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Econ-331 · economics · Economics

Uncovered-interest-parity deviations

LLM status Open unverified

Statement

Original subject group: International finance and exchange rates

Audit classification: Background; proposed extension. Fama supplies conditional evidence about forward premiums; the recent IMF source supplies FX identification/policy background. The exact risk/belief/constraint decomposition is not verified as a source-stated open problem. Log and gross parity are distinguished.

Definitions and precise research target

Let $s_t$ be log domestic-currency units per foreign currency, and $i_t,i_t^*$ comparable one-period domestic and foreign log gross interest rates. Define the log return difference from borrowing domestically and investing abroad as $x_{t+1}=i_t^*-i_t+s_{t+1}-s_t$. Let $\mathbb P$ be the objective probability law and $\mathcal F_t$ dated available information. Under this conditional law, its expectation is the currency premium $\pi_t=E_{\mathbb P}[x_{t+1}\mid\mathcal F_t]$; the log-parity convention requires $\pi_t=0$; exact gross-return parity is separately defined below. An investor's subjective law $\mathbb Q_t$ gives an expectation wedge \[b_t=E_{\mathbb Q_t}[s_{t+1}\mid\mathcal F_t]-E_{\mathbb P}[s_{t+1}\mid\mathcal F_t].\] The task is to identify which part of predictable excess returns is attributable to compensation for priced risks, belief errors, or trading and financing constraints. Specify a discount-factor model, a survey measurement model for $\mathbb Q_t$, and the mechanism relating intermediary constraints to attainable portfolios. Determine the identifying restrictions separating these components and derive bounds when objective expectations or marginal investors are unobserved. Evaluate variation by currency, horizon and financial state. Realized excess returns may remain predictable under rational expectations with time-varying risk compensation; a regression slope inconsistent with parity does not by itself identify irrationality. Exact gross-return parity and log parity are distinct; use the chosen convention consistently.

Additional formal definitions and scope (editorial). Specify gross domestic and foreign rates $R_t^d,R_t^f>0$, exchange rate $S_t=e^{s_t}>0$, and net currency payoff $X_{t+1}=R_t^f S_{t+1}/S_t-R_t^d$ per unit initially invested. Exact expected-return parity is $E_t[X_{t+1}]=0$. If $i_t=\log R_t^d$ and $i_t^*=\log R_t^f$, then $x_{t+1}=i_t^*-i_t+\Delta s_{t+1}$ is the exact log return difference; $E_t[x]=0$ is a log-parity convention and differs from exact gross parity by Jensen terms. Under unconstrained common-belief pricing, $E_t[M X]=0$, so $E_t[X]=-\operatorname{Cov}_t(M,X)/E_t[M]$. With tastes, constraints or different subjective laws the Euler equation changes. Define the survey kernel and marginal investor before interpreting $b_t$ as priced belief error.

For the identification part, declare an admissible class $\Theta$ of structural models, including preferences, technologies, shock laws, measurement/sampling rules and any equilibrium selector. If $O$ denotes the complete observable history and $T(\theta)$ the target defined above, the identified set is $\mathcal I_T=\{T(\theta):\theta\in\Theta,\ \mathcal L_\theta(O)=\mathcal L_{\rm obs}(O)\}$; point identification means this set is a singleton. A $do(a)$ comparison replaces stated policy/technology equations while fixing the declared exogenous law and re-solving the remaining equations. These definitions do not assert that an identification theorem holds without further restrictions.

Variants and assumption changes

  1. Rational expectations and exact gross parity (editorial). Fix a common pricing kernel and identify risk compensation for $X$, retaining Jensen effects when reporting logs.
  2. Surveys and financing constraints (editorial). Add a measurement model for subjective expectations and a financing wedge; identify or bound their separate contributions rather than inferring beliefs from a forward-regression slope.

References and source audit

  1. Primary journal scan verifies Eugene F. Fama, 1984, Journal of Monetary Economics 14:319--338; DOI matches journal PII. Locator: Introduction, p.319; Section 2, pp.320--323. Support: Forward-premium evidence under maintained assumptions. Full PDF accessible. URL: https://eml.berkeley.edu/~craine/EconH195/Fall_14/webpage/Fama_Forward%20Discount.pdf.
  2. Official IMF/elibrary indexed records verify five authors, 2026 SDN/2026/003 and DOI. Official release is September 17, 2026; URL contains September 16. Locator: Official indexed abstract and currency-premium/decomposition passages; no exact full-page locator certified. Support: Fundamental/financial FX identification and policy background. Landing page initially opened; later full-text fetches failed. Indexed primary passages inspected only. URL: https://www.imf.org/en/publications/staff-discussion-notes/issues/2026/09/16/drivers-of-exchange-rates-in-emdes-implications-for-foreign-exchange-intervention-578715.
  1. Eugene F. Fama. 1984. Forward and Spot Exchange Rates. Introduction, p.319; Section 2, pp.320--323. https://eml.berkeley.edu/~craine/EconH195/Fall_14/webpage/Fama_Forward%20Discount.pdf. DOI: https://doi.org/10.1016/0304-3932(84)90046-1.
  2. Ece Ozge Emeksiz; Andres Fernandez; Nikhil Patel; Ivan Petrella; Tatjana Schulze. 2026. Drivers of Exchange Rates in EMDEs: Implications for Foreign Exchange Intervention. Official indexed abstract and currency-premium/decomposition passages; no exact full-page locator certified. https://www.imf.org/en/publications/staff-discussion-notes/issues/2026/09/16/drivers-of-exchange-rates-in-emdes-implications-for-foreign-exchange-intervention-578715. DOI: https://doi.org/10.5089/9798229055208.006.

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.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-331-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-331

Uncovered-interest-parity deviations: a research attempt

Status: unresolved. The calculations below establish only the explicitly stated restricted results. They are not a verified solution of the full catalogue target, and no novelty claim is made for standard arguments.

Target and choice of return convention

The target is a separately interpretable decomposition of predictable currency returns into risk compensation, beliefs and portfolio constraints, with identification or bounds. Fix one information cell at time $t$; all expectations below are conditional on it. Let $R^d,R^f>0$ be gross rates and let $S'/S>0$ be the exchange-rate change. Use the zero-initial-cost gross payoff \[ X=R^fS'/S-R^d,\qquad \mu=\mathbb E_P X. \] Its log counterpart is $x=\log(R^f/R^d)+\log(S'/S)$, so $X=R^d(e^x-1)$. The statistical observation of $P$ identifies $\mu$, but does not specify a marginal investor or that investor's preferences. Time-series estimation of this conditional law is a further problem.

Fama's primary paper, particularly Sections 2 and 4, was inspected for its forward-premium decomposition. Its regressions are not a procedure identifying three structural causes. The attempt below supplies an explicit normalized Euler decomposition instead of interpreting a regression slope as a belief error.

An exact, convention-dependent decomposition

Let $L=dQ/dP>0$ be the marginal investor's subjective density, with $\mathbb E_P L=1$. Let $m>0$ be that investor's marginal-utility discount factor, normalized by $\mathbb E_Pm=1$. Define the signed trading wedge $\kappa=\mathbb E_P[LmX]$. Under an interior, unconstrained choice this wedge is zero. At a position limit it may be nonzero, with sign determined by the limit and payoff convention. It is not automatically a nonnegative welfare cost.
Proposition: For integrable products $LmX$ and $mX$, \[ \mu=\underbrace{-\operatorname{Cov}_P(m,X)}_{r} +\underbrace{-\mathbb E_P[(L-1)mX]}_{b} +\underbrace{\kappa}_{c}. \]
Proof: Expand $\kappa=\mathbb E_P[mX]+\mathbb E_P[(L-1)mX]$ and use $\mathbb E_P[mX]=\mu+\operatorname{Cov}_P(m,X)$. ∎
The belief component here is a pricing-weighted expectation distortion, not merely the survey mean of next period's log exchange rate. Allocating the interaction between $L$ and $m$ differently gives a different valid decomposition. Thus even an identified triplet requires an announced normalization and attribution convention.

For a representative investor maximizing a differentiable concave objective over a scalar position $z\leq\bar z$, the derivative at the optimum is proportional to $\mathbb E_Q[mX]$. At an upper binding limit it equals a nonnegative multiplier, whereas a lower binding limit gives the opposite sign. Specifying the position, wealth and constraint is therefore necessary to interpret $c$ as a financing mechanism.

Three observationally identical explanations

Take $R^d=2$, $R^f=1$, $S=1$ and $S'\in\{3,1\}$ with probabilities $(3/5,2/5)$. Then $X\in\{1,-1\}$ and $\mu=1/5$. Consider three models with this identical observable payoff law:
  1. Common beliefs, $L=1$, and $m=(5/6,5/4)$. Then $\mathbb E_Pm=1$, $\mathbb E_PmX=0$, and $r=1/5$.
  2. Constant $m=1$, and $L=(5/6,5/4)$. Then $Q$ assigns probability $1/2$ to each state, $\kappa=0$, and $b=1/5$.
  3. Constant $m=L=1$, with the investor at an upper position limit. Then $\kappa=1/5$ and the whole return is assigned to the constraint component.
Positive discount factors in the first model can be implemented by appropriate state-contingent consumption and strictly concave utility; a locally linear marginal utility supplies the latter models. These are one-asset pricing examples, not claimed complete calibrated currency equilibria. Their implication is precise: the payoff law alone cannot distinguish the triplet even when all state probabilities and returns are observed. Additional equilibrium assets or consumption restrictions could reject some examples.

What surveys identify and what they do not

Suppose a survey records $Z=\mathbb E_Q[\log S']+\epsilon$, where $\mathbb E[\epsilon\mid\mathcal F_t]=0$, and represents the actual marginal investor. In the two-state case the mean identifies $Q(S'=3)$, since $\log3\neq\log1$. It separates the belief-only example from the other two. With at least three exchange-rate states, a single survey mean supplies one linear restriction plus probability normalization, so subjective probabilities generally remain set identified. Even complete subjective probabilities leave the risk-versus-constraint distinction unresolved when $m$ is unobserved.

In a finite $K$-state experiment with objective probabilities $p_s>0$, suppose additional data identify a pricing kernel $m_s$ and subjective probabilities $q_s$. Then the displayed decomposition is point identified: substitute $L_s=q_s/p_s$ and infer $\kappa$. This inference labels the residual but does not establish a particular financing mechanism; the multiplier must also agree with observed constraints and portfolio optimality. If $m$ and $q$ belong instead to announced compact positive sets, optimize the three displayed formulas jointly over these sets and the Euler inequalities. Continuous objectives attain extrema. These extrema are sharp for that finite admissible class, provided each retained pair is structurally implementable; imposing only moment bounds gives outer bounds, not automatic sharpness.

Useful sensitivity inequalities and a log check

If $\operatorname{sd}(m)\leq a$, $\mathbb E(L-1)^2\leq d^2$, and $\mathbb E(mX)^2\leq V^2$, Cauchy--Schwarz yields \[ |r|\leq a\operatorname{sd}(X),\qquad |b|\leq dV, \qquad |\mu-\kappa|\leq a\operatorname{sd}(X)+dV. \] These are transparent necessary bounds, not a rectangular sharp identified set: all components share the same latent $(L,m)$. If $x$ is conditionally Gaussian with mean $a_x$ and variance $v_x$, then \[ \mu=R^d\{\exp(a_x+v_x/2)-1\}. \] Thus zero expected log return gives a positive gross premium whenever $v_x>0$. In the finite-state example gross parity under $Q$ does not assert log parity. This is an essential specification check before attaching labels to data.

Completion Estimate

40%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives an exact normalized gross-return risk-belief-constraint identity, observationally equivalent explanations, and survey and sensitivity restrictions. It does not identify marginal investors, pricing kernels, portfolio feasibility, or objective conditional laws across currencies and financial states in a complete currency equilibrium.

Remaining gap and sources

The exact identity, equivalence construction and sensitivity bounds are established within their stated domains. The original task still requires identifying marginal investors, a justified survey kernel, financing feasibility, and credible objective laws across currencies and states. No such empirical information is supplied here, and no decomposition has been estimated.

Eugene F. Fama (1984), Forward and Spot Exchange Rates, Journal of Monetary Economics 14, 319--338, Sections 2 and 4, primary scan inspected: https://eml.berkeley.edu/~craine/EconH195/Fall_14/webpage/Fama_Forward%20Discount.pdf.

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