17. Markovian Asset Pricing

• State vector \(\mathbf X_t\) (\(n\)-dim), \(\mathcal X=\{\mathbf X_t\}\) an \(n\)-dim Markov process (usually but not necessarily stationary and ergodic).
• Shock vector \(\mathbf W_t\) (\(k\)-dim), \(\mathcal W=\{\mathbf W_t\}\) the shock process (usually but not necessarily multi-Gaussian, i.i.d.).
• Recursive evolution: \(\mathbf X_{t+1}=\boldsymbol\psi(\mathbf X_t,\mathbf W_{t+1})\) for some vector function \(\boldsymbol\psi\).
• Filtration \(\mathcal F_t\) generated by \(\{\mathbf W_1,\dots,\mathbf W_t\}\) and \(\mathbf X_0\).

17.1.2 Multiplicative Functional

Let the scalar process \(\{\exp(A_t)\}\) (17.1) have log increment (17.2): \(A_{t+1}-A_t=\kappa(\mathbf X_t,\mathbf W_{t+1})\). In general (17.3):

\(\beta\) is a scalar function of the state, \(\boldsymbol\alpha\) a \(k\)-dim vector function. The coefficient \(\boldsymbol\alpha(\mathbf X_t)\) is stochastic, so (17.3) has stochastic volatility and is unconditionally non-Gaussian; if \(\boldsymbol\psi\), \(\beta\) are linear and \(\boldsymbol\alpha\) constant, \(\exp(A_t)\) is unconditionally log-normal. \(A_t\) is a scalar, modellable as log consumption, cash flow, etc. \(\exp(A_t)\) is a multiplicative functional because the product of two such processes satisfies the same property (log increments add, \(\kappa_m=\kappa_1+\kappa_2\)).

Two applications: the SDF process \(\{S_t\}\) (17.4): \(\ln S_{t+1}-\ln S_t=\kappa_S(\mathbf X_t,\mathbf W_{t+1})\) (increments usually negative); the cash-flow (CF) process \(\{G_t\}\) (17.6): \(\ln G_{t+1}-\ln G_t=\kappa_G(\mathbf X_t,\mathbf W_{t+1})\) (usually positive, growing cash flows).

17.1.3 Proportional Risk Compensation

A security pays \(G_{t+\tau}\) only at \(t+\tau\), with time-\(t\) price \(Q_t(G_{t+\tau})\) defined by the SDF (17.5), (17.7): \(Q_t(G_{t+\tau})=G_t\,\mathbb E[\frac{S_{t+\tau}}{S_t}\frac{G_{t+\tau}}{G_t}\mid\mathcal F_t]\) (where \(\frac{S_{t+\tau}}{S_t}\frac{G_{t+\tau}}{G_t}\) is still a multiplicative functional). Define the proportional risk compensation (17.8):

Interpretation. Part A ≈ the net average asset return: it can be rewritten as \(\ln((\frac{\mathbb E[G_{t+\tau}\mid\mathcal F_t]}{Q_t(G_{t+\tau})})^{1/\tau})\), the log of the one-period gross asset return (≈ net one-period return). Part B ≈ the (negative) net average risk-free rate: rewritten as \(-\ln((\frac{1}{\mathbb E[S_{t+\tau}/S_t\mid\mathcal F_t]})^{1/\tau})\). It is "proportional" because Part A minus negative Part B is the log of the ratio "asset gross return / risk-free gross return" — taking the ratio cancels levels and focuses on multiples. So (17.8) ≈ the risk premium.

17.2 Two Ways to Characterize State Vector

A general \(\boldsymbol\psi(\cdot,\cdot)\) may be nonlinear and hard to model; structure helps.

17.2.1 Small Noise Approximation around Steady State

Redefine the state vector as a family of processes scaled by the shock magnitude \(h\) (17.9): \(\mathbf X_{t+1}(h)=\boldsymbol\psi(\mathbf X_t(h),h\mathbf W_{t+1},h)\). As \(h\to0\), \(\mathbf X_{t+1}(0)=\boldsymbol\psi(\mathbf X_t(0),0,0)=\mathbf X_t(0)\) (steady state, \(\mathbf X_t(0)\) constant). Assume \(\mathbf W_{t+1}\sim\mathcal N(\mathbf 0,\mathbf I)\) i.i.d. A second-order Taylor expansion in \(h\) at \(h=0\) (17.10): \(\mathbf X_t(h)\approx\mathbf X_t(0)+h\mathbf X_t'(0)+\frac{h^2}2\mathbf X_t''(0)\).

First-derivative dynamics (17.11) (chain rule):

where \(\boldsymbol\psi_{\mathbf X^T}\) is the Jacobian of \(\boldsymbol\psi\) w.r.t. \(\mathbf X\) (\(n\times n\)), \(\boldsymbol\psi_{\mathbf W^T}\) w.r.t. \(\mathbf W\) (\(n\times k\)), \(\boldsymbol\psi_h\) w.r.t. \(h\) (\(n\times1\)), all at \(0\). In steady state \(\mathbf X_{t+1}'(0)=\mathbf X_t'(0)\), with mean \(\boldsymbol\mu_1=(\mathbf I-\boldsymbol\psi_{\mathbf X^T})^{-1}\boldsymbol\psi_h\) and variance \(\boldsymbol\Sigma_1=\boldsymbol\psi_{\mathbf X^T}\boldsymbol\Sigma_1\boldsymbol\psi_{\mathbf X^T}'+\boldsymbol\psi_{\mathbf W^T}\boldsymbol\psi_{\mathbf W^T}'\). The first derivative is normally distributed.

Second-derivative dynamics (17.12), (17.13): \(\mathbf X_{t+1}''(0)\) is built from the second partials of \(\boldsymbol\psi\) (\(\boldsymbol\psi_{\mathbf X\mathbf X^T}\), \(\boldsymbol\psi_{\mathbf X\mathbf W^T}\), \(\boldsymbol\psi_{\mathbf W\mathbf W^T}\), \(\boldsymbol\psi_{h\mathbf X^T}\), \(\boldsymbol\psi_{h\mathbf W^T}\), \(\boldsymbol\psi_{hh}\)) and quadratic forms in the first-order terms \(\mathbf X_t'(0),\mathbf W_{t+1}\); its mean and variance follow from (17.13) (which contains only quadratic terms in the already-characterized \(\mathbf X_t'(0)\)). Substituting (17.11), (17.13) back into (17.10) gives the dynamics of \(\mathbf X_t(h)\): recursively linear in \(\mathbf X_t'(0)\), \(\mathbf W_{t+1}\) under this approximation, but nonlinear in them.

17.2.2 Discrete State Markov Chain

Second way: let \(\mathcal X=\{\mathbf X_t\}\) be an \(n\)-state Markov chain, with \(\mathbf X_t\) a state-coordinate vector (entry \(j\) is 1 when state \(j\) realizes, others 0). The \(n\times n\) transition matrix \(\mathbf P=[p_{ij}]\), \(p_{ij}\) the probability of going from state \(i\) to \(j\). State vector (form 17.3): \(\mathbf X_{t+1}=\mathbb E[\mathbf X_{t+1}\mid\mathbf X_t]+\underbrace{\mathbf X_{t+1}-\mathbb E[\mathbf X_{t+1}\mid\mathbf X_t]}_{\text{unexpected shock}}\). Then \(A_{t+1}-A_t=\kappa(\mathbf X_t,\mathbf W_{t+1})\) is conditionally normal given \(\mathbf X_t\) but not unconditionally normal.

17.3 Long Horizon Valuation

17.3.1 Motivating Example: Eigenvalue and Eigenvector

Let the \(n\times n\) matrix \(\mathbf F\) have all non-negative entries. If \(\mathbf e\) is an eigenvector with eigenvalue \(\varepsilon\) (17.14): \(\mathbf F\mathbf e=\varepsilon\mathbf e\), then \(\mathbf F^j\mathbf e=\varepsilon^j\mathbf e\).

17.3.2 Eigenfunction and Eigenvalue

Extend §17.3.1 to the functional world (under Hansen-Scheinkman 2009 conditions): there exists a unique eigenfunction \(e(\cdot)\) and eigenvalue \(\exp(\eta)\) such that (17.15):

where \(\exp(A_t)\) is the multiplicative functional of (17.1), (17.2) (conditioning on \(\mathbf X_t=\mathbf x\) equals \(\mathcal F_t\)). Define the iteration operator \(\mathbb M\) (17.18): \(\mathbb M f(\mathbf x)\equiv\mathbb E[\frac{\exp(A_{t+1})}{\exp(A_t)}f(\mathbf X_{t+1})\mid\mathbf X_t=\mathbf x]\); (17.15) is \(\mathbb M e(\mathbf x)=e(\mathbf x)\exp(\eta)\). The iteration property (17.19) \(\mathbb E[\frac{\exp(A_{t+2})}{\exp(A_t)}f(\mathbf X_{t+2})\mid\mathbf X_t=\mathbf x]=\mathbb M^2 f(\mathbf x)\) gives, iterating, (17.16): \(\mathbb M^j e(\mathbf x)=e(\mathbf x)\exp(j\eta)\), i.e. \(\mathbb E[\frac{\exp(A_{t+j})}{\exp(A_t)}e(\mathbf X_{t+j})\mid\mathbf X_t=\mathbf x]=e(\mathbf x)\exp(j\eta)\). From (17.15) with \(j=1\), rearranging gives (17.20):

17.3.3 Long Term Proportional Risk Compensation

Define \(M_t=\exp(A_t)e(\mathbf X_t)\exp(-t\eta)\); substituting into (17.20) gives \(\mathbb E[\frac{M_{t+1}}{M_t}\mid\mathbf X_t]=1\), i.e. \(\mathbb E[M_{t+1}\mid\mathbf X_t]=M_t\) (17.21): \(\{M_t\}\) is a martingale. From the definition of \(M_t\), rewrite the multiplicative functional (17.22):

The permanent part is a martingale (accumulates past changes, no future conditional change); the transient part is stationary when \(\{\mathbf X_t\}\) is stationary and \(e(\cdot)\) regular (does not accumulate); plus the constant \(\exp(\tau\eta)\). To remove the correlation between permanent and transient parts, define a change of measure \(\tilde{\mathbb E}\) (the "tilde" measure, via the martingale \(M_t\)) (17.24): \(\mathbb E[\frac{M_{t+\tau}}{M_t}Y_{t+m}\mid\mathcal F_t]=\tilde{\mathbb E}[Y_{t+m}\mid\mathcal F_t]\). The new density is non-negative and integrates to 1 (guaranteed by the definition of \(M_t\) and (17.21)). Then (17.25): \(\mathbb E[\frac{M_{t+\tau}}{M_t}\frac{e(\mathbf X_t)}{e(\mathbf X_{t+\tau})}\mid\mathcal F_t]=\tilde{\mathbb E}[\frac{e(\mathbf X_t)}{e(\mathbf X_{t+\tau})}\mid\mathcal F_t]\). By stationarity, as \(\tau\to\infty\), \(\frac1\tau\ln(\tilde{\mathbb E}[\frac{e(\mathbf X_t)}{e(\mathbf X_{t+\tau})}\mid\mathcal F_t])\to0\), so (17.26):

Applying (17.26) to the SDF \(\{S_t\}\) and CF \(\{G_t\}\) gives (17.27), (17.28): \(\frac{S_{t+\tau}}{S_t}=\frac{M_{S,t+\tau}}{M_{S,t}}\frac{e_S(\mathbf X_t)}{e_S(\mathbf X_{t+\tau})}\exp(\tau\eta_S)\), \(\frac{G_{t+\tau}}{G_t}=\frac{M_{G,t+\tau}}{M_{G,t}}\frac{e_G(\mathbf X_t)}{e_G(\mathbf X_{t+\tau})}\exp(\tau\eta_G)\).

Long-term proportional risk compensation. Substituting the three limits (17.30), (17.31), (17.32) (\(\lim\frac1\tau\ln\mathbb E[\frac{G_{t+\tau}}{G_t}]=\eta_G\), \(\lim\frac1\tau\ln\mathbb E[\frac{S_{t+\tau}}{S_t}\frac{G_{t+\tau}}{G_t}]=\eta_G+\eta_S\), \(\lim\frac1\tau\ln\mathbb E[\frac{S_{t+\tau}}{S_t}]=\eta_S\)) into (17.8) gives (17.33): long-term proportional risk compensation \(=\eta_G-(\eta_G+\eta_S)+\eta_S=0\).

17.3.4 Long Term Holding Period Return of Bond

A bond pays 1 after \(\tau\) periods, priced at \(t\) by \(\mathbb E[\frac{S_{t+\tau}}{S_t}\mid\mathcal F_t]\) and at \(t+1\) by \(\mathbb E[\frac{S_{t+\tau}}{S_{t+1}}\mid\mathcal F_{t+1}]\). The one-period holding-period return \(\mathrm{HPR}_1(\tau)\) (17.34):

Assume (17.27) (SDF has a permanent part). Simplifying with the tilde measure \(\tilde{\mathbb E}\) (17.35), (17.36), and assuming as \(\tau\to\infty\) that \(\tilde{\mathbb E}[\frac{1}{e_S(\mathbf X_{t+\tau})}\mid\mathcal F_{t+1}]\to\tilde{\mathbb E}[\frac{1}{e_S(\mathbf X_{t+\tau})}\mid\mathcal F_t]\) (\(\mathbf X_{t+\tau}\) is independent of \(t,t+1\) info in the far future) (17.37), gives (17.38): \(\lim_{\tau\to\infty}\mathrm{HPR}_1(\tau)=\frac{e_S(\mathbf X_{t+1})}{e_S(\mathbf X_t)}\exp(-\eta_S)\). Substituting back into (17.27) gives (17.39):

17.4 Literature on Long Horizon Valuation

17.4.1 Kazemi (1992)

Kazemi (1992) is exactly §17.3.4 but assumes the SDF has no permanent part (17.29). Then \(\frac{S_{t+1}}{S_t}=\frac{e_S(\mathbf X_t)}{e_S(\mathbf X_{t+1})}\exp(\eta_S)=(\mathrm{HPR}_1(\infty))^{-1}\) (in continuous time): the (inverse of the) long bond's one-period holding return is itself the SDF, a one-factor pricing model. Stationarity of the state can be tested with data.

17.4.2 Alvarez and Jermann (2005)

Consider (17.27) (SDF has a permanent part), \(\frac{S_{t+1}}{S_t}=\frac{M_{S,t+1}}{M_{S,t}}\frac{e_S(\mathbf X_t)}{e_S(\mathbf X_{t+1})}\exp(\eta_S)\), with a martingale component. They find: the permanent part \(\frac{M_{S,t+1}}{M_{S,t}}\) is hugely volatile, the transient part relatively unimportant. Measuring the permanent part's importance: define a measurement \(\mathbb E[\phi(m)]\) of the size of \(m\equiv\frac{M_{S,t+1}}{M_{S,t}}\), requiring \(\phi(1)=0\) (reports 0 when the permanent part degenerates to 1) and \(\phi\) convex (by Jensen \(\mathbb E[\phi(m)]\geq\phi(\mathbb E[m])=\phi(1)=0\), always reports a positive value). Alvarez-Jermann use \(\phi(m)=-\ln m\) (17.40): the measurement \(\mathbb E[-\ln\frac{M_{S,t+1}}{M_{S,t}}]\). By stationarity and Birkhoff's LLN (17.41): \(\mathbb E[-\ln\frac{M_{S,t+1}}{M_{S,t}}]=-\lim_{t\to\infty}\frac1t\mathbb E[\ln\frac{M_{S,t}}{M_{S,0}}]\). Define \(L(\frac{S_t}{S_0})\) (17.42): \(\equiv\frac1t\ln(\mathbb E[\frac{S_t}{S_0}])-\frac1t\mathbb E[\ln\frac{S_t}{S_0}]\) (Part A − Part B); combined with (17.27), \(\lim_{t\to\infty}L(\frac{S_t}{S_0})=\mathbb E[-\ln\frac{M_{S,t+1}}{M_{S,t}}]\). Pricing any asset \(i\) via \(\mathbb E[\frac{S_t}{S_0}R_{0,t}^i]=1\) (17.43) and the risk-free asset (17.45), with Jensen, gives (17.44), (17.46):

Interpretation (entropy bound): the RHS \(\frac1t[\mathbb E[\ln R_{0,t}^i]-\ln R_{0,t}^f]\) is asset \(i\)'s average one-period long-run risk premium. (17.46) says the measurement of the permanent part is no less than the maximum average long-run risk premium of any asset (clearly positive in data), proving the importance of the SDF's permanent part.

17.4.3 Recovery Theorem: Ross (2015)

Ross (2015) assumes a complete market and no martingale part of the SDF (17.29), and shows the physical distribution and the SDF can be recovered from asset prices alone. An \(n\)-state Markov chain reduces the degrees of freedom. Let the transition matrix \(\mathbf P=[p_{ij}]\) (\(n^2-n\) d.o.f., each row sums to 1), the SDF matrix \(\mathbf S=[s_{ij}]\) (\(s_{ij}\) the state-price density of one unit of cash flow when today is state \(i\) and tomorrow state \(j\)), and the Arrow-Debreu price matrix \(\mathbf A=[a_{ij}]\) with \(a_{ij}=p_{ij}s_{ij}\). By (17.29), the d.o.f. of \(\mathbf S\) drops to \(n\) (\(e_S(\mathbf X_i)\) takes at most \(n\) values; \(\frac{S_{t+1}}{S_t}=\frac{e_S(\mathbf X_t)}{e_S(\mathbf X_{t+1})}\exp(\eta_S)\) has at most \(n-1\) ratios + 1 \(\eta_S\) = \(n\)).

Recovery strategy: ① estimate \(\mathbf A\) from option data; ② assume all entries of \(\mathbf P\) strictly positive (no-arbitrage then makes \(\mathbf A\) all-positive); ③ by Theorem 17.1, \(\mathbf A\)'s largest-absolute-value eigenvalue \(\varepsilon>0\) with all-positive eigenvector \(\mathbf e=(e_1,\dots,e_n)'\); ④ construct \(\tilde{\mathbf P}=[\tilde p_{ij}]\), \(\tilde p_{ij}\equiv\frac{a_{ij}e_j}{\varepsilon e_i}\). Then \(\tilde p_{ij}>0\), \(\sum_j\tilde p_{ij}=1\) (using \(\mathbf A\mathbf e=\varepsilon\mathbf e\)), a valid transition probability.

Ross (2015) claims \(\tilde{\mathbf P}=\mathbf P\) (the physical measure). But this holds only when the SDF has no permanent part (17.29). Verify: if \(\tilde{\mathbf P}=\mathbf P\), then \(\tilde s_{ij}=\frac{a_{ij}}{\tilde p_{ij}}=\frac{e_i}{e_j}\varepsilon\) (17.47), (17.48), matching the (17.29) form \(\frac{S_{t+1}}{S_t}=\frac{e_S(\mathbf X_t)}{e_S(\mathbf X_{t+1})}\exp(\eta_S)\) (\(e_i\)↔\(e_S(\mathbf X_t)\), \(\varepsilon\)↔\(\exp(\eta_S)\)).

17.5 Shock Elasticities

Study how a one-time shock affects pricing over a period — shock elasticities.

17.5.1 One-Period Shock Elasticities

State \(\{\mathbf X_t:t=0,1\}\) (\(n\times1\) Markov), shock \(\mathbf W_1\sim\mathcal N(\mathbf 0,\mathbf I)\) (\(k\times1\)). SDF process (17.49): \(\ln S_1-\ln S_0=\beta_S(\mathbf X_0)+\boldsymbol\alpha_S(\mathbf X_0)\cdot\mathbf W_1\); CF process (17.50): \(\ln G_1-\ln G_0=\beta_G(\mathbf X_0)+\boldsymbol\alpha_G(\mathbf X_0)\cdot\mathbf W_1\). CF gross return \(R^G_{0,1}\) (17.51): \(\frac{G_1}{\mathbb E[G_1\frac{S_1}{S_0}\mid\mathcal F_0]}\). The log expected return (≈ net return) (17.52), the risk-free (17.53); their difference is the risk premium:

Call \(\boldsymbol\alpha_G(\mathbf X_0)\) the shock exposures (the CF process's loadings on each shock component) and \(-\boldsymbol\alpha_S(\mathbf X_0)\) the shock prices (the impact of each unit of exposure on the return, closely related to prices).

Modified cash-flow process. Define a scalar modification factor \(H_1(r)\) for the \(t=1\) cash flow (17.54): \(\ln H_1(r)=r\boldsymbol\nu(\mathbf X_0)\cdot\mathbf W_1-\frac12 r^2\boldsymbol\nu(\mathbf X_0)\cdot\boldsymbol\nu(\mathbf X_0)\), \(\boldsymbol\nu(\mathbf X_0)\) being \(k\times1\). The modified \(G_1^{\text{modified}}=G_1 H_1(r)\) changes the shock exposure from \(\boldsymbol\alpha_G(\mathbf X_0)\) to \(\boldsymbol\alpha_G(\mathbf X_0)+r\boldsymbol\nu(\mathbf X_0)\) (17.55). Assume \(\boldsymbol\nu(\mathbf X_0)\cdot\boldsymbol\nu(\mathbf X_0)=1\) (17.56); properties: \(\mathrm{Var}(\boldsymbol\nu(\mathbf X_0)\cdot\mathbf W_1)=1\), \(\mathbb E[H_1(r)\mid\mathbf X_0]=1\) (17.57), \(H_1(r)>0\) (17.58). So \(H_1(r)\) can serve as a change-of-measure density, with the new measure \(\tilde{\mathbb E}^{H_1(r)}\) satisfying \(\tilde{\mathbb E}^{H_1(r)}[\mathbf W_1\mid\mathbf X_0]=r\boldsymbol\nu(\mathbf X_0)\) (Remark 17.2).

Taking the gross return \(R^{G,\text{modified}}_{0,1}\) of the modified asset, substituting (17.55) into (17.52) gives (17.59). Elasticity definitions: differentiating \(\ln\mathbb E[R^{G,\text{modified}}_{0,1}\mid\mathcal F_0]\) w.r.t. \(r\) at \(r=0\) (one-period version of (17.64)) gives three elasticities:

• Shock-price elasticity \(\varepsilon_R\): \(\frac{d}{dr}\ln\mathbb E[R^{G,\text{modified}}_{0,1}\mid\mathcal F_0]\big|_{r=0}=-\boldsymbol\alpha_S(\mathbf X_0)\cdot\boldsymbol\nu(\mathbf X_0)\) — the percentage change in expected return when the original \(G_1\)'s shock exposure moves by \(\boldsymbol\nu(\mathbf X_0)\,dr\).
• Shock-exposure elasticity \(\varepsilon_O\): \(\frac{d}{dr}\ln\mathbb E[\frac{G_1}{G_0}H_1(r)\mid\mathcal F_0]\big|_{r=0}\) — the percentage change in expected cash flow.
• Shock-cost elasticity \(\varepsilon_C\): \(\frac{d}{dr}\ln\mathbb E[\frac{G_1}{G_0}\frac{S_1}{S_0}H_1(r)\mid\mathcal F_0]\big|_{r=0}\) — the percentage change in the cost (price) of bearing the shock.

17.5.2 Multi-Period Shock Elasticities

Extend to multiple periods \(\{\mathbf X_t,\mathbf W_t:t=0,1,2,\dots\}\). SDF (17.60), CF (17.61) are multiplicative functionals; the modification happens only at \(t=1\), factor \(H_1(r)\) (17.62), modified cash flow \(G_t^{\text{modified}}=G_t H_1(r)\) (\(t\geq1\), since by the (17.61) multiplicative structure the \(t=1\) multiple propagates to all \(G_{t+\tau}\)). Gross return \(R^{G,\text{modified}}_{0,t}\) (17.63). Differentiating \(\ln\mathbb E[R^{G,\text{modified}}_{0,t}\mid\mathbf X_0=\mathbf x]\) at \(r=0\) (17.64): shock-price = shock-exposure − shock-cost elasticity. Generally, letting \(m_t\) represent \(G_t\) or \(G_t S_t\), the derivative (17.65): \(=\boldsymbol\nu(\mathbf x)\cdot\frac{\mathbb E[\frac{m_t}{m_0}\mathbf W_1\mid\mathbf X_0=\mathbf x]}{\mathbb E[\frac{m_t}{m_0}\mid\mathbf X_0=\mathbf x]}\). By (17.23) set \(\frac{m_t}{m_0}=\frac{M_t}{M_0}\frac{e(\mathbf X_0)}{e(\mathbf X_t)}\exp(t\eta)\) (17.66) (permanent × transient × constant); using the change of measure (17.67) and zero far-future covariance, the long-run limit (17.69): \(\lim_{t\to\infty}\frac{d}{dr}\ln\mathbb E[\frac{m_t}{m_0}H_1(r)\mid\mathbf X_0=\mathbf x]\big|_{r=0}=\boldsymbol\nu(\mathbf x)\cdot\tilde{\mathbb E}[\mathbf W_1\mid\mathbf X_0=\mathbf x]\).

Example 17.1 (vector autoregression). \(\mathbf X_{t+1}=\mathbf A\mathbf X_t+\mathbf B\mathbf W_{t+1}\) (\(\mathbf W\sim\mathcal N(\mathbf 0,\mathbf I)\) i.i.d.) (17.70), \(\ln C_{t+1}-\ln C_t=\mu_c+\mathbf U_c\cdot\mathbf X_t+\mathbf F_c\cdot\mathbf W_{t+1}\) (17.71), \(\ln H_1(r)=r\boldsymbol\nu(\mathbf X_0)\cdot\mathbf W_1-\frac12 r^2\boldsymbol\nu(\mathbf X_0)\cdot\boldsymbol\nu(\mathbf X_0)\) (17.62). Using (12.10) in He (2019a) §12.4.4 to compute \(\frac{d}{dr}\ln\mathbb E[\frac{C_t}{C_0}H_1(r)\mid\mathbf X_0=\mathbf x]\big|_{r=0}\) (17.72) gives (17.73): \(=(\mathbf F_c'+\mathbf U_c'(\mathbf I-\mathbf A)^{-1}(\mathbf I-\mathbf A^{t-1})\mathbf B)\boldsymbol\nu(\mathbf X_0)\).

17.5.3 Compare Shock Elasticity to Impulse Response Function

Let \(\mathbf X_{t+1}=\mathbf A\mathbf X_t+\mathbf B\mathbf W_{t+1}\) (17.74), \(\ln H_1(r)\) as before. Define the pseudo-IRF \(\widehat{\mathrm{IRF}}_{1,t}(\mathbf W_1)=\mathbb E[\frac{Y_t}{Y_0}\mid\mathbf X_0,\mathbf W_1]\) (17.75) (not the strict IRF, but related). The shock elasticity of \(\frac{Y_t}{Y_0}\), \(\mathrm{SE}_{1,t}\) (17.76):

i.e. the shock elasticity is a weighted average of the pseudo-IRF (weights being probabilities under the \(H_1(r)\) measure). Special case: if \(\ln(\frac{Y_t}{Y_0})\) is conditionally normal with \(\mathbb E[\ln\frac{Y_t}{Y_0}\mid\mathbf X_0,\mathbf W_1]=\mathbf k\cdot\mathbf W_1\), then (17.77) \(\mathbb E[H_1(r)\mathbb E[\frac{Y_t}{Y_0}\mid\mathbf X_0,\mathbf W_1]\mid\mathbf X_0]=e^{r\mathbf k\cdot\boldsymbol\nu(\mathbf X_0)+\text{constant}}\), and (17.76) gives \(\mathrm{SE}_{1,t}=\mathbf k\cdot\boldsymbol\nu(\mathbf X_0)\). If \(\mathbf W_1=\boldsymbol\nu(\mathbf X_0)\), the shock elasticity of \(\frac{Y_t}{Y_0}\) exactly equals the pseudo-IRF of \(\ln(\frac{Y_t}{Y_0})\).

17.6 Approximation under Recursive Utility

Endowment economy, exogenous random consumption sequence \(\{C_t\}\) (\(C_t\) analogous to the state vector \(\mathbf X_t\) above).

17.6.1 Recursive Utility

Rewrite the (9.1) Epstein-Zin utility equivalently (17.78): \(V_t=[(1-\beta)C_t^{1-\rho}+\beta(R_t^{1-\rho})]^{\frac1{1-\rho}}\), \(R_t=\mathbb E[V_{t+1}^{1-\gamma}\mid\mathcal F_t]^{\frac1{1-\gamma}}\) (17.79). \(\gamma\) governs risk aversion, \(\rho\) governs IES, \(\beta\) is the discount factor. Log objects \(v_t\equiv\ln V_t\), \(c_t\equiv\ln C_t\), \(r_t\equiv\ln R_t\), \(s_t\equiv S_t\). In logs (17.80): \(v_t=\frac1{1-\rho}\ln[(1-\beta)e^{(1-\rho)c_t}+\beta(e^{(1-\rho)r_t})]\), (17.81): \(r_t=\frac1{1-\gamma}\ln\mathbb E[e^{(1-\gamma)v_{t+1}}\mid\mathcal F_t]\).

17.6.2 Approximation of Value Function

Let \(v_t,c_t,r_t\) all be functions of \(h\) (\(h\) measures exposure to uncertainty), \(C_{t+1}(h)=\psi(C_t(h),h\mathbf W_{t+1},h)\). At \(h=0\) there is no uncertainty. A second-order expansion of \(v_t(h)\) at \(h=0\): \(v_t(h)\approx v_t(0)+hv_t'(0)+\frac{h^2}2 v_t''(0)\) (17.108).

• Zeroth order (17.82): \(v_t(0)=\frac1{1-\rho}\ln[(1-\beta)e^{(1-\rho)c_t(0)}+\beta(e^{(1-\rho)r_t(0)})]\), \(r_t(0)=v_{t+1}(0)\) (17.83). Assume constant certainty consumption growth \(c_{t+1}(0)-c_t(0)=\eta_c\) (17.84); by homogeneity of degree 1 of (17.78) in \((C_t,V_{t+1})\), \(v_t(0)-c_t(0)=\eta_{v-c}\) constant (17.85), solving \(e^{(1-\rho)\eta_{v-c}}=\frac{1-\beta}{1-\beta e^{(1-\rho)\eta_c}}\).
• First order (17.87) (define \(\lambda\equiv\frac{\beta e^{(1-\rho)(\eta_{v-c}+\eta_c)}}{(1-\beta)+\beta e^{(1-\rho)(\eta_{v-c}+\eta_c)}}\)):

• Second order (17.88): \(v_t''(0)=(1-\lambda)c_t''(0)+\lambda r_t''(0)+(1-\lambda)\lambda(c_t'(0)-r_t'(0))^2\).

It remains to characterize the log continuation value \(r_t(h)\) (17.81). Two approximations (two assumptions):

First: \(\gamma\) fixed (independent of \(h\)). Zeroth order \(r_t(0)=v_{t+1}(0)\) (17.89); first order \(r_t'(0)=\mathbb E[v_{t+1}'(0)\mid\mathcal F_t]\) (17.90); second order (17.91): \(r_t''(0)=\mathbb E[v_{t+1}''(0)\mid\mathcal F_t]+(1-\gamma)\underbrace{(\mathbb E[(v_{t+1}'(0))^2\mid\mathcal F_t]-(\mathbb E[v_{t+1}'(0)\mid\mathcal F_t])^2)}_{\text{conditional variance of 1st-order approx}}\).

Second: \(\gamma\) related to \(h\) via \(1-\gamma=\frac{1-\gamma_0}h\) (\(\gamma\) varies positively with exposure \(h\)). Define \(\tilde v_t(h)\equiv\frac{v_t(h)-v_t(0)}h\), \(\tilde r_t(h)\equiv\frac{r_t(h)-v_{t+1}(0)}h\), giving properties (17.93)–(17.100): \(\tilde v_t(0)=v_t'(0)\), \(2\tilde v_t'(0)=v_t''(0)\), \(\tilde r_t(0)=r_t'(0)\), \(2\tilde r_t'(0)=r_t''(0)\). First order (17.102): \(r_t'(0)=\frac1{1-\gamma_0}\ln\mathbb E[e^{(1-\gamma_0)(v_{t+1}'(0))}\mid\mathcal F_t]\); define \(\frac{\tilde M_{t+1}}{\tilde M_{t,t+1}}\equiv\frac{e^{(1-\gamma_0)(v_{t+1}'(0))}}{\mathbb E[e^{(1-\gamma_0)(v_{t+1}'(0))}\mid\mathcal F_t]}\) (17.104), second order (17.105): \(r_t''(0)=\mathbb E[v_{t+1}''(0)\frac{\tilde M_{t+1}}{\tilde M_{t,t+1}}\mid\mathcal F_t]\).

Combine (taking the second way) (17.106), (17.107), with (17.85), giving (17.108): \(v_t(h)=v_t(0)+hv_t'(0)+\frac{h^2}2 v_t''(0)\), which depends only on the exogenous consumption dynamics (since \(v_t'(0),v_t''(0)\) are in turn determined by \(c_t'(0),c_t''(0),v_{t+1}'(0),v_{t+1}''(0)\), iteratively all by consumption dynamics). Key point: \(v_t(h)\) depends on consumption dynamics in all future periods, reflecting the forward-looking (non-time-separable) property of Epstein-Zin recursive utility.

By (9.12), the EZ SDF satisfies (17.109): \(\frac{S_{t+1}}{S_t}=\beta(\frac{C_{t+1}}{C_t})^{-\rho}(\frac{V_{t+1}}{R_t})^{\rho-\gamma}\). In logs (17.110): \(s_{t+1}-s_t=\ln\beta-\rho\underbrace{(c_{t+1}-c_t)}_{\text{Term A}}+(\rho-\gamma)\underbrace{(v_{t+1}-r_t)}_{\text{Term B}}\). Term A expands directly (17.111); Term B expands in \(h\) (17.112), (17.113). At \(h=1\) (i.e. \(\gamma=\gamma_0\)), combining (17.111), (17.113) into (17.110) gives the SDF-increment approximation (17.114), which again depends only on the exogenous consumption dynamics (as for the value function (17.108)).