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Portfolio Theory and Quantitative Investment / CAPM and Equilibrium Models

Non-Standard CAPM Forms: Zero-Beta, Taxes, and Multi-Beta

The standard CAPM rests on ten assumptions. Several of those assumptions are demonstrably false in practice: there is no truly riskless borrowing rate available to individuals (only institutions can borrow at or near RFR_F); investors face taxes that treat dividends and capital gains differently; some assets (human capital, private equity, restricted stock) are non-marketable; and investors have heterogeneous expectations. The empirical failures of the standard CAPM — documented in the tests chapter — are at least partially traceable to these violations.
This note develops three non-standard equilibrium models derived by relaxing the standard assumptions one at a time, using the same Lagrangian framework from the CAPM derivation:
  1. Zero-beta CAPM (Black 1972): no riskless asset. The riskless rate is replaced by the return on the minimum-variance zero-beta portfolio.
  2. After-tax CAPM (Brennan 1970): differential taxation of dividends vs capital gains shifts the SML.
  3. Non-marketable assets and multi-beta CAPMs: human capital and other non-tradeable wealth enter the pricing equation as additional risk factors.
The economic significance: each modification changes what the SML intercept means and, potentially, what the slope means. The zero-beta CAPM predicts a higher intercept than the standard form (the zero-beta portfolio earns more than the riskless rate). The after-tax form predicts a slope that depends on dividend yield. Multi-beta forms predict multiple betas per stock — exactly the structure APT formalises.
Learning objectives. By the end of this note you should be able to:
  • state the zero-beta CAPM equation and identify the zero-beta portfolio;
  • explain why RˉZ>RF\bar{R}_Z > R_F when there is riskless lending but no riskless borrowing;
  • describe the efficient frontier when (a) there is no riskless asset, and (b) there is riskless lending but not riskless borrowing;
  • state Brennan's after-tax CAPM and explain the role of dividend yield in the modified SML;
  • describe the multi-beta / consumption CAPM and state the modified pricing equation.

In this lesson

    1. Zero-beta CAPM (Black 1972)
    1. Riskless lending but no riskless borrowing
    1. After-tax CAPM (Brennan 1970)
    1. Non-marketable assets: human capital CAPM
    1. Multi-beta CAPM and the general framework

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