Risk-premium foundations

Original subject group: Asset pricing and financial markets

Audit classification: Published research agenda. The sources establish narrow benchmark puzzles and motivate a joint-risk-premium agenda. The catalogue moment-feasibility model is editorial; no universal impossibility or current unsolvability certificate is established.

Definitions and precise research target

Fix a discrete-time exchange economy with countries, households and assets. Household $i$ consumes $c_{it}>0$, receives labor income, and trades equity, housing claims and a one-period safe bond subject to specified borrowing and trading constraints. Let $R^e_{t+1}$, $R^h_{t+1}$, and $R^f_t$ denote their real gross returns; housing returns include net rents. A model specifies preferences, a probability law for consumption and dividends, beliefs and feasible trading sets. An equilibrium requires household optimality and market clearing. Pricing uses the Euler equations or inequalities of marginal investors under their stated subjective laws; a common stochastic discount factor is required only under additional assumptions justifying it. The representative-agent benchmark imposes \[M_{t+1}=\beta(c_{t+1}/c_t)^{-\gamma},\qquad E_t[M_{t+1}R^j_{t+1}]=1.\] Here $E_t$ means expectation conditional on date-$t$ information, $\beta\in(0,1)$ is the time-discount factor, $\gamma>0$ is relative risk aversion, and $j$ indexes unconstrained traded claims. The research task is to find a quantitatively disciplined extension that jointly reproduces equity and housing excess-return means, their variances, cross-covariance and the safe rate, using preference and friction parameters independently compatible with household evidence. Fix the target population, measurement conventions and acceptable moment tolerances before calibration. Report which restrictions fail in the benchmark, which mechanism relaxes them, and whether distinct mechanisms remain observationally equivalent. Existence of a fitted model is not uniqueness of its explanation.

Additional formal definitions and scope (editorial). Time is $t\in\mathbb N_0$ on a filtered probability space $(\Omega,\mathcal F,(\mathcal F_t),\mathbb P)$; all prices and choices are adapted. A gross return is $R^j_{t+1}=(P^j_{t+1}+D^j_{t+1})/P^j_t$, with positive initial price and distributions net of declared operating costs. For housing, distinguish a traded diversified housing claim from an owner's illiquid dwelling and imputed consumption services. The safe return is $\mathcal F_t$-measurable. In the benchmark, utility is $E\sum_t\beta^t u_\gamma(c_t)$, with $u_\gamma(c)=(c^{1-\gamma}-1)/(1-\gamma)$ if $\gamma\ne1$ and $\log c$ otherwise, and the expectation and utility sums must be finite. Let $\Theta$ be a declared class of economies, $m(\theta)\in\mathbb R^d$ their moment vectors, $m^{\rm data}$ an independently fixed target, and $\epsilon_r\geq0$ the tolerance for moment $r$. The precise feasibility question is whether $\{\theta\in\Theta:|m_r(\theta)-m_r^{\rm data}|\leq\epsilon_r\ \forall r\}$ is nonempty. Equilibrium requires individual optimality and clearing; a common positive discount factor is imposed only where common-belief no-arbitrage or representative-agent assumptions justify it. Constrained households generally satisfy Euler inequalities, and subjective pricing uses their own probability laws.

Variants and assumption changes

1. Restricted benchmark (source-motivated). Restrict $\Theta$ to the stated CRRA representative-agent economy with an independently calibrated consumption process and parameter bounds; characterize the achievable equity/safe-rate moment region. This narrower benchmark already has published negative results for historical calibrations.
2. Joint housing/equity extension (editorial). Add housing service utility, household heterogeneity and trading restrictions, and retain a fixed joint moment target. Compare existence of a fit with identification of its mechanism using household data.

References and source audit

1. Primary journal PDF cover and Sections 1 and 5 inspected; authors, title and 1985 date match. DOI corresponds to the journal PII. Locator: Section 1, journal pp.145--146; Section 5, pp.158--159. Support: Equity/safe-return puzzle; benchmark answered. Full PDF accessible. URL: https://rajnishmehra.com/wp-content/uploads/2023/09/The-Equity-Premium-A-Puzzle.pdf.
2. Primary FRBSF cover verifies authors, 2019 WP 2019-10 and DOI. Locator: Section 7, printed p.40 (PDF p.43). Support: Housing/equity benchmark challenge. Full PDF accessible. URL: https://www.frbsf.org/wp-content/uploads/wp2019-10.pdf.

1. Rajnish Mehra; Edward C. Prescott. 1985. The Equity Premium: A Puzzle. Section 1, journal pp.145--146; Section 5, pp.158--159. https://rajnishmehra.com/wp-content/uploads/2023/09/The-Equity-Premium-A-Puzzle.pdf. DOI: https://doi.org/10.1016/0304-3932(85)90061-3.
2. Òscar Jordà; Moritz Schularick; Alan M. Taylor. 2019. The Total Risk Premium Puzzle. Section 7, printed p.40 (PDF p.43). https://www.frbsf.org/wp-content/uploads/wp2019-10.pdf. DOI: https://doi.org/10.24148/wp2019-10.

Integrated refinement: ECON-024

Joint household-portfolio, consumption, and asset-price discipline

This refinement adopts the additional source conventions (audited:conventions), subject to its local restrictions. Joint household-portfolio, consumption and pricing discipline. Strengthen the joint moment target to include micro portfolios and consumption responses in the same equilibrium that generates asset returns. An illustrative two-asset budget is \[ c_t+a_{t+1}+q_tk_{t+1}+\Phi(k_t,k_{t+1}) =(1+r_t)a_t+(q_t+d_t)k_t+w_tz_t, \quad a_{t+1}\geq\underline a,\quad k_{t+1}\geq0. \] Specify earnings, dividends, adjustment costs, production, net asset supplies, solvency, equilibrium existence and selection. Clear both asset markets, for example $\int a_{t+1}d\mu_t=B_t$ and $\int k_{t+1}d\mu_t=K_t$. For one common parameter vector $\vartheta\in\mathcal H$ and joint empirical confidence region $\mathcal C$, characterize $\{\vartheta:m(\vartheta)\in\mathcal C\}$. Account for dependence across moment blocks; separately fitted parameter vectors do not satisfy this target. Distinguish compatibility, identification, some-equilibrium versus all-equilibria fit, and validation on later observations.

Supplied source audit for ECON-024

Local [S1], [S2], etc. in this audit and the references immediately below belong to ECON-024, independently of the original entry's reference numbering. [S1], PDF pp. 4--5, supports the household-finance/asset-price agenda. This editorial economy restores price risk missing from the original budget. The URL's 2019 directory does not establish the lecture's creation date.

References for integrated refinement ECON-024

[S1] Benjamin Moll (n.d.). Lecture 11: Good Research Topics and Open Questions. Distributional Macroeconomics lecture slides; publicly archived at a 2019 URL.

Verified locator / source role: PDF pp. 4–5: multiple assets, portfolios, and asset pricing.

https://benjaminmoll.com/wp-content/uploads/2019/07/Lecture11_2149.pdf

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.