Diversification of Risks:• Total Risk = Firm Specific Risk + Market Wide Risk• Unique Risk = 1. Management Risk

a) acts of godb) Product Obsolescencec) Errors

2. Default Riska) Financial Leverageb) Operating Leverage

3. Industry Risk4. Liquidity Risk

• Market Risks = 1. Interest Rate Riska) Inflation Riskb) Real Rate Risk

2. Market Riskc) Business cyclesd) Economic Policy Dec. Tax Code Change

Minimum Variance Portfolio

• The maximum return any one will be willing to receive for a certain level of risk.

• Adding the risk free modifies the efficient set to be linear now.

Capital market line• Risk added by new asset is proportional to σim.

Covariance is standardized by dividing it with σm².

Beta = σim/ σm² and E(Ri)-Rf = β[E(Rm)-Rf]

Computing Beta:

• Using regression:Ri = ά +βRm + εi

σi² = β² σm² + σε² (The variation terms are related but not the standard deviation terms)Divide both sides by σ i²: 1=β² σm²/ σi²+ σε²/ σi² (proportion of market risk + proportion of unique risk)The relationship between coefficient of determination (R²) and correlation coefficient ρ.βi=Covariance(i,m)/var.(m)= σim/ σm²= ρim σiσm/σm² = Pim(σ i/ σ m)We know Ri²=β² σm²/ σi²= ρim²(σi²/ σm²)(σm²/σi²)= ρim²Ri = ρim

Two Basic ideas About Risk and ReturnMarket Model/Single Factor Model

• CAPM: Ri=Rf+β(Rm-Rf)1. Investors require compensation for risk2. They care only about a stock’s contribution to portfolio.

• Capital Asset Pricing Model: ExampleIf Treasury bill rate = 5.6%Bristol Myers Squibb beta = .81Expected market risk premium = 8.4%Ri=Rf+ β(Rm-Rf) =5.6+.81(8.4)=12.4%β=1 Ri=Rm

β=0 Ri=Rf

β>1 Ri>Rm

β<1 Ri<Rm

Testing the Capital Asset Pricing Model

• If a portfolio is efficient, there must be a straight line relationship between the expected return of any share and its beta relative to that portfolio. So testing whether the market portfolio is efficient.

• Problems of testing:1. measuring expected returns2. measuring the market portfolio3. measuring beta

Black, Jensen and Scholes’s Test of CAPM

Beta

rf

rm

AverageMonthlyreturn

Fitted line

Theoreticalline

Validity of Capital Asset Pricing ModelEvidence is mixed 1. Long-run average returns are significantly related to beta 2. But beta is not a complete explanation. Low beta stocks have earned higher rates of return than predicted by the model. So have small company stocks.

CAPM• Capital Asset Pricing Model is Attractive

Because:1. It is simple and usually gives sensible answers.2. It distinguishes between diversifiable and non-

diversifiable risk.

• Testing of CAPM is Controversial1. No one knows for sure how to define and measure the market portfolio – and using the wrong

index could lead to the wrong answer.2. The model is hard to prove or disprove3. The model has competitors e.g: arbitrage Pricing

Model. Note: you can reject CAPM without rejecting the main points of modern portfolio theory.

The standard CAPM concentrates on how stocks contributeto the level and uncertainty of investor’s wealth.Consumption is outside the model.

STOCKS (And Other Risky Assets)

WEALTH = MARKET PORTFOLIO

Standard CAPM assumes investors areConcerned with the amount and uncertainty of futurewealth

Market riskMakes wealth

Uncertain.

The consumption CAPM defines risk as a stock’scontribution to uncertainty about consumption. Wealth (theintermediate step between stock returns and consumption)drops out of the model.

STOCKS (And Other Risky Assets)

Consumption

Consumption CAPM connects uncertaintyabout stock returnsdirectly to uncertaintyabout consumption.

Wealth is uncertain.

WEALTH

Consumptionis uncertain

Arbitrage pricing theory (APT)

• Suppose returns depend on more than one factor i.e many factors:Return = a+b1(rfactor – rf)+b2(rfactor-rf)+…..+diversifiable noiseA diversified portfolio that is not exposed to any factors must offer the risk-free rate:Return = a=rf when all b’s=0But in general expected return depends on factor exposure Return – rf+b1(rfactor-rf)+b2(rfactor-rf)+…Where rfactor +expected return on “pure play” portfolio exposed only to factor i.

Arbitrage pricing Theory (APT)

• Preserves distinction between diversifiable and non-diversifiable risk.

• CAPM and APT can both hold-e.g. CAPM implies one factor APT, with rfactor + rm

• But APT is more general – e.g. unlike CAPM, market portfolio doesn’t have to be efficient.

• But usefulness of APT requires heavy-duty statistics to

Identify factorsMeasure factor returns

Arbitrage Pricing Theory:Three ways to construct Factors:

1. Factor Analysis (FA)2. Macro Economic Variables (MC)3. Firm Characteristics (FC)

Factor Analysis (FA): Developed using a statistical procedure called Factor Analysis

ADVANTAGE: Best estimate of FactorDISADVANTAGE: No economic intuition

Macro Economic Variables (MC): Use macro variables as factors e.g. productivity, Interest rates, inflation rates

ADVANTAGE: Best economic interpretationDISADVANTAGE: Difficult to measure unanticipated changes.

Firm Characteristics (FC): Use micro variables for firm as factors e.g. size, P/E, Book to market, etc.

ADVANTAGE: More intuitive than FADISADVANTAGE: Historical data? Future changes?

Chen, Roll and Ross (1986)

The economic factors:

3. Industrial production : Survey of current4. Consumption : Business3. Oil prices : B of Labor Stats.4. Inflation : Consumer Price Index5. Treasury Bill Rate : 1 month T. bill6. Long term Govt. Bonds :Treasury Bonds 10/15/20 yrs.7. Low Grade Bonds : Baa or Lower8. Equally Weighted Equities : NYSE9. Value Weighted Equities : NYSE

Derived Series in Chen, Roll and Ross (1986):

The variables used are:

μP(t), γP(t) + Monthly or Annual Growth rate of industrial prod.E(I(t)) = expected InflationUI(t) = unexpected InflationRHO(t) = Real Int. Rate = Tb(t-1)-I(t)DEI(t) = Change in expected inflation

E[I(t+1)]-E[I(t/t-1]URP(t) = Risk Premium

Baa(t)-LGB(t)UTS(t) = Term structure

LGB(t)-TB(t-1)Regression Reformed:R=a+βMP*MP+βDEI*DEI+βURP*URP+βUTS*UTS+Rm+Σj