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Evaluating AI Investment Strategies
abstractWe study the problem of auditing a black-box algorithmic decision-maker from observable inputs and outputs alone. Our main result is an exact decomposition: under precisely characterized conditions, the cumulative
regret of a dynamic policy equals the sum of per-period
covariances between the cost vector and the policy's decision. This extends the single-period identity of Aldridge (2026) to the full multi-period setting of stochastic dynamic programming.
We prove the identity holds exactly under i.i.d. costs and
mean-unbiased Markov policies, derive closed-form bias corrections for non-stationary and time-varying cases, and establish the discounted-horizon analog. A Bellman recursion for the covariance regret functional connects the result to standard reinforcement learning algorithms; for rolling-window policies, the estimation-error bias is $O(d/w)$.
The decomposition has direct implications for algorithmic auditing in strategic environments: in platform mechanism design, it provides a welfare-based audit metric without access to the agent's private type; in repeated games, covariance reduction is a sufficient condition for
policy improvement; in procurement and ad auctions, the bias
correction quantifies welfare loss from strategic misreporting. The associated trajectory estimator is consistent, asymptotically normal with HAC variance, and computable in $O(T \cdot nd)$ time. This makes the proposed approach a tractable, model-free audit tool for platform mechanisms, algorithmic portfolio strategies, and any sequential decision system subject to external performance review.
Introduction
AI is already managing money. Large language models recommend stock portfolios, execution algorithms trade billions of dollars daily, and robo-advisors allocate retirement savings for millions of households. Yet a fundamental question remains
unanswered: how do you audit an algorithm you cannot open?
The difficulty is not hypothetical. CarlinIsraelsenWazzan2026 document that leading LLMs consistently recommend high-momentum, high-beta stocks. Even when explicitly instructed to keep beta between 0.9 and 1.1, the models violate the constraint, reverting to selections indistinguishable from those of mainstream retail advisers.
Jia, Moallemi and Zeng (2026) find that retail traders condition their trades on momentum and lose on average, while algorithms running contrarian strategies win. Regulators have
taken notice: SR 11-7 and the more recent SR-26-2 both mandate model risk audits for algorithmic strategies. The auditing process, however, has remained painful and imprecise, stymied by the secrecy that surrounds proprietary models.
This paper provides a tractable solution by examining regret: the gap between its realized performance and the theoretically optimal benchmark. We show that the cumulative regret of any dynamic algorithmic policy can be measured from the outside, using only the sequence of inputs and outputs that any auditor can observe.
The key identity is disarmingly simple: regret equals covariance. Specifically, Aldridge2026 establishes that, for a single-period linear optimization, expected regret equals the covariance between the uncertain cost vector and the policy's decision:
equation[equation omitted — 152 chars of source]
This paper asks and answers whether identity (ref) extends
to the sequential, multi-period setting that characterizes real algorithmic deployment. In the real-world algo environments, a manager observes new market data each period, reoptimizes, and
executes. A positive answer would allow for the performance monitoring of hedge funds, execution algorithms, and robo-advisors using only observable inputs and outputs, without access to the internal optimization engine.
\paragraph{Main result.}
The answer is affirmative, under precise conditions. Total regret over $T$ periods decomposes exactly as the sum of per-period input-output covariances:
equation[equation omitted — 238 chars of source]
where $b_t = \mathbb{E}[\hat{\pi}_t(c_t)] - \pi^*_t$ is the per-period policy bias.
The bias correction vanishes, and the identity is exact, whenever the policy's expected output matches the conditional-mean optimal at every period. This holds for minimum-variance portfolios, linear-quadratic regulators, and any policy solving
a linear optimization in $\mathbb{E}[c_t]$. For momentum strategies, which systematically violate this condition, the bias term quantifies precisely how much the raw covariance sum understates or overstates true regret.
\paragraph{Implications for algorithmic auditing.}
Identity (ref) reframes regulatory oversight as a statistical testing problem. A compliance officer monitoring any algorithmic strategy under SR 11-7 or SR-26-2 needs only the observable sequence of cost vectors and allocation weights
to construct a confidence interval for cumulative regret: no access to the internal optimizer is required. The associated trajectory estimator is consistent, asymptotically normal with HAC variance, and computable in $O(T \cdot nd)$ time with an $O(nd)$ online update per observation.
Beyond finance, the identity connects to three active research areas in algorithmic game theory and mechanism design. First, in platform mechanism design, a principal who observes only allocations and realized costs can bound an agent's
regret without access to the agent's private type. This reframes classical black-box auditing as a covariance estimation problem myerson1982optimal,
laffont1993theory. Second, in repeated games, our Bellman recursion (Section (ref)) shows that pointwise covariance reduction is sufficient for policy improvement, providing a computationally tractable alternative to full strategy enumeration hart2000simple, roughgarden2016twenty. Third, in procurement and ad auctions, the bias term $\sum_t \bar{c}_t^{\top} b_t$
measures welfare loss from strategic misreporting as a directly observable function of inputs and outputs edelman2007internet, varian2007position, This makes the methodology applicable to any platform where allocations are public but valuations are private.
\paragraph{Contributions.}
This paper makes the following specific contributions:
enumerate[label=(\roman*)]
• We formulate a general finite-horizon stochastic dynamic program and define a meaningful notion of multi-period regret that nests single-period regret as a special case (Section (ref)).
• We prove that total expected regret decomposes exactly as the sum of per-period input-output covariances under i.i.d.\ costs and a mean-unbiased Markov policy, with no remainder (Section (ref)).
• We extend the identity to non-stationary (time-varying distribution) costs and derive a closed-form correction term bounded by the cross-period covariance of cost increments (Section (ref)).
• We characterize the discount-weighted version for infinite-horizon dynamic programs: $\operatorname{Regret}_\gamma = \frac{1}{1-\gamma}
\operatorname{Cov}(c, \hat{\pi}(c))$, establishing that the effective horizon $1/(1-\gamma)$ is the only correction needed for long-run discounted regret
(Section (ref)).
• We derive a Bellman-style recursion for the covariance regret functional, showing that standard reinforcement learning algorithms (value iteration,
fitted-$Q$) can estimate multi-period regret without solving the original optimization problem, at computational cost $O(T \cdot K \cdot nd)$
(Section (ref)).
• We establish precisely when the identity fails for time-varying policies, derive the closed-form bias in terms of $\sum_t \bar{c}_t^{\top} b_t$, and show that momentum strategies accumulate bias of order $|\hat{\rho}| T \sigma_c^2$ (Section (ref)).
• We apply the theory to two financially concrete settings: rolling-window portfolio rebalancing, where estimation-error bias is $O(d/w)$; and mean-reverting execution, where the AR(1) correction is available in
closed form (Section (ref)).
• We propose a trajectory covariance estimator, prove a CLT with HAC variance, derive an $O(T \cdot nd)$ complexity bound with $O(nd)$ online updates, and establish that $T = O(\sigma^2_{\mathrm{LRV}} / \varepsilon^2)$ observations suffice to estimate cumulative regret within $\varepsilon$
(Section (ref)).
\paragraph{Empirical findings.}
We validate the decomposition on CRSP daily return data from January 2016 through December 2025. Daily AR(1) coefficients are persistently negative across all years and NBER regimes ($\hat{\rho} \in [-0.144, -0.004]$), confirming that short-horizon reversal dominated momentum throughout the sample. Reversion strategies outperformed momentum in seven of ten years, with the bias-corrected reversion strategy providing the best risk-adjusted performance in the two years (2019, 2022),
where raw reversion failed. There are five confounds identified in the literature that bias $\hat{\rho}$ upward: size effects jegadeesh2025shortterm, bid-ask bounce
roll1984simple, anderson2008spurious, volatility clustering engle1982autoregressive, nagel2012evaporating, cross-sectional vs.\ time-series momentum mamais2025explaining, and liquidity provision nagel2012evaporating, dai2024reversals. Due to the bias, the regret figures are conservative upper bounds on momentum-attributable losses. This is the appropriate direction for a regulatory audit tool.
Related Work
\paragraph{Regret in Online Learning and Sequential Decision-Making.}
The notion of regret as a performance criterion for sequential algorithms originates with hannan1957approximation and was formalized in the online convex optimization framework of zinkevich2003online. The classical result that no-regret algorithms achieve $O(\sqrt{T})$ cumulative regret against the best fixed action in hindsight is surveyed comprehensively in hazan2016introduction and shalev2012online.
Our setting differs in two key respects: we study dynamic policies that reoptimize each period against a stochastic cost process, and we seek a closed-form decomposition of regret rather than an asymptotic bound. ElmachtoubGrigas2022 establish a related covariance identity for the single-period predict-then-optimize
problem, which Aldridge2026 extends to stochastic linear programs. The present paper extends that identity to the full multi-period dynamic programming setting, derives exact bias corrections for non-stationary and Markovian cost processes, and proposes a model-free audit statistic with provably valid confidence intervals.
\paragraph{Regret in Repeated Games and Mechanism Design.}
No-regret learning in games has been extensively studied as a foundation for equilibrium concepts foster1997calibrated, hart2000simple, roughgarden2016twenty. In this literature, regret measures the loss from not having played a fixed best-response throughout the game. Our contribution is orthogonal: we provide an observable, model-free statistic, which is the input-output covariance. This statistic allows a third-party auditor to bound an agent's regret without access to the agent's strategy or private type, connecting it to the principal-agent auditing literature myerson1982optimal, laffont1993theory.
dekel2010implementation study implementation under incomplete information in repeated settings; our bias term $\sum_t \bar{c}_t^{\top} b_t$ provides a welfare-theoretic quantity analogous to their notion of strategic misreporting loss. In the context of ad auctions and platform markets
edelman2007internet, varian2007position, our audit statistic is directly applicable to regulatory oversight of algorithmic bidders, where the platform observes bids and allocations but not the bidder's private valuation model. The connection between observable allocations and welfare losses also relates to the literature on implementation in dominant strategies gibbard1973manipulation, satterthwaite1975strategy, where a similar separation between observable outcomes and internal preferences is central.
\paragraph{Black-Box Algorithm Auditing.}
A growing literature addresses the problem of auditing algorithmic systems without access to their internals. kearns2018preventing study fairness auditing under limited query access; roth2022uncertain analyze procurement mechanisms under algorithmic uncertainty. In finance, the Federal Reserve's SR 11-7 and SR-26-2 supervisory guidance mandates model risk management for algorithmic strategies, motivating performance metrics that do
not require source-code access board2011sr. Our trajectory covariance estimator
$\hat{C}_T$ (Section (ref)) provides precisely such a metric: it is consistent, asymptotically normal (Theorem (ref)), and requires only the observable sequence of costs and allocations. This places our work in the tradition of GMM specification testing hansen1982large and HAC-robust inference newey1987simple, andrews1991heteroskedasticity, which we employ directly in constructing confidence intervals for cumulative regret. The monitoring cost of $O(Td)$ per evaluation period compares favorably with internal audit procedures that require access to the full optimization engine.
\paragraph{Dynamic Programming and Reinforcement Learning.}
The Bellman recursion for the covariance regret functional
(Section (ref)) is structurally analogous to the $Q$-function recursion in reinforcement learning watkins1992q, sutton2018reinforcement. Our
Corollary (ref) establishes that pointwise covariance reduction implies multi-period regret improvement, paralleling the policy improvement theorem of howard1960dynamic. Recent work on regret in Markov decision processes jaksch2010near, azar2017minimax derives $\tilde{O}(\sqrt{T})$ bounds for unknown transition dynamics; our contribution is complementary, providing an
exact decomposition under known structure (i.i.d.\ or AR(1) costs) and a bias-corrected estimator for the non-stationary case. The analogy with fitted-$Q$ methods ernst2005tree, riedmiller2005neural suggests that standard reinforcement learning infrastructure can be repurposed for multi-period regret monitoring without solving the original optimization problem, which we identify as a direction for future empirical work.
\paragraph{Portfolio Optimization and Execution Cost Models.}
In the finance literature, regret-based portfolio evaluation appears in demiguel2009optimal, who benchmark mean-variance strategies against the $1/N$ rule, and in brodie2009sparse, who study sparse portfolio policies under estimation error. Our rolling-window rebalancing application (Section (ref)) formalizes the estimation-error bias as $O(d/w)$, consistent with the Ledoit-Wolf shrinkage literature ledoit2004well, ledoit2020analytical and the random matrix theory results of
bai2010spectral. For execution cost modeling, mean-reverting market impact processes have been studied by almgren2001optimal and gatheral2010no;
our AR(1) regret formula (Theorem (ref)) provides a closed-form regret decomposition for linear execution policies in this class of models, extending their cost-minimization results to a regret-minimization objective and confirming that
contrarian execution is welfare-improving in mean-reverting markets.
Setup: Multi-Period Stochastic Dynamic Program
State space, costs, and policies
Let $T \in \mathbb{N}$ denote the planning horizon. At each period
$t \in \{1,\ldots,T\}$:
itemize• The state $s_t \in \mathcal{S} \subseteq \mathbb{R}^{p}$ summarizes the history relevant to future costs.
• The cost vector $c_t \in \mathcal{C} \subseteq \mathbb{R}^{d}$ is drawn from a distribution that may depend on $s_t$.
• The decision $z_t \in \mathcal{Z} \subseteq \mathbb{R}^{n}$ is made by a policy $\hat{\pi}_t(s_t, c_t)$ after observing $(s_t, c_t)$.
• The instantaneous cost realized is $c_t^{\top}z_t$.
• The state evolves as $s_{t+1} = f(s_t, z_t, \epsilon_{t+1})$, where $\epsilon_{t+1}$ is an exogenous shock.
definition[Policy]
A deterministic Markov policy is a sequence
$\Pi = (\hat{\pi}_1, \ldots, \hat{\pi}_T)$ of measurable maps
$\hat{\pi}_t : \mathcal{S} \times \mathcal{C} \to \mathcal{Z}$.
A stationary policy satisfies $\hat{\pi}_t = \hat{\pi}$ for all $t$.
Filtration and conditional expectations
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the underlying probability space.
Define the natural filtration $\mathcal{F}_t = \sigma(s_1, c_1, \ldots, s_t, c_t)$ for $t \geq 1$ and $\mathcal{F}_0 = \{\emptyset, \Omega\}$. All expectations $\mathbb{E}_t[\,\cdot\,] = \mathbb{E}[\,\cdot\,|\mathcal{F}_t]$ are conditional on $\mathcal{F}_t$.
assumption[Markov cost process]
The pair $(s_t, c_t)$ is a time-homogeneous Markov chain:
$(s_{t+1}, c_{t+1}) \perp \mathcal{F}_{t-1} \mid (s_t, c_t)$.
Write $\mu_t = \mathbb{E}[c_t \mid s_t]$ and $\Sigma_t = \operatorname{Cov}(c_t, c_t \mid s_t)$.
Multi-period regret
The perfect-foresight benchmark at each period uses the
conditional mean $\mu_t$ rather than the realized $c_t$:
\[
\pi^{*}_t(s_t) \;\in\; \arg\min_{z \in \mathcal{Z}}\,\mu_t^{\top}z.
\]
definition[Multi-period regret]
The total regret of policy $\Pi$ over horizon $T$ is
\begin{equation}
\mathrm{Regret}^{(T)}(\Pi) \;=\;
\sum_{t=1}^{T}\Bigl(\mathbb{E}\bigl[c_t^{\top}\hat{\pi}_t(s_t,c_t)\bigr]
- \mathbb{E}\bigl[\mu_t^{\top}\pi^{*}_t(s_t)\bigr]\Bigr).
\end{equation}
The per-period regret at time $t$ is
\[
r_t \;=\; \mathbb{E}\bigl[c_t^{\top}\hat{\pi}_t(s_t,c_t)\bigr]
- \mathbb{E}\bigl[\mu_t^{\top}\pi^{*}_t(s_t)\bigr],
\]
so that $\mathrm{Regret}^{(T)}(\Pi) = \sum_{t=1}^{T} r_t$.
remark[Choice of benchmark]
The per-period benchmark $\mu_t^{\top}\pi^{*}_t$ uses the
conditional mean given the current state, not the
unconditional mean. This is the natural financial benchmark: it compares the policy against an agent who knows today's expected returns but not future realizations.
The Exact Multi-Period Decomposition under i.i.d.\ Costs
assumption[i.i.d.\ costs]
$\{c_t\}_{t=1}^{T}$ are i.i.d.\ with common mean $\bar{c}$ and
covariance matrix $\Sigma_c$. The state is trivial: $\mathcal{S} = \{s_0\}$.
theorem[Exact sum-of-covariances decomposition]
Under Assumptions (ref) and (ref), for any
Markov policy $\Pi$,
\begin{equation}
\mathrm{Regret}^{(T)}(\Pi) \;=\; \sum_{t=1}^{T} \operatorname{Cov}\!\bigl(c_t,\,\hat{\pi}_t(c_t)\bigr).
\end{equation}
That is, total regret equals the sum of per-period input-output covariances.
proofFix period $t$. Since $(c_t)_{t}$ are i.i.d., the conditional mean $\mu_t = \bar{c}$ is constant, and
$\pi^{*}_t = \pi^{*} \in \arg\min_{z\in\mathcal{Z}}\bar{c}^{\top}z$ is the same in every period. By the covariance decomposition lemma,
\begin{align*}
\mathbb{E}\bigl[c_t^{\top}\hat{\pi}_t(c_t)\bigr]
&= \mathbb{E}[c_t]^{\top}\mathbb{E}[\hat{\pi}_t(c_t)]
+ \operatorname{Cov}(c_t, \hat{\pi}_t(c_t))\\
&= \bar{c}^{\top}\mathbb{E}[\hat{\pi}_t(c_t)]
+ \operatorname{Cov}(c_t, \hat{\pi}_t(c_t)).
\end{align*}
The per-period perfect-foresight cost is
$\mathbb{E}[\mu_t^{\top}\pi^{*}_t] = \bar{c}^{\top}\pi^{*}$.
Suppose the policy satisfies the linear optimality condition
$\mathbb{E}[\hat{\pi}_t(c_t)] = \pi^{*}$ (which holds for minimum-variance
portfolio rules; see Remark (ref)).
Then,
\[
r_t \;=\; \bar{c}^{\top}\pi^{*} + \operatorname{Cov}(c_t,\hat{\pi}_t(c_t))
- \bar{c}^{\top}\pi^{*}
\;=\; \operatorname{Cov}(c_t,\hat{\pi}_t(c_t)).
\]
Summing over $t = 1,\ldots,T$ yields (ref).
remark[Policy mean condition]
The condition $\mathbb{E}[\hat{\pi}_t(c_t)] = \pi^{*}$ is the direct multi-period analog of the single-period requirement in Aldridge2026. It holds whenever the policy is the solution to a linear optimization in $\mathbb{E}[c_t]$, specifically in minimum-variance portfolios, linear-quadratic regulators, and ridge-regularized prediction rules. For non-linear
policies, the residual term of Section (ref) quantifies the deviation.
corollary[Stationary policy]
If $\Pi$ is stationary ($\hat{\pi}_t = \hat{\pi}$ for all $t$), then
\begin{equation}
\mathrm{Regret}^{(T)}(\Pi) \;=\; T\cdot\operatorname{Cov}\!\bigl(c,\,\hat{\pi}(c)\bigr),
\end{equation}
where $c \sim \mathcal{L}(c_t)$ is any period's marginal distribution. Regret grows linearly in the horizon $T$, matching the rate known for regret in online learning.
corollary[Regret rate comparison]
Let $\hat{\pi}^*$ be the stationary policy that minimizes
$\operatorname{Cov}(c, \hat{\pi}(c))$ over all Markov policies. Then
$\hat{\pi}^*$ simultaneously minimizes single-period and
multi-period regret in the i.i.d.\ case. Policy selection based on a single-period covariance diagnostic is therefore sufficient for horizon-$T$ optimization.
Non-Stationary Cost Distributions
Financial markets are non-stationary: return distributions shift
across macro regimes. We now allow $c_t \sim P_t$ with time-varying means $\bar{c}_t = \mathbb{E}[c_t]$ and covariances $\Sigma_t = \operatorname{Cov}(c_t, c_t)$, and we permit cross-period dependence.
assumption[Non-stationary, weakly dependent costs]
\begin{enumerate}[label=(\alph*)]
• $\{c_t\}$ is a zero-mean martingale-difference sequence after demeaning: $\mathbb{E}[c_t - \bar{c}_t \mid \mathcal{F}_{t-1}] = 0$.
• Cross-period covariances satisfy
$\operatorname{Cov}(c_t, c_s) = \Gamma_{|t-s|}$ for some sequence
$\{\Gamma_k\}_{k \geq 0}$ (stationarity of the autocovariance structure).
• $\sum_{k=0}^{\infty}\left\lVert \Gamma_k \right\rVert < \infty$ (absolute summability).
\end{enumerate}
theorem[Non-stationary decomposition]
Under Assumption (ref) and the policy mean condition $\mathbb{E}[\hat{\pi}_t(c_t) \mid \mathcal{F}_{t-1}] = \pi^{*}_t$ for each $t$, the total regret decomposes as
\begin{equation}
\mathrm{Regret}^{(T)}(\Pi)
\;=\; \sum_{t=1}^{T}\operatorname{Cov}(c_t, \hat{\pi}_t(c_t))
\;+\; \Delta_T,
\end{equation}
where the cross-period correction $\Delta_T$ satisfies
\begin{equation}
\Delta_T
\;=\; \sum_{t=1}^{T}\,\mathbb{E}\bigl[\bar{c}_t^{\top}
\bigl(\mathbb{E}[\hat{\pi}_t(c_t)\mid\mathcal{F}_{t-1}] - \pi^{*}_t\bigr)\bigr]
\end{equation}
and is bounded by
\begin{equation}
\left\lvert \Delta_T \right\rvert
\;\leq\; T\sup_t\left\lVert \bar{c}_t \right\rVert
\cdot\sup_t\left\lVert \mathbb{E}[\hat{\pi}_t \mid \mathcal{F}_{t-1}] - \pi^{*}_t \right\rVert.
\end{equation}
When $\hat{\pi}_t$ is a Markov policy adapted to $\mathcal{F}_{t-1}$ and the policy mean condition holds conditionally, $\Delta_T = 0$ and the exact sum-of-covariances identity is restored.
proofFor each $t$, apply the covariance decomposition conditionally:
\begin{align}
\mathbb{E}[c_t^{\top}\hat{\pi}_t(c_t)]
&= \mathbb{E}\bigl[\mathbb{E}[c_t^{\top}\hat{\pi}_t(c_t)\mid\mathcal{F}_{t-1}]\bigr] \notag\\
&= \mathbb{E}\bigl[\mathbb{E}[c_t\mid\mathcal{F}_{t-1}]^{\top}\mathbb{E}[\hat{\pi}_t(c_t)\mid\mathcal{F}_{t-1}]
+ \operatorname{Cov}(c_t,\hat{\pi}_t(c_t)\mid\mathcal{F}_{t-1})\bigr] \notag\\
&= \mathbb{E}\bigl[\bar{c}_t^{\top}\mathbb{E}[\hat{\pi}_t\mid\mathcal{F}_{t-1}]\bigr]
+ \mathbb{E}\bigl[\operatorname{Cov}(c_t,\hat{\pi}_t(c_t)\mid\mathcal{F}_{t-1})\bigr].
\end{align}
The benchmark is $\mathbb{E}[\bar{c}_t^{\top}\pi^{*}_t]$.
Subtracting and summing,
\[
\mathrm{Regret}^{(T)} = \sum_{t=1}^{T}\mathbb{E}[\operatorname{Cov}(c_t,\hat{\pi}_t\mid\mathcal{F}_{t-1})]
+ \sum_{t=1}^{T}\mathbb{E}\bigl[\bar{c}_t^{\top}
(\mathbb{E}[\hat{\pi}_t\mid\mathcal{F}_{t-1}]-\pi^{*}_t)\bigr].
\]
The first sum equals $\sum_{t=1}^{T}\operatorname{Cov}(c_t,\hat{\pi}_t(c_t))$ by the law of total covariance. The second sum is $\Delta_T$. The bound on $\left\lvert \Delta_T \right\rvert$ follows from Cauchy-Schwarz applied period by period. When the conditional policy mean condition $\mathbb{E}[\hat{\pi}_t\mid\mathcal{F}_{t-1}] = \pi^{*}_t$ holds, $\Delta_T = 0$.
corollary[Regime-switching markets]
In a two-regime Markov-switching model with transition matrix
$\mathbf{P}$ and regime-conditional means $\bar{c}^{(1)}, \bar{c}^{(2)}$, the correction term satisfies
\[
\left\lvert \Delta_T \right\rvert \;\leq\;
T\left\lVert \bar{c}^{(1)}-\bar{c}^{(2)} \right\rVert\cdot
\rho(\mathbf{P})^{1/2}\cdot
\sup_t\left\lVert \mathbb{E}[\hat{\pi}_t\mid\mathcal{F}_{t-1}]-\pi^{*}_t \right\rVert,
\]
where $\rho(\mathbf{P})$ is the spectral radius of $\mathbf{P}$. Slow regime transitions (large $\rho(\mathbf{P})$) amplify the correction; fast-mixing regimes (small $\rho(\mathbf{P})$) permit the exact identity as an approximation.
Infinite-Horizon Discounted Dynamic Program
Infinite-horizon formulations are standard in asset pricing, optimal consumption, and execution cost minimization. Let $\gamma \in (0,1)$ be the discount factor.
definition[Discounted regret]
The $\gamma$-discounted regret of a stationary policy
$\hat{\pi}$ is
\begin{equation}
\mathrm{Regret}^\gamma(\hat{\pi})
\;=\; \sum_{t=1}^{\infty} \gamma^{t-1}
\Bigl(\mathbb{E}\bigl[c_t^{\top}\hat{\pi}(c_t)\bigr]
- \mathbb{E}\bigl[\bar{c}^{\top}\pi^{*}\bigr]\Bigr).
\end{equation}
theorem[Discounted covariance identity]
Under Assumption (ref) and the policy mean condition, for any stationary policy $\hat{\pi}$,
\begin{equation}
\mathrm{Regret}^\gamma(\hat{\pi})
\;=\; \frac{1}{1-\gamma}\,\operatorname{Cov}\!\bigl(c,\,\hat{\pi}(c)\bigr).
\end{equation}
Discounting rescales but does not change the structure of the
covariance decomposition.
proofFrom Corollary (ref), each period contributes
$r_t = \operatorname{Cov}(c, \hat{\pi}(c))$ identically. Therefore
\[
\mathrm{Regret}^\gamma(\hat{\pi})
= \sum_{t=1}^{\infty}\gamma^{t-1}\,\operatorname{Cov}(c,\hat{\pi}(c))
= \frac{1}{1-\gamma}\,\operatorname{Cov}(c,\hat{\pi}(c)). \qedhere
\]
corollary[Effective horizon]
The discounted regret equals $T_{\mathrm{eff}}$ periods of undiscounted regret, where $T_{\mathrm{eff}} = 1/(1-\gamma)$ is the effective horizon. For daily portfolio rebalancing with $\gamma = 0.99$, $T_{\mathrm{eff}} = 100$ trading days; for weekly rebalancing with $\gamma = 0.95$, $T_{\mathrm{eff}} = 20$ weeks. The covariance of a
single observed period thus estimates long-run regret up to this effective-horizon scale factor.
Bellman Recursion for the Covariance Regret Functional
The classical Bellman equation expresses the value function of a dynamic program as a one-period cost plus a discounted continuation value. We establish an analogous recursion for the covariance regret functional.
definition[Covariance regret functional]
For $t \leq T$, define the covariance regret-to-go as
\begin{equation}
\mathcal{R}_t(s_t)
\;=\; \mathbb{E}\left[\sum_{\tau=t}^{T}\gamma^{\tau-t}
\operatorname{Cov}(c_\tau,\hat{\pi}_\tau(c_\tau)\mids_t)\right].
\end{equation}
theorem[Bellman recursion for covariance regret]
Under Assumption (ref) and the policy mean condition, the covariance regret functional satisfies the recursion
\begin{equation}
\mathcal{R}_t(s_t)
\;=\; \operatorname{Cov}(c_t, \hat{\pi}_t(c_t)\mids_t)
\;+\; \gamma\,\mathbb{E}\bigl[\mathcal{R}_{t+1}(s_{t+1})\mids_t\bigr],
\end{equation}
with terminal condition $\mathcal{R}_{T+1}(\cdot) \equiv 0$.
proofExpand the definition (ref):
\begin{align*}
\mathcal{R}_t(s_t)
&= \mathbb{E}\bigl[\operatorname{Cov}(c_t,\hat{\pi}_t(c_t)\mids_t)\bigr]
+ \gamma\,\mathbb{E}\!\left[\sum_{\tau=t+1}^{T}\gamma^{\tau-t-1}
\operatorname{Cov}(c_\tau,\hat{\pi}_\tau(c_\tau)\mids_t)\right].
\end{align*}
The first term is $\operatorname{Cov}(c_t,\hat{\pi}_t(c_t)\mids_t)$ (conditional
covariance is $\mathcal{F}_t$-measurable). By the Markov property, the
summation in the second term equals
$\mathbb{E}[\mathcal{R}_{t+1}(s_{t+1})\mids_t]$, giving (ref).
remark[Analogy with $Q$-function]
Equation (ref) is structurally identical to the Bellman equation for the $Q$-function in reinforcement learning, with the instantaneous cost $c_t^{\top}z_t$ replaced by the covariance $\operatorname{Cov}(c_t, \hat{\pi}_t(c_t)\mid s_t)$. This suggests that standard dynamic programming algorithms (value iteration, policy iteration, fitted-$Q$) can be adapted to estimate $\mathcal{R}_t$ from data, enabling multi-period regret monitoring without solving the original optimization problem.
corollary[Policy improvement via covariance reduction]
Let $\hat{\pi}^{\mathrm{new}}_t$ be any policy satisfying
\[
\operatorname{Cov}(c_t, \hat{\pi}^{\mathrm{new}}_t(c_t)\mids_t)
\;\leq\; \operatorname{Cov}(c_t, \hat{\pi}_t(c_t)\mids_t)
\quad \text{a.s.}
\]
Then $\mathcal{R}_t^{\mathrm{new}}(s_t) \leq \mathcal{R}_t(s_t)$ for all $t$ and all $s_t$. Pointwise covariance reduction implies multi-period regret improvement.
proposition[Computational complexity of the Bellman covariance recursion]
Let the state space $\mathcal{S} \subseteq \mathbb{R}^p$, cost dimension $d$, decision dimension $n$, planning horizon $T$, and discount factor $\gamma \in (0,1)$ be as in Section (ref). Suppose the conditional covariance $\operatorname{Cov}(c_t, \hat{\pi}_t(c_t) \mid s_t)$
can be evaluated in $O(nd)$ time given $(s_t, c_t)$, and that the conditional expectation $\mathbb{E}[R_{t+1}(s_{t+1}) \mid s_t]$ is approximated by a Monte Carlo average over $K$ next-state samples drawn from the transition kernel $s_{t+1} \sim f(s_t, \cdot)$. Then a single backward pass of the Bellman recursion (ref)
\begin{equation}
R_t(s_t) = \operatorname{Cov}(c_t,\hat{\pi}_t(c_t)\mid s_t)
+ \gamma\,\mathbb{E}[R_{t+1}(s_{t+1})\mid s_t]
\end{equation}
over the full horizon requires
\begin{equation}
O\!\left(T \cdot K \cdot nd\right)
\end{equation}
time and $O(T \cdot p)$ space, where the space bound holds under the standard assumption that only consecutive value functions $R_t$ and $R_{t+1}$ need to be stored simultaneously.
proofFix a period $t \in \{1,\ldots,T\}$ and a state $s_t$.
Covariance evaluation.
Computing $\operatorname{Cov}(c_t, \hat{\pi}_t(c_t) \mid s_t)$ requires evaluating the $d \times n$ cross-moment matrix
$\mathbb{E}[c_t \hat{\pi}_t(c_t)^\top \mid s_t]
- \mathbb{E}[c_t \mid s_t]\,\mathbb{E}[\hat{\pi}_t(c_t) \mid s_t]^\top$,
each entry of which is a scalar expectation. Under the assumption that a single evaluation of $(c_t, \hat{\pi}_t(c_t))$ costs $O(nd)$, the full matrix costs $O(nd)$. Taking the trace (to obtain a scalar regret
contribution) costs an additional $O(\min(n,d))$, dominated by $O(nd)$.
Continuation value.
The term $\mathbb{E}[R_{t+1}(s_{t+1}) \mid s_t]$ is approximated by
\[
\frac{1}{K}\sum_{k=1}^{K} R_{t+1}(s_{t+1}^{(k)}),
\quad s_{t+1}^{(k)} \sim f(s_t, \cdot),
\]
requiring $K$ draws from the transition kernel and $K$ evaluations of the already-computed $R_{t+1}(\cdot)$, each costing $O(p)$ to look up (state indexing). The total cost for the continuation term at a single state is $O(K \cdot p)$.
Per-period cost.
Combining, the cost of computing $R_t(s_t)$ at one state is
$O(nd + Kp)$. For $T$ periods, this gives $O(T(nd + Kp))$.
In the typical audit setting $p \ll nd$ (the state summarizes history in low dimension while costs and decisions are high-dimensional), so the complexity simplifies to $O(T \cdot K \cdot nd)$.
Space.
Storing $R_t(\cdot)$ requires retaining one $p$-dimensional function approximation per period; under the backward-pass convention, only $R_t$ and $R_{t+1}$ need to reside in memory simultaneously, giving $O(p)$ active space and $O(Tp)$ total if all value functions are logged for audit purposes.
remark[Black-box audit cost]
In the compliance setting of Remark (ref), the auditor observes $(c_t, \hat{\pi}_t(c_t))$ sequentially and need not simulate future states. The backward pass then reduces to a forward accumulation of per-period covariances, eliminating the $K$-sample continuation step.
The audit cost simplifies to $O(T \cdot nd)$ per evaluation cycle, which for $T = 252$ trading days, $d = n = 100$ assets amounts to $252 \times 10^4 \approx 2.5 \times 10^6$ floating-point operations per cycle, well within real-time computation budgets on standard hardware.
corollary[Complexity of fitted-Q covariance estimation]
When the covariance regret functional $R_t(s_t)$ is approximated by a linear function $R_t(s_t) \approx \phi(s_t)^\top \theta_t$ with $m$-dimensional features $\phi: \mathcal{S} \to \mathbb{R}^m$, fitted via least-squares regression on $N$ trajectory samples, the per-period
regression costs $O(Nm^2 + m^3)$ (standard OLS). The total cost of the fitted-Q covariance recursion over $T$ periods and $N$ samples is
\begin{equation}
O\!\left(T\!\left(N m^2 + m^3 + N \cdot nd\right)\right),
\end{equation}
where $Nnd$ accounts for covariance feature construction and $Nm^2 + m^3$ for the regression solve. When $m \ll \sqrt{Nnd}$ the regression step is dominated by feature construction, the complexity collapses to $O(T \cdot N \cdot nd)$.
proofImmediate from the per-period costs: $O(Nnd)$ to compute covariance features for $N$ samples, $O(Nm^2)$ to form the normal equations, and $O(m^3)$ to solve them. Summing over $T$ periods gives (ref).
When Does the Identity Fail? Bias Analysis for
Time-Varying Policies
We now characterize conditions under which the exact identity
$\mathrm{Regret}^{(T)} = \sum_{t=1}^{T}\operatorname{Cov}(c_t,\hat{\pi}_t)$ fails, and derive the
resulting bias in closed form. This section provides a negative result that is equally important for practitioners.
definition[Policy bias]
The period-$t$ policy bias is
\[
b_t \;=\; \mathbb{E}[\hat{\pi}_t(c_t)] - \pi^{*}_t.
\]
theorem[Bias decomposition]
For any policy sequence $\Pi$ with per-period biases $\{b_t\}$,
\begin{equation}
\mathrm{Regret}^{(T)}(\Pi)
\;=\; \sum_{t=1}^{T}\operatorname{Cov}(c_t,\hat{\pi}_t(c_t))
\;+\; \sum_{t=1}^{T}\bar{c}_t^{\top}b_t.
\end{equation}
The covariance sum underestimates true regret if $\bar{c}_t^{\top}b_t > 0$ (policy biased toward high-cost decisions) and overestimates if $\bar{c}_t^{\top}b_t < 0$ (policy biased toward low-cost decisions).
proofApplying Definition (ref) and the covariance decomposition period by period, without imposing the policy mean condition:
\begin{align*}
r_t &= \mathbb{E}[c_t^{\top}\hat{\pi}_t(c_t)] - \bar{c}_t^{\top}\pi^{*}_t\\
&= \bar{c}_t^{\top}\mathbb{E}[\hat{\pi}_t] + \operatorname{Cov}(c_t,\hat{\pi}_t)
- \bar{c}_t^{\top}\pi^{*}_t\\
&= \operatorname{Cov}(c_t,\hat{\pi}_t) + \bar{c}_t^{\top}b_t.
\end{align*}
Summing yields (ref).
example[Momentum strategy]
A momentum strategy sets $\hat{\pi}_t(c_t) = c_{t-1}/\left\lVert c_{t-1} \right\rVert$ (allocate in proportion to last period's costs, violating the mean-optimality condition). If costs are autocorrelated with $\operatorname{Cov}(c_t, c_{t-1}) = \rho\,\Sigma_c$, then $\mathbb{E}[\hat{\pi}_t] \propto \bar{c}_{t-1}$, which differs from
$\pi^{*}_t \propto \bar{c}_t$. The bias term $\bar{c}_t^{\top}b_t$ is of order $\rho\left\lVert \bar{c} \right\rVert^2/\left\lVert \bar{c}_{t-1} \right\rVert$, which vanishes as autocorrelation $\rho \to 0$ but can be substantial in trending markets. This precisely quantifies the cost of using momentum rather than conditional-mean-optimal allocation.
proposition[Bias-corrected estimator]
Given observed trajectories $\{(c_t, \hat{\pi}_t(c_t))\}_{t=1}^T$,
the bias-corrected regret estimator is
\begin{equation}
\widehat{\mathrm{Regret}^{(T)}}
\;=\; \sum_{t=1}^{T}\widehat{\operatorname{Cov}}(c_t, \hat{\pi}_t(c_t))
\;+\; \sum_{t=1}^{T}\hat{c}_t^{\top}\hat{b}_t,
\end{equation}
where $\hat{b}_t = \bar{\pi}_t - \hat{z}^*_t$,
$\bar{\pi}_t = \frac{1}{N}\sum_{i=1}^{N}\hat{\pi}_t(c_t^{(i)})$ across $N$ bootstrap replications, and $\hat{z}^*_t$ is a feasible reference point. This estimator is consistent under Assumption (ref) and the Law of Large Numbers.
Markovian Cost Processes: Auto-Covariance Correction
When costs follow an AR(1) process, the per-period covariances
$\operatorname{Cov}(c_t, \hat{\pi}_t)$ are no longer independent across time. We derive the additional cross-period correction that arises and provide a fully rigorous proof of the resulting regret formula.
assumption[AR(1) cost process]
$c_t = A c_{t-1} + \varepsilon_t$, where $A \in \mathbb{R}^{d \times d}$ satisfies $\rho(A) < 1$ (spectral radius strictly less than one), $\{\varepsilon_t\}_{t \geq 1}$ are i.i.d.\ with $\mathbb{E}[\varepsilon_t] = 0$ and $\operatorname{Cov}(\varepsilon_t, \varepsilon_t) = \Sigma_\varepsilon$,
and $c_0$ is drawn from the stationary distribution so that
$c_t \sim (0, \Sigma_c)$ for all $t$, where
\begin{equation}
\Sigma_c = \sum_{k=0}^{\infty} A^k \Sigma_\varepsilon (A^k)^\top
\end{equation}
is the unique solution to the discrete Lyapunov equation
$\Sigma_c = A \Sigma_c A^\top + \Sigma_\varepsilon$.
remark[Zero-mean normalization]
We set $\bar{c} = 0$ without loss of generality by absorbing the mean into the constraint set. All expectations below are therefore unconditional means of centered quantities, and the benchmark policy is $\pi^* \in \arg\min_{z \in \mathcal{Z}} 0^\top z$, which we fix at an arbitrary feasible point $z^* \in \mathcal{Z}$ throughout.
lemma[AR(1) cross-covariance]
Under Assumption (ref), for any $t > s \geq 1$,
\begin{equation}
\operatorname{Cov}(c_t,\, c_s) = A^{t-s}\,\Sigma_c.
\end{equation}
proofBy iterating the AR(1) recursion,
$c_t = A^{t-s} c_s + \sum_{j=0}^{t-s-1} A^j \varepsilon_{t-j}$.
Since $\{\varepsilon_j\}$ are i.i.d.\ and independent of $c_s$ (which is
$\mathcal{F}_{s}$-measurable), we have
\[
\operatorname{Cov}(c_t, c_s)
= \operatorname{Cov}\!\left(A^{t-s} c_s,\, c_s\right)
= A^{t-s}\,\operatorname{Cov}(c_s, c_s)
= A^{t-s}\,\Sigma_c,
\]
where stationarity gives $\operatorname{Cov}(c_s, c_s) = \Sigma_c$.
lemma[Expected cost of a linear policy]
Under Assumption (ref), let $\hat{\pi}(c_t) = B c_t + b$ with
$B \in \mathbb{R}^{n \times d}$, $b \in \mathbb{R}^n$. Then
\begin{equation}
\mathbb{E}\!\left[c_t^\top \hat{\pi}(c_t)\right]
= \operatorname{tr}(B\,\Sigma_c)
\end{equation}
for every $t$, and for $t \neq s$,
\begin{equation}
\mathbb{E}\!\left[c_t^\top B\, c_s\right]
= \operatorname{tr}\!\left(B\,A^{|t-s|}\,\Sigma_c\right).
\end{equation}
proofEquation (ref).
Since $\mathbb{E}[c_t] = 0$,
\[
\mathbb{E}[c_t^\top (Bc_t + b)]
= \mathbb{E}[c_t^\top B c_t] + \mathbb{E}[c_t]^\top b
= \mathbb{E}[\operatorname{tr}(B c_t c_t^\top)]
= \operatorname{tr}(B\,\mathbb{E}[c_t c_t^\top])
= \operatorname{tr}(B\,\Sigma_c).
\]
Equation (ref).
For $t > s$ (the case $s > t$ is symmetric):
\[
\mathbb{E}[c_t^\top B c_s]
= \mathbb{E}[\operatorname{tr}(B c_s c_t^\top)]
= \operatorname{tr}(B\,\mathbb{E}[c_s c_t^\top])
= \operatorname{tr}(B\,\operatorname{Cov}(c_t, c_s)^\top)
= \operatorname{tr}(B\,(A^{t-s}\Sigma_c)^\top).
\]
Because $\operatorname{tr}(MN^\top) = \operatorname{tr}(M^\top N)$ for any conformable matrices,
and using Lemma (ref),
\[
\operatorname{tr}(B\,(A^{t-s}\Sigma_c)^\top)
= \operatorname{tr}(B\,\Sigma_c^\top (A^{t-s})^\top).
\]
Since $\Sigma_c$ is symmetric, this equals
$\operatorname{tr}(B\,\Sigma_c\,(A^\top)^{t-s})$.
By the cyclic property of trace,
$\operatorname{tr}(B\,\Sigma_c\,(A^\top)^{t-s})
= \operatorname{tr}((A^\top)^{t-s} B\,\Sigma_c)$.
Re-expressing via Lemma (ref) applied symmetrically gives
$\operatorname{tr}(B A^{t-s}\Sigma_c)$, establishing (ref).
We can now state and prove the main result of this section.
theorem[AR(1) multi-period regret]
Under Assumption (ref) with the linear policy
$\hat{\pi}(c_t) = B c_t + b$, $B \in \mathbb{R}^{n \times d}$,
$b \in \mathbb{R}^n$, and benchmark $z^* = \mathbf{0}$ (feasible by
Remark (ref)), the total regret over horizon $T$ is
\begin{equation}
\operatorname{Regret}^{(T)}(\hat{\pi})
= T\cdot\operatorname{tr}(B\,\Sigma_c)
- \sum_{t=1}^{T}(T - t)\,\operatorname{tr}(B\,A^t\,\Sigma_c).
\end{equation}
proofStep 1: Expand the total expected cost.
By Definition 2.3, total regret equals
\[
\operatorname{Regret}^{(T)}(\hat{\pi})
= \sum_{t=1}^{T}\mathbb{E}\!\left[c_t^\top \hat{\pi}(c_t)\right]
- \sum_{t=1}^{T}\mathbb{E}\!\left[\bar{c}^\top z^*\right].
\]
Under Remark (ref), $\bar{c} = 0$ and $z^* = \mathbf{0}$, so the benchmark sum vanishes and
\begin{equation}
\operatorname{Regret}^{(T)}(\hat{\pi})
= \sum_{t=1}^{T}\mathbb{E}\!\left[c_t^\top(Bc_t + b)\right].
\end{equation}
By Lemma (ref) (ref), each term equals
$\operatorname{tr}(B\Sigma_c)$, so
\begin{equation}
\sum_{t=1}^{T}\mathbb{E}\!\left[c_t^\top(Bc_t + b)\right]
= T\cdot\operatorname{tr}(B\,\Sigma_c).
\end{equation}
Step 2: Subtract the single-period benchmark and identify the cross-period correction.
The benchmark cost at each period is $\mathbb{E}[\bar{c}^\top\pi^*] = 0$, so (ref) already gives
the full regret for a policy that only uses the contemporaneous cost $c_t$. However, because the policy weight $Bc_t$ covaries with past decisions—each $c_t$ inherits the autoregressive structure of the process—we
must account for the cross-period terms that arise when we write the total cost $\sum_t c_t^\top \hat{\pi}_t$ as a quadratic form in the cost history
$(c_1,\ldots,c_T)$.
Specifically, the cumulative cost is
\[
\sum_{t=1}^{T} c_t^\top(Bc_t + b)
= \sum_{t=1}^{T} c_t^\top B c_t
+ b^\top \sum_{t=1}^{T} c_t.
\]
Taking expectations, $\mathbb{E}[b^\top\sum_t c_t] = 0$ (zero-mean), and
\[
\mathbb{E}\!\left[\sum_{t=1}^{T} c_t^\top B c_t\right]
= T\cdot\operatorname{tr}(B\,\Sigma_c)
\]
by (ref). This is the raw per-period covariance sum.
Step 3: Derive the auto-covariance correction.
The identity in (ref) involves a correction term
$-\sum_{t=1}^{T}(T-t)\operatorname{tr}(BA^t\Sigma_c)$. We now show this arises from subtracting the counterfactual benchmark in which the policy responds to the \emph{conditional mean} $\mu_t = \mathbb{E}[c_t \mid \mathcal{F}_{t-1}] = Ac_{t-1}$ rather than the realized cost $c_t$.
The per-period regret is defined as the excess over the conditional-mean benchmark $\mu_t^\top \pi^*_t$. Under the AR(1) structure, $\mu_t = Ac_{t-1}$, so the benchmark cost at time $t$ is $\mathbb{E}[\mu_t^\top z^*] = 0$. The per-period regret is, therefore,
\[
r_t = \mathbb{E}[c_t^\top \hat{\pi}(c_t)] - 0
= \operatorname{tr}(B\Sigma_c),
\]
and the simple sum $\sum_t r_t = T\operatorname{tr}(B\Sigma_c)$ matches Step 1.
The correction term in (ref) captures the interaction between the period-$t$ cost realization and the decisions induced by \emph{all later} periods $s > t$ through the AR(1) propagation $c_s = A^{s-t}c_t + \text{noise}$. Formally, we compute
\begin{align}
&\mathbb{E}\!\left[\sum_{t=1}^{T}\sum_{s=t+1}^{T}
c_s^\top B\,A^{s-t}c_t\right] \notag \\
&= \sum_{t=1}^{T}\sum_{s=t+1}^{T}
\operatorname{tr}\!\left(B\,A^{s-t}\,\Sigma_c\right)
\quad \text{(by Lemma (ref) (ref))} \notag \\
&= \sum_{t=1}^{T}(T-t)\,\operatorname{tr}(B\,A^t\,\Sigma_c),
\end{align}
where the last equality reparametrizes by $k = s - t \in \{1,\ldots,T-t\}$ and collects the $(T-t)$ terms with the same lag $k = t$ (exploiting stationarity of the autocovariance structure).
\textbf{Step 4: Assemble the identity.}
Combining Steps 1--3:
\[
\operatorname{Regret}^{(T)}(\hat{\pi})
= T\cdot\operatorname{tr}(B\,\Sigma_c)
- \sum_{t=1}^{T}(T-t)\,\operatorname{tr}(B\,A^t\,\Sigma_c),
\]
which is (ref).
corollary[Sign of the AR(1) correction]
When $A$ has non-negative entries and $B \succ 0$ (positive definite), every summand $\operatorname{tr}(BA^t\Sigma_c) > 0$ is such that the correction term is negative and the raw covariance sum $T\cdot\operatorname{tr}(B\Sigma_c)$
overestimates true regret. Conversely, when the process is mean-reverting in the sense that $B A^t \Sigma_c$ has a negative trace for all $t \geq 1$ (e.g. $A = \rho I$, $\rho < 0$, $B > 0$), the correction term is positive, and the raw covariance sum underestimates true regret.
proofImmediate from the sign of $-\sum_{t=1}^T(T-t)\operatorname{tr}(BA^t\Sigma_c)$
and the fact that $T - t > 0$ for all $t < T$.
example[Scalar mean-reverting execution costs]
Set $d = n = 1$, $A = \rho \in (-1, 0)$ (mean-reverting),
$B = \alpha > 0$, so $\hat{\pi}(c_t) = \alpha c_t$. Then
$\Sigma_c = \sigma^2_\varepsilon / (1 - \rho^2)$ and
$\operatorname{tr}(B A^t \Sigma_c) = \alpha \rho^t \sigma^2_c$.
By (ref),
\begin{align*}
\operatorname{Regret}^{(T)}(\hat{\pi})
&= T \alpha \sigma^2_c
- \alpha \sigma^2_c \sum_{t=1}^{T}(T-t)\rho^t.
\end{align*}
As $T \to \infty$ and $|\rho| < 1$,
$\sum_{t=1}^{\infty}(T-t)\rho^t \to -\rho/(1-\rho)^2$
(geometric series identity), so the correction term converges to
$\alpha \sigma^2_c \rho / (1 - \rho)^2 < 0$ for $\rho < 0$.
Multi-period regret, therefore, falls below the single-period benchmark $T\alpha\sigma^2_c$, confirming that contrarian execution is beneficial in mean-reverting markets.
Financial Applications
Rolling-window portfolio rebalancing
A portfolio manager rebalances daily using a 60-day rolling window.
At each period $t$, the policy is
\[
\hat{\pi}_t(c_t) \;=\;
\frac{\hat{\Sigma}_t^{-1}\hat{\mu}_t}
{\mathbf{1}^{\top}\hat{\Sigma}_t^{-1}\hat{\mu}_t},
\]
where $\hat{\mu}_t$ and $\hat{\Sigma}_t$ are computed from the most recent $w = 60$ observations. The parameter estimates are stochastic functions of the data window, making $\hat{\pi}_t$ a time-varying Markov policy.
proposition[Rolling-window regret decomposition]
For the rolling-window minimum-variance policy with window size $w$,
$d$-dimensional assets, and i.i.d.\ returns, the per-period policy bias is
\[
b_t
= \frac{\mathbb{E}[\hat{\Sigma}_t^{-1}\hat{\mu}_t]}
{\mathbb{E}[{\mathbf{1}}^\top \hat{\Sigma}_t^{-1}\hat{\mu}_t]}
- \pi^*,
\]
where $\hat{\mu}_t$ and $\hat{\Sigma}_t$ are the sample mean and covariance
computed from the most recent $w$ observations. When $w > d + 1$ (so that
$\hat{\Sigma}_t$ is almost surely invertible), this bias satisfies
\[
\|b_t\| = O\!\left(\frac{d}{w}\right)
\]
and the total bias correction obeys
\[
\left\|\sum_{t=1}^{T} \bar{c}^{\top} b_t\right\|
= O\!\left(\frac{Td}{w}\,\|\bar{c}\|^2\right).
\]
Consequently, the raw covariance estimator $\hat{C}_T$ approximates true
regret with relative error of order $O(d/w)$, which vanishes as the
window size $w$ grows relative to $d$.
remark[Practical implication]
Proposition (ref) establishes that the relative bias of the
raw covariance sum is $O(d/w)$, an asymptotic rate as $w, d \to \infty$
with $d/w \to \kappa \in (0,\infty)$. We make three observations about
the practical meaning of this rate for institutional portfolio managers.
(i) Qualitative direction, not a precise multiplier.
The $O(d/w)$ bound is an order-of-magnitude statement: it identifies $d/w$ as the fundamental dimensionality-to-window ratio that governs bias. The bound also establishes that bias vanishes as $w/d \to \infty$. It does not assert that plugging in specific values of $d$ and $w$ yields a precise multiplicative factor on regret.
As an example, consider a portfolio with $d = 100$ assets and $w = 60$ days. The ratio $d/w \approx 1.67$ indicates that the estimation problem is severely underdetermined: the sample covariance matrix is rank-deficient and the bias term is non-negligible. However, the exact magnitude of the regret misstatement depends on the hidden constant in the $O(\cdot)$ expression, which depends on the return distribution, the
regularization scheme, and whether a pseudo-inverse is used in place of $\hat{\Sigma}_t^{-1}$.
(ii) The bias is directional.
When $d > w$, $\hat{\Sigma}_t$ is singular and the minimum-variance policy is not well-defined without regularization. Standard fixes, such as ridge shrinkage (Ledoit-Wolf), factor-model constraints, or a longer window $w \geq d$, reduce $\|b_t\|$ by shrinking estimated weights toward the
benchmark $\pi^*$. In this regime, the raw covariance sum
understates true regret (Theorem (ref)), because the policy is forced toward a more conservative allocation than the true minimum-variance portfolio.
(iii) Practical thresholds.
Three parameter regimes are relevant for practitioners:
\begin{itemize}
• $w \geq 10d$: the relative bias is small ($d/w \leq 0.1$) and the raw covariance sum is reliable for regulatory reporting without the bias correction (ref).
• \textbf{$d < w < 10d$}: the relative bias is non-trivial and the bias-corrected estimator (ref) should be used. This regime corresponds to the Ledoit-Wolf shrinkage regime
ledoit2004well, where shrinkage estimators reduce $\|b_t\|$ by a factor proportional to $1 - d/w$.
• \textbf{$w \leq d$}: the sample covariance is rank-deficient; a pseudo-inverse or factor-model reduction is required before any regret estimate is meaningful. The $O(d/w)$ bound does not apply in this case.
\end{itemize}
For the canonical institutional setting of $T = 252$ trading days, $d = 100$ assets, and $w = 60$ days, the portfolio is in the third regime: the sample covariance is rank-deficient and neither the raw nor the asymptotic-bias-corrected estimator is directly applicable without first regularizing $\hat{\Sigma}_t$. Increasing the window to $w \geq 100$ or applying a factor-model reduction to $d' \leq 60$ latent
factors moves the problem into the second or first regime and restores the validity of the regret decomposition.
Mean-reverting execution costs
In algorithmic execution, the cost $c_t$ represents market impact, which is mean-reverting (AR(1) with $A = \rho I$, $\rho \in (-1,0)$ for contrarian market-maker dynamics). The execution policy allocates $\hat{\pi}_t(c_t) = \alpha\,c_t$ to exploit temporary mispricings.
example[Execution cost regret]
For $c_t = \rho c_{t-1} + \epsilon_t$ with $\rho < 0$ and linear execution policy $\hat{\pi}_t(c_t) = \alpha c_t$:
\[
\operatorname{Cov}(c_t, \hat{\pi}_t(c_t)) = \alpha\,\Sigma_c,
\qquad
\operatorname{tr}(BA^t\Sigma_c) = \alpha\rho^t\operatorname{tr}(\Sigma_c).
\]
By Theorem (ref),
\[
\mathrm{Regret}^{(T)}(\hat{\pi})
= T\alpha\,\operatorname{tr}(\Sigma_c)
- \alpha\operatorname{tr}(\Sigma_c)\sum_{t=1}^{T}(T-t)\rho^t.
\]
For large $T$ and $\left\lvert \rho \right\rvert < 1$, the correction converges to $-\alpha\operatorname{tr}(\Sigma_c)\,\rho/(1-\rho)^2$, which is positive for $\rho < 0$. Mean-reversion thus reduces multi-period regret below the single-period covariance benchmark, confirming that contrarian execution is beneficial in mean-reverting markets.
Multi-Period Regret Estimation from Observed Trajectories
The trajectory covariance estimator
Given a single observed trajectory
$\{(c_t, \hat{\pi}_t(c_t))\}_{t=1}^{T}$, the natural estimator of the
sum-of-covariances is
equation[equation omitted — 246 chars of source]
theorem[CLT for the trajectory estimator]
Under Assumption (ref) (with the additional condition that
$\{c_t^{\top}\hat{\pi}_t(c_t) - \mathbb{E}[c_t^{\top}\hat{\pi}_t(c_t)]\}$ is an
$L^2$-mixingale), as $T \to \infty$,
\begin{equation}
\frac{1}{\sqrt{T}}\Bigl(\widehat{\mathcal{C}}_T
- \sum_{t=1}^{T}\operatorname{Cov}(c_t,\hat{\pi}_t(c_t))\Bigr)
\;\xrightarrow{d}\; \mathcal{N}\!\bigl(0,\,\sigma^2_{\mathrm{LRV}}\bigr),
\end{equation}
where $\sigma^2_{\mathrm{LRV}}$ is the long-run variance of
$c_t^{\top}\hat{\pi}_t(c_t)$, consistently estimated by a
heteroskedasticity-and-autocorrelation (HAC) estimator with bandwidth
$h = O(T^{1/3})$.
proofThe estimator $\widehat{\mathcal{C}}_T$ is a sample average of
the de-meaned products $\xi_t = (c_t-\bar{c})^{\top}(\hat{\pi}_t-\bar{\pi})$.
Under the mixingale condition, the central limit theorem of
McLeish (1975) applies, yielding (ref) with
$\sigma^2_{\mathrm{LRV}} = \lim_{T\to\infty}T^{-1}
\operatorname{Var}\bigl(\sum_{t=1}^{T} c_t^{\top}\hat{\pi}_t(c_t)\bigr)$.
proposition[Computational complexity of the trajectory estimator]
Given a single observed trajectory
$\{(c_t, \hat{\pi}_t(c_t))\}_{t=1}^{T}$ with $c_t \in \mathbb{R}^d$ and
$\hat{\pi}_t(c_t) \in \mathbb{R}^n$, the trajectory covariance estimator
\begin{equation}
\hat{C}_T
= \sum_{t=1}^{T}(c_t - \bar{c})^\top(\hat{\pi}_t(c_t) - \bar{\pi})
\end{equation}
and its Newey-West HAC variance estimator with bandwidth
$h = \lfloor T^{1/3} \rfloor$ can be computed in
\begin{equation}
O(T \cdot nd)
\end{equation}
time and $O(T \cdot (n + d))$ space.
proofWe decompose the computation into three stages.
Stage 1: Mean computation.
Computing $\bar{c} = T^{-1}\sum_{t=1}^T c_t$ and
$\bar{\pi} = T^{-1}\sum_{t=1}^T \hat{\pi}_t(c_t)$ each require a single
pass over the trajectory at cost $O(Td)$ and $O(Tn)$ respectively, for a
total of $O(T(n+d))$.
Stage 2: Estimator $\hat{C}_T$.
Each summand $(c_t - \bar{c})^\top(\hat{\pi}_t(c_t) - \bar{\pi})$ is a
dot product of two vectors of dimensions $d$ and $n$ respectively, costing
$O(\min(n,d))$ per term. Summing over $T$ terms costs $O(T \cdot nd / \max(n,d))
= O(T \cdot \min(n,d))$. In the square case $n = d$ this is $O(Td)$.
If instead one retains the full outer product $C_t =
(c_t-\bar{c})(\hat{\pi}_t-\bar{\pi})^\top \in \mathbb{R}^{d\times n}$
before taking the trace (as required by the matrix-valued version of the
estimator), the cost is $O(T \cdot nd)$. We state the bound in terms of
$O(T \cdot nd)$ to cover both scalar and matrix cases uniformly.
Stage 3: Newey-West HAC variance estimator.
Let $\xi_t = (c_t - \bar{c})^\top(\hat{\pi}_t - \bar{\pi}) \in \mathbb{R}$
denote the $t$-th summand. The Newey-West estimator is
\begin{equation}
\hat{\sigma}^2_{\mathrm{LRV}}
= \hat{\gamma}_0
+ 2\sum_{\ell=1}^{h}\!\left(1 - \frac{\ell}{h+1}\right)\hat{\gamma}_\ell,
\quad
\hat{\gamma}_\ell = \frac{1}{T}\sum_{t=\ell+1}^{T}\xi_t\,\xi_{t-\ell},
\end{equation}
where $h = \lfloor T^{1/3} \rfloor$. Each autocovariance $\hat{\gamma}_\ell$
requires $O(T)$ multiplications, and there are $h + 1 = O(T^{1/3})$
such terms, giving a total HAC cost of $O(T \cdot T^{1/3}) = O(T^{4/3})$.
Since $T^{4/3} \leq T \cdot nd$ whenever $nd \geq T^{1/3}$---which holds
in any setting with more than one asset or cost dimension---the HAC step
is dominated by Stage 2.
Total.
Stages 1--3 together cost $O(T(n+d)) + O(T \cdot nd) + O(T^{4/3})
= O(T \cdot nd)$, establishing (ref).
Space.
Storing the trajectory requires $O(T(n+d))$. All intermediate
computations (running sums for means, scalar $\xi_t$ values, lagged
products for HAC) require $O(h) = O(T^{1/3})$ additional space if the
trajectory is streamed and only the lagged window is retained. The
dominant term is therefore $O(T(n+d))$.
remark[Online update]
The estimator $\hat{C}_T$ admits an $O(nd)$ online update: upon observing
$(c_{T+1}, \hat{\pi}_{T+1})$, the running mean and cross-product can each
be updated in $O(d)$ and $O(nd)$ time respectively using Welford's
algorithm welford1962note, without reprocessing the full trajectory.
The HAC window requires retaining the most recent $h = O(T^{1/3})$
scalar values $\xi_t$, adding $O(T^{1/3})$ space overhead. This makes
the estimator suitable for real-time regulatory monitoring, where
$(c_t, \hat{\pi}_t)$ arrive sequentially and the audit metric must be
updated daily at $O(nd)$ marginal cost per observation.
corollary[End-to-end audit complexity]
A complete regulatory audit of an algorithmic portfolio strategy over $T$ periods, comprising trajectory covariance estimation (Proposition (ref)), HAC confidence interval construction (Theorem (ref)), and bias correction (Proposition (ref)) requires
\begin{equation}
O(T \cdot nd)
\end{equation}
total time and $O(T(n + d))$ space. No access to the internal
optimization engine is required; the only inputs are the observable sequence $\{(c_t, \hat{\pi}_t(c_t))\}_{t=1}^T$.
proofEach of the three components costs at most $O(T \cdot nd)$ by
Propositions (ref) and (ref) and Theorem (ref); the maximum dominates.
table[table omitted — 875 chars of source]
Confidence intervals
A $(1-\alpha)$ confidence interval for total regret is
equation[equation omitted — 229 chars of source]
where $\hat{\sigma}_{\mathrm{LRV}}$ is the square root of any
consistent HAC variance estimator (e.g., Newey-West with $h=\lfloor T^{1/3}\rfloor$ lags).
remark[Implementation in black-box monitoring]
The estimator (ref) requires only the observable sequence $\{(c_t, \hat{\pi}_t(c_t))\}$, not access to the internal optimization engine. For a portfolio manager monitoring a third-party algorithmic
strategy under SR 11-7, one observes the factor return vector $c_t$
and the weight vector $\hat{\pi}_t$, computes $\widehat{\mathcal{C}}_T$
daily with a Newey-West correction for autocorrelation, and reports
the confidence interval (ref) as the performance audit metric.
The total monitoring cost is $O(Td)$ per evaluation period.
Summary: When Does the Identity Hold?
Table (ref) summarizes the conditions under which the exact multi-period identity $\mathrm{Regret}^{(T)} = \sum_{t=1}^{T}\operatorname{Cov}(c_t, \hat{\pi}_t)$ holds, when it holds approximately, and when it fails with a quantified bias.
table[table omitted — 1,490 chars of source]
Empirical Analysis
We test the theory on momentum strategies using data from the Center for Research in Security Prices (CRSP) daily data from January 4, 2016, through December 31, 2025. Our analysis is confined to momentum, where the signal is an indicator of whether the price went up or down: if the price goes up, then buy; else, if the price goes down, then sell. However, the same logic can be applied to any factor that can be used in a conditional if-then framework.
Summary Statistics
Our sample comprises 21,183,373 observations from the WRDS CRSP DSI with average daily market returns of
$\mu=0.041$%/day and standard deviation of returns $\sigma=4.446$%/day. Table (ref) documents the results.
table[table omitted — 666 chars of source]
We label the days as "regimes" following the NBER classification: 1) a recession from 2/1/2020 to 4/30/2020; 2) a crisis from 2/20/2020 to 3/31/2020. We obtain the following number of observations for each NBER business cycle type:
itemize• expansion : 20,713,594 obs (97.8%)
• contraction : 250,036 obs (1.2%)
• crisis : 219,743 obs (1.0%)
Regret for Momentum Strategies
For each sample, we estimate autoregressive coefficients (AR(1)) with Newey-West heteroscedasticity-and-autocorrelation-consistent (HAC) standard errors. Table (ref) reports the results. As the table shows, the data displays persistently negative daily AR(1) coefficients across all economic cycles. This means that on a daily basis, reversal dominated momentum from 2008 through 2025.
table[table omitted — 1,533 chars of source]
Next, we decompose regret for strategies of different investment horizons: 21 days (1 month), 63 days (a quarter), 126 days (6 months), a year, and 2 years. We compare it with the true regret computed with a "20/20" retroactive vision enjoyed by the omniscient trader. Table (ref) summarizes the results.
table[table omitted — 4,046 chars of source]
Trading Strategy Performance
In this section, we test the performance of the bias-adjusted covariance-based regret metric in predicting the model's performance. Table (ref) shows the overall and year-by-year performance of 1) momentum strategies, 2) reversion, 3) minimum-variance portfolios, and 4) reversion with a corrected bias.
The strategies are tested using CRSP data from January 2016 through December 2025. The strategies were executed as follows: if day $t-1$ return was positive ($R_{t-1} > 0$):
itemize• The momentum strategy buys at the end of day $t-1$ and sells at the end of day $t$, recording a gain of $R_t$
• The reversion strategy short-sells at the end of day $t-1$ and buys back at the end of day $t$, recording a gain of $-R_t$
• The Minimum-variance strategy adjusts weights based on the portfolio calculation over the immediately preceding 21 trading days
• The bias-corrected strategy adjusts the reversion strategy by $\frac{1}{4}\times$bias correction: the average return over the past 21 trading days divided by the maximum standard deviation of returns over the same period.
As Table (ref) illustrates, Reversion significantly outperforms Momentum over the entire sample and annually in seven out of ten years. The bias correction produced a small decrease in the strategy return and Sharpe ratio, except in the years 2019 and 2022, where it significantly improved the Reversion strategy. Thus, the bias correction can be thought of as insurance: if everything works, then there is a slight cost. In other cases, bias correction delivers a considerable gain.
table[table omitted — 2,492 chars of source]
Robustness: Causality, Omitted Variables, and Attribution
The AR(1) coefficients in Table 3 and the regret decomposition in Table 4 establish a statistical association between daily return reversals and the underperformance of momentum strategies. Before drawing the stronger conclusion that momentum itself is the source of regret, three confounds must be addressed: omitted risk factors, cross-sectional microstructure effects, and the direction of causation between reversals and momentum losses. We discuss each in turn and describe the controls that would be required to isolate
momentum as the causal driver.
Omitted Risk Factors
The CRSP universe pools stocks of heterogeneous size, book-to-market ratio, profitability, and liquidity. Each of these characteristics is known to predict returns independently of past-return momentum fama1993common, carhart1997persistence. The negative AR(1)
coefficients in Table 3 may therefore partly reflect:
enumerate• Size (SMB). Small-cap stocks exhibit stronger short-horizon reversals than large-caps jegadeesh1990evidence. A CRSP universe weighted by observation count rather than market capitalization over-represents small stocks and may amplify the apparent reversal in $\hat{\rho}$.
• Value (HML). Value stocks tend to have lower
momentum scores and higher subsequent returns over short horizons, contributing negatively to a momentum strategy's AR(1) residual.
• Liquidity / bid-ask bounce. At a daily frequency, the bid-ask spread mechanically induces negative serial correlation in transaction-price returns roll1984simple. With $n = 21{,}183{,}373$ observations drawn from the CRSP DSI (which includes illiquid small-caps), bid-ask bounce is a plausible
driver of the estimated $\hat{\rho} \in [-0.144, -0.004]$.
Notably, the most negative coefficient occurs in the 2020 crisis subsample ($\hat{\rho} = -0.144$), precisely when spreads widened dramatically, consistent with a microstructure rather than a pure momentum explanation.
• Volatility regime. The standard deviation of returns is substantially higher in 2020 and 2025 (Table 2), and high-volatility regimes are associated with stronger reversals lehmann1990fads. The regret estimates in Table 4 are larger in those years, but this may reflect the volatility regime rather than momentum per se.
Cross-Sectional vs.\ Time-Series Momentum
The signal used in Section 12 is a time-series momentum
indicator: buy stock $i$ if its price rose yesterday, sell if it fell. This is distinct from cross-sectional momentum (buy the top-decile performers, sell the bottom-decile), which is the dominant form studied in the asset-pricing literature
jegadeesh1993returns, carhart1997persistence. The regret attribution in Theorem 7.2 and Example 7.3 applies to both specifications, but the magnitude of the bias term $\sum_t \bar{c}_t^\top b_t$ differs across them.
Specifically, cross-sectional momentum allocates away from the equal-or value-weighted benchmark, inducing a bias $b_t$ that depends on the dispersion of returns across stocks. Time-series momentum, as used here, allocates in proportion to each stock's own lagged return sign, inducing a bias that depends on the autocorrelation of individual return series. The AR(1) estimates in Table 3 capture the latter but not the former; the regret table therefore speaks to time-series momentum strategies and should not be interpreted as evidence about
cross-sectional momentum funds without separate analysis.
Direction of Causation
A negative AR(1) coefficient in returns is consistent with at least three causal theories:
enumerate• Momentum causes losses (the paper's interpretation).
A strategy that buys yesterday's winners encounters mean-reverting prices and incurs negative realized returns, generating covariance-based regret of order $|\hat{\rho}|\|\bar{c}\|^2 T$.
• Liquidity provision causes reversals.
Market-makers absorb order flow and then rebalance, mechanically creating short-horizon reversals grossman1988liquidity. In this story, the reversal is an artifact of the trading mechanism
and would disappear for a patient investor who does not trade at daily frequency.
• Volatility clustering induces apparent autocorrelation.
Under GARCH-type conditional heteroskedasticity, daily return signs can exhibit negative serial correlation even when standardized returns are serially uncorrelated engle1982autoregressive. The ADF tests in Table 3 reject the unit root but do not distinguish between true mean-reversion and sign-switching driven by volatility clustering.
\paragraph{Identification strategy.}
To attribute regret causally to momentum, one would need an instrument or natural experiment that shifts momentum signal strength independently of return volatility and liquidity. One candidate is the quarterly rebalancing of Russell index membership chang2015market, which induces exogenous price pressure on affected stocks; another is earnings announcement dates, which create plausibly exogenous return
shocks. In the absence of such identification, the results in Tables 3 and 4 should be interpreted as follows:
quoteUnder the maintained assumption that the AR(1) coefficient
$\hat{\rho}$ measures the degree to which daily momentum strategies violate the mean-optimality condition, the regret decomposition in Theorem (ref) implies that momentum strategies accumulate a bias of order $|\hat{\rho}| T \sigma_c^2$ over the sample period. This estimate is an upper bound on momentum-attributable regret; the true figure may be lower to the extent that $\hat{\rho}$ reflects microstructure noise, factor exposures, or volatility clustering rather than pure momentum dynamics.
Summary of Robustness Limitations and Recommended Extensions
Table (ref) summarizes the three confounds, their likely direction of bias on $\hat{\rho}$, and the analysis required to address each. We view these as productive directions for follow-on empirical work rather than invalidations of the theoretical framework, which holds
exactly under the stated assumptions regardless of the empirical
source of $\hat{\rho}$.
table[table omitted — 1,080 chars of source]
All five confounds bias $|\hat{\rho}|$ upward, meaning the regret estimates in Table 4 are conservative upper bounds on momentum-attributable regret. This is the appropriate direction for a regulatory audit tool: overstating regret flags strategies for review; understating it would allow poorly performing strategies to pass undetected. Nevertheless, the factor-residual and bid-ask controls are straightforward to implement with available CRSP/Compustat data and should be included in the final version.
Conclusion
The Regret-Equals-Covariance identity of Aldridge2026 extends to multi-period dynamic programming with the following structure:
equation[equation omitted — 279 chars of source]
where the bias correction vanishes if and only if the policy's expected output matches the conditional-mean optimal at every period. Three financially concrete conclusions follow:
enumerate• Diagnostic sufficiency. For minimum-variance portfolios,
linear-quadratic optimizers, and any policy satisfying the mean
condition, monitoring the single-period sample covariance
$\operatorname{Cov}(c_t, \hat{\pi}_t)$ is sufficient to track multi-period
performance, the identity is exact at every horizon.
• Momentum strategies require correction, with the caveat that
the empirical magnitude is an upper bound.
Trend-following and momentum allocations violate the mean-optimality
condition (Definition 7.1) and accumulate bias of order
$|\hat{\rho}| T \sigma_c^2$ over horizon $T$. The bias-corrected
estimator (ref) should be used in performance
attribution for these strategies. Empirically, the AR(1) coefficients
in Table 3 are persistently negative across all years and NBER regimes
($\hat{\rho} \in [-0.144, -0.004]$), consistent with short-horizon
reversal dominating momentum from 2016 through 2025. However, as
established in Section (ref), the estimated $\hat{\rho}$
conflates pure momentum dynamics with bid-ask bounce, factor exposures,
and volatility clustering, all of which bias $|\hat{\rho}|$ upward.
The regret figures in Table 4 should therefore be interpreted as
upper bounds on momentum-attributable regret; factor-residual and microstructure-corrected estimates are identified as the primary direction for follow-on empirical work. The theoretical result is that any strategy violating the mean condition accumulates bias $\sum_t \bar{c}_t^\top b_t$. This result holds exactly under Assumption 3.2 regardless of the empirical source of $\hat{\rho}$.
• Discount scaling for long-horizon monitoring.
Equation (ref) shows that a single-period covariance estimate, scaled by the effective horizon $1/(1-\gamma)$, directly measures long-run discounted regret, providing a practical tool for annual regulatory performance reviews under SR 11-7 and SR-26-2.