Corporate Finance

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Corporate Finance

• Risk and Return
• Prof. André FarberSOLVAY BUSINESS
  SCHOOLUNIVERSITÉ LIBRE DE BRUXELLES

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Risk and return

• Objectives for this session
• 1. Review
• 2. Efficient set
• 3. Optimal portfolio
• 4. CAPM

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Review Risk and expected returns for porfolios

• In order to better understand the driving force
  explaining the benefits from diversification, let
  us consider a portfolio of two stocks (A,B)
• Characteristics
• Expected returns
• Standard deviations
• Covariance
• Portfolio defined by fractions invested in each
  stock XA , XB XA XB 1
• Expected return on portfolio
• Variance of the portfolio's return

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The efficient set for two assets correlation 0
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Example
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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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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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Some intuition
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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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Choosing between 2 risky assets

• Choose the asset with the highest ratio of excess
  expected return to risk
• Example RF 6
• Exp.Return Risk
• A 9 10
• B 15 20
• Asset Sharpe ratio
• A (9-6)/10 0.30
• B (15-6)/20 0.45

• A

Expected return
B
A
Risk
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Optimal portofolio with borrowing and lending
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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Optimal portfolio property
Slope
M
x j
RF
Slope
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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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Pratical implications

• Efficient market hypothesis CAPM passive
  investment
• Buy index fund
• Choose asset allocation