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Théorie FinancièreRisk and expected returns (2)

• Professeur André Farber

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Risk and return

• Objectives for this session
• 1. Review 2 risky assets
• 2. Many risky assets
• 3. Beta
• 4. Optimal portfolio
• 5. Equilibrium CAPM

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Review The efficient set for two assets
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Formulas
Returns normal distribution
Expected return
Variance
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Choosing portfolios from many stocks

• Porfolio composition
• (X1, X2, ... , Xi, ... , XN)
• X1 X2 ... Xi ... XN 1
• Expected return
• Risk
• Note
• N terms for variances
• N(N-1) terms for covariances
• Covariances dominate

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Using matrices
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Some intuition
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Example

• Consider the risk of an equally weighted
  portfolio of N "identical  stocks
• Equally weighted
• Variance of portfolio
• If we increase the number of securities ?
• Variance of portfolio

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Diversification
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Conclusion

• 1. Diversification pays - adding securities to
  the portfolio decreases risk. This is because
  securities are not perfectly positively
  correlated
• 2. There is a limit to the benefit of
  diversification the risk of the portfolio can't
  be less than the average covariance (cov) between
  the stocks
• The variance of a security's return can be broken
  down in the following way
• The proper definition of the risk of an
  individual security in a portfolio M is the
  covariance of the security with the portfolio

Portfolio risk
Total risk of individual security
Unsystematic or diversifiable risk
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The efficient set for many securities

• Portfolio choice choose an efficient portfolio
• Efficient portfolios maximise expected return for
  a given risk
• They are located on the upper boundary of the
  shaded region (each point in this region
  correspond to a given portfolio)

Expected Return
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Risk
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Optimal portofolio with borrowing and lending
M
Optimal portfolio maximize Sharpe ratio
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Capital asset pricing model (CAPM)

• Sharpe (1964) Lintner (1965)
• Assumptions
• Perfect capital markets
• Homogeneous expectations
• Main conclusions Everyone picks the same
  optimal portfolio
• Main implications
• 1. M is the market portfolio a market value
  weighted portfolio of all stocks
• 2. The risk of a security is the beta of the
  security
• Beta measures the sensitivity of the return of
  an individual security to the return of the
  market portfolio
• The average beta across all securities, weighted
  by the proportion of each security's market value
  to that of the market is 1

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Market equilibrium illustration
Wealth Risk free asset Market Portfolio Firm 1 Firm 2 Firm 3
Optimal portfolio Optimal portfolio Optimal portfolio 100 20 50 30
Alan 10 -10 20 4 10 6
Ben 20 -5 25 5 12.5 7.5
Clara 30 15 15 3 7.5 4.5
Market 60 0 60 12 30 18
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Capital Asset Pricing Model
Expected return
RM
Rj
Risk free interest rate
ßj
1
Beta
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Beta

• Prof. André FarberSOLVAY BUSINESS
  SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

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Measuring the risk of an individual asset

• The measure of risk of an individual asset in a
  portfolio has to incorporate the impact of
  diversification.
• The standard deviation is not an correct measure
  for the risk of an individual security in a
  portfolio.
• The risk of an individual is its systematic risk
  or market risk, the risk that can not be
  eliminated through diversification.
• Remember the optimal portfolio is the market
  portfolio.
• The risk of an individual asset is measured by
  beta.
• The definition of beta is

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Beta

• Several interpretations of beta are possible
• (1) Beta is the responsiveness coefficient of Ri
  to the market
• (2) Beta is the relative contribution of stock i
  to the variance of the market portfolio
• (3) Beta indicates whether the risk of the
  portfolio will increase or decrease if the weight
  of i in the portfolio is slightly modified

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Beta as a slope
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A measure of systematic risk beta

• Consider the following linear model
• Rt Realized return on a security during period t
• ? A constant a return that the stock will
  realize in any period
• RMt Realized return on the market as a whole
  during period t
• ? A measure of the response of the return on the
  security to the return on the market
• ut A return specific to the security for period
  t (idosyncratic return or unsystematic return)- a
  random variable with mean 0
• Partition of yearly return into
• Market related part ß RMt
• Company specific part a ut

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Beta - illustration

• Suppose Rt 2 1.2 RMt ut
• If RMt 10
• The expected return on the security given the
  return on the market
• ERt RMt 2 1.2 x 10 14
• If Rt 17, ut 17-14 3

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Measuring Beta

• Data past returns for the security and for the
  market
• Do linear regression slope of regression
  estimated beta

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Decomposing of the variance of a portfolio

• How much does each asset contribute to the risk
  of a portfolio?
• The variance of the portfolio with 2 risky assets
• can be written as
• The variance of the portfolio is the weighted
  average of the covariances of each individual
  asset with the portfolio.

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Example
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Beta and the decomposition of the variance

• The variance of the market portfolio can be
  expressed as
• To calculate the contribution of each security to
  the overall risk, divide each term by the
  variance of the portfolio

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Marginal contribution to risk some math

• Consider portfolio M. What happens if the
  fraction invested in stock I changes?
• Consider a fraction X invested in stock i
• Take first derivative with respect to X for X 0
• Risk of portfolio increase if and only if
• The marginal contribution of stock i to the risk
  is

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Marginal contribution to risk illustration
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Beta and marginal contribution to risk

• Increase (sightly) the weight of i
• The risk of the portfolio increases if
• The risk of the portfolio is unchanged if
• The risk of the portfolio decreases if

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Inside beta

• Remember the relationship between the correlation
  coefficient and the covariance
• Beta can be written as
• Two determinants of beta
• the correlation of the security return with the
  market
• the volatility of the security relative to the
  volatility of the market

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Properties of beta

• Two importants properties of beta to remember
• (1) The weighted average beta across all
  securities is 1
• (2) The beta of a portfolio is the weighted
  average beta of the securities

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Risk premium and beta

• 3. The expected return on a security is
  positively related to its beta
• Capital-Asset Pricing Model (CAPM)
• The expected return on a security equals
• the risk-free rate
• plus
• the excess market return (the market risk
  premium)
• times
• Beta of the security

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CAPM - Illustration
Expected Return
Beta
1
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CAPM - Example

• Assume Risk-free rate 6 Market risk
  premium 8.5
• Beta Expected Return ()
• American Express 1.5 18.75
• BankAmerica 1.4 17.9
• Chrysler 1.4 17.9
• Digital Equipement 1.1 15.35
• Walt Disney 0.9 13.65
• Du Pont 1.0 14.5
• ATT 0.76 12.46
• General Mills 0.5 10.25
• Gillette 0.6 11.1
• Southern California Edison 0.5 10.25
• Gold Bullion -0.07 5.40

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CAPM two formulations
Consider a future uncertain cash flow C to be
received in 1 year. PV calculation based on CAPM
See Brealey and Myers Chap 9
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Risk-adjusted expected cash flow
Using risk-adjusted discount rates is OK if you
know beta.
The adjusted risk-adjusted discount rate does not
work for OPTIONS or projects with unknown
betas. To understand how to proceed in that case,
we need to go deeper into valuation theory.
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Example (see Introduction)
You observe the following data
Value Up market (u) Proba 0.75 Down market (d) Proba 0.25 Expected return
Bond 95.24 105 105 5
Market Portfolio 100 1.2 0.80 10
What is the value of the following asset? What
are its expected returns?
NewAsset ? 200 100 ?
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Valuation of project with CAPM
Step 1 calculate statistics for the market
portfolio
Up mkt Proba .75 Down mkt Proba .25
Return 20 -20
Expected return
Market risk premium
Variance
Price of covariance
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Valuation of project with CAPM (2)
Step 2 Calculate statistics for the project
Expected cash flow
Covariance with market portfolio
)
(Reminder
Step 3 Value the project
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Valuation of project with CAPM (3)
Once the value of the project is known, the beta
can be calculated.
Value Up mkt Proba .75 Down mkt Proba .25
Cash flow 154.76 200 100
Returns 29.23 -35.38
Expected return
Beta