## Annex 2.1 – Glossary ## Source details **Canonical URL:** [Annex 2.1 – Glossary](https://www.imf.org/-/media/files/publications/gfsr/2026/october/english/ch2annex.pdf) ## Other formats - [Markdown version](/-/media/files/publications/gfsr/2026/october/english/ch2annex.pdf.md) - [Structured JSON version](/-/media/files/publications/gfsr/2026/october/english/ch2annex.pdf.json) --- ### Key technical terms and definitions - Arbitrage strategies - Trading strategies that seek to profit from pricing inconsistencies between related securities. Arbitrage generally involves buying an undervalued asset and selling a related overvalued asset, expecting the price gap to narrow over time. - Hedge fund - Describes a business model rather than a uniform legal category. Generally refers to a privately offered pooled vehicle for institutional or high-net-worth investors with broad flexibility to use leverage, derivatives, short selling, concentrated positions, and less-liquid investments. Contrasted with public mutual funds that serve retail investors and face prospectus disclosure, tighter leverage requirements, and more frequent redemptions. Hedge funds may charge performance fees or impose lockups, notice periods, gates, and side pockets. Legal perimeters vary by jurisdiction (United States: private funds limited to accredited investors or qualified purchasers and classified under Form PF; European Union and United Kingdom: part of alternative investment fund category; Cayman Islands: classification depends on redemption rights). - Hedge fund strategies - Main strategy groups: credit funds, equity funds, event-driven funds, macro funds, managed futures/CTA funds, relative value funds, and multi-strategy funds. - Margin / haircut - Margin is collateral required to support a leveraged position; a haircut is the discount applied to collateral value in secured funding. - Prime brokers - Financial intermediaries (typically broker-dealer units of large global banking groups) providing financing and operational services: margin loans, financing, securities lending, custody, clearing, and execution services. - Synthetic leverage - Market exposure obtained through derivatives rather than direct borrowing, measured as derivatives positions’ equivalent exposure to underlying assets relative to funds’ gross assets. ### Return-implied leverage measure (monthly fund-level leverage construction) - Objective: recover variation in leverage across funds and over time while matching aggregate strategy-level leverage from the OFR and reported fund leverage categories. - Step 1 — Return-implied leverage score: - Score_{i,t} = f(X_{i,t}) (Equation (2.4)), where X_{i,t} denotes return- and risk-based characteristics for fund i in month t and f(·) is estimated from strategy-specific logistic models. - Fund characteristics: mean excess return; volatility; downside deviation; skewness; excess kurtosis; expected shortfall; maximum drawdown; share of near-zero returns; factor-exposure measures (equity, volatility, interest-rate, credit, foreign-exchange, commodity, liquidity, intermediary, trend-following, Treasury basis-risk). - Periods of large factor movements defined as months when average absolute standardized movement across the factor set exceeds its 80th percentile. - Step 2 — Translate score into unconstrained leverage target: - Center the score within each strategy-month using AUM weights. - L*_{i,t} = L^{OFR}_{s,t} + κ_s Scorẽ_{i,t} (Equation (2.5)), where L^{OFR}_{s,t} is OFR leverage for strategy s, Scorẽ_{i,t} is strategy-month centered score, and κ_s is a strategy-specific scaling parameter. - Step 3 — Impose consistency with reported leverage categories and OFR aggregates: - Funds with below 5 times leverage (1-2 and 2-5 categories) constrained to reported ranges; higher categories provide lower-bound restrictions. - Funds with non-standard or missing leverage labels retained without category-specific restrictions beyond non-negativity. - Projection: L_{i,t} = P(L*_{i,t}) subject to Σ_i w_{i,t} L_{i,t} = L^{OFR}_{s,t} (Equation (2.6)), where w_{i,t} denotes fund share of strategy AUM and P(·) denotes the projection operator. ### Equity-level hedge fund vulnerability measures (quarterly stock-level measures) - Notation: - m_{h,j,t} = dollar position of hedge fund manager h in stock j at end of quarter t. - θ_{j,t} = market capitalization of stock j. - HF_{j,t} = Σ_h m_{h,j,t}; pso_{j,t} = HF_{j,t} / θ_{j,t} (Equation (2.7)). - Crowdedness (Crowd-ADV): - Days-ADV_{j,t} = HF_{j,t} / ADV_{j,t} (Equation (2.8)), where ADV_{j,t} is the 20-day moving average daily dollar trading volume averaged over the quarter. - FundDaysADV_{h,t} = Σ_j w_{h,j,t} Days-ADV_{j,t} with w_{h,j,t} = m_{h,j,t} / Σ_q m_{h,q,t} (Equation (2.9)). - CrowdADV_{j,t} = Σ_h (m_{h,j,t} / θ_{j,t}) × FundDaysADV_{h,t} (Equation (2.10)). - Interpretation: Crowd-ADV rises with hedge fund ownership and with how crowded those owners’ portfolios are. - Investor Flow Fragility: - Fragility: G_{j,t} = 1/θ_{j,t} × w_{j,t}^T Ω_t w_{j,t} (Equation (2.11)), where w_{j,t} is vector of dollar positions in stock j across hedge fund owners and Ω_t is covariance matrix of owner-level investor flow. - Owner flow: f_{h,t} = (A_{h,t} − A_{h,t−1}(1 + r_{h,t})) / A_{h,t−1} (Equation (2.12)); Ω_t estimated via factor structure: Ω_t = B_t Σ_{f,t} B_t^T + diag(σ^2_{ε,h,t}) (Equation (2.13)). - Regressions use √G_{j,t} (units of a return) and within-quarter percentile ranks. - Leverage Exposure: - LevExposure_{j,t} = Σ_h (m_{h,j,t} / θ_{j,t}) L_{h,t} (Equation (2.15)), where L_{h,t} is estimated leverage of manager h (AUM-weighted average across funds mapped to manager). ### Vulnerability regression timing and interpretation - Measures constructed at end of quarter t and related to outcomes over quarter t+1 (predetermined). - Each measure enters as a within-quarter percentile rank; coefficient represents effect of moving a stock from bottom to top of that quarter’s cross-sectional distribution. - Key empirical findings: - Both large and small hedge funds significantly amplify tail risk in equity markets during stress. - Contribution of small funds is more pronounced, especially through the leverage channel. - The top 50 hedge funds account for roughly half of the sector’s AUM. ### Redemption restrictions and hedge fund capital stability - Redemption channels: initial lockup period, notice period, and redemption frequency. - Stricter restrictions adopted by hedge funds that are smaller or less leveraged. - Empirical flow patterns during stress (months when VIX is in top decile): - Funds with lockup periods of twelve months or less record average net outflows of about 82 basis points of assets under management. - Funds with longer lockup periods record average net outflows of about 12 basis points. - Fund-level net flows regression (basis points) over horizon h ∈ {0,1}: - Y_{i,t+h} = α_i + μ_{s,t} + β_1 Rigidity_{i,t−1} + β_2 (Stress_t × Rigidity_{i,t−1}) + β_3 Controls_{i,t−1} + ε_{i,t+h} (Equation (2.16)). - Stress_t = 1 when the VIX or MOVE1M index is in the top decile. - Controls: lagged log AUM, past 12-month returns, return volatility, fund age; μ_{s,t} are strategy-time fixed effects; α_i are fund fixed effects; standard errors clustered at fund and time levels. - Regression findings on redemption restrictions during stress: - Funds with lockup periods of at least twelve months record net flows that are 61 basis points higher than otherwise similar funds. - Funds with redemption frequencies longer than one month record net flows that are 87 basis points higher than otherwise similar funds. - Longer notice periods have no additional effect once lockups and redemption frequency are controlled for. - Similar qualitative pattern (smaller magnitudes) when stress measured using Treasury market volatility instead of equity volatility. ### Additional empirical and data notes - Flows expressed in basis points of AUM and winsorized at the 1st and 99th percentiles. - Trend: AUM share of funds with a rigidity index of at least 3 has declined since the mid-2000s; share with lockups of at least 12 months fell from about 40 percent in the early 2000s to about 15 percent recently. - For non-prime-broker banks, GIV shocks constructed from the full universe of hedge funds show non-prime-broker bank returns are not significantly affected by hedge fund shocks at any horizon. - Data sources: HFR; Bloomberg Finance L.P.; CFTC; EUROPACE AG/Haver Analytics; FactSet; Kenneth French Data Library; Haver Analytics; and IMF staff calculations. --- ### 1. Small and Large Hedge Funds ### Effect of hedge fund ownership on bid-ask spreads - Regression setup: - Panels estimate effect of a 10-percentage-point increase in hedge fund ownership on bid-ask spreads in the subsequent quarter (basis points). - Estimates from security-level panel regressions of subsequent changes in bid-ask spreads on changes in aggregate hedge fund ownership, controlling for market capitalization, turnover, price-to-book ratios, dividends, profitability, and initial bid-ask spread, with security and quarter fixed effects. - Classifications: - Large (small) funds: managers with above- (below-) median assets under management. - High- (low-) turnover funds: above- (below-) median portfolio turnover, where turnover = one-half of sum of absolute changes in portfolio weights between consecutive reporting dates. - All classifications relative to hedge fund median in each quarter. - Inference: - Error bars denote 90 percent confidence intervals. - Stress periods: quarters in which average VIX exceeds its 90th percentile. - Qualitative implications: - Strategies with larger size or longer holding horizons associated with subsequent price adjustment (information discovery role). - High-turnover strategies associated with improvements in trading efficiency during normal periods. - High leverage of macro, relative value, and managed futures funds may increase potential to amplify market strains when positions are unwound during stress. ### Strategy heterogeneity, leverage, and market effects - Key qualitative findings: - Relative value funds: high leverage and rapid portfolio turnover. - Macro and managed futures/CTA funds: elevated leverage. - Multi-strategy funds: among the largest managers. - Event-driven funds: more concentrated portfolios. --- ### 1. Fund Flow Response to VIX Stress, by Funding-Term ### Fund flow regressions and measurement details - Interaction between lagged contractual redemption-term indicators and indicator for months in top decile of VIX and MOVE distributions, estimated on monthly hedge fund flows. - Three contractual terms estimated jointly: lockup ≥ twelve months; redemption less frequent than monthly; notice period ≥ 45 days. Each coefficient is incremental effect holding other two constant. - Flows winsorized at 1st and 99th percentiles and expressed in basis points of AUM. - Positive coefficient = smaller net outflows during stress. - Standard errors two-way clustered by fund and month; bars show 90 percent confidence intervals. ### Macrofinancial effects of hedge fund stress — spillovers to prime brokers - Granular instrumental variable (GIV) approach (two steps, Gabaix and Koijen (2024) framework): - Step 1: For each prime broker, identify connected hedge funds and purge returns of common macrofinancial drivers via PCA: R_{i,t} = α_i + Σ λ_{i,k} f_{k,t} + ε_{i,t}. - Step 2: Construct broker-specific aggregate hedge-fund shock as AUM-weighted average of idiosyncratic innovations ε_{i,t} minus unweighted average; use as instrument in 2SLS estimating prime broker equity-return responses. - Second-stage regression: R'_{q,t+h} = γ_q + θ_h ĤAggHFreturns_{q,t} + β_h Controls_t + η_{q,t+h}, with h = 0,1,6. - Sample: February 2010 – February 2026. Standard errors clustered by broker and month. - First three principal components account for over 51 percent of variance in connected-fund returns across broker-specific estimations. - Constructed shocks not statistically significantly serially correlated; first-stage F-statistics exceed conventional thresholds. - Controls: changes in 10-year Treasury yield, log VIX, changes in US dollar index, Moody’s Baa corporate bond spread, and US equity total returns. - Findings: - Spillovers larger for prime brokers with riskier client profiles (client riskiness = average variance of past returns across connected hedge funds). - Client leverage measured as AUM-weighted share of clients with estimated leverage above strategy median. - Brokers split at median of time-averaged leverage; contrast sharper along leverage dimension. Effect on more leveraged-client brokers statistically significantly different from less leveraged-client brokers. ### Macrofinancial effects of hedge fund ownership on domestic equity returns under stress - Regression: Stock returns_{c,t} = α_c + α_t + β HFOwnership_{c,t-1} × GlobalShock_t + θ Controls_{c,t} + ε_{c,t}. - Dependent variable: percentage change in total equity return index of country c at quarter t. - HFOwnership_{c,t-1} = first lag of aggregate hedge fund ownership share (percentage points). - GlobalShock_t = 1 if VIX > its 90th percentile. - Controls: country and quarter fixed effects; policy rate, real GDP growth, inflation, credit-to-GDP gap; HFOwnership_{c,t-1}. - Standard errors clustered by country and quarter. - Sample: 2010:Q1–2025:Q4, covering 22 economies. - Aggregate hedge fund footprint from FactSet securities holdings data (equities); analogous mutual fund ownership measure constructed for comparison. - Liquidity heterogeneity: countries split at median of time-averaged Amihud (2002) illiquidity ratio. ### FX carry trades, crowdedness, and tail-risk local projections (Box 2.1) - Strategy index s regressions (36-month rolling windows): - r_{s,t} − r_{f,t} = α_{s,w} + β_{s,w} Carry_t + γ_{s,w} MktRF_t + ε_{s,t}. - Carry factor: S&P FX Carry G10 index in baseline. - Crowdedness indicator (per month): net share of 27 HFRI strategy classes with statistically significant positive minus negative carry loadings at 95 percent confidence level. - Difference-in-differences used to estimate post-unwind changes in Nikkei and carry loadings around August 2024 and January 2015 episodes. - Tail-risk local projections (10th-percentile quantile): - Q_{τ}(y_{t+h} | s_t, X_t) = α_h(τ) + β_h(τ) s_t + γ(τ)' X_t + ε_t, where y_{t+h} is average weekly percent change over next h weeks and s_t is standardized gross short position in funding currency. - Control vector X_t: 1-month OIS interest differential versus US dollar (basis points), 1-month at-the-money implied volatility, VIX, 4-week exchange-rate momentum, weekly broad-dollar return, MOVE index, and own-asset lagged weekly return. - Weekly sample: January 2013–June 2026. - Crowding amplification: interact positioning s_t with crowding dummy c_t (equals 1 when crowdedness indicator > historical upper tercile). - Robustness: pooled funding currencies, longer-horizon crowded-positioning states, replacing gross shorts with net shorts (similar results but net-short coefficients smaller), pooled high-yielding EM investment currencies (one-standard-deviation build-up in funding-currency shorts associated with worse lower-tail outcomes). --- ### BOX 2.2. IMPLICATIONS OF US TREASURY CASH-FUTURES BASIS OF HEDGE FUNDS ### Overview and objective - Examines hedge funds’ role in US Treasury cash-futures basis trade and implications for Treasury market functioning and financial stability. - Measures hedge fund exposure to basis trading using sensitivity of hedge fund returns to Treasury net-basis volatility and assesses heterogeneity across strategies, leverage, fund size, and domicile. - Evaluates whether leveraged basis traders absorb Treasury dislocations in normal conditions but amplify them during stress. ### Measurement of Treasury net basis and volatility - 10-year Treasury cash-futures basis measured using Bloomberg’s Cheapest-to-Deliver (CTD) Net Basis (captures price dislocation between CTD Treasury security and corresponding futures contract after adjusting for carry). - Daily basis changes calculated only within the same futures contract. - Monthly basis volatility = square root of sum of squared daily basis changes within each month; series standardized as proxy for Treasury basis-trade condition fluctuations. - Bloomberg’s carry calculation uses bond-specific term repo rate for futures delivery horizon and incorporates coupon and accrued-interest effects. ### Estimating hedge fund footprint in basis trading - Rolling regressions (36-month window) of monthly hedge fund strategy returns on Treasury basis volatility and controls: - R_t = α + β1 BasisVol_t + β2 BasisVol_{t-1} + Γ X_t + ε_t (equation 2.25). - Strategy returns from HFRI indices (aggregate hedge fund, relative value, macro). - Controls: broad equity returns; trend-following bond strategy returns; changes in high-yield credit spreads; changes in short-term interest rates; commodity returns; emerging market equity returns. - Treasury basis-trade footprint measured by partial R²: Partial R² = 1 − SSR_full / SSR_restricted (equation 2.26). - Joint F-test on basis-related coefficients; periods with joint significance highlighted in main-text figures. ### Relation between footprint and leverage, fund size, domicile - Rolling partial R² aggregated to quarterly frequency and compared with contemporaneous OFR leverage measures for relative value and macro strategies. - Offshore and asset-weighted HFR relative value sub-indices proxy for offshore-domiciled funds and larger funds. ### Three exercises testing state-dependent effects of leveraged positioning 1) Distributional effects via quantile local projections - Quantile specification: - Q_τ(Y_{t+h} | F_t, X_t) = α_{τ,h} + β_{τ,h} F_t + γ′_{τ,h} X_t + ε_{t+h} (equation 2.27). - Y_{t+h} = change in 10-year net basis; F_t = standardized four-week build-up in leveraged positioning. - Controls: standardized level of leveraged book, lagged basis changes, lagged VIX, MOVE level and change, issuance-wave control, auction-size surprise. - Hedge fund activity measured using CFTC leveraged-fund Treasury futures positions converted into DV01 and scaled by privately held marketable Treasury debt. 2) Interaction with stress and Federal Reserve footprint - Specification: - Y_{t+h} = α_h + β_h F_t + δ_h D_t + κ_h fp_t + θ_h F_t D_t + λ_h F_t fp_t + μ_h D_t fp_t + ν_{h} F_t D_t fp_t + γ′_h X_t + ε_{t+h} (equation 2.28). - D_t = VIX-based stress indicator. - fp_t = Federal Reserve footprint in Treasury market (share of SOMA Treasury holdings relative to marketable debt). - Compare effect of positioning build-up in calm periods, in stress at average footprint, and in stress when footprint is one standard deviation smaller. 3) Daily supply shocks and dealer capacity - Specification: - Y_{t+h} = α_h + β_h S_t + ρ_h ΔE_t + χ_h C_t + φ_h (S_t × ΔE_t) + ψ_h (S_t × ΔE_t × C_t) + ζ′_h Z_t + γ′_h X_t + ε_{t+h} (equation 2.29). - S_t = auction-size surprise (DV01-weighted change in announced offering size). - ΔE_t = change in standardized positioning-to-debt ratio. - C_t = 1 when dealer utilization in top tercile. - Z_t includes remaining pairwise interactions. - Supply shocks defined as DV01-weighted change in offering size relative to previous auction of same tenor and reopening status; effects cumulative over horizon. ### Definitions and data construction - Treasury net-basis volatility: monthly realized volatility of Bloomberg’s 10-year Treasury futures net basis, computed from daily same-contract net basis changes. - Positioning-to-debt ratio: DV01-weighted leveraged-fund net short Treasury futures positions scaled by privately held marketable Treasury debt outstanding. ### Empirical presentation - Results presented in main-text figures (Figure 2.2.2, panels 1–3) and a figure reporting 36-month rolling partial R² estimates for relative value hedge fund returns (Figure 2.3.11). - Solid (dotted) lines denote periods when Treasury net-basis volatility factors are jointly significant (insignificant) at the 10 percent level. *Global Financial Stability Report — Chapter 2 Online Annex, IMF | October 2026* ### Annex 2.1 – Glossary ### Annex 2.1 – Glossary ### Key technical terms and definitions - Arbitrage strategies - Trading strategies that seek to profit from pricing inconsistencies between related securities. Arbitrage generally involves buying an undervalued asset and selling a related overvalued asset, expecting the price gap to narrow over time. - Hedge fund - The term hedge fund describes a business model rather than a uniform legal category. It generally refers to a privately offered pooled vehicle for institutional or high-net-worth investors with broad flexibility to use leverage, derivatives, short selling, concentrated positions, and less-liquid investments. By contrast, public mutual funds generally serve retail investors and are subject to prospectus disclosure, tighter leverage requirements, and more frequent redemptions. Hedge funds may also face fewer product-level restrictions and may charge performance fees or impose lockups, notice periods, gates, and side pockets. These are typical rather than defining features, and the precise legal perimeter varies across jurisdictions. In the United States, for instance, hedge funds are private funds generally limited to accredited investors or qualified purchasers and classified under Form PF based on performance-fee arrangements and their permitted use of leverage or short selling; in the European Union and United Kingdom, they form part of the broader alternative investment fund category; and in the Cayman Islands, fund classification depends primarily on redemption rights, so an open-ended hedge fund falls within the legal definition of a “mutual fund,” without implying a retail offering. - Hedge fund strategies - Main hedge fund strategy groups include credit funds, which focus on fixed-income securities; equity funds, which take long and short positions in equities; event-driven funds, which trade on mergers, restructurings, distress, and corporate events; macro funds, which take positions across rates, foreign exchange, commodities, and equity markets; managed futures/CTA funds, which trade futures contracts, options, and foreign exchange derivatives; relative value funds, which exploit pricing gaps between related securities; and multi-strategy funds, which allocate capital across several strategies and asset classes. - Margin / haircut - Margin is collateral required to support a leveraged position; a haircut is the discount applied to collateral value in secured funding. - Prime brokers - Financial intermediaries, typically broker-dealer units of large global banking groups, which provide hedge funds and other institutional investors with a range of financing and operational services. These can include leverage through margin loans, financing, securities lending, custody, clearing, and execution services. - Synthetic leverage - Synthetic leverage refers to market exposure obtained through derivatives rather than direct borrowing. It is generally measured using derivatives positions expressed in terms of their equivalent exposure to the underlying assets relative to funds’ gross assets. *Global Financial Stability Report — Chapter 2 Online Annex, IMF | October 2026* ### 1. Small and Large Hedge Funds ### 1. Small and Large Hedge Funds ### Effect of Hedge Fund Ownership on Bid-Ask Spreads - Panels estimate the effect of a 10-percentage-point increase in hedge fund ownership on bid-ask spreads in the subsequent quarter (basis points). - Estimates come from security-level panel regressions of subsequent changes in bid-ask spreads on changes in aggregate hedge fund ownership, controlling for market capitalization, turnover, price-to-book ratios, dividends, profitability, and the initial bid-ask spread, with security and quarter fixed effects. - Large (small) funds are those managed by hedge fund managers with above- (below-) median assets under management. - High- (low-) turnover funds are funds with above- (below-) median portfolio turnover, where turnover is measured as one-half of the sum of absolute changes in portfolio weights between consecutive reporting dates (captures the share of the portfolio reallocated between reporting periods). - All classifications are determined relative to the hedge fund median in each quarter. - Error bars denote 90 percent confidence intervals. - Stress periods are defined as quarters in which the average VIX exceeds its 90th percentile. ### Strategy Heterogeneity, Leverage, and Market Effects - Hedge fund strategies differ markedly in size, leverage, trading horizons, portfolio concentration, and portfolio liquidity; these differences imply strategy-specific effects on market functioning. - Key qualitative findings: - Relative value funds stand out for their high leverage and rapid portfolio turnover. - Macro and managed futures/CTA funds exhibit elevated leverage. - Multi-strategy funds are among the largest managers. - Event-driven funds tend to hold more concentrated portfolios. - Implications: - Strategies characterized by larger size or longer holding horizons tend to be more closely associated with subsequent price adjustment, consistent with a greater role in information discovery. - High-turnover strategies appear more closely associated with improvements in trading efficiency during normal periods. - High leverage of macro, relative value, and managed futures funds may increase their potential to amplify market strains when positions are unwound during periods of stress. ### Return-Implied Leverage Measure (monthly fund-level leverage construction) - Objective: recover variation in leverage across funds and over time while ensuring consistency with aggregate strategy-level leverage estimates from the OFR and reported fund leverage categories. - Step 1 — Return-implied leverage score: - Constructed from fund returns and a broad set of risk factors. - Score is based on fund return behavior characteristics, including volatility, downside risk, tail losses, drawdowns, and factor exposures. - Formally: Score_{i,t} = f(X_{i,t}) (Equation (2.4)), where X_{i,t} denotes return- and risk-based characteristics for fund i in month t and f(·) is estimated from strategy-specific logistic models. - Fund characteristics include mean excess return, volatility, downside deviation, skewness, excess kurtosis, expected shortfall, maximum drawdown, share of near-zero returns, and factor-exposure measures (equity, volatility, interest-rate, credit, foreign-exchange, commodity, liquidity, intermediary, trend-following, Treasury basis-risk). - Periods of large factor movements are defined as months in which the average absolute standardized movement across the factor set exceeds its 80th percentile. - Step 2 — Translate score into unconstrained leverage target: - Center the score within each strategy-month using assets under management as weights (so the score determines distribution across funds within a strategy, not the aggregate level). - Construct unconstrained leverage target: L*_{i,t} = L^{OFR}_{s,t} + κ_s Scorẽ_{i,t} (Equation (2.5)), where L^{OFR}_{s,t} denotes OFR leverage for strategy s, Scorẽ_{i,t} is the strategy-month centered leverage score, and κ_s is a strategy-specific scaling parameter. - Step 3 — Impose consistency with reported leverage categories and OFR aggregates: - Funds with below 5 times leverage (1-2 and 2-5 categories) are constrained to remain within their reported leverage ranges. - For higher leverage categories, the reported category provides only a lower-bound restriction. - Funds with non-standard, unspecified, or missing leverage labels are retained but are not assigned category-specific restrictions beyond non-negativity. - Project unconstrained targets onto the feasible set defined by leverage-category restrictions and the requirement that the AUM-weighted average leverage within each strategy-month equals OFR leverage: - L_{i,t} = P(L*_{i,t}) subject to Σ_i w_{i,t} L_{i,t} = L^{OFR}_{s,t} (Equation (2.6)), where w_{i,t} denotes the fund’s share of strategy AUM and P(·) denotes the projection operator. ### Equity-Level Hedge Fund Vulnerability Measures Three quarterly stock-level measures constructed from security-level institutional holdings, each reported as within-quarter percentile ranks and entering regressions as effects of moving a stock from bottom to top of the cross-sectional distribution. - Notation: - m_{h,j,t} denotes dollar position of hedge fund manager h in stock j at end of quarter t. - θ_{j,t} denotes market capitalization of stock j. - HF_{j,t} = Σ_h m_{h,j,t}; pso_{j,t} = HF_{j,t} / θ_{j,t} (Equation (2.7)). - Crowdedness (Crowd-ADV): - Purpose: measure a stock’s exposure to crowded liquidation by the hedge funds that own it (following Brown, Howard, and Lundblad (2022)). - Step 1: Days-ADV_{j,t} = HF_{j,t} / ADV_{j,t} (Equation (2.8)), where ADV_{j,t} is the 20-day moving average daily dollar trading volume averaged over the quarter; Days-ADV is the number of trading days the hedge fund industry would need to exit its position at average daily volume. - Step 2: FundDaysADV_{h,t} = Σ_j w_{h,j,t} Days-ADV_{j,t} with w_{h,j,t} = m_{h,j,t} / Σ_q m_{h,q,t} (Equation (2.9)). - Aggregate to stock: CrowdADV_{j,t} = Σ_h (m_{h,j,t} / θ_{j,t}) × FundDaysADV_{h,t} (Equation (2.10)). - Interpretation: Crowd-ADV rises with hedge fund ownership of the stock and with how crowded those owners’ portfolios are. - Investor Flow Fragility: - Purpose: measure how large a swing in demand a stock could face if its hedge fund owners were hit by redemptions simultaneously (adapts Greenwood and Thesmar (2011)). - Let w_{j,t} be vector of dollar positions in stock j across hedge fund owners; Ω_t is the covariance matrix of owner-level investor flow. - Fragility: G_{j,t} = 1/θ_{j,t} × w_{j,t}^T Ω_t w_{j,t} (Equation (2.11)). - Owner flow f_{h,t} = (A_{h,t} − A_{h,t−1}(1 + r_{h,t})) / A_{h,t−1} (Equation (2.12)), where A denotes assets and r returns. - Ω_t estimated via factor structure: common flow factors are AUM-weighted mean flow rates within each investment strategy; Ω_t = B_t Σ_{f,t} B_t^T + diag(σ^2_{ε,h,t}) (Equation (2.13)). - Resulting measure used as √G_{j,t} in regressions (units of a return) and as a within-quarter percentile rank. - Leverage Exposure: - Purpose: measure a stock’s exposure to levered funds by weighting each owner’s stake by that owner’s estimated leverage. - Leverage exposure: LevExposure_{j,t} = Σ_h (m_{h,j,t} / θ_{j,t}) L_{h,t} (Equation (2.15)), where L_{h,t} is the estimated leverage of manager h. - Note: leverage is aggregated to the manager as an AUM-weighted average across funds mapped to that manager; since leverage is estimated, standard errors on its coefficients may be underestimated. ### Vulnerability Regression Analysis (timing and interpretation) - All three measures are constructed from information available at end of quarter t and related to outcomes over quarter t+1 (predetermined). - Each measure enters as a within-quarter percentile rank; a coefficient represents the effect of moving a stock from the bottom to the top of that measure’s cross-sectional distribution in the quarter. - Figures referenced report effects on equity returns and volatility across Advanced Economies (AEs) and Emerging Markets (EMs), and separately for largest 50 managers and others: - Both large and small hedge funds significantly amplify tail risk in equity markets during stress. - The contribution of small funds is more pronounced, especially through the leverage channel. - The top 50 hedge funds account for roughly half of the sector’s AUM. ### Redemption Restrictions and Hedge Fund Capital Stability - Hedge funds limit investor redemptions through three contractual channels: initial lockup period, notice period, and redemption frequency. - Redemption terms vary systematically with management firm characteristics; stricter restrictions are adopted by hedge funds that are smaller or less leveraged. - Empirical flow patterns during stress (months when VIX is in top decile): - Funds with lockup periods of twelve months or less record average net outflows of about 82 basis points of assets under management. - Funds with longer lockup periods record average net outflows of about 12 basis points. - Regression specification for fund-level net flows (basis points) over horizon h ∈ {0,1}: - Y_{i,t+h} = α_i + μ_{s,t} + β_1 Rigidity_{i,t−1} + β_2 (Stress_t × Rigidity_{i,t−1}) + β_3 Controls_{i,t−1} + ε_{i,t+h} (Equation (2.16)). - Stress_t is binary equal to one when the VIX or MOVE1M index is in the top decile of its sample distribution. - Controls include lagged log AUM, past 12-month returns, return volatility, and fund age; μ_{s,t} are strategy-time fixed effects; α_i are fund fixed effects; standard errors clustered at fund and time levels. - Regression findings on redemption restrictions during stress: - Funds with lockup periods of at least twelve months record net flows that are 61 basis points higher than otherwise similar funds (i.e., they experience smaller outflows). - Funds with redemption frequencies longer than one month record net flows that are 87 basis points higher than otherwise similar funds. - Longer notice periods have no additional effect once lockups and redemption frequency are controlled for. - The same qualitative pattern holds, with somewhat smaller magnitudes, when stress is measured using Treasury market volatility rather than equity market volatility, indicating the mechanism is not specific to equity market disruptions. _Global Financial Stability Report Chapter 2 Online Annex, IMF | October 2026_ ### 1. Fund Flow Response to VIX Stress, by Funding-Term ### ch2annex - 1. Fund Flow Response to VIX Stress, by Funding-Term ### Fund flow regressions and key measurement details - Panels report interaction between lagged indicators of contractual redemption terms and an indicator for months in the top decile of the VIX and MOVE distributions, estimated on monthly hedge fund flows. - Three contractual terms estimated jointly: a lockup period of at least twelve months; redemption less frequent than monthly; and a notice period of at least 45 days. Each coefficient is the incremental effect of that term holding the other two constant. - Flows are winsorized at the 1st and 99th percentiles and expressed in basis points of assets under management. - Positive coefficient interpretation: smaller net outflows during stress. - Standard errors are two-way clustered by fund and month. Bars show 90 percent confidence intervals. ### Macrofinancial effects of hedge fund stress — spillovers to prime brokers - Spillover analysis implemented using a granular instrumental variable (GIV) approach in two steps (Gabaix and Koijen (2024) framework): - Step 1: For each prime broker, identify connected hedge funds and purge their returns of common macrofinancial drivers via factor model using principal component analysis (PCA): R_{i,t} = α_i + Σ λ_{i,k} f_{k,t} + ε_{i,t}. - Step 2: Construct broker-specific aggregate hedge-fund shock as the difference between an AUM-weighted average of the idiosyncratic innovations ε_{i,t} and the unweighted average; use this GIV shock as instrument in 2SLS estimating prime broker equity-return responses. - Second-stage regression: R'_{q,t+h} = γ_q + θ_h ĤAggHFreturns_{q,t} + β_h Controls_t + η_{q,t+h}, with h = 0,1,6. - The coefficient θ_h measures the strength of the spillover from the hedge fund sector to prime brokers. - Sample period: February 2010 – February 2026. Standard errors clustered by broker and month. - First three principal components account for over 51 percent of the variance in connected-fund returns across broker-specific estimations. - Constructed shocks are not statistically significantly serially correlated; first-stage F-statistics exceed conventional thresholds (instrument strength satisfied). - Regression controls include changes in the 10-year Treasury yield, the log of the VIX, changes in the US dollar index, Moody’s Baa corporate bond spread, and US equity total returns. - Spillovers are larger for prime brokers with a riskier client profile: - Client riskiness measured as average variance of past returns across connected hedge funds. - Client leverage measured as AUM-weighted share of clients whose estimated leverage exceeds the median for their strategy. - Brokers split at the median of time-averaged leverage; contrast sharper along the leverage dimension. The effect on more leveraged-client brokers is statistically significantly different from less leveraged-client brokers. ### Macrofinancial effects of hedge fund ownership on domestic equity returns under stress - Regression estimated: Stock returns_{c,t} = α_c + α_t + β HFOwnership_{c,t-1} × GlobalShock_t + θ Controls_{c,t} + ε_{c,t}. - Dependent variable: percentage change in total equity return index of country c at quarter t. - HFOwnership_{c,t-1} is the first lag of aggregate hedge fund ownership share (both in percentage points). - GlobalShock_t is a dummy equal to 1 if the VIX is above its 90th percentile. - Controls: country and quarter fixed effects; time-varying domestic macrofinancial factors (policy rate, real GDP growth, inflation, credit-to-GDP gap) and HFOwnership_{c,t-1}. - Standard errors clustered by country and quarter. - Sample period: 2010:Q1–2025:Q4, covering 22 economies. - Aggregate hedge fund footprint constructed from FactSet securities holdings data focusing on equities; an analogous ownership measure is constructed for mutual funds for comparison. - Liquidity heterogeneity: countries split at the median of the time-averaged Amihud (2002) illiquidity ratio (average absolute daily return per dollar of trading volume, market-cap-weighted across listed stocks); subsample regressions retain same fixed effects and macrofinancial controls. ### FX carry trades, crowdedness, and tail-risk local projections (Box 2.1) - Empirical approach for strategy index s: regress monthly excess returns over 36-month rolling windows on the carry factor and the market factor: - r_{s,t} − r_{f,t} = α_{s,w} + β_{s,w} Carry_t + γ_{s,w} MktRF_t + ε_{s,t}. - Carry factor: S&P FX Carry G10 index in baseline. - Crowdedness indicator (per month): net share of 27 HFRI strategy classes with statistically significant positive minus negative carry loadings at the 95 percent confidence level; formula provided (s = 27). - Difference-in-differences specification used to estimate post-unwind changes in Nikkei and carry loadings for offshore and onshore funds around August 2024 and January 2015 episodes. - Tail-risk local projections (10th-percentile quantile) estimated as: - Q_{τ}(y_{t+h} | s_t, X_t) = α_h(τ) + β_h(τ) s_t + γ(τ)' X_t + ε_t, where y_{t+h} is average weekly percent change over next h weeks and s_t is standardized gross short position in funding currency. - Control vector X_t includes: 1-month OIS interest differential versus the US dollar (basis points), 1-month at-the-money implied volatility, the VIX, 4-week exchange-rate momentum, weekly broad-dollar return, the MOVE index, and own-asset lagged weekly return. - Weekly sample: January 2013–June 2026. - Crowding amplification tested by interacting positioning s_t with a crowding dummy c_t that equals one when the crowdedness indicator is above its historical upper tercile; equation augmented accordingly. - Robustness: pooled funding currencies, longer-horizon crowded-positioning states, replacing gross shorts with net shorts (similar results but net-short coefficients smaller), and pooled high-yielding EM investment currencies as outcomes (one-standard-deviation build-up in funding-currency shorts associated with worse lower-tail outcomes for investment currencies). ### Additional empirical and data notes - Flows expressed in basis points of assets under management. - Winsorization at the 1st and 99th percentiles applied to flow data. - Redemption-restriction trends: the AUM share of funds with a rigidity index of at least 3 has declined since the mid-2000s; the share with lockups of at least 12 months fell from about 40 percent in the early 2000s to about 15 percent recently. - For comparisons, analysis repeated for non-prime-broker banks by constructing the GIV shock from the full universe of hedge funds; non-prime broker bank returns are not significantly affected by hedge fund shocks at any horizon. - Data sources cited: HFR; Bloomberg Finance L.P.; CFTC; EUROPACE AG/Haver Analytics; FactSet; Kenneth French Data Library; Haver Analytics; and IMF staff calculations. *Global Financial Stability Report Chapter 2 Online Annex, IMF | October 2026* ### BOX 2.2. IMPLICATIONS OF US TREASURY CASH-FUTURES BASIS OF HEDGE FUNDS ### BOX 2.2. IMPLICATIONS OF US TREASURY CASH-FUTURES BASIS OF HEDGE FUNDS ### Overview and objective - The box examines the growing role of hedge funds in the US Treasury cash-futures basis trade and its implications for Treasury market functioning and financial stability. - It measures evolution of hedge fund exposure to basis trading using the sensitivity of hedge fund returns to Treasury net-basis volatility and assesses heterogeneity across strategies, leverage, fund size, and domicile. - It evaluates market implications of hedge fund positioning, focusing on whether leveraged basis traders absorb Treasury market dislocations in normal conditions but amplify them during periods of stress. ### Measurement of the Treasury net basis and volatility - The 10-year Treasury cash-futures basis is measured using Bloomberg’s Cheapest-to-Deliver (CTD) Net Basis that captures price dislocation between the CTD Treasury security and the corresponding futures contract after adjusting for the carry associated with holding and financing the cash security to delivery. - Daily basis changes are calculated only within the same futures contract. - Monthly basis volatility is measured as the square root of the sum of squared daily basis changes within each month. - The resulting series is standardized and used as a proxy for fluctuations in Treasury basis-trade conditions. - Bloomberg’s carry calculation uses a bond-specific term repo rate corresponding to the futures delivery horizon and incorporates the relevant coupon and accrued-interest effects. ### Estimating hedge fund footprint in basis trading - Hedge fund exposure is estimated with rolling regressions of monthly hedge fund strategy returns on Treasury basis volatility and a set of control variables. - Strategy returns are measured using HFRI indices, including aggregate hedge fund, relative value, and macro strategy benchmarks. - Control variables include: - broad equity market returns, - trend-following bond strategy returns, - changes in high-yield credit spreads, - changes in short-term interest rates, - commodity returns, - emerging market equity returns. - Regressions use a 36-month rolling window and the specification is: - R_t = α + β1 BasisVol_t + β2 BasisVol_{t-1} + Γ X_t + ε_t (equation 2.25) - The Treasury basis-trade footprint is the incremental explanatory power of Treasury basis volatility for hedge fund returns, measured with the partial R²: - Partial R² = 1 − SSR_full / SSR_restricted (equation 2.26) - A joint F-test on the basis-related coefficients is conducted within each rolling window; periods when basis variables are jointly significant are highlighted in main-text figures. ### Relation between footprint and leverage, fund size, domicile - Rolling partial R² estimates are aggregated to quarterly frequency and compared with contemporaneous leverage measures for relative value and macro strategies from OFR. - This assesses whether higher leverage periods are associated with greater sensitivity of strategy returns to Treasury basis-trade conditions, consistent with the use of leverage to scale Treasury basis positions. - Offshore and asset-weighted HFR relative value sub-indices are used to proxy for offshore-domiciled funds and larger funds, respectively. ### Three exercises testing state-dependent effects of leveraged positioning - Central hypothesis: hedge fund arbitrage capital is stabilizing when dealer balance sheets have room to finance the trade, but destabilizing when that capacity is constrained. 1) Distributional effects via quantile local projections - Tests whether positioning affects the full distribution of subsequent basis changes by estimating quantile local projections at the 10th, 50th, and 90th percentiles over horizons from one to eight weeks. - Baseline model (quantile specification): - Q_τ(Y_{t+h} | F_t, X_t) = α_{τ,h} + β_{τ,h} F_t + γ′_{τ,h} X_t + ε_{t+h} (equation 2.27) - Y_{t+h} is the change in the 10-year net basis and F_t is the standardized four-week build-up in leveraged positioning. - X_t controls include standardized level of the leveraged book, lagged basis changes, lagged VIX, MOVE level and change, an issuance-wave control, and the auction-size surprise. - Hedge fund activity in the basis trade is measured using CFTC leveraged-fund Treasury futures positions, converted into DV01 and scaled by privately held marketable Treasury debt. 2) Interaction with stress and Federal Reserve footprint - Tests whether the effect of leveraged positioning is stronger under stress and amplified when the Federal Reserve’s Treasury-market footprint is smaller. - Specification: - Y_{t+h} = α_h + β_h F_t + δ_h D_t + κ_h fp_t + θ_h F_t D_t + λ_h F_t fp_t + μ_h D_t fp_t + ν_{h} F_t D_t fp_t + γ′_h X_t + ε_{t+h} (equation 2.28) - D_t is a VIX-based stress indicator. - fp_t is the Federal Reserve footprint in the Treasury market, measured by the share of SOMA Treasury holdings relative to marketable debt. - The plotted paths compare the effect of a positioning build-up in calm periods, in stress at the average footprint, and in stress when the footprint is one standard deviation smaller. 3) Daily supply shocks and dealer capacity - Focuses on Treasury supply shocks and dealer capacity at a daily frequency. - Specification: - Y_{t+h} = α_h + β_h S_t + ρ_h ΔE_t + χ_h C_t + φ_h (S_t × ΔE_t) + ψ_h (S_t × ΔE_t × C_t) + ζ′_h Z_t + γ′_h X_t + ε_{t+h} (equation 2.29) - S_t is the auction-size surprise (DV01-weighted change in announced offering size). - ΔE_t is the change in the standardized positioning-to-debt ratio. - C_t = 1 when dealer utilization is in its top tercile. - Z_t includes the remaining pairwise interactions. - Supply shocks are defined as DV01-weighted change in offering size relative to the previous auction of the same tenor and reopening status. - Effects are cumulative over the horizon. ### Definitions and data construction - Treasury net-basis volatility: monthly realized volatility of Bloomberg’s 10-year Treasury futures net basis, computed from daily changes in the same-contract net basis to exclude mechanical variation associated with futures contract rolls. - Positioning-to-debt ratio: DV01-weighted leveraged-fund net short Treasury futures positions scaled by privately held marketable Treasury debt outstanding. ### Empirical presentation - Results are presented in main-text figures (Figure 2.2.2, panels 1–3) and in a figure reporting 36-month rolling partial R² estimates for relative value hedge fund returns (Figure 2.3.11). - Solid (dotted) lines denote periods when Treasury net-basis volatility factors are jointly significant (insignificant) at the 10 percent level. --- _Source: https://www.imf.org/-/media/files/publications/gfsr/2026/october/english/ch2annex.pdf_