harry Markowitz’s portfolio theory” to include the notions of specific and systematic risk. CAPM is a very useful tool that has enabled financial analysts or the independent investors to evaluate the risk of a specific investment while...

...pricing model (CAPM)
Using the Capital Asset Pricing Model, we need to keep three things in mind. 1 there is a basic reward for waiting, the risk free rate. 2 the greater the risk, the greater the expected reward. 3 there is a consisted trade off between risk and reward.
In finance, It 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...

...ECON 405: Quantitative Finance
CAPM and APT
In this document, I use the package ”gmm”. You can get it the usual way through R or though the development website RForge for a more recent version. For the latter, you can install it by typing the following in R: > install.packages("gmm", repos="http://R-Forge.R-project.org") The data I use come with the package and can be extracted as follows: > > > > library(gmm) data(Finance) R > > > >
Rm F) 0.70956 0.70956 0.70956...

...Is CAPM Beta Dead or Alive? Depends on How you Measure It
Jiri Novak*
* Uppsala University, Sweden E-mail: jiri.novak@fek.uu.se October 2007 Abstract: The CAPM beta is arguably the most common risk factor used in estimating expected stock returns. Despite of its popularity several past studies documented weak (if any) association between CAPM beta and realized stock returns, which led several researchers to proclaim beta “dead”. This paper shows...

...estimated to testify that the CAPM works in practice.
The capital asset pricing model (CAPM) provides us with an insight into the relationship between the risk of an asset and its expected return. This relationship serves two significant functions. First, it provides a benchmark rate of return for evaluating possible investments. Second, the model helps us to make an educated guess as to the expected return on asset that have not yet been traded in the...

...European Journal of Social Sciences – Volume 21, Number 2 (2011)
From Regular-Beta CAPM to Downside-Beta CAPM
Qaiser Abbas
Corresponding Author, Professor Department of Management Sciences COMSATS Institute of
Information Technology Chak Shahzad, Park Road, Islamabad
E-mail: qaisar@comsats.edu.pk
Usman Ayub
Assistant Professor and PhD Scholar COMSATS Institute of Information Technology
Chak Shahzad, Park Road, Islamabad
E-mail: usman_ayub@comsats.edu.pk...

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