The Capital Asset Pricing Model Ming Liu Industrial Engineering and Management Sciences, Northwestern University Winter 2009

Returns to financial securities  P 0 : security price at time 0  P 1 : security price at time 1  DIV 1 : dividend at time 1  r = total return = dividend yield + capital gain rate r = DIV 1 /P 0 +(P 1 -P 0 )/P 0 (random variable) r i :the return on security i,

 Decompose this return r i into that part correlated with the market and that part uncorrelated with the market r m = the return on the market portfolio ε i = the specific return of firm i

Systemic and Idiosyncratic Risk r i =α i + β i r m + ε i systemic risk undiversifiable risk beta risk market risk idiosyncratic risk diversifiable risk non-systematic risk

"Beta" (β) an asset market risk parameter, represents straight-line inclination degree. E is average "residual" yield, describing an average asset yield deviation from "fair" yield as shown by the central line.

 The larger is β i, the more subject to market risk is this firm.  The larger is σ[  i ] the more important is firm- specific risk. r i =α i +β i r m +ε i

Example Decomposing the Total Risk of a Stock Considering two stocks: A: An automobile stock with β A =1.5, B: An oil exploration company with β B =0.5, The variance of the market return is What is the total risk of each stock? Which has a higher expected rate of return?

Portfolio risk 1.Decompose each security return into systematic and idiosyncratic risk: r i =α i +β i r m +ε i 2.Form a portfolio of these securities, with portfolio weights w 1, w 2, …, w n. (sum to one) 3.The portfolio rate of return is a weighted average of the individual returns r p = w 1 r 1 +w 2 r 2 +…+w n r n

r p = w 1 [α 1 +β 1 r m +ε 1 ] + w 2 [α 2 +β 2 r m +ε 2 ] + … + w n [α n +β n r m +ε n ] Rearrange to get r p = α * +β * r m +ε *, where α * := w 1 α 1 +w 2 α 2 +…+w n α n β * := w 1 β 1 +w 2 β 2 +…+w n β n ε * := w 1 ε 1 +w 2 ε 2 +…+w n ε n zero

Conclusions β of portfolio is weighted-average β Well diversified -> risk only from βr m term The standard deviation of a well diversified portfolio:

Construct the market portfolio The market portfolio includes every security in the market The weight of each security in the portfolio is proportional to its relative size in the economy A common proxy measure for the market portfolio is the S&P 500 index html

The Capital Asset Pricing Model Market model r i =α i +β i r m +ε i with α i =( 1 - β i ) r f r i =(1-β i ) r f +β i r m +ε i

Does this restricted case make sense?  What does it imply for the return on a risk-free asset ( β i =0 )?  What does it imply about the return on an asset that has the same market risk as the market portfolio ( β i =1 )? r i =(1-β i ) r f +β i r m +ε i The CAPM equation can be rewritten as r i -r f =β i (r m –r f )+ε i

The CAPM can also be written as a linear relationship between the β of a security and its expected rate of return, E(r i )-r f =β i (E (r m )–r f ) E(r i ) : expected rate of return on the security E (r m ): expected rate of return on the market portfolio r f : the risk free rate β i : the security’s beta

The Security Market Line E(r i ) βiβi rfrf E(r i )=r f +β i (E (r m )–r f ) E(r i )=(1- β i )r f +β i E (r m ) β A =1.5β B =0.5 E(r A ) E(r B )

Example Using the Security Market Line (SML) The β of Cisco Systems is about The risk free rate r f = 0.07 Expected risk premium on market E (r m )–r f = 0.06 The expected rate of return on CSCO:

How to get β ? If we know  σ[r i ] standard deviation of r i  σ[r m ] standard deviation of r m  ρ im correlation between r i and r m

How to get β ? Estimate beta: r i =α i + β i r m + ε i

CAPM serves as a benchmark – Against which actual returns are compared – Against which other asset pricing models are compared Advantages: – Simplicity – Works well on average Disadvantages: – What is the true market portfolio and risk free rate? – How do you estimate beta? – Standard deviation not a good measure of risk.