Return and Risk: The Asset-Pricing Model: CAPM and APT

Portfolio Theory Ex1, Two portfolio  Portfolio 1: a single investment A: with expected return : 1 and variance 25  Portfolio 2: equally weighted combination of two uncorrected investments: with expected return of 1, and variances 25  Both have the same expected return : 1, but the variance of portfolio 2 is.5 2 * *25 = 12.5  Portfolio is preferred than portfolio 1.  It is because the zero covariance diversifies some of the return volatility.

A portfolio of two risky assets Ex2:  Asset 1 has expected return of.22 and SD of.32  Asset 2 has expected return of.13 and SD of.23  Covariance is X 1 X 2 E(r p ~ )  (r p ~ )

Efficient Portfolio Frontier  E(R)  

Efficient Sets and Diversification  E(R) -1 <   = 1  = -1

What about portfolio of n assets? Expected return of portfolio Standard deviation of portfolio’s return.

Markowitz Portfolio Theory  Combining stocks into portfolios can reduce standard deviation below the level obtained from a simple weighted average calculation.  Less than perfect correlation coefficients make this possible. efficient portfolios  The various weighted combinations of stocks that create this standard deviations constitute the set of efficient portfolios.

Combination of risk-free and risky asset   p 2 = x 1 2  1 2, Thus, :  p = x 1  1, x 1 =  p /  1  E(r p ~ ) = x 1 r 1 + x 2 r f =  p *(r 1 /  1 ) + x 2 r f rfrf. Asset 1 Expected return of portfolio Standard deviation of portfolio’s return.

Which risky asset to choose? Standard deviation of portfolio’s return. Risk-free rate (R f ) S. Capital market line. X Y

Which risky asset to choose? Expected return of portfolio Standard deviation of portfolio’s return. Risk-free rate (R f ) 4 S. 5.. Capital market line. X Y Lending Borrowing

The Chosen Portfolio, M  Will different individual have different choice of different risky portfolio asset?  What if different individual holds the same expectation (homogeneous expectation), that is,.market reflects all the information?  What does portfolio S look like?  All investors will invest in portfolio S, regardless of their risk aversion. But they may NOT have the same portion of their wealth in the two assets.

Security Market Line Expected return on security (%) Beta of security RmRm RfRf 0.81 S M T... Security market line (SML)

Security Market Line  For a well diversified portfolio, the risk measure of individual stock is not the SD of return, it is beta.  Investors are not rewarded with any return for bearing any unsystematic risk.  Why should equilibrium prices of securities fall on SML?  If point A lies above the Security market line, then investors will bid up the price until the return goes back on line  If point B lies below the security market line, then investors will sell the security, push down the price until it goes back on line.

Capital Asset Pricing Model  If investors hold market portfolio, how do they measure the risk of individual securities? The covariance with the markets, that is Beta.  CAPM, for any security i, E (r i ~ ) = r f +  i [E(r m ~ - r f )], where, E(r m ~ - r f ) : expected market risk premium  i = COV(r i ~, r m ~ )/  m 2

Testing the CAPM Avg Risk Premium Portfolio Beta 1.0 SML Investors Market Portfolio Beta vs. Average Risk Premium

Testing the CAPM Avg Risk Premium Portfolio Beta 1.0 SML Investors Market Portfolio Beta vs. Average Risk Premium

Testing the CAPM Average Return (%) Company size SmallestLargest Company Size vs. Average Return

Testing the CAPM Average Return (%) Book-Market Ratio Highest Lowest Book-Market vs. Average Return

About CAPM  Why does CAPM not hold ?  Is CAPM dead?  Expected return vs. real return  Short term or long term effect?  The contribution of CAPM  How the financial markets may price risky assets  How to measure a risky asset’s risk  How to calculate expected rate of return.

The measurement of beta  Choice of market proxy  The time period  Measurement error: the problem of overestimate for high beta and underestimate for low beta stocks  Instability over time

Arbitrage Pricing Theory Alternative to CAPM Expected Risk Premium = r - r f = B factor1 (r factor1 - r f ) + B f2 (r f2 - r f ) + … Return= a + b factor1 (r factor1 ) + b f2 (r f2 ) + …

Arbitrage Pricing Theory Estimated risk premiums for taking on risk factors ( )