The Capital Asset Pricing Model Ming Liu Industrial

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

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

Ø 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 εi = the specific return of firm i

Systemic and Idiosyncratic Risk ri=αi+ βi rm + εi 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.

ri=αi+βi rm +εi Ø The larger is βi , the more subject to market risk is this firm. Ø The larger is σ[ i] the more important is firmspecific risk.

Example Decomposing the Total Risk of a Stock Considering two stocks: A: An automobile stock with βA=1. 5, B: An oil exploration company with βB=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: ri=αi+βirm+εi 2. Form a portfolio of these securities, with portfolio weights w 1, w 2, …, wn. (sum to one) 3. The portfolio rate of return is a weighted average of the individual returns rp = w 1 r 1+w 2 r 2+…+wnrn

rp = w 1[α 1+β 1 rm+ε 1] + w 2[α 2+β 2 rm+ε 2] + … + wn[αn+βnrm+εn] Rearrange to get zero rp = α*+β*rm+ε*, where α*: = w 1α 1+w 2 α 2+…+wn α n β*: = w 1 β 1+w 2 β 2+…+wn β n ε*: = w 1 ε 1+w 2 ε 2+…+wn εn

Conclusions • β of portfolio is weighted-average β • Well diversified -> risk only from βrm 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. http: //www. indexarb. com/index. Component. Wts. SP 5 00. html

The Capital Asset Pricing Model Market model ri=αi+βirm+εi with αi=(1 -βi) rf ri=(1 -βi) rf+βirm+εi

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

The CAPM can also be written as a linear relationship between the β of a security and its expected rate of return, E(ri )-rf=βi (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 βi : the security’s beta

The Security Market Line E(ri )=rf+βi (E (rm )–rf ) E(ri )=(1 - βi)rf+βi E (rm ) E(r. A ) E(r. B ) rf βB=0. 5 βA=1. 5 βi

Example Using the Security Market Line (SML) The β 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:

How to get β? • If we know Ø σ[ri] ----- standard deviation of ri Ø σ[rm] ----- standard deviation of rm Ø ρim----- correlation between ri and rm

How to get β? • Estimate beta: ri=αi+ βi rm + εi • http: //finance. yahoo. com/

• 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.