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Week 7_EMH - Markets and Information Efficient Market...

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Markets and Information Efficient Market Hypothesis and Performance Measures
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2 CAPM and APT Risk-return trade- off Single market portfolio Unequivocal statement on the E(r)- β relationship for ALL assets Strong conditions No-arbitrage condition Set of factors Cannot rule out a violation of the relationship for any particular asset Factors are not specified
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3 An Example: IBM versus Dell Assuming the risk-free rate is 6%, are IBM and Dell priced correctly? IBM Dell Boom Price (Prob=0.5) 140 160 Bust Price (Prob=0.5) 100 80 E(price) 120 120 Current Price 100 90 E(return) 20% 33.3%
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4 Business Cycle Factor Set the business cycle indicator, F BC , equal one if the economy is booming, and zero if the economy is in recession Given the 50/50 probability of boom/bust, E(F BC ) = 0.5 The unexpected change, f BC = F BC -E(F BC ), is 0.5 when booming, and –0.5 when in recession
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5 Factor Loadings The factor model r i = E(r i ) + β i,BC f BC + e i , i = IBM, Dell Conditional expectation in two states: 0.4 = E(r IBM ) + β IBM,BC × 0.5 0 = E(r IBM ) + β IBM,BC × (-0.5) E(r IBM )=20%, β IBM,BC =0.4 E(r Dell )=33.3%, β Dell,BC =0.89
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6 Factor Risk Premium The APT pricing based on IBM E(r IBM ) = r f + λ BC β IBM,BC 0.2 = 0.06 + λ BC × 0.4 ⇒ λ BC = 0.35 E(r Dell ) = 0.06+0.35 × 0.89 = 0.3715 > 0.33 Dell is overpriced. The APT pricing based on Dell E(r Dell ) = r f + λ BC β Dell,BC 0.333 = 0.06 + λ BC × 0.89 ⇒ λ BC = 0.307 E(r IBM ) = 0.06+0.307 × 0.4 = 0.183 < 0.2 IBM is underpriced.
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APT relative pricing example APT relative pricing 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 beta (business cycle) E(r) IBM Dell
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8 Arbitrage Tracking portfolio T: risk-free plus IBM w IBM × β IBM,BC + w f × 0 = β Dell,BC w IBM = 2.225, w f = -1.225 Buy T and sell Dell What is the β BC of the total portfolio? r T -r Dell = 3.87% Bust: r T -r Dell = 3.76%
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Position ($100 portfolio) CF now CF Boom CF Bust buy IBM Now: $100 Boom: $140 Bust: $100 2.225 shares -$222.50 +$311.50 $222.50 [email protected] 6% +$122.50 -$129.85 -$129.85 sell Dell Now: $90 Boom: $160 Bust: $80 1.1 shares +$100.00 -$177.78 -$88.89 Total $0 +$3.87 +$3.76 Arbitrage Example
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10 Equilibrium Price Price equation: E(r Dell ) = 0.06 + 0.35 × β Dell,BC Factor model: r Dell = E(r Dell ) + β Dell,BC f BC + e Dell Boom and bust 160/P –1 = 0.06+0.35 β Dell,BC + β Dell,BC × 0.5 80/P –1 = 0.06+0.35 β Dell,BC + β Dell,BC × (-0.5) β Dell,BC = 0.92 and p Dell = $86.8 [E(r)=38%] Relative to IBM, investors underestimated Dell’s exposure to business cycle ( β Dell,BC = 0.89) and were willing to pay a higher price (p Dell = $90). What if the true premium λ BC = 0.307, or a weighted average of 0.35 and 0.307?
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11 APT Pricing Relative pricing: assets are priced relative to each other. For n-factor models, need n+1 assets: E(r 1 ) = λ 0 + λ 1 β 1,1 + λ 2 β 1,2 + ... + λ n β 1,n E(r 2 ) = λ 0 + λ 1 β 2,1 + λ 2 β 2,2 + ... + λ n β 2,n ......
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