Portfolio Theory Capital Asset Pricing Model and Arbitrage Pricing Theory

Contribution of MPT  Establish diversifiable versus nondiversifiable risks  Quantify diversifiable and nondiversifiable risk

Market Equilibrium Condition  Law of one price  Price of risk = Reward-to-risk ratio  For well diversified portfolios, the only remaining risks are systematic risk  Hence,

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

Market Risk Premium  Market Risk Premium: r M - r f = A  2 M –depends on aggregate investors’ risk aversion (A) –and market’s volatility (  2 M )  Historically: – r M - r f = 12.5% % = 8.74% –  M = 20.39% –  2 M = =  Implying an average investor has: – A = 2.1

Reward and Risk in CAPM  Reward –Risk Premium: [E(r i ) - r F ]  Risk –Systematic Risk:  i =  iM /  M 2  Ratio of Risk Premium to Systematic Risk = [E(r i ) - r F ] /  i

Equilibrium in a CAPM World  This condition must apply to all assets, including the market portfolio  Define  M = 1  CAPM equation:  E(r i ) = r F +  i x [E(r M ) - r F ]

Systematic Risk of a Portfolio  Systematic Risk of a Portfolio is a weighted average   =  w i  i

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

Security Market Line (SML) M Stock i SML r f =7% Market Risk premium=8%

Uses of CAPM  Benchmarking  Capital Budgeting  Regulation

CAPM and Index Models  Index models - uses actual portfolios  Test for mean-variance efficiency of the index  Bad index or bad model?

Security Characteristic Line (SCL) (A Scatter Diagram)  =  =  =

Estimating Beta  Past does not always predict the future  Regression toward the mean  Is Beta and CAPM dead?

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

APT Equilibrium Condition  Law of One Price  If two portfolios, A and B, both only have one systematic factor (k),  There can be many risk factors. The equilibrium condition holds for each risk factor.

APT example EconomyStock AStock B Good10%12% Bad5%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 = $0  Arbitrage return EconomyPortfolio Good = 100 =2% Bad = 50 = 1%

Creating A Zero-beta (risk-free) Portfolio  The case of 2 well-diversified portfolios, v and u  Weights for v and u in the Zero-beta Portfolio: w v = -  u / (  v -  u ) w u =  v / (  v -  u ) Note: w v + w u = 1  The Zero-beta Portfolio has no risk (why?)  Excess return on the Zero-beta Portfolio  v+u =  v w v +  u w u  Arbitrage opportunity? –As long as  v+u  0 there is an arbitrage opportunity –The market is out of equilibrium

Arbitrage: An Example  Portfolio v:  v = 1.3,  v = 2%  Portfolio u:  u = 0.8,  u = 1%  Since alpha  0, there is an arbitrage opportunity –Create a zero-beta (risk-free) portfolio: w v = -.8/( ) = -1.6 w u = 1.3/( ) = 2.6 –  v+u = 2.6 * * 1.3 = 0 –  v+u = 2.6 * 1% * 2% = - 0.6%  Sell the portfolio and buy the risk-free asset

Multi-factor Models  Factor Portfolio (R MK ) –A well-diversified portfolio with beta=1 on one factor and beta=0 on any other factor  R i = r fi +  i1 R M1 +  i2 R M2 + e i –r fi is the risk-free rate –R M1 is the excess return on factor portfolio 1 –R M2 is the excess return on factor portfolio 2

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