Portfolio Theory Capital Asset Pricing Model and Arbitrage Pricing Theory.
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Transcript of Portfolio Theory Capital Asset Pricing Model and Arbitrage Pricing Theory.
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Portfolio Theory
Capital Asset Pricing Model and Arbitrage Pricing Theory
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Contribution of MPT
Establish diversifiable versus nondiversifiable risks
Quantify diversifiable and nondiversifiable risk
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Market Equilibrium Condition
Law of one pricePrice of risk = Reward-to-risk ratioFor well diversified portfolios, the only
remaining risks are systematic riskHence,
j Fi F
i j
r rr r
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CAPM
Assumptions (see recommended textbook)The Equilibrium World
– The Market Portfolio is the Optimal Risky Portfolio
– the Capital Market Line is the Optimal CAL
The Separation Theorem– aka Mutual Fund Theorem
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Market Risk Premium
Market Risk Premium: rM - rf = A 2M
– depends on aggregate investors’ risk aversion (A)– and market’s volatility (2
M)Historically:
– rM - rf = 12.5% - 3.76% = 8.74%– M = 20.39%– 2
M = 0.20392 = 0.0416 Implying an average investor has:
– A = 2.1
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Reward and Risk in CAPM
Reward– Risk Premium: [E(ri) - rF]
Risk– Systematic Risk: i = iM/M
2
Ratio of Risk Premium to Systematic Risk= [E(ri) - rF] / i
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Equilibrium in a CAPM World
This condition must apply to all assets, including the market portfolio
Define M = 1CAPM equation:E(ri) = rF + i x [E(rM) - rF]
( )( ) j Fi F
i j
E r rE r r
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Systematic Risk of a Portfolio
Systematic Risk of a Portfolio is a weighted average
= wi i
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The Security Market Line
The Security Market Line (SML)– return-beta () relationship for individual
securities
The Capital Market(Allocation) Line (CML/CAL)– return-standard deviation relationship for
efficient portfolios
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Security Market Line (SML)
0%
5%
10%
15%
20%
25%
0.0 0.5 1.0 1.5 2.0 2.5
beta ()
Exp
ecte
d R
etu
rn
MStock i
SML
rf=7%
Market Risk premium=8%
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Uses of CAPM
BenchmarkingCapital BudgetingRegulation
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CAPM and Index Models
Index models - uses actual portfoliosTest for mean-variance efficiency of the
indexBad index or bad model?
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Security Characteristic Line (SCL) (A Scatter Diagram)
-20%
-15%
-10%
-5%
0%
5%
10%
15%
-15% -10% -5% 0% 5% 10% 15%
Market Excess Return (RM)
Sto
ck's
Excess R
etu
rn (
R i)
= -0.0006
= 1.0177
= 0.5715
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Estimating Beta
Past does not always predict the futureRegression toward the meanIs Beta and CAPM dead?
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Arbitrage Pricing Theory (APT)
Assumption– Risk-free arbitrage cannot exist in an efficient
market– Arbitrage
• A zero-investment portfolio with sure profit– e.g. violation of law of one price
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APT Equilibrium Condition
Law of One PriceIf two portfolios, A and B, both only have
one systematic factor (k),
, ,
A F B F
k A k B
r r r r
There can be many risk factors. The equilibrium condition holds for each risk factor.
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APT example
Economy Stock A Stock BGood 10% 12%Bad 5% 6%
Stock A sells for $10 per share Stock B sells for $50 per share Arbitrage strategy
– Short sell 500 shares of stock A ($5000)– Buy 100 shares of stock B ($5000)
• Net investment = $5000 - $5000 = $0
Arbitrage returnEconomy PortfolioGood -500+600 = 100 =2%Bad -250+300 = 50 = 1%
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Multi-factor Models
Factor Portfolio (RMK)
– A well-diversified portfolio with beta=1 on one factor and beta=0 on any other factor
Ri = rfi + i1RM1 + i2RM2 + ei
– rfi is the risk-free rate
– RM1 is the excess return on factor portfolio 1
– RM2 is the excess return on factor portfolio 2
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Summary
CAPM– Empirical application of CAPM needs a proxy for the
market portfolio– Empirical evidence lacks support
• Could be due to poor proxy or poor model
APT– Difficult to apply empirically– The model does not identify systematic risk factors
Empirical Models– Lacks economic intuition– E.g. Market-to-book ratio as a systematic risk factor