Asset Pricing Theory
Paris Dauphine University-PSL
Focus: Production economies where firms make investment decisions
Key Insight: Asset prices reflect both consumption and investment opportunities
Topics:
\[ Y_t = F(K_t, Z_t L_t),\qquad F_K>0,\;F_L>0,\;F_{KK}<0,\;F_{LL}<0, \] \[ K_{t+1} = (1-\delta)K_t + \Phi(I_t,K_t),\qquad \Phi_I>0,\;\Phi_{II}\le 0, \]
where \(\Phi(I,K)\) converts investment goods \(I_t\) into installed capital; with no adjustment costs, \(\Phi(I,K)=I\).
Preferences: \[ \max_{\{C_t,L_t,\{x^i_t\}\}} E_0 \sum_{t=0}^{\infty} \beta^t u(C_t,1-L_t),\qquad 0<\beta<1, \] with \(u_C>0\), \(u_{CC}<0\), \(u_{1-L}>0\), \(u_{(1-L)(1-L)}<0\).
Budget constraint in units of consumption: \[ C_t + \sum_i p_t^i x_t^i \le w_t L_t + \sum_i x_{t-1}^i (d_t^i+p_t^i) + \Pi_t, \]
\[ 1 = E_t\big[M_{t+1} R_{t+1}^i\big],\qquad M_{t+1} \equiv \beta\,\frac{u_C(C_{t+1},1-L_{t+1})}{u_C(C_t,1-L_t)}. \] Proof: The Lagrangian for the household problem includes: (since utilities are increasing on consumption, the budget constraint is always binding) \[ \begin{aligned} \mathcal{L} &= E_0 \sum_{t=0}^{\infty} \beta^t \left[u(C_t,1-L_t) + \mu_t\left(w_t L_t + \sum_i x_{t-1}^i(p_t^i+d_t^i) + \Pi_t - C_t - \sum_i p_t^i x_t^i\right)\right] \\ &= E_0 \sum_{t=0}^{\infty} \color{blue}E_t\color{black} \beta^t \left[u(C_t,1-L_t) + \mu_t\left(w_t L_t + \sum_i x_{t-1}^i(p_t^i+d_t^i) + \Pi_t - C_t - \sum_i p_t^i x_t^i\right)\right] \end{aligned} \] First-order condition (FOC) w.r.t. \(x_t^i\): \[ -\beta^t u_C(C_t,\cdot)\,p_t^i + \beta^{t+1} E_t\!\left[u_C(C_{t+1},\cdot)(p_{t+1}^i+d_{t+1}^i)\right]=0. \] Dividing by \(\beta^t u_C(C_t,\cdot)p_t^i\): \[ 1 = E_t\left[\beta\frac{u_C(C_{t+1},\cdot)}{u_C(C_t,\cdot)}\frac{p_{t+1}^i+d_{t+1}^i}{p_t^i}\right] = E_t[M_{t+1}R_{t+1}^i]. \]
The representative firm maximizes its value (priced by \(M_{t+1}\)): \[ \max_{\{I_t,L_t\}} E_0 \sum_{t=0}^{\infty} \left[\prod_{s=0}^{t-1} M_{s+1}\right] D_t, \] with dividends
\[ D_t \equiv Y_t - I_t - \mathcal{G}(I_t,K_t) - w_t L_t, \]
where \(\mathcal{G}\) captures (convex) adjustment costs in units of consumption, typically \(\mathcal{G}(I,K)=\frac{\kappa}{2}\left(\frac{I}{K}-\delta\right)^2 K\).
Capital accumulation: \[ K_{t+1}=(1-\delta)K_t+\Phi(I_t,K_t). \]
Let \(\lambda_t\) denote the Lagrange multiplier on capital accumulation in units of consumption (the marginal value of installed capital).
Define the lagrangian: \[ \mathcal{L} = E_0 \sum_{t=0}^{\infty} \left[\prod_{s=0}^{t-1} M_{s+1}\right] \left[D_t - \lambda_t\,(K_{t+1}-(1-\delta)K_t-\Phi(I_t,K_t))\right]. \]
Define \(S_t \equiv \prod_{s=0}^{t-1} M_{s+1}\) for notational convenience. The Lagrangian becomes
\[ \mathcal{L} = E_0 \sum_{t=0}^{\infty} S_t \left[D_t - \lambda_t\,(K_{t+1}-(1-\delta)K_t-\Phi(I_t,K_t))\right]. \]
\(I_t\)-FOC: \[ -1 - \mathcal{G}_I(I_t,K_t) + \lambda_t \Phi_I(I_t,K_t)=0 \quad \Rightarrow \quad \lambda_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\Phi_I(I_t,K_t)} \equiv q_t. \]
\(L_t\)-FOC: \[ F_L(K_t,Z_t L_t)\,Z_t = w_t. \]
Envelope condition:
\[ \begin{aligned} \frac{\partial \mathcal{L}}{\partial K_{t+1}} &= E_0\left[S_t \left( - \lambda_t\right) + S_{t+1}\left(\frac{\partial D_{t+1}}{\partial K_{t+1}} + \lambda_{t+1}(1-\delta) + \lambda_{t+1}\Phi_K(I_{t+1},K_{t+1})\right)\right] = 0. \end{aligned} \]
Since \(K_{t+1}\) is chosen at time \(t\), we can condition on \(\mathcal{F}_t\)
\[ \begin{aligned} 0 &= E_t\left[-\lambda_t + M_{t+1}\left(\frac{\partial D_{t+1}}{\partial K_{t+1}} + \lambda_{t+1}(1-\delta+\Phi_K(I_{t+1},K_{t+1}))\right)\right]. \end{aligned} \]
Noting that \(\frac{\partial D_{t+1}}{\partial K_{t+1}} = F_K(K_{t+1},Z_{t+1}L_{t+1}) - \mathcal{G}_K(I_{t+1},K_{t+1})\):
\[ \lambda_t = E_t\left[M_{t+1}\left(F_K(K_{t+1},Z_{t+1}L_{t+1}) - \mathcal{G}_K(I_{t+1},K_{t+1}) + \lambda_{t+1}(1-\delta+\Phi_K(I_{t+1},K_{t+1}))\right)\right]. \]
Consider the payoff of one marginal unit of installed capital at \(t\) (priced at \(q_t\)). Holding this unit from \(t\) to \(t+1\) delivers:
a cash flow (dividend) equal to the marginal contribution to profits: \[ d^K_{t+1} = F_K(K_{t+1},Z_{t+1}L_{t+1}) - \frac{\partial \mathcal{G}(I_{t+1},K_{t+1})}{\partial K_{t+1}}, \]
plus a resale value equal to its continuation marginal value after depreciation and spillovers: \[ q_{t+1}\cdot\big(1-\delta+\Phi_K(I_{t+1},K_{t+1})\big). \]
Thus, the gross return on installed capital is \[ R^K_{t+1} \equiv \frac{d^K_{t+1} + q_{t+1}\big(1-\delta+\Phi_K(I_{t+1},K_{t+1})\big)}{q_t}. \]
By the firm FOC and no-arbitrage: \[ 1=E_t\big[M_{t+1} R^K_{t+1}\big]. \]
From the \(I_t\)-FOC: \[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\Phi_I(I_t,K_t)}. \]
This connects marginal q (a valuation object) to average q and investment rates, delivering testable implications.
For any asset \(i\): \[ E_t[R_{t+1}^i] - R_{t+1}^f = -\frac{\text{Cov}_t(R_{t+1}^i,M_{t+1})}{E_t[M_{t+1}]},\qquad R_{t+1}^f \equiv \frac{1}{E_t[M_{t+1}]}. \]
For installed capital: \[ 1=E_t[M_{t+1}R^K_{t+1}]\quad \Longrightarrow \quad E_t[R^K_{t+1}]-R_{t+1}^f = -\frac{\text{Cov}_t(R^K_{t+1},M_{t+1})}{E_t[M_{t+1}]}. \]
In equilibrium \(M_{t+1}=\beta \dfrac{u_C(C_{t+1},1-L_{t+1})}{u_C(C_t,1-L_t)}\).
Cochrane shows that innovations in investment opportunities—measured by \(q_t\), investment-to-capital \(I_t/K_t\), or related marginal conditions—serve as simple linear proxies for the SDF in the data: \[
M_{t+1}\approx a - b \cdot \Delta \log q_{t+1} \quad \text{or} \quad
M_{t+1}\approx a' - b'\cdot \frac{I_{t}}{K_{t}} + \varepsilon_{t+1},
\] yielding the investment CAPM: assets that covary positively with good investment states (high \(q\), high \(I/K\)) have low expected returns.
Intuition: Good times for investment (high \(q\)) are times when the marginal utility of consumption is low (low \(M\)). Assets that pay off in those states hedge poorly and must offer higher premia.
This is the key insight of production-based asset pricing: investment opportunities serve as a state variable that proxies for the SDF.
Start from \(1=E_t[M_{t+1}R^K_{t+1}]\).
Using the standard asset pricing equation, for any asset \(i\): \[ 1 = E_t[M_{t+1}R_{t+1}^i] = E_t[M_{t+1}]E_t[R_{t+1}^i] + \text{Cov}_t(M_{t+1}, R_{t+1}^i). \]
Rearranging: \[ E_t[R_{t+1}^i] = R_{t+1}^f - \frac{\text{Cov}_t(M_{t+1}, R_{t+1}^i)}{E_t[M_{t+1}]}. \]
If we use the return on capital as a factor proxy (linear SDF approximation), \[ M_{t+1}\approx a - b\,R^K_{t+1}, \] then \[ E_t[R_{t+1}^i]-R_{t+1}^f \approx b\;\text{Cov}_t(R_{t+1}^i,R^K_{t+1}), \] which is the investment-based CAPM form.
Suppose the investment good has a relative price \(Q_t\) in consumption units, or—equivalently—an efficiency \(\mu_t \equiv 1/Q_t\) so that one unit of consumption buys \(\mu_t\) units of effective investment:
\[ K_{t+1}=(1-\delta)K_t+\mu_t\,\Phi(I_t,K_t), \qquad \mu_t>0. \]
Output is still in consumption units: \[ Y_t=F(K_t,Z_t L_t). \]
Dividends (in consumption units): \[ D_t = Y_t - I_t - \mathcal{G}(I_t,K_t) - w_t L_t. \]
Let \(q_t\) again denote the marginal value of installed capital in consumption units.
\(L_t\)-FOC (unchanged): \[ F_L(K_t,Z_tL_t)\,Z_t = w_t. \]
\(I_t\)-FOC: \[ -1 - \mathcal{G}_I(I_t,K_t) + q_t\,\mu_t\,\Phi_I(I_t,K_t)=0 \quad \Rightarrow \quad q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\mu_t\,\Phi_I(I_t,K_t)}. \]
Envelope / shadow value: \[ q_t = F_K(K_t,Z_t L_t) - \mathcal{G}_K(I_t,K_t) + E_t\!\left[M_{t+1}\,q_{t+1}\left(1-\delta+\mu_{t+1}\Phi_K(I_{t+1},K_{t+1})\right)\right]. \]
Return on installed capital: \[ R^K_{t+1} \equiv \frac{d^K_{t+1} + q_{t+1}\left(1-\delta+\mu_{t+1}\Phi_K(I_{t+1},K_{t+1})\right)}{q_t}, \] with \[ d^K_{t+1} \equiv F_K(K_{t+1},Z_{t+1}L_{t+1}) - \mathcal{G}_K(I_{t+1},K_{t+1}). \]
Pricing: \[ 1=E_t\!\left[M_{t+1} R^K_{t+1}\right]. \]
Key IST implication: shocks to \(\mu_t\) (or \(Q_t=1/\mu_t\)) move \(q_t\) and \(R^K_{t+1}\), altering investment opportunities and thereby risk premia through \(1=E_t[M_{t+1}R^K_{t+1}]\).
In competitive complete-markets equilibrium the exact SDF remains the MRS: \[ M_{t+1}=\beta\,\frac{u_C(C_{t+1},1-L_{t+1})}{u_C(C_t,1-L_t)}. \]
IST does not change the MRS definition; it changes returns via \(\mu_t\) (i.e., it changes what is being priced by the same \(M_{t+1}\)). A helpful linear proxy is: \[ M_{t+1}\approx a - b_1 \cdot \Delta \log q_{t+1} - b_2 \cdot \Delta \log \mu_{t+1}, \] so that \[ E_t[R_{t+1}^i]-R_{t+1}^f \approx b_1 \,\text{Cov}_t(R_{t+1}^i,\Delta \log q_{t+1}) + b_2 \,\text{Cov}_t(R_{t+1}^i,\Delta \log \mu_{t+1}). \]
Interpretation: assets that pay off when IST improves (high \(\mu_{t+1}\), cheap/effective investment) hedge low-\(M\) states poorly and earn higher premia.
Goods: \[ C_t + I_t + \mathcal{G}(I_t,K_t) = Y_t. \]
Labor and asset markets: \[ L_t \in [0,1], \qquad \text{and all asset holdings clear with net supply given by firms}. \]
An equilibrium is \(\{C_t,L_t,I_t,K_{t+1},q_t,w_t\}_{t\ge 0}\) and prices \(\{M_{t+1},R_{t+1}^i\}\) such that:
\[ 1=E_t\!\left[M_{t+1} \frac{d^K_{t+1} + q_{t+1}\left(1-\delta+\Phi_K^{\text{(IST)}}\right)}{q_t}\right], \] with \(\Phi_K^{\text{(IST)}} \equiv \mu_{t+1}\Phi_K(I_{t+1},K_{t+1})\) (equals \(\Phi_K\) without IST).
With IST, the capital accumulation constraint is: \[ K_{t+1}=(1-\delta)K_t+\mu_t\,\Phi(I_t,K_t). \]
The firm’s FOC w.r.t. \(I_t\) gives: \[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\mu_t\,\Phi_I(I_t,K_t)}. \]
The envelope condition w.r.t. \(K_{t+1}\) gives: \[ q_t = E_t\left[M_{t+1}\left(F_K(K_{t+1},Z_{t+1}L_{t+1}) - \mathcal{G}_K(I_{t+1},K_{t+1}) + q_{t+1}(1-\delta+\mu_{t+1}\Phi_K(I_{t+1},K_{t+1}))\right)\right]. \]
This can be written as: \[ 1 = E_t\left[M_{t+1}\frac{d^K_{t+1} + q_{t+1}(1-\delta+\mu_{t+1}\Phi_K(I_{t+1},K_{t+1}))}{q_t}\right] = E_t[M_{t+1}R^K_{t+1}]. \]
\[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\Phi_I(I_t,K_t)}\quad\text{(no IST)}, \] \[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\mu_t\,\Phi_I(I_t,K_t)}\quad\text{(with IST)}. \]
The firm’s Lagrangian (with IST) includes: \[ \mathcal{L} = E_0 \sum_{t=0}^{\infty} \left[\prod_{s=0}^{t-1} M_{s+1}\right] \left[D_t - \lambda_t\,(K_{t+1}-(1-\delta)K_t-\mu_t\Phi(I_t,K_t))\right], \] where \(D_t = Y_t - I_t - \mathcal{G}(I_t,K_t) - w_t L_t\).
FOC w.r.t. \(I_t\): \[ -1 - \mathcal{G}_I(I_t,K_t) + \lambda_t \mu_t\Phi_I(I_t,K_t)=0. \]
Solving for \(\lambda_t = q_t\): \[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\mu_t\,\Phi_I(I_t,K_t)}. \]
Without IST, \(\mu_t=1\), so: \[ q_t = \frac{1+\mathcal{G}_I(I_t,K_t)}{\Phi_I(I_t,K_t)}. \]
For any asset \(i\), \[ E_t[R_{t+1}^i]-R_{t+1}^f = -\frac{\text{Cov}_t(R_{t+1}^i,M_{t+1})}{E_t[M_{t+1}]}, \] and using linear SDF proxies in \(q\) and IST shocks yields investment-based factor models.
Start from the fundamental pricing equation: \[ 1 = E_t[M_{t+1}R_{t+1}^i] = E_t[M_{t+1}]E_t[R_{t+1}^i] + \text{Cov}_t(M_{t+1},R_{t+1}^i). \]
Rearranging: \[ E_t[R_{t+1}^i] = \frac{1 - \text{Cov}_t(M_{t+1},R_{t+1}^i)}{E_t[M_{t+1}]}. \]
Since \(R_{t+1}^f = 1/E_t[M_{t+1}]\): \[ E_t[R_{t+1}^i] - R_{t+1}^f = -\frac{\text{Cov}_t(M_{t+1},R_{t+1}^i)}{E_t[M_{t+1}]}. \]
To obtain factor models, project \(M_{t+1}\) onto investment-related variables (e.g., \(\Delta \log q_{t+1}\), \(\Delta \log \mu_{t+1}\)): \[ M_{t+1} = a + \sum_k b_k f_{k,t+1} + \varepsilon_{t+1}, \] where \(f_{k,t+1}\) are factors and \(E_t[\varepsilon_{t+1}f_{j,t+1}]=0\). Substituting into the risk premium equation yields: \[ E_t[R_{t+1}^i] - R_{t+1}^f \approx -\sum_k \frac{b_k}{E_t[M_{t+1}]}\text{Cov}_t(R_{t+1}^i,f_{k,t+1}). \]
Asset Pricing Theory