# Equilibrium: The Sharpe-Lintner CAPM

# Equilibrium: The Sharpe-Lintner CAPM

Sharpe [1964] and Lintner [1965a] examined the equilibrium implications of Tobin's two-fund separation theorem deriving the CAPM. In addition to the assumptions outlined above, they assumed that all investors have a common time horizon and homogeneous beliefs about * μ* and

**Σ**.

Equation (5.2) now implies that all investors demand combinations of the risk-free asset and the same portfolio of risky assets, **Σ**^{-1}( * μ* -

*). Since in equilibrium demand equals supply, the supply of each risky asset in the market portfolio must be proportional to*

**r**_{f}**Σ**

^{-1}(

*-*

**μ***). Formally, multiplying equation (5.2) by*

**r**_{f}(the fraction obtained from the total wealth of individual *i* over the total wealth of individuals), gives:^{7}

where

is a measure of the aggregated risk tolerance in the economy, and **x**_{m}, is the vector of asset weights in the market portfolio. Since *θ*_{m }> 0, equation (5.3) implies that the market portfolio is mean-variance efficient.

Multiplying equation (5.3) by **Σ** shows that the equilibrium-expected excess returns on assets are proportional to the vector of their covariances with the market return, **Σ x**

_{m}:

The market risk tolerance parameter, *θ*_{m}, can be eliminated from the previous relationship by multiplying equation (5.4) by x and rearranging. Letting:

and

this provides

Substituting this into equation (5.4) gives the Sharpe-Lintner CAPM pricing model:

where

is the vector of assets' * β*s relative to the market portfolio.

The main conclusion drawn from the Sharpe-Lintner CAPM is, therefore, that expected excess returns across assets are proportional to their *β* relative to the market portfolio. Alternatively, as shown in the above derivation, it predicts that the market portfolio is mean-variance efficient. The equivalence of these two implications was first shown by Fama [1976] and Ross [1977], and was central to Roll's [1977] critique (further discussed in section 5.3.8). Due to the restrictive nature of the assumptions underlying the model presented above, several attempts have been made to apply it to a more general context. The main results are reviewed below. However, the focus has been on those factors of particular relevance to real estate investment.

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^{7}Note that, is defined as `in equilibrium'.