From SDF to CAPM and Factor Models¶
The SDF framework is the most general representation of pricing. Classic models like CAPM and multi-factor models are special cases that arise by restricting the form of \(M_T\). This section traces those restrictions and reveals a unification: the Girsanov kernel \(\boldsymbol{\theta}\) and the factor risk premia \(\boldsymbol{\lambda}\) are the same mathematical object.
Scope
This material belongs to equilibrium asset pricing theory, not derivative pricing. It is not required for computing option values but shows how the measure-change machinery of Girsanov's theorem connects to the risk-return tradeoffs studied in portfolio theory.
Linear SDF Gives CAPM¶
The general SDF \(M_T = e^{-rT} u'(C_T)/u'(C_0)\) is nonlinear in consumption. CAPM arises when we approximate it as linear in market returns---an assumption that is exact under quadratic utility or jointly Gaussian returns:
Since \(\operatorname{Cov}(M_T, R_i) = -b\,\operatorname{Cov}(R_M, R_i)\), the risk premium formula \(\mathbb{E}[R_i] - R_f = -R_f \operatorname{Cov}(M_T, R_i)\) becomes proportional to market covariance. Applying it to the market itself and dividing yields:
where \(\beta_i = \operatorname{Cov}(R_i, R_M) / \operatorname{Var}(R_M)\). CAPM is an SDF driven by a single factor.
Multi-Factor SDF Gives Factor Models¶
Allowing the SDF to depend on \(k\) factors gives:
Then expected returns satisfy:
where \(\beta_{ij}\) is the exposure to factor \(j\) and \(\lambda_j\) is the price of risk for factor \(j\). Factor models are multi-dimensional SDFs.
The Unification¶
The continuous-time market price of risk \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\) and the factor pricing equation \(\mathbb{E}[\mathbf{R}] - R_f\mathbf{1} = \beta\boldsymbol{\lambda}\) are the same structure in different notation:
| Factor model | Measure change | Role |
|---|---|---|
| \(\beta_{ij}\) | \(\sigma_{ij}\) | Exposure to risk source \(j\) |
| \(\lambda_j\) | \(\theta_j\) | Price per unit of risk source \(j\) |
| \(\sum_j \beta_{ij}\lambda_j\) | \(\sum_j \sigma_{ij}\theta_j\) | Total risk premium on asset \(i\) |
Risk premia exist because investors dislike states where marginal utility is high. Everything else---SDF, CAPM, factor models, measure changes---is a representation of this single idea at different levels of generality.
Exercises¶
Exercise 1. Starting from \(\mathbb{E}[M_T R_T] = 1\) and the linear SDF \(M_T = a - bR_M\), derive the CAPM equation \(\mathbb{E}[R_i] - R_f = \beta_i(\mathbb{E}[R_M] - R_f)\).
Solution to Exercise 1
From \(\mathbb{E}[(a - bR_M)R_i] = 1\), expand:
Apply to the risk-free asset (\(R_i = R_f\)):
Subtract the risk-free equation from the general one and use \(\mathbb{E}[R_M R_i] = \operatorname{Cov}(R_M, R_i) + \mathbb{E}[R_M]\mathbb{E}[R_i]\) to get:
Applying the same equation to the market (\(R_i = R_M\)) gives \(\mathbb{E}[R_M] - R_f = b\,R_f\,\operatorname{Var}(R_M)\). Dividing:
Exercise 2. Explain why a linear SDF \(M_T = a - bR_M\) is economically restrictive. Under what conditions on preferences or return distributions is it exact?
Solution to Exercise 2
A linear SDF is exact under either of two conditions:
-
Quadratic utility: \(u(C) = C - \frac{\gamma}{2}C^2\) gives \(u'(C) = 1 - \gamma C\), which is linear in consumption. If consumption is proportional to market wealth, \(M_T \propto u'(C_T)\) is linear in \(R_M\).
-
Gaussian returns: If all returns are jointly normal, then \(\mathbb{E}[R_i \mid R_M]\) is linear in \(R_M\) for any SDF. The conditional expectation structure makes the pricing equation behave as if the SDF were linear.
In general, \(M_T = e^{-rT}u'(C_T)/u'(C_0)\) is nonlinear, and the linear approximation fails when returns are skewed, heavy-tailed, or utility is not quadratic. This motivates multi-factor models.
Exercise 3. In a two-factor model with \(M_T = a - b_1 F_1 - b_2 F_2\), derive the pricing equation \(\mathbb{E}[R_i] - R_f = \beta_{i1}\lambda_1 + \beta_{i2}\lambda_2\).
Solution to Exercise 3
From \(\mathbb{E}[M_T R_i] = 1\):
Applying to the risk-free asset: \(a - b_1\,\mathbb{E}[F_1] - b_2\,\mathbb{E}[F_2] = 1/R_f\). Subtracting and decomposing \(\mathbb{E}[F_j R_i]\) into mean and covariance terms:
Defining \(\beta_{ij}\) via multivariate regression of \(R_i\) on \((F_1, F_2)\) and \(\lambda_j\) as the corresponding factor risk premia, this gives \(\mathbb{E}[R_i] - R_f = \beta_{i1}\lambda_1 + \beta_{i2}\lambda_2\), where each \(\lambda_j\) represents the excess return per unit of exposure to factor \(j\).
Exercise 4. Explain how the Girsanov kernel \(\boldsymbol{\theta}\) in the measure-change framework corresponds to the factor risk premia \(\boldsymbol{\lambda}\) in the factor model framework. Why does non-uniqueness of \(\boldsymbol{\theta}\) correspond to market incompleteness?
Solution to Exercise 4
The market price of risk \(\boldsymbol{\theta}\) is the Girsanov kernel that removes drift from discounted prices. In matrix form, \(\boldsymbol{\mu} - r\mathbf{1} = \Sigma\boldsymbol{\theta}\) says excess returns equal volatility exposure times prices of risk. Comparing with \(\mathbb{E}[\mathbf{R}] - R_f\mathbf{1} = \beta\boldsymbol{\lambda}\), the volatility matrix \(\Sigma\) encodes factor exposures and \(\boldsymbol{\theta}\) encodes risk premia per unit of Brownian shock.
When the number of Brownian motions exceeds the number of traded assets, \(\Sigma\) has a non-trivial null space and \(\boldsymbol{\theta}\) is not uniquely determined. Each valid \(\boldsymbol{\theta}\) defines a different equivalent local martingale measure and different derivative prices. This is market incompleteness: the pricing kernel is not fully pinned down by traded assets, and additional information (calibration, equilibrium, or preferences) is needed to select a unique measure.
Exercise 5. In a single-factor market with \(M_T = a - bR_M\) and \(\mathbb{E}[M_T] = 1/R_f\), identify which quantity plays the role of the market price of risk \(\theta\) in § Risk Premium Decomposition. Express \(\theta\) explicitly in terms of \(b\), \(R_f\), and the market's volatility.
Solution to Exercise 5
Comparing \(\mathbb{E}[R_M] - R_f = -R_f \operatorname{Cov}(M_T, R_M)\) with \(M_T = a - bR_M\) gives \(\operatorname{Cov}(M_T, R_M) = -b\,\operatorname{Var}(R_M)\), so
The Sharpe ratio of the market is
The SDF coefficient \(b\) governs the price of market risk, so \(bR_f\sigma_M\) plays the role of the Girsanov kernel \(\theta\) in the continuous-time pricing framework. \(\square\)
Exercise 6. A claim has zero exposure to every priced factor: \(\beta_{ij} = 0\) for all \(j\). Use the factor pricing equation to determine its expected return. Connect the result to the canonical drift-removal statement in § Risk Premium Decomposition.
Solution to Exercise 6
From \(\mathbb{E}[R_i] - R_f = \sum_j \beta_{ij}\lambda_j = 0\), the claim earns the risk-free rate: \(\mathbb{E}[R_i] = R_f\).
The continuous-time analogue is that an asset with zero exposure to every Brownian shock (\(\sigma_{ij} = 0\) for all \(j\)) has \(\mu_i = r\) under \(\mathbb{P}\) already, and the Girsanov substitution leaves its dynamics unchanged. In both representations, a position carrying no priced risk earns no risk premium.