CAPITAL ASSET PRICING MODEL

INTRODUCTION

The Capital Asset Pricing Model (CAPM) is a model developed in an attempt to explain variation in yield rates on various types of investments

CAPM is based on the idea that investors demand additional expected return (called the risk premium) if they are asked to accept additional risk

The CAPM model says that this expected return that these investors would demand is equal to the rate on a risk-free security plus a risk premium

INTRODUCTION

The model was the work of financial economist (and, later, Nobel laureate in economics) William Sharpe, set out in his 1970 book "Portfolio Theory And Capital Markets"

His model starts with the idea that individual investment contains two types of risk: Systematic Risk (or Market risk) Unsystematic Risk (or Specific risk)

CAPM considers only systematic risk and assumes that unsystematic risk can be eliminated by diversification

DIVERSIFICATION

A risk management technique that mixes a wide variety of investments within a portfolio

The rationale behind this technique contends that a portfolio of different kinds of investments will, on average, yield higher returns and pose a lower risk than any individual investment found within the portfolio

This only works for unsystematic risks

ASSUMPTIONS

The Capital Asset Pricing Model is a ceteris paribus model. It is only valid within a special set of assumptions: Investors are risk averse individuals who maximize

the expected utility of their end of period wealth Investors have homogenous expectations (beliefs)

about asset returns Asset returns are distributed by the normal

distribution There exists a risk free asset and investors may

borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate

ASSUMPTIONS

The Capital Asset Pricing Model is a ceteris paribus model. It is only valid within a special set of assumptions: There is a definite number of assets and their

quantities are fixed within the one period world All assets are perfectly divisible and priced in a

perfectly competitive marked Asset markets are frictionless and information is

costless and simultaneously available to all investors

There are no market imperfections such as taxes, regulations, or restrictions on short selling

CAPM

Formulation of CAPM is given by:

where: rk- yield rate on a specific security k

rf- risk-free rate of interest

rp- yield rate on the market portfolio

k- a measure of systematic risk for security k

CAPM: DerivationStep 1. The derivation of the CAP-model starts by assuming that all assets are stochastic and follow a normal distribution. This distribution is described completely by its two parameters: mean value (m) and variance (s2). The mean value is a measure of location among many such as median and mode. Likewise, the variance value is a measure of dispersion among many such as range, semiinterquartile range, semivariance, mean absolute deviation. In the hypothetical world of the CAPM theory all that the investor bothers about is the values of the normal distribution. In the real world asset return are not normally distributed and investors do find other measures of location and dispersion relevant. However, the assumption may be seen as a reasonable approximation and it is needed in order to simplify matters.

Figure 1. Optimal portfolio choice for a risk averse investor in a world with risky assets

CAPM: DerivationStep 2. The next assumption is that investors are risk averse and maximize expected utility. They perceive variance as a bad and mean as a good. This is also illustrated in figure I where tree risk-averse indifference curves are drawn.

CAPM: DerivationProposition 1: An individual investor will maximize expected utility of his end of period wealth where his subjective marginal rate of substitution between risk and return represented by his indifference curves is equal to the objective marginal rate of transformation offered by the minimum variance opportunity set: MRSspmp = MRTspmp.

CAPM: DerivationStep 3. Assume now that there in addition to the many risky assets exist a risk free asset and that investors may borrow or lend unlimited amounts of this asset at a constant rate: the risk free rate (rf). Furthermore, capital markets are assumed to be frictionless. The effect on the shape of the portfolio production possibility area is profound as illustrated in figure 11 below.

CAPM: DerivationStep 4. Assume that all investors have homogeneous beliefs about the expected distribution of returns offered by all assets. Also, capital markets are frictionless and information is costless and simultaneously available to all investors. Furthermore, there are no market imperfections. Taken together this implies that all investors calculate the same equation for the market capital line and that the borrowing rate equals the lending rate.

CAPM: DerivationStep 5. Assume further that all assets are perfectly divisible and priced in a perfectly competitive marked. Furthermore, there is a definite number of assets and their quantities are fixed within the one period world. Then the portfolio M turns out to be the market portfolio of all risky assets. The reason is that equilibrium requires all prices to be adjusted so that the excess demand for any asset is zero. That is, each asset is equally attractive to investors. Theoretically the reduction of variance from diversification increases as the number of risky assets included in the portfolio M rise. Therefore, all assets will be hold in the portfolio M in accordance to their market value weight: wi = Vi/SVi, where Vi is the market value of asset i and SVi is the market value of all assets.

CAPM: DerivationProposition 2: With all the above assumptions in mind (step 1-5) the capital market line (8) shows the relation between mean and variance of portfolios (consisting of the risk free asset and the market portfolio) that are efficiently priced and perfectly diversified. The capital market line equation could rightly be called the capital portfolio pricing model (CPPM) since it prices efficient portfolios. What is more interesting is to develop an equation for pricing of individual assets. This is exactly what the capital asset pricing model (CAPM) does. The CAP-model does not requires any new assumptions only new algebraic manipulations within the framework of the CPP-model.

CAPM: DerivationStep 6. From CPPM to CAPM. What is wanted is a model for efficient pricing of capital for individual assets (E[k], the CAPM), not one for efficient cost of capital for portfolios (mp, the CPPM). Now, imagine a portfolio consisting of a% in a risky asset I and (1 - a)% in the market portfolio M from the CPP-model.

Alternative Ways of Reflecting Risk

Through a risk premium in the interest rate, and

Through an adjustment in the expected payments

Formulas

Statistical definition of measure of systematic risk

Called the market price of risk

Example #1A common stock is currently selling for $50

and will pay a dividend of $2 at the end of one year. The value of for this stock in the recent past has been 1.5. The current risk-free rate of interest is 5.4%. It is assumed that the market risk premium for common stocks from the table is applicable. Find the implicit expected value of the price of this stock at the end of one year.

Table

Type of Security

Average nominal

yield rate

Average real yield

rate

Average risk

premium

Common stocks 12.0% 8.8% 8.4%

Corporate bonds 5.1 2.1 1.7

Treasury bills 3.5 .4 0

Example #2

a)Use the dividend discount model to find the implied annual rate of dividend increase for the common stock in the previous example, if present values are computed at the rate of interest produced by the Capital Asset Pricing Modelb)Is this implied annual rate of dividend increase

consistent with the answer to the previous example?c)Assuming the real risk-free rate of interest is

3%, find the excess of the annual rate of dividend increase over the annual rate of inflation

Example #3

A business firm decides to use the CAPM to evaluate two projects A and B. Project A has normal risk with , while project B has high risk with . Each project is expected to return the same dollar amount at the end of one year and nothing thereafter. The risk-free rate of interest is 5% and the market risk premium is 7%. If the two projects are combined into one project, find for the combined project

Example #4

Stock A has and investors expect it to return 7%. Stock B has and investors expect it to return

Find the risk free rate of interestFind A/B, assuming equal correlation coefficients between the market portfolio and Stocks A and B

Sources The Theory of Interest. Kellison, Stephen G. CAPM Assumptions and Limitations: CAPM – Where Market

Theories Converge and Clash. Ivkovic, Inya. http://investment.suite101.com/article.cfm/capm_assumptions_and_limitations

Understanding the concept of CAPM. Ivkovic, Inya. http://investment.suite101.com/article.cfm/understanding_the_concept_of_capm

The Capital Asset Pricing Model. Mathiesen, H. http://www.encycogov.com/A2MonitorSystems/AppA2MonitorSystems/AppBtoA2CAP_model/CAP_Model.asp