Papers

The Log S-fBM model, introduced by Wu et al., is a stochastic volatility model whose log volatility is a stationary fractional Brownian motion (S-fBM): a stationary Gaussian process with power-decaying autocovariance driven by the Hurst exponent $H$, and variance scaled by an intermittency coefficient. A key property is that it reconciles rough volatility, where $H$ is typically near $0.1$ (see Gatheral et al.), with multifractal volatility, where $H$ is close to $0$ as in Bacry, Muzy et al.: the model's volatility measure converges to a multifractal random measure as $H\to0$. Numerical findings in Wu et al. show intermittency of order $0.02$ across financial assets, motivating a small intermittency approximation of log volatility moments for calibration via the general method of moments (GMM). In this work, we conduct a statistical analysis of the Log S-fBM model. We derive scaling properties of the S-fBM process and the Log S-fBM integrated volatility measure, present deviation inequalities with tail distributions sensitive to $H$ and intermittency, and develop a hypothesis test for the null Hurst exponent, i.e.\ rough versus multifractal dynamics. Finally, we revisit scale invariance of the log volatility increment process via explicit small-intermittency formulas, reproducing analogous properties in both regimes.

Whatever the policy question under consideration, reasoned and realistic evaluation requires credible policy analysis under uncertainty. This holds particularly to the study of climate policy in the United States, where science and ideology have become increasingly entangled. Welfare economics provides a transparent formal framework to describe and evaluate the subtle tradeoffs that must be addressed when comparing alternative climate policies. The largely qualitative discussions of climate policy in public-facing reports suffer from failure to use welfare economics to assess tradeoffs. I use 2025 reports by the National Academies and by the US Department of Energy as case studies.

Buy Now, Pay Later (BNPL) has moved from a novelty product to mainstream consumer finance in the space of a few years. Before 2020, BNPL was a niche offering at the checkouts of online fashion merchants. BNPL then experienced substantial growth (Consumer Financial Protection Bureau, 2022; Salem & Udis, 2025), coinciding with increased online shopping since the onset of the COVID-19 pandemic in 2020. BNPL is now a pervasive payment option. In 2026, you can use BNPL to delay and split up payments for anything ranging from a pizza to your rent, and many items in between. As the market has grown, so has the academic literature. There is now sufficient research into BNPL to take stock and review what we have learned. In this article, we summarize insights from the economics, finance, and marketing academic literature. While we have learned a great deal about BNPL in the last few years, we also discuss many remaining open questions and current concerns impacting policy decisions. We start by explaining what BNPL products are and the business models of BNPL lenders. Then, we summarize the economics and psychology behind why some consumers use these products. Next, we describe the customer base that uses BNPL. We show the effects of BNPL on consumers, an area of research with substantial policy interest. Another area of policy interest is whether BNPL loans should be reported in a consumer's credit report. We explain why BNPL is typically not on credit reports and its implications. Finally, we offer some concluding thoughts.

In-group favouritism and out-group hostility are well-documented, but real-world group settings rarely fit a simple dichotomy. Often, a "middle-group" shares some identity markers with the in-group without fully belonging to it. How do people treat such intermediate groups? We address this question using a formal identity marker framework and a multi-lab online experiment (N = 376) with a private allocation task immune to reputation effects and demand characteristics. We find that a middle-group can be treated neutrally, i.e., distinct from both in-group favouritism and out-group hostility, but only under specific conditions. When the middle-group shares two identity markers with the in-group, including university affiliation, a complete hierarchy emerges: the in-group is favoured, the middle-group is treated according to the allocation rule, and the out-group bears the loss. However, when the middle-group shares only one marker, it is treated indistinguishably from the out-group, regardless of which marker is shared. Furthermore, differentiated behaviour evolves only after participants have re-encountered the experimental instructions and group constellation in a second phase, not gradually across rounds. These findings demonstrate that participants can perceive beyond binary in-/out-group categorisation, but such perception requires both sufficient identity overlap and repeated exposure to the group setting. Our identity marker framework provides a tool for systematically studying group relations in more complex settings.

We define the variance swap delta as the sensitivity of the price of variance to a change in underlying price. We use Carr-Madan spanning formulas to analyze this sensitivity when the implied volatility smile curve may depend on the underlying price. We show that the variance swap total delta is zero for the class of smile curves that are pure functions of (log) moneyness, which goes against the empirical observation that variance is up when the market is down. We propose a simple modification of the smile to correct this issue.

As US power markets contend with growing demand for firm generation, the nuclear industry has offered Small Modular Reactors (SMRs). However, how these concepts would fare in a rapidly evolving power grid is unclear, given the paucity of operational examples. Current literature, informed by substantial cost escalations for traditional nuclear plants, focuses on the investment costs SMRs need to achieve for private investment feasibility. However, this work finds that the operating and marginal costs of SMRs are more critical to economic feasibility in market environments. This work dispatches SMRs using a flexible operations model, considering revenue from two main electric markets, capacity and wholesale energy markets, with and without policy support. Manufacturer advertised costs for investment and operating costs are used, with fuel costs calculated from manufacturer provided design parameters. Results indicate that SMRs are uneconomical primarily because investment cost reductions are offset by increased marginal costs. As such, an environment of prices and subsidies beyond historic norms are necessary to attract private investment at manufacturer advertised cost benchmarks. Current SMRs are as profitable as advanced estimates of the AP1000 traditional nuclear reactor, and if investment costs escalate at the average rate for nuclear projects, they are similar to Vogtle 3 & 4. In projected future power markets, reductions in marginal cost may be more beneficial than those in investment costs.

We develop a theoretical model that endogenizes the regulator's decision to impose short-selling bans to prevent large stock price declines. Empirically, we test the model's predictions using the cross-sectional variation in short-selling restrictions implemented across European countries in 2020. Consistent with our model, we find that bans had a detrimental effect on liquidity and failed to support the average price levels, but were effective in limiting large price drawdowns. Finally, we show that the effectiveness of the bans depends on the share of informed stockholders, a central variable in our framework, thus informing the design of more effective regulatory responses.

Let \(P\) be a fixed physical law and let \(Q\) be an equivalent martingale measure selected from the martingale-measure set associated with a chosen numeraire. A change of numeraire maps \(Q\) to \(T_LQ\), where \(d(T_LQ)=L\,dQ\) and \(L\) is the terminal likelihood ratio. The forward relative-entropy projection minimizing \(D_{\mathrm{KL}}(P\Vert Q)\) commutes with this transform because its objective changes only by the constant \(-E_P\log L\). The minimal entropy martingale measure (MEMM) orientation \(D_{\mathrm{KL}}(Q\Vert P)\) does not have this property, and a trinomial counterexample shows that independently recomputed MEMMs need not be likelihood compatible. We make two economic consequences explicit. First, the two entropy orientations are precisely the \(Q\)-dependent terms in the classical convex-dual objectives for logarithmic and exponential utility, respectively. Second, likelihood compatibility is equivalent to equality of the pricing functionals obtained in the two numeraires. Hence the forward selectors value every integrable claim consistently across numeraires, whereas the two MEMMs in the counterexample assign different prices to a nonreplicable digital claim. We also prove a finite-state class-level characterization: uniform invariance over the elementary one-period likelihood-ratio families forces a smooth convex \(f\)-divergence to be logarithmic, up to scaling and affine equivalence. Finally, in finite-state markets, the forward projection exists under the usual strictly positive feasible-point condition; its density \(dP/dQ^*\) is attainable log-optimal terminal wealth, and the minimum forward entropy equals maximal expected log growth.

Formulaic alpha discovery is a pool-dependent symbolic search problem in which informative feedback is observed primarily when a complete expression is evaluated. This delayed feedback creates two coupled difficulties: the retained alpha pool does not preserve the full history of realized evaluation feedback, and the value of an intermediate construction action is uncertain because its consequence depends on the formula eventually completed. We introduce AlphaRJM, which addresses these difficulties through Reward-Jump Memory, an event-driven latent state that remains fixed during token construction and updates only at terminal evaluation events using the realized pool reward and evaluation outcome, and an action-conditioned SDE return critic that represents future discounted discovery returns with stochastic particles. The particles guide action selection through their mean and uncertainty and are learned using a distributional Bellman objective combining energy-distance matching, mean calibration, and jump regularization. Empirically, AlphaRJM delivers strong and stable gains across multiple equity universes, forecasting horizons, and random seeds, while ablations confirm the complementary roles of persistent evaluation history, stochastic return modeling, and distributional supervision.

We study variance-optimal stock hedging and the convergence of approximate strategies in the rough Hawkes–Heston model. Starting from the model's affine conditional transform and the affine Volterra jump framework, we obtain semi-explicit hedges for European calls and a representation of the minimum quadratic error through the Galtchouk–Kunita–Watanabe projection. Our main approximation result keeps the original stock, variance driver, and information flow fixed while regularizing the kernel used to evaluate the hedge. To handle singular memory and common marked jumps, we construct the approximate holdings from histories available before trading and preserve the conditional transform's random modulus envelope. Riccati–Volterra stability and weighted truncation then yield convergence in the original stock's trading norm on compact Fourier intervals. For calls, a joint choice of kernel regularization and Fourier cutoff gives convergence of the initial capitals and strategies, uniform-in-time square-mean convergence of continuous-time gains, and convergence of the terminal mean-square error to the variance-optimal value. A numerical experiment with shifted fractional kernels illustrates the construction on common original-market paths.

Whether artificial intelligence pays off for the firms that build it into their products is hard to establish, because AI innovation is itself hard to observe. The medical technology sector is a rare exception: an AI-enabled device must obtain clearance from a national health authority before it can reach a patient, leaving a dated, firm-attributable record of AI innovation output that can be observed directly rather than proxied. We exploit this setting with a three-stage recursive model estimated on a novel firm-level dataset linking FDA premarket clearances, USPTO patents, Scopus publications, and Orbis financials, tracing the full innovation chain from external collaboration through AI device introduction to firm performance. We find that external AI research collaboration is a robust driver of AI device introduction across firm sizes and estimators, with a larger effect for small firms, consistent with external knowledge ties substituting for limited internal R&D capacity. Decomposing by partner type, the effect is largest for industry and clinical collaborations and smallest for academic ties, consistent with the former being closer to the regulatory and commercialisation process. Firms that bring AI devices to market display higher labour productivity, an effect robust for small firms and the full sample that holds under both sequential and joint maximum-likelihood estimation and accumulates across successive device introductions. Effects on profit margins are present but weaker and do not survive all specifications, a pattern consistent with competitive entry eroding pricing power as AI devices diffuse through the sector.

Bouchaud and Farmer (2023) argue that self-fulfilling beliefs generate a realistic wealth distribution. Their reported Gini coefficient, quantiles, and Pareto exponent are computed from dynasty-level wealth averaged over 250 dates, not from a cross section. Correcting the estimand strengthens their qualitative result: with $N=1{,}000{,}000$ dynasties, the terminal cross-sectional Gini is 0.883 rather than 0.7, and median wealth is 0.0052 rather than 0.39 of average wealth. The model therefore generates more inequality, but not the reported quantitative fit. Appendix F analyzes a different process and does not establish the claimed Pareto tail or exponent.

The twenty-first century's transformative technology, artificial intelligence, is increasingly constrained by the twentieth century's transformative technology, the electricity grid. Rapid growth in electricity demand from data centers is leading to higher electricity prices, without a compensating supply-side response. We develop a framework linking data-center load growth, available generation capacity, and market-clearing prices to understand this phenomenon. We first analyze a deterministic model to show how differing estimates of demand and supply growth rates affect prices. We then model the expansion of new data centers and their associated electricity demand, together with build-outs of new electricity supply, as stochastic processes,resulting in probabilistic distributions of supply, demand, and prices rather than a single forecast. Finally, we formulate generation expansion as a stochastic control problem in which a revenue-maximizing investor dynamically chooses the intensity of supply-side investments. The analysis highlights a central challenge of the data-center build-out: even when rapid demand growth increases the need for new generation, the uncertainties related to load forecasts, development execution risks, and value cannibalization from overbuilding capacity may weaken incentives to invest at the pace required to keep electricity prices stable.

MASTER's inter-stock multi-head attention – the module responsible for modeling cross-sectional stock relationships – accounts for 42.5% of model parameters and 25% of predictive value. We systematically decompose this module and uncover a surprising structure: the learned attention is near-uniform (perplexity 278/300), yet forcing exact uniformity eliminates all cross-sectional discrimination. Spectral analysis resolves this paradox: the deviation from uniformity is low-rank (effective rank ~65, top-10 modes capture 96.5% of energy), explaining why sparse approximations consistently fail while Nystrom low-rank attention (m=32 landmarks) matches full O(N^2) attention at O(mN) cost – certified equivalent via TOST at both N=300 (5 seeds, Rank IC p=0.003) and N=800 (10 seeds, Rank IC p=0.034). Additional findings include: (i) attention anti-correlates with return similarity (Spearman rho = -0.614; on the industry-labeled subset, -0.645 unconditionally and -0.627 after controlling for industry, beta, and volatility), suggesting complementarity-seeking rather than correlation mining; (ii) all graph-based alternatives degrade performance, with hard masking worse than complete module removal; and (iii) at N ~ 3,500 with adapted architectures, no cross-stock module (GCN, Nystrom, or MASTER-style pipeline) significantly outperforms a per-stock LSTM baseline (n=4 seeds), indicating that the benefits observed at smaller scales do not trivially transfer. These results establish that the inter-stock attention's value resides in a compressible, dynamic, near-global redistribution that rewards low-rank approximation but resists sparsification.

We introduce a probability-wave framework for modeling the collective behavior of interacting adaptive agents, deriving testable eigenmodes through a generalized behavioral intelligence (GBI) nonlocal probability-wave equation. This framework captures a broad range of human intelligence behaviors with analytical mechanisms and offers an indirect method to examine the Liu-Chen-Ao (LCA) hypothesis of nonlocal entangled nerve fibers in the brain through collective trader behaviors. Our empirical analysis of Chinese intraday stock market data demonstrates that adaptive entangled game modes explain 82-94% (89% overall) of observed decision patterns, a sharp contrast to the predictions of neoclassical finance based on independent rational agents. Moreover, 2-12% of behaviors show adaption to intraday news, events, and environments, characterized by dual equilibrium states and abrupt reference point shifts, while purely independent modes occur in less than 5% of cases. These findings empirically support the LCA hypothesis, as observable trading behaviors reflect underlying brain mechanisms and internal intelligence decision-making in behavioral psychology. Our results highlight the necessity of incorporating adaptive entangled game modules into artificial general intelligence (AGI) architectures, addressing the limitations of conventional artificial neural network (ANN)-based AI, which relies on trillions of opaque parameters. By integrating ANN-based AI with probability-wave-based entangled-brain simulations, machine learning can enrich AGI foundation models (FMs) and facilitate the development of human-like processing units (HPUs) that leverage brain-inspired mechanisms. Such HPUs may ultimately create more compact, efficient, and robust AGI systems, particularly for embodied intelligence and robotics.

We build and evaluate a pre-game win-probability forecaster for individual maps (“games”) in professional League of Legends (LoL). The proposed model is a one-stage logistic regression fit end-to-end on the win/loss log-loss: each team's exponentially-weighted moving average of past same-side results, a ridge-shrunk stable strength that is the maximum-a-posteriori estimate of a logistic mixed model, and a first-pick draft covariate, natively calibrated out of sample (walk-forward slope $0.995$). It augments a purely dynamic Bradley–Terry specification with stable team strengths. A second, independently built two-stage composite mixed model under restricted maximum likelihood (REML) and best linear unbiased prediction (BLUP) shrinkage, with Platt calibration, serves as the strongest rival the authors could build. On $5{,}135$ games across six regional leagues and three international events (2024–2026), under paired per-game Diebold–Mariano inference, the two architectures are statistically indistinguishable on every protocol and window (global holdout $0.2230$ vs.\ $0.2257$; walk-forward $0.2207$ vs.\ $0.2215$), so the simpler model is preferred on parsimony, not accuracy; both improve on the classical dynamic benchmark ($0.2351$) by a clear margin and on the static fits ($0.2301$/$0.2268$) more modestly. Against Polymarket on $928$ matched maps, the forecasts are statistically indistinguishable from the market on its own per-game contracts, with a modest market edge concentrated on cross-region Worlds and series-decider maps.

Financial markets alternate between periods of relative stability and instability, with structural breaks marking the transitions between these regimes. Identifying such breaks in real time is a central requirement for any trading or risk system operating at high frequency. This report studies Bayesian Online Changepoint Detection (BOCPD) and two extensions proposed in the literature, and applies them to the signed order flow of NASDAQ-listed equities.

This text grew out of a historical introduction initially written for a study of interest rates in cryptocurrency markets. The difficulty of defining a term structure for a currency without a conventional bond market led naturally to a more fundamental question: under what historical conditions does a yield curve become observable at all? Credit existed long before modern money, and interest-bearing loans are documented as early as ancient Mesopotamia. For much of history, the surviving evidence lacks the institutional features that facilitate reliable comparisons of interest rates by maturity: standardised debt instruments, sufficiently homogeneous borrowers, regular issuance over a range of maturities, observable market prices, and liquid secondary markets. We trace the gradual emergence of these conditions from ancient Mesopotamia, Greece, and Rome, through medieval and early modern Europe, to the development of modern sovereign debt markets in the nineteenth and twentieth centuries.

For more than four decades, the 60/40 stock/bond portfolio has served as a benchmark for delivering reasonable returns without excessive risk. More recently, a 50/30/20 stock/bond/alternative portfolio has been suggested. We use gold as the alternative and as an inflation hedge. In this paper we ask: how much improvement over these benchmark fixed-weight portfolios can be obtained using widely available public data and standard methods from quantitative finance? We restrict ourselves to long-only dynamic portfolios of stocks, bonds, and gold, plus cash, rebalancing monthly, using only publicly available data. We evaluate portfolios on the conventional metrics: return, volatility, Sharpe ratio (computed in excess of the federal funds rate), drawdown, and turnover, in addition to consistency of performance over time, judged by the consistency of the realized annual volatility. Over the 20–year period 2006–2026, using a conservative estimate of trading costs, we show that all risk-adjusted and drawdown metrics are improved using simple volatility control, where we dynamically mix the fixed-weight portfolios with cash so as to target a fixed volatility. This method relies on a simple estimate of portfolio volatility derived from past returns. We also demonstrate that more sophisticated portfolios based on convex optimization—similar to those used in quantitative hedge funds—yield further substantial improvement in return and risk-adjusted return. We consider two such portfolios, one that uses a simple estimate of future returns based on past returns, and one that forecasts future returns based on past returns and just a handful of widely available public economic data. These portfolios also outperform a suite of standard risk-based allocation methods, such as risk parity and minimum variance, evaluated on the same assets and data.

Decentralized finance (DeFi) lending has grown from nonexistent in 2017 to nearly 40 billion US Dollars in deposited funds in May 2022. Using cryptocurrency as collateral, the platforms match speculative margin trading with yield-seeking depositors lending coins pegged to the dollar (stable coins). Depositors receive claims guaranteed by a basket of collateral, akin to new stable coins. We develop a framework requiring only knowledge of aggregate deposits and borrowings to measure overall system risks to lenders and borrowers. Using evidence from major protocols, the measures identify an increase in system fragility beyond prudent levels around mid 2021, with a potential loss of peg for extreme variations in coin prices. Overall, the model offers an easily implementable aggregate risk metric capturing the perspectives of synthetic investors and offers early warning signals as the industry is moving from deposits guaranteed by collateral to fiat money.

We study voluntary disclosure when investors observe firm reports through noisy information intermediaries such as auditors, analysts, rating agencies, or data providers. Any processing noise overturns the standard prediction of a unique partial-disclosure equilibrium. With low disclosure costs, the model unravels to full disclosure despite positive costs. With higher costs, the game admits two threshold equilibria featuring different disclosure probabilities. We characterize how the cost threshold for unraveling and the equilibrium set respond to changes in noise and fundamental uncertainty. In settings with high disclosure, both uncertainty and processing noise reduce disclosure, while higher certification costs can counterintuitively increase it. Endogenizing disclosure costs as optimal fees shows how profit-maximizing intermediaries select among equilibria, potentially generating a high-fee, high-disclosure regime. Extensions with bounded support, uncertain information endowment, endogenous noise, and competing information sources apply the insights to general information environments. The results caution against interpreting greater frictions as necessarily reducing disclosure.

We show that ordinary business language — "maximize profitability" — induces profit-oriented ambiguity resolution: LLMs systematically dismiss ambiguous signals of potential safety violations to serve business objectives. In 3,600 controlled trials across eight reasoning-capable LLMs, adding a profit mandate to otherwise identical prompts increases risk-dismissing judgments by 6.8 percentage points (p < 0.0001), suppresses board escalation recommendations by 13.9pp (p < 0.0001), and shifts severity assessments downward (p < 0.0001). The mandate never instructs models to downplay risks; instead, chain-of-thought traces reveal motivated reasoning: models acknowledge concerns, then invoke profit logic to justify dismissing them. We characterize these findings as the Profit Alignment Problem: when AI systems are given ordinary business objectives, they develop systematic strategies for suppressing inconvenient information that no designer intended or specified.

Cyber insurance requires accurate modeling of worst-case catastrophic (cat) events, but the field lacks robust quantitative approaches for estimating upper-bound losses. Building on a recent dataset of 24 cyber cat events over 30 years, this work tests whether cyber economic losses follow a power law distribution. We analyze "cyber kittens" - sub-1B USD events distinguished from cat events (1B+ USD) only by magnitude - extracted via LLM from cyber insurance claims data (2020-2024). Using victim count (weighted by claim year) as a proxy for economic loss, we link kitten-sized events to known cat events to estimate losses. The kitten distribution proved consistent with the cat dataset, and power laws were statistically plausible: each order-of-magnitude increase in event size corresponds to a 5-7x drop in probability. Extrapolating, an event 100x the largest 2020-2024 cat event is expected roughly every 206 years, translating to 100-250B USD in losses - catastrophic, but not extraordinary relative to other insurance lines.

Generative AI has become an everyday advisor, and the systems people consult are live and interactive, not pre-scripted. We ask whether access to such a system changes behavior under risk. In an incentivized experiment (N = 158), participants made lottery choices with an optional decision aid presented as a conventional pre-written tool, a live one-shot AI, or a live interactive AI they could query, with information format held equivalent across conditions. Risk preferences are elicited via DOSE. We find no evidence that access to a live AI advisor changes risk aversion.

We establish existence, uniqueness, stability, and convergence results for one-dimensional $q$-Bass martingales, characterized as the martingales with prescribed initial and terminal marginals whose transition kernels are closest to a reference measure $q$. Their existence is equivalent to the solvability of a fixed-point problem for probability distributions. Building on Acciaio and Marini (2026), that requires the first marginal to be supported on finitely many points, we study the case of general marginals in convex order. Under the assumption that $q\ll\lambda$, we prove existence, uniqueness and stability of fixed-point distributions, $\mathcal{W}_\infty$-convergence of the fixed-point iteration, and support-diameter estimates. We also extend the martingale Benamou-Brenier formula from Brownian motion to any additive reference process $X$ and show that the corresponding $X$-Bass martingale is optimal whenever it exists, with an interpretation as an adapted Wasserstein projection of $X$.

We develop a filter for time series, defined at each time $t$ as the minimizer of a discounted convex combination of observed and expected losses. The filter can be estimated by simulation to an arbitrary level of accuracy in $O(1)$ flops at each time point $t$ and can be run for all values $t=1,...,T$ in parallel. These methods are applied to robustly compute a preaveraged price process from the more than 1.5 million trades made on a single financial asset in a single day where the noise's variance is infinite. It yields a flat "volatility signature" plot, down to the 1 second level, so the microstructure noise no longer biases the volatility estimate. This is not true when linear methods are employed.

We develop semi-closed pricing formulas and lifted-model hedging methods for discretely monitored geometric and arithmetic Asian options in the Volterra-Heston stochastic volatility model. Exploiting the affine Volterra structure, we derive a tractable transform for the joint law of the terminal log-price and the discretely monitored geometric average. This transform yields semi-closed pricing formulas for geometric Asian options, which in turn provide effective control variates for Monte Carlo valuation of arithmetic Asian options. Under the stated real-moment and affine-transform hypotheses, we also derive the Galtchouk-Kunita-Watanabe decomposition for Fourier-representable payoffs and obtain a variance-optimal hedge in terms of the Riccati-Volterra equation and the forward-variance curve. Using N-factor Markovian approximations, we obtain a finite-dimensional numerical implementation for hedging Asian options. Our numerical experiments document factor convergence for a regular non-Markovian kernel and the effect of rebalancing frequency on hedging error. In the Heston benchmark, geometric Asian controls substantially reduce the variance of arithmetic-Asian price estimates and improve the finite-sample stability of regression-based hedging relative to direct regression.

We review the evidence on subsidized job training programs in industrial countries. We provide critical, narrative summaries of randomized studies of major job training in the US; of two "judge randomization" studies in the US and Denmark; and of previous meta-analyses. We then perform a meta-analysis of 56 randomized trials of job training in the US since 1973, the most exhaustive meta-analysis in this niche. Here, average impacts are positive but modest: employment rises 1.7 percentage points in years 3-5, and annual pre-tax earnings by roughly \$800 per person offered training. At an average cost of \$13,598 per participant, programs roughly break even under (debatable) assumptions about how long benefits persist. They recoup about three-quarters of their cost for government through higher tax revenue and lower benefit use. Evaluations of the largest federal efforts–JTPA, the Workforce Investment Act, and Job Corps–return small or null effects. One cluster of programs stands out: "sector programs," which screen applicants intensively, design curricula with employers, and track local skill demand. They raise earnings roughly ten times as much. They have proven difficult to replicate.

We develop an expected shortfall factor model (ESFM) to estimate and price common variation in the severity of lower-tail losses in large panels of asset returns. Mean factor models describe common variation in average returns, while quantile factor models describe common movements in tail thresholds. ESFM instead captures common variation in the average severity of losses below those thresholds. The model combines observed risk exposures with latent common factors. We estimate ESFM using an orthogonalized two-step procedure under which first-stage quantile estimation error has no first-order effect on the ES coefficient estimates. We establish nonasymptotic error bounds for the ES coefficients, a finite-sample Gaussian approximation, and consistent selection of the number of latent factors. Applied to a large panel of equities, ESFM uncovers common factors that react sharply to market stress and contain information not captured by mean and quantile factors. Average returns increase across portfolios sorted on ESFM exposure; high-minus-low portfolios earn annualized returns of 8.0%–11.7% and Fama–French five-factor alphas of 10.3%–15.0%. These spreads remain positive and statistically significant after conditioning separately and jointly on mean- and quantile-factor exposures. Tail-by-tail spanning tests show that ESFM factors retain significant alphas after controlling for standard traded factors and the corresponding mean and quantile factors. Adding ESFM to these benchmark factor sets increases the maximum attainable Sharpe ratio. These findings identify common loss severity as a distinct and priced dimension of downside risk.

Sports betting moves money continuously from a large public to a small number of firms. The most influential account of that flow, due to Levitt (2004), holds that books price away from the market-clearing point to exploit predictable public biases. It computes profit from two numbers (the probability that a side wins the proposition and the fraction of handle it attracts), treating the share on a side as independent of the outcome. Here we relax that assumption and derive a profit-bias identity: profit is affine and increasing in the expected share of handle on the losing side, with Levitt's expression as the special case of independence. The identity resolves the book's margin into exactly three channels: its hold, the product of its price shading and the public's lean, and the covariance between bet share and outcome. A lean is thus worthless without shading, and shading is worthless without a lean. Under a public-belief model, the profit driver is the public's Bayes error: the probability of a representative bettor selecting the losing side. We provide necessary and sufficient conditions for a "Goldilocks Zone": prices at which book and bettor both profit. Testing these predictions on 1,139 Major League Baseball games, we find that the apparent dependence between bet share and outcome is a Simpson's paradox: present when games are pooled, but absent once they are separated by which side the book favored. The public leans heavily toward favorites, but we detect no matching shading, and the realized margin is indistinguishable from the hold.