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 undiversifiable risk of security j. βj can be measured as follows:
βj = Cov (Rj, Rm)
Var (Rm)
= σj σm Cor jm
σ2 m
= σj Cor jm
σm
(Equation 2)
In terms of Equation 2, the undiversifiable (systematic) risk (βj) of a security is the product of its standard deviation (σj) and its correlation with the market portfolio divided by the market portfolio’s standard deviation. It can be noted that if a security is perfectly positively correlated with the market portfolio, then CML totally coincides with SML.
Equation 1 shows that the expected rate of return on a security is equal to a risk-free rate plus the risk-premium. The risk-premium equals to the difference between the expected market return and the risk-free rate multiplied by the security’s beta. The risk premium varies directly with systematic risk measured by beta.
The figure above illustrates the security market line. For a given amount of systematic risk (β), SML shows the...

...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 at the same time setting a specific rate of return with respect to the amount of the risk of a portfolio or an individual investment. The CAPM method takes into consideration the factor of time and does not get wrapped up over by the systematic risk factors, which are rarely controlled. In this research paper, I will look at the implications of CAPM in the light of the recent development. I will start by attempting to explain and discuss the various assumptions of the CAPM. Secondly, I will discuss the main theories and moreover, the whole debate that is surrounding this area more specifically through the various critics of the CAPM assumptions.
When Sharpe (1964) and Lintner (1965) proposed CAPM, it was majorly seen as the leading tool in measuring and determining whether an investment will yield negative or positive return. The model attempts to expound the relationship between expected reward/return and the investment risk of very risky...

...Capital asset 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 non-diversifiable risk. The CAPM says that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat the required return, then the investment should not be undertaken. The security market line plots the results of the CAPM for all different risks (betas - a model that calculates the expected return of an asset based on its beta and expected market returns.)
Using the CAPM model and the following assumptions, we can compute the expected return of a stock in this CAPM example: if the risk-free rate is 3%, the beta (risk measure) of the stock is 2 and the expected market return over the period is 10%, the stock is expected to return 17%=(3%+2(10%-3%)).
Risk of a Portfolio
We all know that investments have risk, so it’s safe to assume that all stocks have risk as well? But did you know that there are different types of risk as well?...

...(CAL).
A simplifying assumption underlying modern portfolio theory is that investors have homogeneous expectations, i.e., they all have the same estimates of risk, return, and correlations with other risky assets for all risky assets. Under this assumption, all investors face the same efficient frontier of risky portfolios and will all have the same optimal risky portfolio and CAL.
Under the assumption of homogeneous expectations, this optimal CAL for all investors is termed the capital market line (CML).
A line used in the capital asset pricing model to illustrate the rates of return for efficient portfolios depending on the risk-free rate of return and the level of risk (standard deviation) for a particular portfolio.
The CAPM is a model for pricing an individual security or a portfolio
expected return = risk-free rate + portfolio beta* (the difference between the expected return on the market as a whole and the risk-free rate).
Efficient frontier:
Efficient frontier is the set of portfolios among all the possible portfolios of combinations of individual risky assets that offers the highest expected return for each level of risk (standard deviation
The concept of an efficient frontier can be used to illustrate the benefits of diversification. An undiversified portfolio can be moved closer to the efficient frontier by diversifying it. Diversification can, therefore, increase returns without increasing risk, or reduce risk without...

...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 that the explanatory power of CAPM beta is highly dependent on the way it is estimated. While the conventional beta proxy is indeed largely unrelated to realized stock returns (in fact the relationship is slightly negative), using forward looking beta and eliminating unrealistic assumptions about expected market returns turns it (highly) significant. In addition, this study shows that complementary empirical factors – size and ratio of book-to-market value of equity – that are sometimes presented as potential remedies to beta’s deficiencies do not seem to outperform beta. This suggests they are not good risk proxies on the Swedish stock market, which casts doubt on the universal applicability of the 3-factor model. Keywords: asset pricing, CAPM, beta, factor pricing models, 3-factor model, market efficiency, Sweden, Scandinavia JEL classification: G12, G14 Acknowledgements: I would like to thank Dalibor Petr, Tomas ... and Johan Lyhagen for their help with...

...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 0.70956
They use a particular test for multivariate linear models. If we look at the p-values, it says that we don’t reject the hypothesis that all αi are zero. We can therefore reestimate the model without the intercept: > res2 res2
Call: lm(formula = Z ~ Zm - 1) Coefficients: WMK UIS Zm 0.4770 1.3438 ZOOM Zm 0.7240
ORB 1.0524
MAT 0.7084
ABAX 0.7218
T 0.8037
EMR 0.9395
JCS 0.4137
VOXX 1.3517
We can then look at the systematic and non systematic risk of each asset: > > > > + + sigm > > > > > > > a > >
b > > > > > D Chisq) 1 2 10 8.2292 0.6065
2
Zero-beta CAPM (Black)
The zero-beta CAPM is based on the properties of the portfolio frontier. One of them tells us that for each eﬃcient portfolio rp of risky assets, there exists a portfolio on the lower part of the portfolio frontier, rzp , which is uncorrelated with it. Its β deﬁned as Cov(rp , rzp )/V ar(rp ) is therefore 0. That’s why the model is called the zero-beta...

...the systematic variance as diversification increases, which means diversifying across industries offer benefit over diversifying within a given industry. Second, using the figures 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 marketplace. Although the CAPM is widely used because of the insight it offers, it does not fully withstand empirical tests. CAPM is a one-period model that treats a security’s beta as a constant, but beta can be changed in respond to firms investment in new industry, change in capital structure and so on. If betas change over time, simple historical estimates of beta are not likely to be accurate. Mismeasuring of betas will not reflect stocks’ systematic risk, so in this case the CAPM does not compute the risk premium correctly. Furthermore, the systematic risk, the source of risk premiums, cannot be confined to a single factor. While the CAPM derived from a single-index market cannot provide any insight on this.
The data we used provides us with 5-year period...

...CAPM Project
1.
| T-bill | S&P 500 | Microsoft | Dell | Exxon | GM | IBM | Ford |
Average | 0.00351 | 0.00838 | 0.02632 | 0.03673 | 0.01284 | 0.00725 | 0.01069 | 0.00690 |
SD | 0.00154 | 0.03996 | 0.10257 | 0.15161 | 0.04580 | 0.09327 | 0.08761 | 0.09430 |
2.
| T-bill | S&P 500 | Microsoft | Dell | Exxon | GM | IBM | Ford |
T-bill | 1.00000 | 0.06922 | 0.13241 | 0.06114 | 0.03865 | -0.00266 | 0.04134 | -0.02359 |
S&P 500 | | 1.00000 | 0.54399 | 0.43517 | 0.45093 | 0.44741 | 0.54018 | 0.50752 |
Microsoft | | | 1.00000 | 0.52712 | 0.13734 | 0.12264 | 0.46422 | 0.22796 |
Dell | | | | 1.00000 | 0.06558 | 0.17795 | 0.36041 | 0.26893 |
Exxon | | | | | 1.00000 | 0.20574 | 0.25684 | 0.24470 |
GM | | | | | | 1.00000 | 0.27395 | 0.56059 |
IBM | | | | | | | 1.00000 | 0.25477 |
Ford | | | | | | | | 1.00000 |
3.
a) Because T-bills are bonds issued by the U.S. government, they are virtually default free. Therefore, they have very low risk and returns on T-bills do not vary a lot over time and its standard deviation is small.
(b) Because Microsoft, Dell and IBM all belong to the same industry; therefore, events that affect that industry will have effects on these three companies as well. If Dell and IBM’s computers don’t sell well or the demand is low, Microsoft will sell less software. Therefore, all of the three companies tend to move together and the correlation coefficient between returns...

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