Capital Market Theory: Assumptions, The Separation Theorem

<< Building a Portfolio Using Markowitz Principles

Risk-Free Asset, Estimating the SML >>

Investment Analysis & Portfolio Management (FIN630)

Lesson # 36

ASSET PRICING MODEL

Capital Market Theory:

Capital market theory is a positive theory in that it hypothesis how investors do behave

rather than, how investors should behave, as, in the case of Modem Portfolio Theory

(MPT). It is reasonable "to view capital market" theory; as an extension of portfolio theory,

but it is important to understand that MPT is not based on the validity, or lack thereof, of

capital market theory.

The equilibrium model of interest to many investors is known as the capital asset pricing

model, typically referred to as the CAPM. It allows us to measure the relevant risk of an

individual security as well as to assess the relationship between relevant risk of and the

returns expected from investing. The CAPM is attractive as an equilibrium model because

of its simplicity and its implications. Because of serious challenges, to the model, however,

alternatives have been developed. The primary alternative to the CAPM is arbitrage pricing

theory, or APT, which allows for multiple sources of risk.

Capital Theory Assumptions:

Capital market theory involves a set of predictions concerning equilibrium expected return

on risky assets. It typically is derived by making some simplifying assumptions in order to

facilitate the analysis and help us to understand the arguments without fundamentally

changing the predictions of asset pricing theory.

Capital market theory builds on Markowitz portfolio theory to diversify his or; her portfolio,

according to the Markowitz model, choosing a location on the efficient frontier that matches

his or her return-risk references. Because of the complexity of the real world, additional

assumptions; are made to make individual more alike.

1. All investors can borrow or lend money at the risk-free rate of return.

2. All investors have identical probability distributions for future rates of return; they

have homogeneous expectations with respect to the three inputs of the portfolio

model i.e. expected returns, the variance of returns, and the correlation matrix.

Therefore, given a set of security prices and a risk-free rate, all investors use the

same information to generate an efficient frontier.

3. All investors have the same one-period time horizon.

4. There are no transaction costs.

5. There are no personal income taxes---investors are indifferent between capital gains

and dividends.

6. There is no inflation.

7. There are many investors, and no single investor can affect the price of a stock

through his or her buying and selling decisions. Investors are price takers and act as

if prices are unaffected by their own trades.

8. Capital markets are in equilibrium.

These assumptions appear to be unrealistic and often disturb investors encountering capital

market theory for the first time. However, the important issue is how well the theory

predicts or describes reality, and not the realism of its assumptions. If CMT does a good job

of explaining the returns on risky .assets, it is very useful, and the assumptions made in

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deriving the theory are of less importance.

Most of these assumptions can be relaxed without significant effects on the CAPM or its

implications in other words, the CAPM is robust. Although the results from such a

relaxation of the assumptions may be less clear-cut and precise no significant damage is

done. Many conclusions of the basic model still hold.

Finally, most investors recognize that all of the assumptions of CMT are not unrealistic. For

example, some institutional investors are tax exempt, and brokerage costs today, as

percentage of the transaction, are quite small. Nor is it too unreasonable to as that for the,

one-period horizon of the model, inflation may be fully (or mostly) anticipated and,

therefore, not a major factor.

The Equilibrium Return-Risk Tradeoff:

Given the previous analysis, we can now derive some predictions concerning equilibrium

expected returns and risk. The CAPM is an equilibrium model that encompasses two

important relationships. The first, the capital market line specifies the equilibrium

relationship between expected return and risk for efficient portfolios. The second, the

security market line specifies the equilibrium relationship between expected return and

systematic risk. It applies to individual securities as well as portfolios.

The Capital Market Line:

We now know that portfolio M is die tangency point to a straight line drawn front RF to the

efficient frontier and that this straight line is the best obtainable efficient set line. All

investors will hold portfolio M as their optimal risky portfolio, and all investors will be

somewhere on this steepest; trade-off line between expected return and risk, because it

represents, those combinations of risk-free investing / borrowing and portfolio M that yield

the highest return obtainable for a given level of risk.

The straight line usually referred to as the capital market line (CML, depicts the equilibrium

conditions that prevail in the market for efficient portfolios consisting of the optimal

portfolio of risky assets and die risk-free assets. All combinations of the risk-free asset and

the risk portfolio M are on the CML, and, in equilibrium, all investors will end up ;with

portfolios somewhere on the CML.

The Market Portfolio:

Portfolio M is called the market portfolio of risky securities. It is the highest point of

tangency between RF and the efficient frontier and is the optimal risky portfolio. All

investors would want to be on the optimal line RF-M-L, and, unless they invested 100

percent of their wealth in the risk-free asset, they would own portfolio M with some portion

of their investable wealth or they would invest their own wealth plus borrowed funds in

portfolio M. This portfolio is the optimal portfolio of risky assets.

Why do all investors hold identical risky portfolios? Based on our assumptions above, all

investors use the same Markowitz analysis on the same set of securities, have the same

expected returns and covariance's and have an identical time horizon Therefore, they will,

arrive at the same optimal risky portfolio, arid it will be the market portfolio, designated M.

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The Separation Theorem:

We have established that each investor will hold combinations of the risk-free asset (either

lending or borrowing) and the tangency portfolio from the efficient frontier, which is the

market portfolio. Because we are assuming homogeneous expectations, in equilibrium all

investors will determine the same tangency portfolio. Further, under the assumptions of

CMT all investors agree on the risk-free rate.

Borrowing and lending possibilities, combined with one portfolio of risky assets M, offer an

investor whatever risk-expected return combination he or she seeks; that is, investors can be

anywhere they choose on this line depending on their risk-return preferences. An investor

could:

1. Invest 100 percent of investable funds in the risk-free asset, providing an expected

return of RF and zero risk.

2. Invest 100 percent of investable funds in risky-asset portfolio-M, offering E (RM),

with its risk σM.

3. Invest in any combination of return and risk between these two points; obtained by

varying the proportion wRF invested in the risk-free asset.

4. Invest more than 100 percent of investable funds in the risky-asset portfolio M by

borrowing money at the rate RF, thereby increasing both the expected return and the

risk beyond that offered by portfolio M.

Different investors will choose different portfolios because of their risk preferences (they

have different indifference curves), but they will choose the same combination of risky

securities as denoted by the tangency point M. Investors will then borrow or lend to achieve

various positions on the linear trade-off, between expected return and risk.

Unlike the Markowitz analysis; it is not necessary to match each client's indifference curves

with a particular efficient portfolio, because only one efficient portfolio is held by all

investors. Rather each client will use his or her indifference curves to determine where

along the new efficient frontier RF-M-L he or she should be; In effect, each client must

determine how much of investable funds should be lend or borrowed at RF and how much

should be invested in portfolio M. This result is referred to as a separation property.

The Security Market Line:

The capital market line depicts the risk-return trade-off in the financial markets in

equilibrium. However, it applies only to efficient portfolios and cannot be used to-assess the

equilibrium expected return for a single security. What about individual securities or

inefficient portfolios?

Under the CAPM all investors will hold the market portfolio, which is the bench-mark

portfolio against which other portfolios are measured. How does an individual security

contribute to the risk of the market portfolio?

Investors should expect a risk premium for buying a risky asset such as a, stock. The greater

the riskiness of that stock, the higher the risk premium should be. If investors hold well-

diversified portfolios, they should be interested in portfolio risk rather than individual

security risk. Different stocks will affect a well-diversified portfolio differently. The

relevant risk for an individual stock is its contribution to the riskiness of a well-diversified

portfolio. And the risk of a well-diversified portfolio is market risk, or systematic risk,

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which is non-diversifiable.

Beta:

We now know that investors should hold diversified portfolios to reduce the portfolio risk.

When an investor adds a security to a portfolio what matters is the security's average

covariance with the other securities in the portfolio. We also now know that under CMT all

investors will hold the same portfolio of risky assets, the market portfolio. Therefore, the

risk that matters when we consider any security is its covariance with the market portfolio.

We could relate the expected return on a stock to its covariance with the market portfolio.

However, it is more convenient to use a standardized measure of the systematic risk that

cannot be avoided through diversification. Beta is a relative measure of risk---the risk of an

individual stock relative to the market portfolio of all stocks. If the security's returns move,

more (less) than, the market's returns as the latter changes, the security's returns have more

(less) volatility (fluctuations in price) than those of the market. For example, a security

whose returns rise or fall on average 15 percent when the market return rises or falls 10

percent is said to be an aggressive or volatile security.

CAPM's Expected Return-Beta Relationship:

The security market line (SML) is the CAPM specification of how risk and required rate of

return for any asset security, or portfolio are related. This theory posits a linear relationship

between an asset's risk and its required rate of return. This linear relationship, called the

security market line (SML). Required rate of return is on the vertical axis and beta, the

measure of risk, is on the horizontal axis. The slope of the line is the difference between the

required rate of return on the market index and RF, the risk-free rate.

The capital asset pricing model (CAPM) formally relates the expected rate of return for any

security or portfolio with .the relevant risk measure. The CAPM's expected return-beta

relationship is the most-often cited form of the relationship. Beta is the relevant measure of

risk that cannot be diversified away in a portfolio of securities and, as such, is the measure

that investors should consider in their portfolio management decision process.

The CAPM in its expected return-beta relationship form is a simple but elegant statement. It

says that the expected rate of return on an asset is a function of the two components of the

required rate of return--the risk-free rate and the risk premium. Thus;

ki = Risk-free rate + Risk premium

= RF + βi [E (RM) - RF]

Where;

= the required rate of return on asset i

E (RM) = the expected rate of return on the market portfolio

βi

= the beta coefficient for asset i

This relationship provides an explicit measure of the risk premium. It is the product of the

beta for a particular security i and the market risk premium, E (RM) - RF. Thus,

Risk premium for security i = βi (market risk premium)

= βi [E (RM) - RF]

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Investment Analysis & Portfolio Management (FIN630)

The CAPM's expected return-beta relationship is a simple but elegant statement about

expected (required) return and risk for any security or portfolio. It formalizes the basis of

investments which is that the greater the risk assumed, the greater the expected (required)

return should be. This relationship states that an investor require (expects) a return on arisky

asset equal to the return on a risk-free asset plus a risk premium and the greater the risk

assumed, the greater the risk premium.

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