An Introduction to Asset Pricing Models

1
An Introduction to Asset Pricing Models
Chapter 9

• Innovative Financial Instruments

Dr. A. DeMaskey
2
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

3
Assumptions of Capital Market Theory

• All investors are Markowitz efficient investors
  who choose investments on the basis of expected
  return and risk.
• Investors can borrow or lend any amount of money
  at the riskfree rate of return (RFR).
• All investors have homogeneous expectations that
  is, they estimate identical probability
  distributions for future rates of return.
• All investors have the same one-period time
  horizon, such as one-month, six months, or one
  year.

4
Assumptions of Capital Market Theory

• All investments are infinitely divisible, which
  means that it is possible to buy or sell
  fractional shares of any asset or portfolio.
• There are no taxes or transaction costs involved
  in buying or selling assets.
• There is no inflation or any change in interest
  rates, or inflation is fully anticipated.
• Capital markets are in equilibrium that is, we
  begin with all investments properly priced in
  line with their risk levels.

5
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.
• Judge a theory on how well it explains and helps
  predict behavior, not on its assumptions.

6
Riskfree Asset

• Provides the risk-free rate of return (RFR)
• An asset with zero variance and standard
  deviation
• Zero correlation with all other risky assets
• Covariance between two sets of returns is
• Will lie on the vertical axis of a portfolio graph

7
Combining a Riskfree Asset with a Risky Portfolio

• Expected return
• The expected variance for a two-asset portfolio
• Because the variance of the riskfree asset is
  zero and the correlation between the riskfree
  asset and any risky asset i is zero, this
  simplifies to

8
Combining a Risk-Free Asset with a Risky
Portfolio

• Given the variance formula
• The standard deviation is
• Therefore, the standard deviation of a portfolio
  that combines the riskfree asset with risky
  assets is the linear proportion of the standard
  deviation of the risky asset portfolio.

9
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 riskfree rate and investing in the
  risky portfolio at point M

10
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
• Since the market is in equilibrium, all assets
  are included in this portfolio in proportion to
  their market value.
• Since 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.

11
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

12
Factors Affecting Systematic Risk

• Variability in growth of money supply
• Interest rate volatility
• Variability in

13
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

14
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?
• Observe what happens as you increase the sample
  size of the portfolio by adding securities that
  have some positive correlation

15
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 riskfree rate and invest the
  remainder in the market portfolio

16
The CML and the Separation Theorem

• Investors preferring more risk might borrow funds
  at the RFR and invest everything in the market
  portfolio
• The decision of both investors is to invest in
  portfolio M along the CML
• The decision to borrow or lend to obtain a point
  on the CML is a separate decision based on risk
  preferences
• Tobin refers to this separation of the investment
  decision from the financing decision as the
  separation theorem

17
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
• Together, this means the only important
  consideration is the assets covariance with the
  market portfolio

18
A Risk Measure for the CML

• Since all individual risky assets are part of
  the M portfolio, an assets rate of return in
  relation to the return of the M portfolio may be
  described using the following linear model

where Rit return for asset i during period
t ai constant term for asset i bi slope
coefficient for asset i RMt return for the M
portfolio during period t e random error
term
19
Variance of Returns for a Risky Asset
Note Var(biRMi) is variance related to market
return Var(e) is the residual return not
related to the market portfolio
20
The Capital Asset Pricing Model Expected Return
and Risk

• The existence of a riskfree asset resulted in
  deriving a capital market line (CML) that became
  the relevant frontier
• An assets covariance with the market portfolio
  is the relevant risk measure
• This can be used to determine an appropriate
  expected rate of return on a risky asset - the
  capital asset pricing model (CAPM)

21
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
• The estimated rate of return can also be compared
  to the required rate of return implied by CAPM to
  determine whether a risky asset is over- or
  undervalued

22
The Security Market Line (SML)

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

23
The Security Market Line (SML)

• The equation for the risk-return line is given as

We then define as beta
24
Determining the Expected Rate of Return for a
Risky Asset

• The expected rate of return of a risky 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)

25
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
• To earn better risk-adjusted rates of return than
  the average investor, a superior investor must
  derive value estimates for assets that are
  consistently superior to the consensus market
  evaluation

26
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

27
Calculating Systematic Risk The Characteristic
Line

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

where Ri,t the rate of return for asset i
during period t RM,t the rate of return for the
market portfolio M during t
e the random error term
28
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
• Weak relationship between VL ML betas due to
  difference in intervals used
• There is no correct interval for analysis
• Interval effect impacts smaller firms more

29
The Effect of the Market Proxy

• Choice of market proxy is crucial
• Proper measure must include all risky assets
• Standard Poors 500 Composite Index is most
  often used
• Large proportion of the total market value of
  U.S. stocks
• Value weighted series
• Weaknesses

30
Arbitrage Pricing Theory (APT)

• CAPM is criticized because of the difficulties in
  selecting a proxy for the market portfolio as a
  benchmark
• An alternative pricing theory with fewer
  assumptions was developed
• Arbitrage Pricing Theory

31
Assumptions of Arbitrage Pricing Theory (APT)

• Capital markets are perfectly competitive
• Investors always prefer more wealth to less
  wealth with certainty
• The stochastic process generating asset returns
  can be presented as K factor model

32
Assumptions of CAPMThat Were Not Required by APT

• APT does not assume
• A market portfolio that contains all risky
  assets, and is mean-variance efficient
• Normally distributed security returns
• Quadratic utility function

33
Arbitrage Pricing Theory (APT)

• For i 1 to N where
• Ri return on asset i during a specified time
  period
• Ei expected return for asset i
• bik reaction in asset is returns to movements
  in a common
• factor
• dk a common factor with a zero mean that
  influences the
• returns on all assets
• ei a unique effect on asset is return that, by
  assumption, is
• completely diversifiable in large
  portfolios and has a
• mean of zero
• N number of assets

34
Arbitrage Pricing Theory (APT)

• Multiple factors, dk, expected to have an impact
  on all assets
• Inflation
• Growth in GNP
• Major political upheavals
• Changes in interest rates
• And many more.
• Contrast with CAPMs insistence that only beta is
  relevant

35
Arbitrage Pricing Theory (APT)

• Bik determine how each asset reacts to this
  common factor
• Each asset may be affected by growth in GNP, but
  the effects will differ
• In applying the theory, the factors are not
  identified
• Similar to the CAPM in that the unique effects
  (ei) are independent and will be diversified away
  in a large portfolio

36
Arbitrage Pricing Theory (APT)

• APT assumes that, in equilibrium, the return on a
  zero-investment, zero-systematic-risk portfolio,
  is zero when the unique effects are diversified
  away
• The expected return on any asset i (Ei) can be
  expressed as

37
Arbitrage Pricing Theory (APT)

• Where
• l0 the expected return on an asset with zero
  systematic risk
• where l0 E0
• l1 the risk premium related to each of the
  common factors,
• with i 1 to k
• bi pricing relationship between the risk
  premium and asset i

38
Example of Two Stocks and a Two-Factor Model

• l1 changes in the rate of inflation. The risk
  premium
• related to this factor is 1 for every
  1 change in the
• rate (l1 0.1)
• l2 percent growth in real GNP. The average
  risk premium
• related to this factor is 2 for every 1
  change in the
• rate (l2 0.02)
• l3 the rate of return on a zero-systematic-risk
  asset (zero
• beta boj 0) is 3 (l3 0.03)

39
Example of Two Stocks and a Two-Factor Model

• bx1 the response of asset X to changes in the
• rate of inflation is 0.50 (bx1 0.50)
• by1 the response of asset Y to changes in the
• rate of inflation is 2.00 (by1 2.00)
• bx2 the response of asset X to changes in the
• growth rate of real GNP is 1.50 (bx2
  1.50)
• by2 the response of asset Y to changes in the
• growth rate of real GNP is 1.75 (by2
  1.75)

40
Example of Two Stocks and a Two-Factor Model

• .03 (.01)bi1 (.02)bi2
• Ex .03 (.01)(0.50) (.02)(1.50)
• .065 6.5
• Ey .03 (.01)(2.00) (.02)(1.75)
• .085 8.5

41
Empirical Tests of the APT

• Studies by Roll and Ross and by Chen support APT
  by explaining different rates of return with some
  better results than CAPM
• Reinganums study did not explain small-firm
  results
• Dhrymes and Shanken question the usefulness of
  APT because it was not possible to identify the
  factors

42
Summary

• When you combine the riskfree asset with any
  risky asset on the Markowitz efficient frontier,
  you derive a set of straight-line portfolio
  possibilities
• The dominant line is tangent to the efficient
  frontier
• Referred to as the capital market line (CML)
• All investors should target points along this
  line depending on their risk preferences

43
Summary

• All investors want to invest in the risky
  portfolio, so this market portfolio must contain
  all risky assets
• The investment decision and financing decision
  can be separated
• Everyone wants to invest in the market portfolio
• Investors finance based on risk preferences

44
Summary

• The relevant risk measure for an individual risky
  asset is its systematic risk or covariance with
  the market portfolio
• Once you have determined this Beta measure and a
  security market line, you can determine the
  required return on a security based on its
  systematic risk

45
Summary

• Assuming security markets are not always
  completely efficient, you can identify
  undervalued and overvalued securities by
  comparing your estimate of the rate of return on
  an investment to its required rate of return
• The Arbitrage Pricing Theory (APT) model makes
  simpler assumptions, and is more intuitive, but
  test results are mixed at this point

46
The InternetInvestments Online

• www.valueline.com
• www.barra.com
• www.stanford.edu/wfsharpe.com
• www.wsharpe.com