Capital Asset Pricing Model

Let I be the set of instruments available for trading over a period of time and Ω be the set of what can happen over the period. The prices at the beginning of the period are x\in\mathbf{R}^I where x_i = x(i)\in\mathbf{R} is the initial price of instrument i\in I. The prices at the end of the period are X\colon\Omega\to \mathbf{R}^I where X(\omega)\in\mathbf{R}^I are the instrument prices if \omega\in\Omega occurred.

A position \xi\in\mathbf{R}^I is the number of shares of each instrument purchased at the beginning of the period. Its value at the beginning of the period is \xi\cdot x and its value at the end of the period is \xi\cdot X(\omega) given \omega\in\Omega occurred. The realized return of \xi is the function R_\xi = \xi\cdot X/\xi\cdot x.

If \xi\cdot x\not=0 there exists t\in\mathbf{R} with t\xi\cdot x = 1 so R_{t\xi} = \xi\cdot X. One unit invested proportional to \xi returns this amount.

Assume a probability measure on \Omega. This note provides no guidence on how to do that.

A portfolio \xi\in\mathbf{R}^I is efficient if \operatorname{Var}(R_\xi) \le \operatorname{Var}(R_\eta) whenever E[R_\xi] = E[R_\eta], \eta\in\mathbf{R}^I.

Given a realized return R we can find an efficient portfolio by minimizing \operatorname{Var}(R_\xi) given E[R_\xi] = R. This can be solved using Lagrange multipliers by minimizing \frac{1}{2}\operatorname{Var}(\xi\cdot X) - \lambda(\xi\cdot x - 1) - \mu(E[\xi\cdot X] - R)

If \Sigma = E[XX'] - E[X]E[X'], where prime denotes transpose, then the first order conditions are 0 = \xi'\Sigma - \lambda x' - \mu E[X'] where 0 = \xi\cdot x - 1 and 0 = E[\xi\cdot X] - R. Assuming \Sigma is invertible \xi = \Sigma^{-1}(\lambda x - \mu E[X]). Note every optimal portfolio is a linear combination of \Sigma^{-1}x and \Sigma^{-1}E[X].

A portfolio \xi\in\mathbf{R}^I is efficient if E[R_\xi] \ge E[R_\eta] whenever \operatorname{Var}(R_\xi) = \operatorname{Var}(R_\eta), \eta\in\mathbf{R}^I.

Given a variance V we can find an efficient portfolio by minimizing E[R_\xi] given \operatorname{Var}(R_\xi) = V. This can be solved using Lagrange multipliers by minimizing E[\xi\cdot X] - \lambda(\xi\cdot x - 1) - \frac{\mu}{2}(\operatorname{Var}(\xi\cdot X) - V) The first order conditions are 0 = E[X'] - \lambda x' - \mu \xi'\Sigma where 0 = \xi\cdot x - 1 and 0 = \operatorname{Var}(\xi\cdot X) - V. Assuming \Sigma is invertible \xi = \Sigma^{-1}(E[X] - \lambda x)/\mu. Note every optimal portfolio is a linear combination of \Sigma^{-1}E[X] and \Sigma^{-1}x so the two definitions of efficient portfolio agree.

Exercise. If \xi_0\not=\xi_1 are any two optimal portfolios then every optimal portfolio is a linear combination of \xi_0 and \xi_1.

Hint: \xi_0 = a\Sigma^{-1}x + b\Sigma^{-1}E[X] and \xi_1 = c\Sigma^{-1}x + d\Sigma^{-1}E[X] for some a,b,c,d\in\mathbf{R} with ad - bc\not=0.

CAPM

The Capital Asset Pricing model assumes there are two optimal portfolios, one having variance 0 and the other is called the market portfolio. If \operatorname{Var}(R_\zeta) = 0 then \zeta\cdot X is a constant which we may assume is 1. This is called a zero coupon bond and \zeta\cdot x = 1/R_\zeta is its initial price, called the discount. Let \mu\in\mathbf{R}^I be the market portfolio with \mu\cdot x = 1.

Every optimal portfolio \xi can be written as \xi = \alpha\zeta + \beta\mu for some \alpha,\beta\in\mathbf{R} and we may assume \xi\cdot x = 1 so 1 = \xi\cdot x = \alpha/R_\zeta + \beta and R_\zeta = \alpha + \beta R_\zeta. Taking a dot product with X in \xi = \alpha\zeta + \beta\mu gives R_\xi = \alpha + \beta R_\mu hence R_\xi - R_\zeta = \beta(R_\mu - R_\zeta).

Exercise. Show \beta = \operatorname{Cov}(R_\xi,R_\mu)/\operatorname{Var}(R_\mu).

Hint: \operatorname{Cov}(R_\xi - R_\zeta, R_\mu) = \operatorname{Cov}(\beta(R_\mu - R_\zeta), R_\mu).

The usual CAPM formula is E[R_\xi] - R_\zeta = \beta(E[R_\mu] - R_\zeta) but a much stronger result holds: there is an equality between random variables, not just their expectations.

CAPM

Remarks