## annex1-1

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---

### A. Financial Conditions Indices (FCIs)
- Purpose and evolution
  - Original methodology developed in the April 2017 GFSR; applied with modifications in October 2017 GFSR.
  - Methodology simplified to enable computation of regional aggregates and identification of contributions of underlying FCI components to country aggregate FCIs.
- Economy coverage
  - Sample includes the 29 systemically important jurisdictions in the IMF’s Financial Sector Assessment programs, the original FCI sample of 21 economies, and the top 20 constituents of the EMBIG index.
  - Final sample totals: 43 AE / 29 EM / 21 SIJ / 20 EMBIG-20 (table shows per-country listing; totals shown as "total43292120" in source).
- FCI components (revised from Oct 2017 price-of-risk components)
  - Additions and changes:
    - Sovereign and corporate spreads on local debt for EMs were added.
    - Realized equity volatility replaced with implied volatility based on option prices (such as the VIX) where available.
    - Equity and house price variables adjusted to measure levels (rather than changes) for consistency.
  - Component measurement matrix (variable — measurement — presence by group):
    - real short-term interest rate — 3 month T-bill yield minus CPI yoy — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: +
    - interbank spread — Interbank rate (Libor) minus T-bill yield — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: +
    - term spread — 5-year govt bond yield minus T-bill yield — US, Germany: 1; Other AE: (blank); EM: +
    - sovereign local debt spread — 5-year yield minus US or Germany yield — US, Germany: 1; Other AE: 1; EM: + for AE
    - sovereign dollar debt spread — EMBI spread — US, Germany: 1; Other AE: (blank); EM: +
    - corporate local currency spread — ICE OAS — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + for AE
    - corporate dollar debt spread — CEMBI spread — US, Germany: 1; Other AE: (blank); EM: +
    - equity prices — MSCI P/B — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + (qoq)
    - equity vol — VIX/V2X/VNKY — US, Germany: 1; Other AE: (blank); EM: + all
    - exchange rate — debt-weighted exchange rate — US, Germany: 1; Other AE: (blank); EM: + (vis-à-vis USD)
    - real house prices — BIS house prices yoy minus CPI yoy — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + (qoq)
- FCI weights and aggregation
  - Time-varying weights from a dynamic factor model replaced with fixed weights from a principal component analysis (PCA).
    - Rationale: improved tractability, simplified computation of contributions of FCI drivers, and greater parameter stability.
  - Global aggregation: PPP GDP weights employed instead of PCA on full sample of country-level FCIs.
    - Advantage: allows additivity of countries to regional aggregates and from regions to global total.
  - Effect of revisions: relatively small changes to the global FCI; revised FCI better captures abrupt changes in components.
- Illustrative series
  - Global Price-of-Risk FCI plotted as Standard deviation from mean, 1996:Q1-2018:Q3 (figure compares Original and Revised series).

### B. Capital-Flows-at-Risk (CaR)
- Framework objective
  - Forward-looking assessment of risks to emerging market (EM) capital flows: what global financial conditions today imply about the expected future distribution of capital flows.
  - Uses quantile regression to quantify downside and upside risks to future capital flows conditional on prevailing global financial conditions.
  - Allows analysis of whether drivers of inflow surges differ from drivers of capital flows reversals.
- Data and scope
  - Focus: portfolio debt flows (gross portfolio debt inflows = net non-resident purchases of EM debt instruments).
  - Sample period and coverage: quarterly balance of payments data from 1997Q2 to 2017Q4 for about 60 emerging market and developing countries (China excluded).
  - Flows measured in US dollars, scaled by EM GDP and aggregated across countries.
  - Time horizons:
    - "Medium-term": period from 5 to 8 quarters ahead, averaging inflows as a share of EM GDP over these four quarters.
    - "Near-term" (also reported in GFSR): average flows over the current and the next two quarters.
- Independent variables (push and pull factors) — preferred specification
  - Push factors:
    - investor risk aversion proxied by US BBB-rated corporate bond spread;
    - market interest rates: US 10-year Treasury yields (de-trended using a Hodrick-Prescott filter);
    - the US dollar: DXY dollar index.
  - Domestic/control variables:
    - real GDP growth in emerging market economies (excluding China).
    - lagged inflows (preceding four quarters) and a constant term.
  - Additional variables tested (not included in preferred specification due to lower significance or similar results): market-implied expectations of the federal funds rate, slope of the US yield curve, VIX, real effective exchange rate, Brent oil price, Bloomberg commodity price index, EM exchange rate index, US growth, Fed balance sheet.
- Model specification
  - Quantile regression setup: y_{t+h} (average portfolio debt inflows to EMs in % of GDP between t and t+h) regressed on x_t (vector of independent variables).
  - Estimation: quantile regression minimizes quantile-weighted absolute errors; predicted value gives conditional quantile.
  - Empirical distribution obtained by considering quantiles from the 1st to the 99th percentile.
  - Capital flows-at-risk (CaR) defined as lower 5th percentile of the predicted distribution:
    - Pr(y_{t+h} ≤ CaR_h(α | Ω_t)) = α with α = 5th percentile (i.e., 5 percent probability that capital flows would be lower than CaR).
  - Estimates for the 5th percentile taken directly from quantile regressions; probability density functions obtained by mapping quantile regression estimates into a smoothed Gaussian (normal) distribution.
- Results and interpretation
  - Three main predictive factors for medium-term portfolio debt flows to EMs:
    - investor risk appetite (risk aversion);
    - US long-term interest rates;
    - US dollar.
  - Relationships:
    - Higher US interest rates and a stronger US dollar → weaker inflows, contemporaneously and in the medium term.
    - Stronger risk appetite → greater inflows contemporaneously, but predicts weaker inflows in the medium term (possible mean-reversion in risk appetite).
  - Extreme movements (surges and reversals):
    - Future reversals and, to a lesser extent, surges are disproportionately explained by investor risk aversion (higher coefficient estimates at lowest and highest quantiles).
    - US interest rates and the dollar have less predictive power for capital flow reversals (typically at lowest percentiles).
  - Current assessment (as of dataset reference):
    - Downside risks to medium-term capital flows are currently high, driven by relatively elevated US interest rates, a strong dollar, and buoyant risk appetite.
    - Estimated medium-term capital flows at risk of 0.6 percent of the combined GDP of EMEs (excluding China), on par with the Global Financial Crisis (measured over a four-quarter period).
    - Comparison example: late 2011:Q4 (height of European sovereign debt crisis) had low US interest rates, weaker dollar, but high risk aversion and a more benign medium-term CaR relative to the current assessment.
- Figures and diagnostics
  - B.1. Portfolio Debt Flows to EMs Excluding China (Percent of GDP).
  - B.2. Drivers of Medium-Term Capital Flows (time series of risk aversion (BBB), US 10-year yields (de-trended), and Dollar index).
  - B.3. Estimated Coefficients by Quantile (shows larger risk-aversion coefficients at extremes).
  - B.4. Medium-Term Capital Flows Forecast Densities (probability density with 5th percentile indicated).

### C. Bank Solvency Simulations
- Purpose and sample
  - Forward-looking assessment of bank solvency via simulations estimating stress capital ratios in the following year using current balance sheets and distributions of changes in capital from historical bank profitability.
  - Sample: about 600 banks headquartered in advanced economies; sample size varies over time (mergers, new banks, failures).
  - Regional sample breakdown:
    - Euro area: 175 banks.
    - Other Europe: 65 banks.
    - Asia and Pacific: 105 banks.
    - North America: 260 banks.
- Profitability distributions and shock construction
  - Historical data from 1990 onwards used to build distributions of bank profitability.
  - Profits divided into four categories (all calculated as percentages of relevant balance sheet items):
    - (i) net interest income to loans and securities;
    - (ii) trading income to securities;
    - (iii) provisions to gross loans;
    - (iv) other net income to assets.
  - Shocks measured as year-over-year changes in profitability; ten different distributions calculated for each profit category based on current profitability level (to capture stage-in-cycle effects).
  - Distributions illustrated in Figure C.2 (shocks to bank profits, percentage points).
- Simulation method
  - For each bank: take 10,000 draws from profitability distributions.
  - Apply shocks to balance-sheet-relevant items and add to current profits to estimate next-year profits.
  - Dividend rule: bank pays dividends at current-year rate if it makes a profit; no dividends if it makes a loss.
  - Retained earnings added to current capital to estimate capital at end of next year.
  - Assumption: average risk weight of assets remains the same as in the current year.
  - Simulations run for each year since 2006 using balance sheet, income, and profitability distributions at that point in time.
- Capital shortfall testing and reporting
  - Estimated capital in each simulation tested against two thresholds:
    - Tier 1 ratio of 4.5 percent;
    - leverage ratio of 3 percent.
  - For GSIBs, capital surcharges are added to the Tier 1 ratio threshold.
  - Each simulation where a bank’s capital ratio falls below either threshold is counted.
  - Outcome metric: estimated probability of a capital need over the following year.
  - Results presented as the proportion of sample banks (by assets) with a probability of a capital need that is 20 percent or higher in the simulations.
  - Note: the same capital ratio thresholds are used over time to enable comparison, though thresholds are higher than regulatory ratios in place in pre-crisis and crisis years.
- Conclusion on method
  - Provides a forward-looking assessment of bank solvency as an alternative to assessing current capitalization.
  - Not intended as a substitute for detailed stress tests but offers a useful risk assessment of bank capitalization based on historical experience.

### Jump risk: overview (Section 2)
- Market microstructure context
  - Market microstructure has shifted in recent years due to regulation, evolving balance sheet capacities of financial intermediaries, and an increase in algorithmic trading.
  - Daily aggregate liquidity measures appear relatively benign, but analysis assesses implications for high-frequency asset price dynamics using intraday data.
  - Aim: gauge the extent to which microstructure changes systematically impact intraday price evolution.
- Model and decomposition
  - Log asset price Y(t) modeled as a semimartingale with three components:
    - drift: μ(t) dt
    - continuous component: σ(t) dW(t), where W(t) ~ iid N(0,1)
    - discontinuous component: JUMPS(t)
    - Full model: dY(t) = μ(t) dt + σ(t) dW(t) + JUMPS(t)
  - JUMPS(t) decomposed as:
    - large jumps (finite activity)
    - small jumps (infinite activity)
    - Expression: JUMPS(t) ≔ large jumps(t) [finite activity] + small jumps(t) [infinite activity].
  - Interpretation:
    - Continuous (diffusive) component: smooth price movements from Brownian motion scaled by stochastic volatility σ(t).
    - Large (finite activity) jumps: rare events related to significant news; follow a Poisson process.
    - Small (infinite activity) jumps: frequent, limited-price-absorption events; characterized by Levy-family jump processes (examples: Cauchy, Normal Inverse Gaussian).
- Tests and constructions used
  - Methodology follows Ait-Sahalia and Jacod (2012) (and Erdemlioglu et al., 2013).
  - Primary metric: truncated power variation constructed using intraday price increments.
  - Huang and Tauchen (2005) test used to identify statistically significant jumps under finite activity assumption.
  - Tests can identify jump type presence (finite vs infinite activity) but do not provide jump intensity.
  - Jump activity index: Blumenthal-Getoor Index (index of jump activity β) constructed to describe jumpiness; spectrum ranges from Brownian motion at one end to finite-activity compound Poisson at the other.
- Preliminary approach and outcome measures
  - Preliminary high-level analysis assumes jumps correspond exclusively to finite activity, disentangling continuous components and finite activity jumps.
  - Statistically significant jumps identified with Huang and Tauchen (2005).
  - Tracked measures:
    - Proportion of daily price variation attributable to continuous component vs jump component.
    - Intensity of jumps defined as number of significant jump days per month.
- Empirical scope and data
  - Asset focus: S&P 500 Index (overall index) and individual economic sector components of the S&P 500.
  - Intraday pricing data cadence: recorded at 15, 30 and 60 second time intervals.
  - Sample start dates:
    - Index-level data start: January 2009.
    - Component-level (sector) data start: January 2018.
  - Trading hours included: truncated to prices recorded between 9.30 AM and 4.00 PM.
- Technical points and limitations
  - Large jumps: modeled as finite activity and Poisson; associated with significant news events.
  - Small jumps: modeled as infinite activity Levy processes; associated with frequent price impacts from transactions.
  - Tests applied:
    - Can detect presence/type (finite vs infinite) of jumps but are not informative about jump intensity; Blumenthal-Getoor index used to characterize jump activity further.
  - Figures referenced:
    - Figure D.1: simulated price paths illustrating continuous Brownian motion, infinite-activity Levy process (normal inverse Gaussian), and simulated share price paths over time.
    - Figure D.2: Index of Jump Activity illustrating placement of Compound Poisson Process, Cauchy Process, NIG Process, Brownian Motion on the β index from 0 to 2 (finite to infinite activity).

*Source: IMF staff analysis.*

### Section 1

### Online Annex 1.1 Technical Note — Section 1

### A. Financial Conditions Indices (FCIs)
- Purpose and evolution
  - Original methodology developed in the April 2017 GFSR; applied with modifications in October 2017 GFSR.
  - Methodology simplified to enable computation of regional aggregates and identification of contributions of underlying FCI components to country aggregate FCIs.
- Economy coverage
  - Sample includes the 29 systemically important jurisdictions in the IMF’s Financial Sector Assessment programs, the original FCI sample of 21 economies, and the top 20 constituents of the EMBIG index.
  - Final sample totals: 43 AE / 29 EM / 21 SIJ / 20 EMBIG-20 (table shows per-country listing; totals shown as "total43292120" in source).
- FCI components (revised from Oct 2017 price-of-risk components)
  - Additions and changes:
    - Sovereign and corporate spreads on local debt for EMs were added.
    - Realized equity volatility replaced with implied volatility based on option prices (such as the VIX) where available.
    - Equity and house price variables adjusted to measure levels (rather than changes) for consistency.
  - Component measurement matrix (variable — measurement — presence by group):
    - real short-term interest rate — 3 month T-bill yield minus CPI yoy — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: +
    - interbank spread — Interbank rate (Libor) minus T-bill yield — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: +
    - term spread — 5-year govt bond yield minus T-bill yield — US, Germany: 1; Other AE: (blank); EM: + 
    - sovereign local debt spread — 5-year yield minus US or Germany yield — US, Germany: 1; Other AE: 1; EM: + for AE
    - sovereign dollar debt spread — EMBI spread — US, Germany: 1; Other AE: (blank); EM: +
    - corporate local currency spread — ICE OAS — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + for AE
    - corporate dollar debt spread — CEMBI spread — US, Germany: 1; Other AE: (blank); EM: +
    - equity prices — MSCI P/B — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + (qoq)
    - equity vol — VIX/V2X/VNKY — US, Germany: 1; Other AE: (blank); EM: + all
    - exchange rate — debt-weighted exchange rate — US, Germany: 1; Other AE: (blank); EM: + (vis-à-vis USD)
    - real house prices — BIS house prices yoy minus CPI yoy — US, Germany: 1; Other AE: 1; EM: 1 — Presence in Oct 2017 GFSR: + (qoq)
- FCI weights and aggregation
  - Time-varying weights from a dynamic factor model replaced with fixed weights from a principal component analysis (PCA).
    - Rationale: improved tractability, simplified computation of contributions of FCI drivers, and greater parameter stability.
  - Global aggregation: PPP GDP weights employed instead of PCA on full sample of country-level FCIs.
    - Advantage: allows additivity of countries to regional aggregates and from regions to global total.
  - Effect of revisions: relatively small changes to the global FCI; revised FCI better captures abrupt changes in components.
- Illustrative series
  - Global Price-of-Risk FCI plotted as Standard deviation from mean, 1996:Q1-2018:Q3 (figure compares Original and Revised series).

### B. Capital-Flows-at-Risk (CaR)
- Framework objective
  - Forward-looking assessment of risks to emerging market (EM) capital flows: what global financial conditions today imply about the expected future distribution of capital flows.
  - Uses quantile regression to quantify downside and upside risks to future capital flows conditional on prevailing global financial conditions.
  - Allows analysis of whether drivers of inflow surges differ from drivers of capital flows reversals.
- Data and scope
  - Focus: portfolio debt flows (gross portfolio debt inflows = net non-resident purchases of EM debt instruments).
  - Sample period and coverage: quarterly balance of payments data from 1997Q2 to 2017Q4 for about 60 emerging market and developing countries (China excluded).
  - Flows measured in US dollars, scaled by EM GDP and aggregated across countries.
  - Time horizons:
    - "Medium-term": period from 5 to 8 quarters ahead, averaging inflows as a share of EM GDP over these four quarters.
    - "Near-term" (also reported in GFSR): average flows over the current and the next two quarters.
- Independent variables (push and pull factors) — preferred specification
  - Push factors:
    - investor risk aversion proxied by US BBB-rated corporate bond spread;
    - market interest rates: US 10-year Treasury yields (de-trended using a Hodrick-Prescott filter);
    - the US dollar: DXY dollar index.
  - Domestic/control variables:
    - real GDP growth in emerging market economies (excluding China).
    - lagged inflows (preceding four quarters) and a constant term.
  - Additional variables tested (not included in preferred specification due to lower significance or similar results): market-implied expectations of the federal funds rate, slope of the US yield curve, VIX, real effective exchange rate, Brent oil price, Bloomberg commodity price index, EM exchange rate index, US growth, Fed balance sheet.
- Model specification
  - Quantile regression setup: y_{t+h} (average portfolio debt inflows to EMs in % of GDP between t and t+h) regressed on x_t (vector of independent variables).
  - Estimation: quantile regression minimizes quantile-weighted absolute errors; predicted value gives conditional quantile.
  - Empirical distribution obtained by considering quantiles from the 1st to the 99th percentile.
  - Capital flows-at-risk (CaR) defined as lower 5th percentile of the predicted distribution:
    - Pr(y_{t+h} ≤ CaR_h(α | Ω_t)) = α with α = 5th percentile (i.e., 5 percent probability that capital flows would be lower than CaR).
  - Estimates for the 5th percentile taken directly from quantile regressions; probability density functions obtained by mapping quantile regression estimates into a smoothed Gaussian (normal) distribution.
- Results and interpretation
  - Three main predictive factors for medium-term portfolio debt flows to EMs:
    - investor risk appetite (risk aversion);
    - US long-term interest rates;
    - US dollar.
  - Relationships:
    - Higher US interest rates and a stronger US dollar → weaker inflows, contemporaneously and in the medium term.
    - Stronger risk appetite → greater inflows contemporaneously, but predicts weaker inflows in the medium term (possible mean-reversion in risk appetite).
  - Extreme movements (surges and reversals):
    - Future reversals and, to a lesser extent, surges are disproportionately explained by investor risk aversion (higher coefficient estimates at lowest and highest quantiles).
    - US interest rates and the dollar have less predictive power for capital flow reversals (typically at lowest percentiles).
  - Current assessment (as of dataset reference):
    - Downside risks to medium-term capital flows are currently high, driven by relatively elevated US interest rates, a strong dollar, and buoyant risk appetite.
    - Estimated medium-term capital flows at risk of 0.6 percent of the combined GDP of EMEs (excluding China), on par with the Global Financial Crisis (measured over a four-quarter period).
    - Comparison example: late 2011:Q4 (height of European sovereign debt crisis) had low US interest rates, weaker dollar, but high risk aversion and a more benign medium-term CaR relative to the current assessment.
- Figures and diagnostics
  - B.1. Portfolio Debt Flows to EMs Excluding China (Percent of GDP).
  - B.2. Drivers of Medium-Term Capital Flows (time series of risk aversion (BBB), US 10-year yields (de-trended), and Dollar index).
  - B.3. Estimated Coefficients by Quantile (shows larger risk-aversion coefficients at extremes).
  - B.4. Medium-Term Capital Flows Forecast Densities (probability density with 5th percentile indicated).

### C. Bank Solvency Simulations
- Purpose and sample
  - Forward-looking assessment of bank solvency via simulations estimating stress capital ratios in the following year using current balance sheets and distributions of changes in capital from historical bank profitability.
  - Sample: about 600 banks headquartered in advanced economies; sample size varies over time (mergers, new banks, failures).
  - Regional sample breakdown given in Table C.1 (Euro area 175 banks; Other Europe 65 banks; Asia and Pacific 105 banks; North America 260 banks) with country listings in source.
- Profitability distributions and shock construction
  - Historical data from 1990 onwards used to build distributions of bank profitability.
  - Profits divided into four categories (all calculated as percentages of relevant balance sheet items):
    - (i) net interest income to loans and securities;
    - (ii) trading income to securities;
    - (iii) provisions to gross loans;
    - (iv) other net income to assets.
  - Shocks measured as year-over-year changes in profitability; ten different distributions calculated for each profit category based on current profitability level (to capture stage-in-cycle effects).
  - Distributions illustrated in Figure C.2 (shocks to bank profits, percentage points).
- Simulation method
  - For each bank: take 10,000 draws from profitability distributions.
  - Apply shocks to balance-sheet-relevant items and add to current profits to estimate next-year profits.
  - Dividend rule: bank pays dividends at current-year rate if it makes a profit; no dividends if it makes a loss.
  - Retained earnings added to current capital to estimate capital at end of next year.
  - Assumption: average risk weight of assets remains the same as in the current year.
  - Simulations run for each year since 2006 using balance sheet, income, and profitability distributions at that point in time.
- Capital shortfall testing and reporting
  - Estimated capital in each simulation tested against two thresholds:
    - Tier 1 ratio of 4.5 percent;
    - leverage ratio of 3 percent.
  - For GSIBs, capital surcharges are added to the Tier 1 ratio threshold.
  - Each simulation where a bank’s capital ratio falls below either threshold is counted.
  - Outcome metric: estimated probability of a capital need over the following year.
  - Results presented as the proportion of sample banks (by assets) with a probability of a capital need that is 20 percent or higher in the simulations.
  - Note: the same capital ratio thresholds are used over time to enable comparison, though thresholds are higher than regulatory ratios in place in pre-crisis and crisis years.
- Conclusion on method
  - Provides a forward-looking assessment of bank solvency as an alternative to assessing current capitalization.
  - Not intended as a substitute for detailed stress tests but offers a useful risk assessment of bank capitalization based on historical experience.

*Source: IMF staff analysis.*

### Section 2

### annex1-1 - Section 2

### Jump risk: overview
- Market microstructure has shifted in recent years due to regulation, evolving balance sheet capacities of financial intermediaries, and an increase in algorithmic trading.
- Daily aggregate liquidity measures appear relatively benign, but the analysis assesses implications for high-frequency asset price dynamics using intraday data.
- Aim: gauge the extent to which microstructure changes systematically impact intraday price evolution.

### Methodology
- Log asset price Y(t) is modeled as a semimartingale with three components:
  - drift: μ(t) dt
  - continuous component: σ(t) dW(t), where W(t) ~ iid N(0,1)
  - discontinuous component: JUMPS(t)
  - Full model: dY(t) = μ(t) dt + σ(t) dW(t) + JUMPS(t)
- JUMPS(t) decomposed as:
  - large jumps (finite activity)
  - small jumps (infinite activity)
  - Expression: JUMPS(t) ≔ large jumps(t) [finite activity] + small jumps(t) [infinite activity].
- Interpretation:
  - Continuous (diffusive) component: smooth price movements from Brownian motion scaled by stochastic volatility σ(t).
  - Large (finite activity) jumps: rare events related to significant news; follow a Poisson process.
  - Small (infinite activity) jumps: frequent, limited-price-absorption events; well characterized by Levy family jump processes (examples: Cauchy, Normal Inverse Gaussian).
- Tests and constructions used:
  - Methodology follows Ait-Sahalia and Jacod (2012) (and Erdemlioglu et al., 2013).
  - Primary metric: truncated power variation constructed using intraday price increments.
  - Huang and Tauchen (2005) test used to identify statistically significant jumps under finite activity assumption.
  - Tests can identify jump type presence (finite vs infinite activity) but do not provide jump intensity.
  - Jump activity index: Blumenthal-Getoor Index (index of jump activity β) constructed to describe jumpiness; spectrum ranges from Brownian motion at one end to finite-activity compound Poisson at the other (illustrated in Figure D.2).
- Preliminary approach:
  - A high-level analysis is first conducted assuming jumps correspond exclusively to finite activity, disentangling continuous components and finite activity jumps (related literature: Barndorff-Nielsen and Shephard (2006); Andersen et al. (2007)).
  - Statistically significant jumps are identified with Huang and Tauchen (2005).
  - Relative proportion of daily price variation attributable to continuous component and jump component is tracked over time.
  - Intensity of jumps defined as number of significant jump days per month.

### Empirical scope and data
- Asset focus: S&P 500 Index (overall index) and individual economic sector components of the S&P 500.
- Intraday pricing data cadence: recorded at 15, 30 and 60 second time intervals.
- Sample start dates:
  - Index-level data start: January 2009.
  - Component-level (sector) data start: January 2018.
- Trading hours included: truncated to prices recorded between 9.30 AM and 4.00 PM.
- Figures referenced:
  - Figure D.1: Example simulated price paths illustrating continuous Brownian motion, infinite-activity Levy process (normal inverse Gaussian), and simulated share price paths over time.
  - Figure D.2: Index of Jump Activity illustrating placement of Compound Poisson Process, Cauchy Process, NIG Process, Brownian Motion on the β index from 0 to 2 (finite to infinite activity).

### Key technical points and limitations
- Large jumps: modeled as finite activity and Poisson; associated with significant news events.
- Small jumps: modeled as infinite activity Levy processes; associated with frequent price impacts from transactions.
- The tests applied:
  - Can detect presence/type (finite vs infinite) of jumps but are not informative about jump intensity; separate index (Blumenthal-Getoor) required to characterize jump activity more specifically.
- Outcome measures tracked:
  - Proportion of daily price variation due to continuous vs jump components.
  - Intensity of jumps measured as number of significant jump days per month.

*This section was prepared by Rohit Goel, Piyusha Khot, and Sheheryar Malik.*

---


_Source: https://www.imf.org/-/media/files/publications/gfsr/2018/oct/ch1/doc/annex1-1.pdf_