The Capital Asset Pricing Model Ming Liu Industrial Engineering and Management Sciences, Northwestern University Winter 2009 Returns to financial securities P0: security price at time 0 P1: security price at time 1 DIV1: dividend at time 1 r = total return = dividend yield + capital gain

rate r = DIV1/P0+(P1-P0)/P0 (random variable) ri : the return on security i, Decompose this return ri into that part correlated with the market and that part uncorrelated with the market rm = the return on the market portfolio i = the specific return of firm i Systemic and Idiosyncratic Risk

ri=i+ i rm + i systemic risk undiversifiable risk beta risk market risk idiosyncratic risk diversifiable risk non-systematic risk "Beta" ) 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. ri=i+i rm +i The larger is i , the more subject to market risk is this firm. The larger is [i] the more important is firmspecific risk. 2 i 2 [ri ] [rm ] ( [ i ])

2 Example Decomposing the Total Risk of a Stock Considering two stocks: 2 [ ] .10

A: An automobile stock with A=1.5, A 2 [ ] .18 B B: An oil exploration company with B=0.5, 2

[ r ] .04 m The variance of the market return is What is the total risk of each stock? [rA ]2 1.52 0.04 0.10 0.19 [rB ]2 0.52 0.04 0.18 0.19 Which has a higher expected rate of

return? Portfolio risk 1. Decompose each security return into systematic and idiosyncratic risk: ri=i+irm+i 2. Form a portfolio of these securities, with portfolio weights w1, w2, , wn . (sum to one) 3. The portfolio rate of return is a weighted average of the individual returns rp = w1r1+w2r2++wnrn

rp = w1[1+1rm+1] + w2[2+2rm+2] + + wn[n+nrm+n] Rearrange to get zero rp = *+*rm+*, where *:= w11+w2 2++wn n *:= w1 1+w2 2++wn n

*:= w1 1+w2 2++wn n Conclusions of portfolio is weighted-average Well diversified -> risk only from rm term The standard deviation of a well diversified portfolio: [rp ] * [ rm ] 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. 500.html The Capital Asset Pricing Model Market model ri=i+irm+i with i=(1-i) rf

ri=(1-i) rf+irm+i ri=(1-i) rf+irm+i Does this restricted case make sense? What does it imply for the return on a risk-free asset (i=0)? What does it imply about the return on an asset that has the same market risk as the market portfolio (i=1)? The CAPM equation can be rewritten as ri-rf=i (rm rf )+i

The CAPM can also be written as a linear relationship between the of a security and its expected rate of return, E(ri )-rf=i (E (rm )rf ) E(ri ) : expected rate of return on the security E (rm): expected rate of return on the market portfolio rf : the risk free rate i : the securitys beta The Security Market Line

E(ri ) E(ri )=rf+i (E (rm )rf ) E(ri )=(1- i)rf+i E (rm ) E(rA ) E(rB ) rf B=0.5 A=1.5

i Example Using the Security Market Line (SML) The of Cisco Systems is about 1.37. The risk free rate rf=0.07 Expected risk premium on market E (rm )rf =0.06 The expected rate of return on CSCO: E (ri ) 0.07 1.37 0.06 0.1522 How to get ?

If we know [ri] ----- standard deviation of ri [rm] ----- standard deviation of rm im----- correlation between ri and rm cov(ri , rm ) im [ri ] [rm ] im [ri ] i 2 2 [rm ]

[rm ] [rm ] How to get ? im [ri ] Estimate beta: i [rm ]

ri=i+ i rm + 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.

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