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The capital asset pricing model

Jamal Munshi, Sonoma State Univesity, 1992
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Note: Please read i, j, and p a s subscripts (the i-th and j-th assets and p for portfolio). Also please read the the number 2 as a superscript eg s2 = s-squared. Also, LT=less than and GT=greater than

The equations for portfolio returns and risk

  • kp = sumi(aiki)
  • s2p = sumi(sumj(aiajsij))

A plot of these relationships produces a curve in k-s space that is quadratic and not linear as shown in this graph

The curved line in the middle is a locus of all portfolios where -1 LT rij LT 1. If we now combine these portfolios with risk-free assets (i.e. s=0) we will obtain a straight line in k-s space. This is because the above equations reduce to:

  • kp = amkm + (1- am)kf
  • sp = amsm

where m is a portfolio of risky assets and p is a portfolio formed with a% in m and the rest (=1-a) in risk free assets f.

The m portfolio at the point of tangency shown in the graph below dominates all other m portfolios. We call this the `market portfolio'. If we now enforce the definition of tangency, we obtain an algebraic result that is truly amazing and for which Bill Sharpe was awarded the Nobel Prize in Economics. At the point of tangency the slope of the curve is the same as the slope of the tangential line.

The slope of the tangent is

  • slope = (km - kf )/ sm

The slope of the curve is dk/ds which can be written as

  • dk/ds = (dk/da) / (ds/da)

evaluated at m with a->0 (Why?), the result is

  • dk/da = ki - km
  • ds/da = (sim - smm)/sm

where sim is the covariance between the market portfolio returns and the i-th asset returns

We therefore set

  • slope of the line = slope of the curve at m (this is the definition of tangency)
  • (km - kf )/sm= (ki - km)/((sim - smm)/sm)

Multiply thru by sm and define a parameter bi = sim/smm

  • km - kf = (ki - km) / (bi - 1)

Collect terms and get

ki - kf = bi(km-kf)

This is the so-called CAPM model of asset pricing and shows how investors evaluate riskiness of assets not being held in isolation but being added to the market portfolio or to a WDP.

The following observations are noteworthy

  • ki -kf= the the risk premium of the risky security i
  • km-kf = the risk premium of the m-portfolio
  • bi = a property of the risky security i that is a measure of risk under certain conditions
  • bi = sim/smm = a regression coefficient; b may also be thought of as the slope of the regression line of y=security i returns and x=market returns
  • bi is dimensionless, therefore it is not risk in the same sense that standard deviation is risk but only a coefficient
  • the analogous risk to standard deviation is bi*Sm
  • the amount of risk added to a wdp when the i-th asset is included is wi*bi*Sm