CAPM as a Special Case of SAPM

CAPM appears when the welfare factor has zero price

The paper's formal claim is a nesting result: under A1-A2, SAPM gives a two-factor pricing relation; under A3, the welfare-price term drops out and the Sharpe-Lintner CAPM remains.

• SAPM: E[R_i] - R_f = β_i^m λ_m + β_i^W λ_W.
• A3 sets b_W = 0, which implies λ_W = 0.
• With λ_W = 0, equation (5.1) reduces to equation (5.2).

A1 gives the unconditional beta representation

The proof begins with the stochastic discount factor condition, not with an added empirical factor.

• A1 uses E[mR_i] = 1 and E[mR_f] = 1.
• These imply E[R_i] - R_f = -Cov(m,R_i) / E.
• The expected excess return is tied to covariance with the pricing kernel m.

A2 expands m into market and welfare components

The paper then substitutes the A2 expansion of m, separating market covariance from covariance with the system-welfare factor f_W.

• The market term is proportional to Cov(R_i,R_m).
• The welfare term is proportional to Cov(R_i,f_W).
• The coefficients are b_m / E and b_W / E.

SAPM prices market beta and welfare beta separately

After dividing by the relevant variances, the proof collects terms into β_i^m, β_i^W, λ_m, and λ_W.

• β_i^m is the asset's exposure to the market factor.
• β_i^W is the asset's exposure to f_W.
• λ_W is the market price attached to system-welfare exposure.

λ_m is pinned by applying the relation to the market portfolio

The paper does not leave the market price free. It applies the SAPM relation to i = m on orthogonalized factors.

• For the market portfolio, β_m^m = 1.
• This pins λ_m = E[R_m] - R_f.
• Identifying f_m with the market portfolio recovers the CAPM market term.

A3 switches off the welfare-price channel

The reduction to CAPM happens only after imposing the Missing System restriction.

• A3 states b_W = 0.
• Because λ_W depends on b_W, A3 implies λ_W = 0.
• The term β_i^W λ_W vanishes for every asset.

The remaining equation is Sharpe-Lintner CAPM

Once the welfare term is zero, SAPM becomes E[R_i] - R_f = β_i^m λ_m, which the paper identifies with β_i(E[R_m] - R_f).

• The reduced equation is E[R_i] - R_f = β_i^m λ_m.
• With λ_m = E[R_m] - R_f, this is the Sharpe-Lintner CAPM.
• CAPM is the non-priced-system case of SAPM.

The bilateral payoff space excludes the system coordinate

The paper links A3 to Proposition 1: the bilateral payoff space satisfies U_cS = 0, so the system coordinate is absent there.

• U_cS = 0 is the payoff-space condition named in the proof.
• That condition is the regime where A3 holds.
• CAPM is described as the asset-pricing model native to that bilateral space.

Return beta and causal welfare beta are different objects

The paper's point is not that CAPM users measured the market portfolio incorrectly. It separates market-priced return exposure from planner-relative welfare exposure.

• β_i measures market return exposure in CAPM.
• β_i^W measures exposure to f_W in SAPM.
• Planner-relative welfare is not treated as a market anomaly.

The claim is nesting and separation relative to CAPM and ICAPM

The paper places the result next to Sharpe, Lintner, Merton, and Breeden, but it makes a narrow theory claim rather than an empirical model comparison.

• Sharpe and Lintner anchor the CAPM reduction.
• Merton and Breeden are named as intertemporal comparison points.
• The welfare factor is kept distinct from market-based factors.

This paper has no empirical β_W tile

The manuscript is a foundational theory statement in the ten-paper slate, not a domain calibration or factor-construction paper.

• Status: draft manuscript.
• Target journal: JFE.
• No domain estimate, no empirical test, and no applied β_W ranking are included.

Teach the proof as a restriction exercise

The clean classroom move is to write CAPM, add the SAPM welfare-price term, and ask which assumptions make that term disappear.

• Finance seminar: show how A1-A2 produce equation (5.1).
• Policy seminar: discuss why unpriced system risk can still matter outside the pricing equation.
• Executive seminar: separate priced risk from causal damage.