Asset Pricing Model

•Investment Analysis and
Portfolio Management
Seventh Edition
by
Frank K. Reilly & Keith C. Brown

•Capital Market Theory:
 An Overview

•Capital market theory extends portfolio theory and develops a model for pricing all risky assets

•Capital asset pricing model (CAPM) will allow you to determine the required rate of return for any risky asset

•Assumptions of
Capital Market Theory

1.  All investors are Markowitz efficient investors who want to target points on the efficient frontier. 

–The exact location on the efficient frontier and, therefore, the specific portfolio selected, will depend on the individual investor’s risk-return utility function.

•Assumptions of
Capital Market Theory

2. Investors can borrow or lend any amount of money at the risk-free rate of return (RFR). 

–Clearly it is always possible to lend money at the nominal risk-free rate by buying risk-free securities such as government T-bills.  It is not always possible to borrow at this risk-free rate, but we will see that assuming a higher borrowing rate does not change the general results.

•Assumptions of
Capital Market Theory

3.  All investors have homogeneous expectations; that is, they estimate identical probability distributions for future rates of return.

–Again, this assumption can be relaxed.  As long as the differences in expectations are not vast, their effects are minor.

•Assumptions of
Capital Market Theory

4.  All investors have the same one-period time horizon such as one-month, six months, or one year. 

–The model will be developed for a single hypothetical period, and its results could be affected by a different assumption.  A difference in the time horizon would require investors to derive risk measures and risk-free assets that are consistent with their time horizons.

•Assumptions of
Capital Market Theory

5.All investments are infinitely divisible.

6.There are no taxes or transaction costs involved in buying or selling assets.

7.There is no inflation or any change in interest rates, or inflation is fully anticipated.

8.Capital markets are in equilibrium.

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•Assumptions of
Capital Market Theory

•Some of these assumptions are unrealistic

•Relaxing many of these assumptions would have only minor influence on the model and would not change its main implications or conclusions.

•A theory should be judged on how well it explains and helps predict behavior, not on its assumptions.

•Risk-Free Asset

•An asset with zero standard deviation

•Zero correlation with all other risky assets

•Provides the risk-free rate of return (RFR)

•Will lie on the vertical axis of a portfolio graph

•Risk-Free Asset

Covariance between two sets of returns is

•Combining a Risk-Free Asset
with a Risky Portfolio

Expected return

the weighted average of the two returns

•Combining a Risk-Free Asset
with a Risky Portfolio

Standard deviation

  The expected variance for a two-asset portfolio is

•Combining a Risk-Free Asset
with a Risky Portfolio

Given the variance formula

•Combining a Risk-Free Asset
with a Risky Portfolio

  Since both the expected return and the standard deviation of return for such a portfolio are linear combinations, a graph of possible portfolio returns and risks looks like a straight line between the two assets.

•Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

•Risk-Return Possibilities with Leverage

  To attain a higher expected return than is available at point M (in exchange for accepting higher risk)

•Either invest along the efficient frontier beyond point M, such as point D

•Or, add leverage to the portfolio by borrowing money at the risk-free rate and investing in the risky portfolio at point M

•Portfolio Possibilities Combining the Risk-Free Asset and Risky Portfolios on the Efficient Frontier

•The Market Portfolio

•Because portfolio M lies at the point of tangency, it has the highest portfolio possibility line

•Everybody will want to invest in Portfolio M and borrow or lend to be somewhere on the CML

•Therefore this portfolio must include ALL RISKY ASSETS

•The Market Portfolio

  Because the market is in equilibrium, all assets are included in this portfolio in proportion to their market value

•The Market Portfolio

  Because it contains all risky assets, it is a completely diversified portfolio, which means that all the unique risk of individual assets (unsystematic risk) is diversified away

•Systematic Risk

•Only systematic risk remains in the market portfolio

•Systematic risk is the variability in all risky assets caused by macroeconomic variables

•Systematic risk can be measured by the standard deviation of returns of the market portfolio and can change over time

•How to Measure Diversification

•All portfolios on the CML are perfectly positively correlated with each other and with the completely diversified market Portfolio M

•A completely diversified portfolio would have a correlation with the market portfolio of +1.00

•Diversification and the
Elimination of Unsystematic Risk

•The purpose of diversification is to reduce the standard deviation of the total portfolio

•This assumes that imperfect correlations exist among securities

•As you add securities, you expect the average covariance for the portfolio to decline

•How many securities must you add to obtain a completely diversified portfolio? (At least 30 stocks for a borrowing investor and 40 for a lending investor)

•Diversification and the
Elimination of Unsystematic Risk

  Observe what happens as you increase the sample size of the portfolio by adding securities that have some positive correlation

•Number of Stocks in a Portfolio and the Standard Deviation of Portfolio Return

•The CML and the Separation Theorem

•The CML leads all investors to invest in the M portfolio

•Individual investors should differ in position on the CML depending on risk preferences

•How an investor gets to a point on the CML is based on financing decisions

•Risk averse investors will lend part of the portfolio at the risk-free rate and invest the remainder in the market portfolio

•The CML and the Separation Theorem

  Investors preferring more risk might borrow funds at the RFR and invest everything in the market portfolio

•The CML and the Separation Theorem

  The decision to borrow or lend to obtain a point on the CML is a separate decision based on risk preferences (financing decision)

•A Risk Measure for the CML

•Covariance with the M portfolio is the systematic risk of an asset

•The Markowitz portfolio model considers the average covariance with all other assets in the portfolio

•The only relevant portfolio is the M portfolio

•A Risk Measure for the CML

  Together, this means the only important consideration is the asset’s covariance with the market portfolio

•A Risk Measure for the CML

Because all individual risky assets are part of the M portfolio, an asset’s rate of return in relation to the return for the M portfolio may be described using the following linear model:

•Variance of Returns for a Risky Asset

•The Capital Asset Pricing Model: Expected Return and Risk

•CAPM indicates what should be the expected or required rates of return on risky assets

•This helps to value an asset by providing an appropriate discount rate to use in dividend valuation models

•You can compare an estimated rate of return to the required rate of return implied by CAPM - over/under valued ?

•The Security Market Line (SML)

•The relevant risk measure for an individual risky asset is its covariance with the market portfolio (Covi,m)

•This is shown as the risk measure

•The return for the market portfolio should be consistent with its own risk, which is the covariance of the market with itself - or its variance:

•Graph of Security Market Line (SML)

•The Security Market Line (SML)

The equation for the risk-return line is

•Graph of SML with
Normalized Systematic Risk

•Determining the Expected
Rate of Return for a Risky Asset

•The expected rate of return of a risk asset is determined by the RFR plus a risk premium for the individual asset

•The risk premium is determined by the systematic risk of the asset (beta) and the prevailing market risk premium   (RM-RFR)

•Determining the Expected
Rate of Return for a Risky Asset

Assume:     RFR =   6%    (0.06) 

                RM = 12%    (0.12)

Implied market risk premium =    6%   (0.06)

•Determining the Expected
Rate of Return for a Risky Asset

•In equilibrium, all assets and all portfolios of assets should plot on the SML

•Any security with an estimated return that plots above the SML is underpriced

•Any security with an estimated return that plots below the SML is overpriced

•A superior investor must derive value estimates for assets that are consistently superior to the consensus market evaluation to earn better risk-adjusted rates of return than the average investor

•Identifying Undervalued and Overvalued Assets

•Compare the required rate of return to the expected rate of return for a specific risky asset using the SML over a specific investment horizon to determine if it is an appropriate investment

•Independent estimates of return for the securities provide price and dividend outlooks

•Price, Dividend, and
Rate of Return Estimates

•Comparison of Required Rate of Return to Estimated Rate of Return

•Plot of Estimated Returns
on SML Graph

•Calculating Systematic Risk:
The Characteristic Line

  The systematic risk input of an individual asset is derived from a regression model, referred to as the asset’s characteristic line with the model portfolio:

•Scatter Plot of Rates of Return

•The Impact of the Time Interval

•Number of observations and time interval used in regression vary

•Value Line Investment Services (VL) uses weekly rates of return over five years

•Merrill Lynch, Pierce, Fenner & Smith (ML) uses monthly return over five years

•There is no “correct” interval for analysis

•Weak relationship between VL & ML betas due to difference in intervals used

•The return time interval makes a difference, and its impact increases as the firm’s size declines

•Relaxing the Assumptions

•Differential Borrowing and Lending Rates

–Heterogeneous Expectations and Planning Periods

•Zero Beta Model

–does not require a risk-free asset

•Transaction Costs

–with transactions costs, the SML will be a band of securities, rather than a straight line 

•Relaxing the Assumptions

•Heterogeneous Expectations and Planning Periods

–will have an impact on the CML and SML

•Taxes

–could cause major differences in the CML and SML among investors  

•Empirical Tests of the CAPM

•Stability of Beta

–betas for individual stocks are not stable, but portfolio betas are reasonably stable.  Further, the larger the portfolio of stocks and longer the period, the more stable the beta of the portfolio

•Comparability of Published Estimates of Beta

–differences exist.  Hence, consider the return interval used and the firm’s relative size

•Relationship Between Systematic Risk and Return

•Effect of Skewness on Relationship

–investors prefer stocks with high positive skewness that provide an opportunity for very large returns

•Effect of Size, P/E, and Leverage

–size, and P/E have an inverse impact on returns after considering the CAPM.  Financial Leverage also helps explain cross-section of returns  

•Relationship Between Systematic Risk and Return

•Effect of Book-to-Market Value

–Fama and French questioned the relationship between returns and beta in their seminal 1992 study.  They found the BV/MV ratio to be a key determinant of returns

•Summary of CAPM Risk-Return Empirical Results

–the relationship between beta and rates of return is a moot point  

•The Market Portfolio: Theory versus Practice

•There is a controversy over the market portfolio.  Hence, proxies are used

•There is no unanimity about which proxy to use

•An incorrect market proxy will affect both the beta risk measures and the position and slope of the SML that is used to evaluate portfolio performance

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