we-solve (beta)

Econ-418 · economics · Economics

Equity-premium puzzle

LLM status Open unverified

Statement

Atlas ID: F1; original number: 91; field: Finance. Classification: Editorial research formulation (theoretical/empirical); not a certified unsolved theorem.

Definitions and assumptions

On a filtered probability space $(\Omega,\mathcal F,(\mathcal F_t),\mathbb P)$, let $R_{t+1}^e$ be an equity gross return and $R_t^f>0$ an $\mathcal F_t$-measurable gross risk-free return. A positive stochastic discount factor $m_{t+1}$ prices both returns if $\mathbb E_t[m_{t+1}R_{t+1}^e]=1$ and $R_t^f\mathbb E_t[m_{t+1}]=1$, where $\mathbb E_t$ denotes conditional expectation. Assume finite second moments and integrable products. Model $\theta$ specifies preferences, consumption risks, market participation and a selected equilibrium of budget-feasible household and firm decisions. For an unconstrained representative household at an interior optimum in a time-separable CRRA benchmark, $m_{t+1}=\beta(C_{t+1}/C_t)^{-\gamma}$ with $\beta\in(0,1)$, $\gamma>0$ and $C_t>0$.

Formal research question

For a prespecified admissible parameter and model class $\Theta$, identify whether the joint law of consumption, portfolios, safe returns, equity returns and option prices is compatible with any $\theta\in\Theta$. The pricing restrictions imply \[ \mathbb E_t[R_{t+1}^e]-R_t^f =-\frac{\operatorname{Cov}_t(m_{t+1},R_{t+1}^e)}{\mathbb E_t[m_{t+1}]}. \] Quantify which restrictions reject a benchmark and which data discriminate between disaster risk, persistent consumption risk and alternative preferences. Require competing explanations to reproduce moments excluded from estimation and report the identified range of disaster probabilities or risk aversion rather than choosing parameters solely to fit the premium.

Interpretation and scope

The puzzle is an empirical difficulty for a restricted benchmark, not a contradiction of no-arbitrage or of all consumption-based models. Specify return frequency, inflation adjustment, sample selection and whether the safe asset is held to maturity; a short nominal bill need not be a real risk-free asset. The observed historical premium is an estimate of a population moment, affected by sampling uncertainty and rare events. Recursive preferences require their own marginal valuation formula; the CRRA expression does not carry over unchanged. Conditional identities above do not justify replacing conditional covariances with unconditional moments without handling the time-varying safe rate. State which moments and countries are included before assessing fit, with confidence regions reflecting serial dependence.

Source audit and status

[S1], also atlas link [48], defines the historical benchmark. [S2] offers disaster risk and [S3] offers long-run risk as potential resolutions. Their existence precludes calling the broad question an unsolved impossibility theorem. This is an editorial model-discrimination agenda; the audit does not certify absence of later resolutions.

References

    },leftmargin=*]

  1. Rajnish Mehra and Edward C. Prescott (1985). The Equity Premium: A Puzzle. Journal of Monetary Economics 15(2), 145–161. https://www.sciencedirect.com/science/article/pii/0304393285900613. Locator and role: Publisher or author abstract / bibliographic record; Historical US moments inconsistent with a restricted benchmark calibration; resolves link [48]. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  2. Thomas A. Rietz (1988). The Equity Risk Premium: A Solution. Journal of Monetary Economics 22(1), 117–131. https://doi.org/10.1016/0304-3932(88)90172-9. Locator and role: Publisher or author abstract / bibliographic record; Low-probability consumption disasters as a potential explanation. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  3. Ravi Bansal and Amir Yaron (2004). Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles. Journal of Finance 59(4), 1481–1509. https://doi.org/10.1111/j.1540-6261.2004.00670.x. Locator and role: Publisher or author abstract / bibliographic record; Long-run consumption risk and recursive-preference asset pricing. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-418-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-418

Exact benchmark restrictions and disaster-risk observational equivalence

Problem ID: G12-414. Status: Unresolved. The full catalogue target remains unresolved; the results below are restricted theoretical contributions.

A model-discrimination target

The catalogue asks whether a prespecified class of equilibrium models can reproduce a joint consumption-and-asset law, including moments excluded from fitting. It is not a request to find one parameter that matches one historical premium. No empirical data are supplied here. We derive exact restrictions for a two-state, one-period CRRA pricing benchmark, explain what option prices add in this benchmark, and quantify one rare-event information limit. These are diagnostics for a restricted class, not a resolution of the historical equity-premium puzzle [mp].

Condition on a fixed date-$t$ information set throughout the first calculation, suppressing the conditioning notation. Let consumption growth be $G>0$, let $m=\beta G^{-\gamma}$, and impose $\beta\in(0,1),\gamma>0$. Suppose a disaster state occurs with probability $p\in(0,1)$ and \[ (G,R^e)=\begin{cases}(g_D,r_D),&D,\$g_N,r_N),&N, \end{cases} \qquad 0<g_D<g_N,\quad 0<r_D<r_N. \] Safe gross return $R^f>0$ and both equity gross returns are specified. All products are finite. This is a necessary pricing block of an equilibrium; a household budget and asset supply would still be needed for a full equilibrium construction.

An exact compatibility theorem

Define the risk-neutral disaster weight \[ q=\frac{r_N-R^f}{r_N-r_D}. \tag{1} \]
Theorem (Two-state CRRA compatibility): There is a CRRA stochastic discount factor in the declared parameter domain pricing the safe asset and equity if and only if $0<q<1$ and the numbers \[ \gamma_*= \frac{\log\{q(1-p)/[p(1-q)]\}}{\log(g_N/g_D)},\qquad \beta_*= \frac{1}{R^f\{pg_D^{-\gamma_*}+(1-p)g_N^{-\gamma_*}\}} \tag{2} \] satisfy $\gamma_*>0$ and $0<\beta_*<1$. When such a pair exists it is unique.
Proof: The safe-asset equation is ${\mathbb{E}} m=1/R^f$. Write $q=pm_D/{\mathbb{E}} m$ and $1-q=(1-p)m_N/{\mathbb{E}} m$. Since both state probabilities and discount factors are positive, $q\in(0,1)$. The equity equation becomes \[ 1=\frac{qr_D+(1-q)r_N}{R^f}, \] which yields (1). Taking the ratio of the two state weights gives \[ \frac{q}{1-q}=\frac{p}{1-p}\left(\frac{g_N}{g_D}\right)^\gamma. \] Its logarithm uniquely yields $\gamma_*$ because $g_N/g_D>1$. The safe-asset equation uniquely yields $\beta_*$. These derivations prove necessity. Conversely substituting (2) recovers the prescribed risk-neutral weights and both pricing equations, proving sufficiency. ∎

Positive risk aversion requires $q>p$. Merely checking that $R^f$ lies strictly between the equity returns is therefore insufficient. A fitted pair with $\beta_*>1$ also fails the declared benchmark, even though it is algebraically a positive discount factor. If $g_D=g_N$, excluded here, the consumption-based discount factor is constant; positive equity premium cannot be priced and $\gamma$ is not identified by state variation.

As a reproducible numerical setup, take $p=1/50$, $g_D=3/5$, $g_N=51/50$, $r_D=2/5$, $r_N=27/25$, and $R^f=101/100$. Then $q=7/68$, $q/p>1$, and \[ \gamma_*={\log(343/61)\over\log(17/10)}. \] The resulting value is approximately $3.25$, and substituting it in (2) gives a discount factor below one. These arbitrary numbers illustrate compatibility; they are not estimated consumption disasters or evidence about an actual market. Any empirical exercise would insert jointly estimated moments and retain their uncertainty.

Why additional securities may still be redundant

For a claim paying $h_D,h_N$ in the two states, any compatible discount factor gives price \[ P_h=\frac{qh_D+(1-q)h_N}{R^f}. \tag{3} \] Because $r_D\ne r_N$, every two-state payoff is an affine combination of the risk-free and equity payoffs. Thus all such option prices are already determined by their span. Observing more strikes can reject the two-state payoff model if they violate (3), but cannot separate $p$ from preferences while all prices continue to depend on the same $q$.

If only prices are known, fix $q$ and $g_D,g_N$. For every $p<q$ producing $\beta_*(p)\in(0,1)$, (2) supplies a different positive risk-aversion coefficient with the same asset prices. On any open interval of such $p$, this is genuine price observational equivalence. The complete population consumption law, in contrast, contains $p$ and distinguishes these models. It would be incorrect to call them observationally equivalent for that richer target. The distinction is between identification from a joint population law and precision in a short sample.

A benchmark rejection bound outside two states

Let $\mu_m={\mathbb{E}} m>0$, and let $\sigma_m,\sigma_R$ denote standard deviations. The two pricing equations imply \[ {\mathbb{E}} R^e-R^f=-{\operatorname{Cov}}(m,R^e)/\mu_m. \] Cauchy--Schwarz therefore yields \[ \frac{|{\mathbb{E}} R^e-R^f|}{\sigma_R}\le\frac{\sigma_m}{\mu_m}, \tag{4} \] when $\sigma_R>0$. If log consumption growth is normal with variance $\sigma_g^2$ and $m=\beta G^{-\gamma}$, direct evaluation of the normal exponential moments gives \[ \frac{\sigma_m^2}{\mu_m^2}=e^{\gamma^2\sigma_g^2}-1. \] Hence a benchmark restricting $\gamma\le\bar\gamma$ must satisfy \[ \frac{|{\mathbb{E}} R^e-R^f|}{\sigma_R} \le\sqrt{e^{\bar\gamma^2\sigma_g^2}-1}. \tag{5} \] The calculation is self-contained: ${\mathbb{E}} e^{aZ}=e^{a\mu+a^2\sigma^2/2}$ gives both moments, and $\beta$ cancels. Failure of (5) rejects this conditional lognormal-CRRA block; satisfaction is only necessary, since the sign and magnitude of the actual covariance must also match. With time-varying safe rates these are conditional statements, so plugging an unconditional safe-rate average into (4) without adjustment is invalid.

Rare events and finite samples

For $T$ independent consumption observations, compare a law with no disaster to one with disaster probability $p$ and identical normal-state outcomes. Their sample laws can be coupled to agree whenever the second law has no disaster, an event of probability $(1-p)^T$. Their total variation distance is exactly $1-(1-p)^T$, because the event "at least one disaster" attains the bound. Thus when $Tp$ is small, no test using only this consumption sample can reliably separate them: the sum of its two error probabilities is at least $(1-p)^T$.

This statement concerns the consumption sample alone. It does not ignore extra option data if a specified model makes those data informative about $p$, nor assert that all disaster models are observationally identical. It explains why fitting disaster probability from a no-disaster sample needs explicit restrictions and uncertainty. The full problem remains the joint empirical discrimination of preferences, persistent risks, disasters, participation and equilibrium portfolios. The exact formulas above make several benchmark claims testable but supply neither those data nor a general equilibrium resolution.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves exact two-state CRRA pricing compatibility, option-payoff redundancy, benchmark rejection bounds, and a rare-event testing limit. Full equilibrium feasibility and empirical discrimination among disasters, persistent risks, preferences, participation, and portfolio responses remain outside these pricing diagnostics.

References

Go to contribution →