## Annex I. Tables

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---

### Glossary
- CAViaR Conditional Autoregressive Value at Risk
- CoFiE Cornish-Fisher Expansions
- EVT Extreme Value Theory
- FFNN Feed-Forward Neural Network
- GFC Global-Financial-Crisis
- GPD Generalized Pareto Distribution
- LSTM Long Short-Term Memory Model
- ML Machine Learning
- NN Neural Network
- PCA Principal Component Analysis
- Pre-GFC Pre-Global-Financial-Crisis
- TRR Tail Risk Ratio
- VaR Value-at-Risk

### Introduction — objectives and methodological contributions
- Objective: Propose CoFiE-NN, a machine-learning-based approach for forecasting VaR that combines a neural network (NN) with Cornish-Fisher expansions (CoFiE).
- Two stated advantages of CoFiE-NN:
  - Flexibility: NN component can represent nonlinear relationships between statistical moments.
  - Interpretability: Cornish-Fisher expansions explicitly link statistical moments with percentiles of the distribution.
- Additional claimed advantages:
  - Cornish-Fisher expansions are simple to implement.
  - Allows examination of skewness and kurtosis effects (e.g., 99 percentile can exceed 2.33 when excess kurtosis is positive).
  - Can predict VaR with relatively small amounts of data (useful for FX in developing/emerging markets).
- NN specifications tested:
  - FFNN (Feed-Forward Neural Network) — simple baseline.
  - LSTM (Long Short-Term Memory) — main specification capturing short- and long-term relationships.
- Benchmark models used for comparison:
  - EGARCH-t model
  - CAViaR model
  - EVT model with the generalized Pareto distribution (GPD)
- Statistical evaluation methods:
  - Kupiec (1995) test — tests whether breaches of VaR forecasts are too many or too few.
  - Christoffersen (1998) test — tests independence/autocorrelation of VaR breaches; joint test with Kupiec (1995).
  - Lopez (1999) quadratic loss function — measures magnitude of VaR breaches.

### Key empirical findings (simulated and real data)
- Simulated data findings:
  - CoFiE-NN with LSTM tends to outperform the EGARCH-t model when the sample period is relatively short.
  - CoFiE-NN underperforms CAViaR in terms of Kupiec (1995) and the joint test.
  - CoFiE-NN outperforms CAViaR in Lopez (1999) loss function under all training-data-size settings.
- Real data (30 assets across asset classes; four sample periods: Pre-GFC, GFC, Pre-Covid, Covid):
  - CoFiE-NN with LSTM outperforms EGARCH-t in terms of Kupiec (1995) test and the joint test in all four sample periods except one case.
  - CoFiE-NN underperforms EGARCH-t in terms of Lopez (1999) quadratic loss function.
  - CoFiE-NN underperforms CAViaR in all criteria across the four sample periods in general.
  - CoFiE-NN outperforms EVT in all criteria for all four sample periods except one case.

### CoFiE-NN framework — technical specification
- Cornish-Fisher expansions used up to fourth order (mean, volatility, skewness, kurtosis).
- Notation and cumulants:
  - Time series of log returns: 푟푟푡
  - Cumulants vector: Κ푡 = �휅휅1,푡, 휅휅2,푡, 휅휅3,푡, 휅휅4,푡�
  - Source text expressions (preserved):
    - "Note that skewness is  
      휅휅3 휅휅2 1.5"
    - "and excess kurtosis is 
      휅휅4 휅휅2 2"
- Neural-network dynamics (equation preserved):
  - Κ푡+1 = 푓푓(Κ푡, Κ푡−1, . . Κ푡−푠푠, 푍푍푡)    (1)
    - 푓푓 is the neural network modeling dynamics of moments/cumulants.
    - 푍푍푡 is a vector of exogenous variables.
    - Κ푡−푠푠 is a vector of cumulants at time 푡−푠푠.
    - LSTM is employed as the main specification in empirical applications.
- Implementation steps (preserved procedure):
  - Compute the mean, variance, skewness, and kurtosis using the historical data of the asset prices (e.g., 10-day mean, variance, skewness, kurtosis).
  - Train the neural network using the historical data of the cumulants defined as training data.
  - Forecast the cumulants with the neural network.
  - Compute VaR based on Cornish-Fisher expansion using the forecast of the cumulants.

### Cornish-Fisher VaR formulation
- VaR alpha-percentile representation:
  - R_{α,t} = μ_t + σ_t w_{α,t}  (equation (2))
- Cornish-Fisher expansion for the standardized percentile w_{α,t} (truncated):
  - w_{α,t} = x_α
    + (κ_{3,t} / κ_{2,t}^{1.5}) ⋅ (1/6) H_2(x_α)
    + (κ_{4,t} / κ_{2,t}^{2}) ⋅ (1/24) H_3(x_α)
    − [ (κ_{3,t} / κ_{2,t}^{1.5})^2 ] ⋅ (1/36) (2 H_3(x_α) + H_1(x_α)) + ⋯  (equation (3))
- Definitions and notation:
  - R_{α,t} is the alpha-percentile of VaR.
  - μ_t is the mean of log return.
  - σ_t is the volatility.
  - x_α is the alpha-percentile of the standard normal distribution.
  - H_n(x) is the n-th order Hermite polynomial (e.g., H_1(x)=x, H_2(x)=x^2−1, H_3(x)=x^3−3x).
  - w_{α,t} is the alpha-percentile of the distribution with volatility standardized to one.
- Implementation note:
  - In simulations and empirical applications the alpha is set equal to 97.5 percentile.
  - If skewness and kurtosis are zero then w_{α,t} = x_α and R_{α,t} reduces to the normal-based VaR R_{α,t} = μ_t + σ_t x_α.

### Extensions (as discussed in the source)
- Expected Shortfall (ES):
  - ES_{α,t} = μ_t + σ_t e_{α,t}  (equation (4))
  - e_{α,t} = φ(w_{α,t}) / α ⋅ [ 1 + (κ_3 / κ_2^{1.5}) ⋅ (1/6) w_{α,t}^3 + (κ_4 / κ_2^{2}) ⋅ (1/24) (w_{α,t}^4 − 2 w_{α,t}^2 − 1) ]  (equation (5))
  - φ(x) is the standard normal density.
- Multivariate (portfolio) approaches:
  - Portfolio return r_{p,t} = ∑_{i=1}^N ω_i r_{i,t}  (equation (6))
  - Portfolio mean: E[r_{p,t}] = ∑_{i=1}^N ω_i E[r_{i,t}]  (equation (7))
  - Portfolio variance: Var[r_{p,t}] = ∑_{i=1}^N ω_i^2 Var[r_{i,t}] + ∑_{i≠j} ω_i ω_j Cov(r_{i,t}, r_{j,t})  (equation (8))
  - Under CoFiE-NN: predict cumulants for each asset and cross-terms (covariance, co-skewness, co-kurtosis), then compute portfolio cumulants from constituent cumulants.
  - Trade-offs: second approach avoids NN re-calibration when weights change but requires predicting many cross-cumulants as N grows.
- Multi-step forecasting:
  - Iterative computation: K_{t+1+q} = f(K_{t+q}, K_{t+q−1}, …, K_{t+q−s}, Z_t) for q = 0,1,…,u−1  (equation (9))
  - After obtaining K_{t+u}, apply Cornish-Fisher to compute VaR at t+u.

### Input features and exogenous variables
- Empirical applications use Z_t = (r_t, r_t^2) to capture relationships between contemporary return and future moments (e.g., leverage effect, squared return analogous to GARCH(1,1)).

### Neural network specifications (CoFiE-NN)
- Feed-Forward Neural Network (FFNN):
  - Structure: input layer, hidden layers, output layer.
  - Example equations (preserved): z_{1,t} = f(W_0 x_t + b_0), z_{l+1,t} = f(W_l z_{l,t} + b_l), x_{t+1} = f(W_{L+1} z_{L+1,t} + b_{L+1}) (equations (10)-(12)).
  - In empirical applications: sigmoid activation; number of hidden layers L = 5; number of units in hidden layers equals input dimension; parameters calibrated by minimizing sum of squared errors via backpropagation.
- Long Short-Term Memory (LSTM):
  - LSTM gate and state equations provided (equations (13)-(19)) with standard nomenclature: u_t, f_t, i_t, C_t, c̃_t, c_t, h_t, y_t.
  - LSTM implemented using machine learning package in Matlab.
- Calibration and architecture notes:
  - Small-scale LSTM found sufficient for empirical applications.
  - Key hyperparameter: number of hidden units — if large (e.g., 250), out-of-sample performance tends to be poor.

### Ensuring monotonicity of Cornish-Fisher VaR
- Issue: Cornish-Fisher expansions may produce non-monotonic VaR as a function of percentile when truncated.
- Two methods discussed:
  - Maillard (2012) analytical condition (main solution in source). For truncation up to fourth order, VaR is cubic in percentile and monotonicity ensured if two inequalities hold:
    - |s_t| < √2 − 1  (equation (20))
    - 9 k_t^2 − (3 + 33 s_t^2) k_t + 30 s_t^4 + 7 s_t^2 ≤ 0  (equation (21))
    - where s_t = E_t / 6 and k_t = K_t / 24 (E_t and K_t are skewness and excess kurtosis respectively).
  - Rearrangement method (Chernozhukov et al. 2016) — changes the shape of entire distribution and is applicable to arbitrary order; not the main choice in source.
- Implementation detail: if predicted skewness and kurtosis fall outside the Maillard region, adjust them to the closest point within the admissible region before computing VaR.

### Conventional benchmark models used for comparison
- EGARCH-t model:
  - r_t = σ_t ⋅ ((ν−2)/ν)^{1/2} ε_t  (equation (22))
  - log σ_t^2 − log σ̄^2 = ρ (log σ_{t−1}^2 − log σ̄^2) + β (|ε_{t−1}| − E[|ε_{t−1}|]) + γ ε_{t−1}  (equation (23))
  - ε_t sampled from Student t with degree of freedom ν.
- CAViaR:
  - VaR_{α,t} − VaR_{α}̄ = ρ (VaR_{α,t−1} − VaR_{α}̄) + β (α − f_CCCIC(r_{t−1}, VaR_{α,t−1}))  (equation (24))
  - f_CCCIC defined as (1 + exp(G (r_{t−1} − VaR_{α,t−1})))^{−1}  (equation (25))
  - Parameters estimated by minimizing min_{G,ρ,β} (1/T) ∑ (I(r_t < VaR_{α,t}) − α) [r_t − VaR_{α,t}]  (equation (26))
- EVT with generalized Pareto distribution:
  - r_t = σ_t ε_t, and full shock CDF F_CCCIC(ε_t) piecewise combining Gaussian bulk and generalized Pareto tails with thresholds θ_u and θ_d (equation (27)).
  - Generalized Pareto CDF: F_GGPC(ε, γ, θ, β) = 1 − (1 + γ (ε − θ)/β)^{−1/γ}  (equation (28))
  - For the empirical application: θ_u = Φ(0.05) and θ_d = φ(0) = −∞ (focus on large positive shocks).

### Data and sample periods (empirical setup)
- Assets (30): 20 currencies (EUR, JPY, CNY, KRW, GBP, MXN, INR, CAD, BRL, AUD, CHF, THB, MYR, ZAR, TWD, SGD, NOK, SEK, NZD, DKK), 2 stock indices (NASDAQ 100, Nikkei 225), 3 commodities (WTI oil, Brent oil, Henry Hub natural gas), 2 interest rates (two-year and five-year US Treasury yields), plus 3 currencies (HUF, PLN, CZK) downloaded against EUR and converted to USD.
- Four sample-period settings (daily frequency) with training and test ranges:
  1. Pre-GFC: training 2004/7/2 to 2005/6/30; test 2005/7/1 to 2007/6/29.
  2. GFC: training 2003/7/1 to 2007/6/29; test 2007/7/2 to 2009/12/31.
  3. Pre-COVID: training 2015/1/6 to 2015/12/31; test 2016//1/4 to 2019/12/31.
  4. COVID: training 2018/3/1 to 2020/2/28; test 2020/3/2 to 2023/7/27.
- Descriptive observations:
  - The three commodities show extremely large kurtosis; several currencies (e.g., Swiss Franc, Korean Won) also show large kurtosis.
  - Skewness varies: Swiss Franc shows negative skewness, Mexican Peso shows positive skewness — FX returns are not well described by normal distributions.

### Testing VaR forecasting performance (metrics used)
- Kupiec (1995) unconditional coverage test (LR_uncond) and approximation z = √T (α̂ − α)/√(α(1−α)).
- Christoffersen (1998) independence test (LR_ind) and joint test LR_joint = LR_uncond + LR_ind (chi-square with 2 degrees of freedom).
- Lopez (1999) quadratic loss function (QLF) designed to capture both frequency and magnitude of VaR breaches.

### Simulation testing (Monte Carlo) — setup and parameters
- Data generated by EGARCH-t model with stochastic parameters (time-varying γ_t and ν_t).
- Parameter values used in simulation:
  - σ̄ = 0.02
  - ρ_σ = 0
  - β_σ = 0
  - γ̄ = −0.2
  - ρ_γ = 0.99
  - β_γ = 0.05
  - ν̄ = 10
  - ρ_ν = 0.99
  - β_ν = 0.2
- Training sample sizes considered: T = 250, T = 500, T = 1000, T = 2000.
- Simulation outcomes summary:
  - CoFiE-NN with FFNN outperforms EGARCH-t in all three criteria across all T settings except one case (T=500) in Lopez loss.
  - FFNN outperforms CAViaR in loss function but underperforms it in Kupiec test and joint test.
  - FFNN outperforms EVT in all criteria.
  - CoFiE-NN with LSTM outperforms EGARCH-t in Kupiec and joint tests when training size is small (T=250, T=500) but underperforms when T is large (T=1000, T=2000).
  - LSTM outperforms EGARCH-t in loss function except one case (T=250).
  - LSTM outperforms CAViaR in loss function but underperforms it in Kupiec and joint tests.
  - LSTM outperforms EVT in all criteria.
- Interpretation: CoFiE-NN can work successfully even with small training samples and tends to outperform EGARCH-t even when data-generating process resembles EGARCH-t with stochastic moments.

### Empirical results across 30 assets (summary)
- Aggregate findings across four sample-period cases (Tables 4 summary):
  - CoFiE-NN with FFNN:
    - Outperforms EVT in most cases.
    - Generally underperforms EGARCH-t and CAViaR.
  - CoFiE-NN with LSTM:
    - Outperforms EGARCH-t in Kupiec (1995) test and joint test in most sample periods (except COVID case) but tends to underperform EGARCH-t on Lopez (1999) loss function.
    - Underperforms CAViaR in general.
    - Outperforms EVT in almost all cases (except Pre-GFC).
  - Overall: CoFiE-NN with LSTM tends to outperform EGARCH-t and EVT but tends to underperform CAViaR.

### Tail risk ratio (TRR) empirical application and PCA results
- Definition:
  - TRR_{α,i,t} = (VaR_{α,i,t} − μ_{i,t}) / σ_{i,t}  (equation (41))
  - Under CoFiE-NN the tail risk ratio equals the volatility-standardized percentile w_{α,t} (equation (42)).
- Properties:
  - If skewness and kurtosis are zero, TRR reduces to x_α.
  - TRR is time-varying when skewness and kurtosis are time-varying even if volatility is constant.
  - Under EGARCH-t with constant ν, TRR is constant and equals ((ν−2)/ν)^{1/2} F_ν^{−1}(α)  (equation (43)).
  - TRR is not defined under CAViaR unless volatility is imported from another model.
- Empirical sample: TRRs constructed for 22 currencies during 2019/8-2023/7; training 2014/8–2019/7.
- PCA findings:
  - Tail risk ratio PCA (proportion of variance explained):
    - Factor 1: 19.9% | 19.9%
    - Factor 2: 9.7% | 29.6%
    - Factor 3: 7.8% | 37.4%
    - Factor 4: 6.6% | 44.0%
    - Factor 5: 6.4% | 50.4%
    - (Factors 6–22 listed with exact percentages and accumulations as in source)
  - Volatility PCA (proportion of variance explained):
    - Factor 1: 60.7% | 60.7%
    - Factor 2: 13.6% | 74.3%
    - Factor 3: 6.3% | 80.6%
    - (Factors 4–22 listed with exact percentages and accumulations as in source)
  - Interpretation:
    - The first principal component explains 20 percent of tail risk dynamics across the 22 currencies.
    - The first principal component explains 60 percent of volatility dynamics across the same set of currencies.
    - First PCA factor interpreted as global factor for both tail risk ratios and volatilities.
    - Second PCA factor: for tail risk ratios related to Chinese Yuan; for volatilities associated with Latin American countries.
    - Third PCA factors: related to Eastern European countries for both measures; third PCA factor of tail risk ratios also related to Asian countries.
  - Time evolution: global volatility factor spikes at onset of COVID-19 and rises after Russian invasion of Ukraine; global tail-risk factor shows more frequent spikes and a peak in June 2021 (after updated Fed dot plot).
- Portfolio optimization note:
  - TRR naturally arises in the context of optimal asset allocation under a VaR constraint.

### Portfolio optimization under VaR constraint (link to TRR)
- VaR-constrained mean maximization problem (source notation):
  - max_{ω_t} ω_t' r_t^e  (equation (44))
  - subject to VaR_α(−ω_t' r_t') ≤ L̄  (equation (45))
  - Under multivariate normal returns, VaR constraint reduces to x_α ⋅ ω_t' Σ ω_t ≤ L̄  (equation (46)) and classic solution ω_t^* = (1/(2 λ κ_α)) Σ^{−1} r_t^e  (equation (47)).
- General (non-normal) case using TRR:
  - VaR constraint becomes w_{p,α,t} ⋅ ω_t' Σ ω_t ≤ L̄  (equation (48)), where w_{p,α,t} (portfolio TRR) depends nonlinearly on asset allocation ω_t and higher moments.
  - Implication: the VaR-constrained investor may reduce exposure when higher moments increase even if portfolio volatility is unchanged.

### Conclusion and suggested future research
- CoFiE-NN summary:
  - Combines neural network (NN) forecasting of cumulants with Cornish-Fisher expansions to produce VaR forecasts.
  - Advantages: captures nonlinear dynamics of high-order moments (via NN) while retaining interpretability (via Cornish-Fisher).
  - LSTM adopted as main NN specification, though any NN can be used.
- Empirical findings recap:
  - CoFiE-NN with LSTM tends to outperform EGARCH-t in Kupiec (1995) test (frequency of breaches) but not in Lopez (1999) quadratic loss (magnitude of breaches).
  - CoFiE-NN underperforms CAViaR but outperforms EVT on average.
  - TRR analysis: only 20 percent of tail risk dynamics across 22 currencies is explained by a single common factor versus 60 percent for volatility.
- Future research directions:
  - Extend CoFiE-NN to multivariate and multi-period forecasting.
  - Explore inclusion of fifth or higher order moments in CoFiE-NN.
  - Conduct more comprehensive performance analyses across more assets and settings.

### Annex I: Selected tables and calibration details
- LSTM: Calibration and Hyperparameters
  - Dimension: 6
  - Number of units: 2
  - Calibration method: Adams
  - Maximum epochs: 250
  - Gradient thresholds: 1
  - Learn rate drop factor: 0.2
- Descriptive Statistics (Mean, Stdev, Skewness, Kurtosis) — selected entries (exact values preserved):
  - EUR: 0.00%, 0.6%, 0.1, 2.5
  - JPY: 0.01%, 0.6%, -0.4, 4.4
  - CNY: 0.00%, 0.2%, 0.0, 15.8
  - KRW: 0.00%, 0.7%, -0.6, 46.9
  - GBP: 0.00%, 0.6%, -0.6, 9.5
  - MXN: 0.01%, 0.7%, 0.7, 10.4
  - INR: 0.00%, 0.6%, -0.6, 9.5
  - CAD: 0.00%, 0.6%, -0.1, 5.3
  - BRL: 0.02%, 1.0%, 0.0, 7.7
  - AUD: 0.00%, 0.8%, -0.6, 10.8
  - CHF: -0.01%, 0.7%, -1.2, 34.1
  - THB: 0.00%, 0.4%, 0.1, 9.2
  - MYR: 0.00%, 0.4%, -0.4, 8.5
  - ZAR: 0.02%, 1.1%, 0.2, 4.2
  - TWD: 0.00%, 0.3%, -0.3, 14.5
  - SGD: 0.00%, 0.3%, 0.0, 4.7
  - NOK: 0.00%, 0.8%, 0.2, 4.1
  - SEK: 0.00%, 0.7%, -0.1, 3.6
  - NZD: 0.00%, 0.8%, -0.4, 4.5
  - DKK: 0.00%, 0.6%, -0.2, 4.5
  - CZK: -0.01%, 0.7%, 0.3, 3.8
  - HUF: 0.01%, 0.9%, 0.3, 4.0
  - PLN: 0.00%, 0.8%, 0.5, 5.9
  - NASDAQ: 0.03%, 1.8%, 0.0, 6.6
  - US 2Y: 0.00%, 4.8%, 0.0, 6.5
  - US 10Y: -0.01%, 2.5%, 0.0, 25.2
  - WTI: 0.02%, 5.6%, -14.2, 1718.1
  - Brent: 0.02%, 2.7%, -2.0, 75.0
  - Nikkei: 0.01%, 1.5%, -0.4, 6.2
  - Henry Hub: 0.00%, 5.4%, -0.2, 48.8
- Winning Rates: CoFiE-NN (FFNN) — Winning rate (Kupiec test), Winning rate (Joint test), Winning rate (Lopez function) — selected period values as in source:
  - Pre-GFC: 47%, 17%, 63%; 47%, 17%, 63%; 20%, 77%, 40%
  - GFC: 10%, 3%, 73%; 7%, 3%, 77%; 3%, 7%, 70%
  - Pre-Covid: 27%, 13%, 63%; 27%, 20%, 67%; 47%, 77%, 80%
  - Covid: 23%, 3%, 70%; 20%, 0%, 77%; 10%, 0%, 63%
- Winning Rates: CoFiE-NN (LSTM) — selected period values as in source:
  - Pre-GFC: 70%, 17%, 80%; 67%, 47%, 80%; 20%, 77%, 43%
  - GFC: 53%, 7%, 93%; 50%, 33%, 93%; 23%, 37%, 80%
  - Pre-Covid: 63%, 17%, 67%; 57%, 27%, 70%; 30%, 70%, 77%
  - Covid: 53%, 3%, 93%; 47%, 27%, 97%; 33%, 27%, 73%

*IMF Working Papers — Forecasting Tail Risk via Neural Networks with Asymptotic Expansions — Annex I. Tables*

### Annex I. Tables ........................................................................................................

### Annex I. Tables

### Glossary
- CAViaR Conditional Autoregressive Value at Risk
- CoFiE Cornish-Fisher Expansions
- EVT Extreme Value Theory
- FFNN Feed-Forward Neural Network
- GFC Global-Financial-Crisis
- GPD Generalized Pareto Distribution
- LSTM Long Short-Term Memory Model
- ML Machine Learning
- NN Neural Network
- PCA Principal Component Analysis
- Pre-GFC Pre-Global-Financial-Crisis
- TRR Tail Risk Ratio
- VaR Value-at-Risk

### Introduction — objectives and methodological contributions
- Objective: Propose CoFiE-NN, a machine-learning-based approach for forecasting VaR that combines a neural network (NN) with Cornish-Fisher expansions (CoFiE).
- Two stated advantages of CoFiE-NN:
  - Flexibility: NN component can represent nonlinear relationships between statistical moments.
  - Interpretability: Cornish-Fisher expansions explicitly link statistical moments with percentiles of the distribution.
- Additional claimed advantages:
  - Cornish-Fisher expansions are simple to implement.
  - Allows examination of skewness and kurtosis effects (e.g., 99 percentile can exceed 2.33 when excess kurtosis is positive).
  - Can predict VaR with relatively small amounts of data (useful for FX in developing/emerging markets).
- NN specifications tested:
  - FFNN (Feed-Forward Neural Network) — simple baseline.
  - LSTM (Long Short-Term Memory) — main specification capturing short- and long-term relationships.
- Benchmark models used for comparison:
  - EGARCH-t model
  - CAViaR model
  - EVT model with the generalized Pareto distribution (GPD)
- Statistical evaluation methods:
  - Kupiec (1995) test — tests whether breaches of VaR forecasts are too many or too few.
  - Christoffersen (1998) test — tests independence/autocorrelation of VaR breaches; joint test with Kupiec (1995).
  - Lopez (1999) quadratic loss function — measures magnitude of VaR breaches.

### Key empirical findings (simulated and real data)
- Simulated data:
  - CoFiE-NN with LSTM tends to outperform the EGARCH-t model when the sample period is relatively short.
  - CoFiE-NN underperforms CAViaR in terms of Kupiec (1995) and the joint test.
  - CoFiE-NN outperforms CAViaR in Lopez (1999) loss function under all training-data-size settings.
- Real data (30 assets across asset classes, emphasis on FX; four sample periods: Pre-GFC, GFC, Pre-Covid, Covid):
  - CoFiE-NN with LSTM outperforms EGARCH-t in terms of Kupiec (1995) test and the joint test in all four sample periods except one case.
  - CoFiE-NN underperforms EGARCH-t in terms of Lopez (1999) quadratic loss function.
  - CoFiE-NN underperforms CAViaR in all criteria across the four sample periods in general.
  - CoFiE-NN outperforms EVT in all criteria for all four sample periods except one case.

### Tail risk ratio (TRR) empirical application and PCA results
- Tail risk ratio: defined as volatility-scaled VaR to facilitate cross-country comparisons of VaR across currencies with different volatility levels.
- Empirical sample: TRRs constructed for 22 currencies during 2019/8-2023/7.
- PCA findings:
  - The first principal component explains 20 percent of tail risk dynamics across the 22 currencies.
  - By contrast, the first principal component explains 60 percent of volatility dynamics across the same set of currencies.
  - Interpretation of PCA factor loadings:
    - First PCA factor: interpreted as a global factor for both tail risk ratios and volatilities.
    - Second PCA factor: for tail risk ratios is related to Chinese Yuan; for volatilities is associated with Latin American countries.
    - Third PCA factors: related to Eastern European countries for both measures; third PCA factor of tail risk ratios also related to Asian countries.
- Portfolio optimization note:
  - TRR naturally arises in the context of optimal asset allocation under a VaR constraint.

### Literature positioning
- CoFiE-NN contributes to three strands:
  - ML for forecasting VaR — distinct from quantile-regression NN and heavy-tailed LSTM approaches by combining NN with Cornish-Fisher expansions and reducing VaR forecasting to moment forecasting; framework can forecast Expected Shortfall (ES) without additional calibration (ES forecasting not the paper’s focus).
  - ML for forecasting volatility — CoFiE-NN primarily targets VaR rather than just volatility.
  - ML applications to FX markets — CoFiE-NN uses LSTM for tail-risk forecasting of FX markets, distinguishing it from earlier ML applications to exchange-rate forecasting.

### CoFiE-NN framework — technical specification
- For exposition, Cornish-Fisher expansions are used up to fourth order (mean, volatility, skewness, kurtosis).
- Notation and cumulants:
  - Time series of log returns: 푟푟푡
  - Cumulants vector: Κ푡 = �휅휅1,푡, 휅휅2,푡, 휅휅3,푡, 휅휅4,푡�
  - Source text expressions (preserved exactly as presented):
    - "Note that skewness is  
      휅휅3 휅휅2 1.5"
    - "and excess kurtosis is 
      휅휅4 휅휅2 2"
- Neural-network dynamics (preserved equation formatting):
  - Κ푡+1 = 푓푓(Κ푡, Κ푡−1, . . Κ푡−푠푠, 푍푍푡)    (1)
    - 푓푓 is the neural network modeling dynamics of moments/cumulants.
    - 푍푍푡 is a vector of exogenous variables.
    - Κ푡−푠푠 is a vector of cumulants at time 푡−푠푠.
    - LSTM is employed as the main specification in empirical applications.
- Implementation steps (preserved procedure):
  - Compute the mean, variance, skewness, and kurtosis using the historical data of the asset prices (e.g., 10-day mean, variance, skewness, kurtosis).
  - Train the neural network using the historical data of the cumulants defined as training data.
  - Forecast the cumulants with the neural network.

*IMF Working Papers — Forecasting Tail Risk via Neural Networks with Asymptotic Expansions — Annex I. Tables*

### 4.   Compute VaR based on Cornish-Fisher expansion using the forecast of the cumulants.

### 4.   Compute VaR based on Cornish-Fisher expansion using the forecast of the cumulants.

### Cornish-Fisher VaR formulation
- VaR alpha-percentile representation:
  - R_{α,t} = μ_t + σ_t w_{α,t}  (equation (2) in source)
- Cornish-Fisher expansion for the standardized percentile w_{α,t} (truncated representation in source):
  - w_{α,t} = x_α
    + (κ_{3,t} / κ_{2,t}^{1.5}) ⋅ (1/6) H_2(x_α)
    + (κ_{4,t} / κ_{2,t}^{2}) ⋅ (1/24) H_3(x_α)
    − [ (κ_{3,t} / κ_{2,t}^{1.5})^2 ] ⋅ (1/36) (2 H_3(x_α) + H_1(x_α)) + ⋯  (equation (3))
- Definitions and notation (as in source):
  - R_{α,t} is the alpha-percentile of VaR.
  - μ_t is the mean of log return.
  - σ_t is the volatility.
  - x_α is the alpha-percentile of the standard normal distribution.
  - H_n(x) is the n-th order Hermite polynomial (e.g., H_1(x)=x, H_2(x)=x^2−1, H_3(x)=x^3−3x).
  - w_{α,t} is the alpha-percentile of the distribution with volatility standardized to one.
- Implementation note from source:
  - In simulations and empirical applications the alpha is set equal to 97.5 percentile.
  - If skewness and kurtosis are zero then w_{α,t} = x_α and R_{α,t} reduces to the normal-based VaR R_{α,t} = μ_t + σ_t x_α.

### Extensions (as discussed in the source)
- Expected Shortfall (ES):
  - ES_{α,t} = μ_t + σ_t e_{α,t}  (equation (4))
  - e_{α,t} = φ(w_{α,t}) / α ⋅ [ 1 + (κ_3 / κ_2^{1.5}) ⋅ (1/6) w_{α,t}^3 + (κ_4 / κ_2^{2}) ⋅ (1/24) (w_{α,t}^4 − 2 w_{α,t}^2 − 1) ]  (equation (5))
  - φ(x) is the standard normal density.
- Multivariate (portfolio) approaches:
  - Portfolio return r_{p,t} = ∑_{i=1}^N ω_i r_{i,t}  (equation (6))
  - Portfolio mean: E[r_{p,t}] = ∑_{i=1}^N ω_i E[r_{i,t}]  (equation (7))
  - Portfolio variance: Var[r_{p,t}] = ∑_{i=1}^N ω_i^2 Var[r_{i,t}] + ∑_{i≠j} ω_i ω_j Cov(r_{i,t}, r_{j,t})  (equation (8))
  - Under CoFiE-NN: predict cumulants for each asset and cross-terms (covariance, co-skewness, co-kurtosis), then compute portfolio cumulants from constituent cumulants.
  - Trade-offs: second approach avoids NN re-calibration when weights change but requires predicting many cross-cumulants as N grows.
- Multi-step forecasting (beyond t+1):
  - Iterative computation: K_{t+1+q} = f(K_{t+q}, K_{t+q−1}, …, K_{t+q−s}, Z_t) for q = 0,1,…,u−1  (equation (9))
  - After obtaining K_{t+u}, apply Cornish-Fisher to compute VaR at t+u.

### Input features and exogenous variables
- In empirical applications, vector Z_t = (r_t, r_t^2) is used to capture relationships between contemporary return and future moments (e.g., leverage effect, squared return analogous to GARCH(1,1)).

### Neural network specifications (CoFiE-NN)
- Feed-Forward Neural Network (FFNN):
  - Structure: input layer, hidden layers, output layer.
  - Hidden layers count L; example equations: z_{1,t} = f(W_0 x_t + b_0), z_{l+1,t} = f(W_l z_{l,t} + b_l), x_{t+1} = f(W_{L+1} z_{L+1,t} + b_{L+1}) (equations (10)-(12)).
  - In empirical applications: sigmoid activation; number of hidden layers L = 5; number of units in hidden layers equals input dimension; parameters calibrated by minimizing sum of squared errors via backpropagation.
- Long Short-Term Memory (LSTM):
  - LSTM gate and state equations provided in source (equations (13)-(19)) with standard nomenclature:
    - u_t input vector, f_t forget gate activation, i_t update gate activation, C_t output gate activation, c̃_t cell input activation, c_t cell state, h_t hidden state, y_t output.
  - LSTM implemented using machine learning package in Matlab.
- Calibration and architecture notes:
  - Small-scale LSTM found sufficient for empirical applications.
  - Key hyperparameter: number of hidden units — if large (e.g., 250), out-of-sample performance tends to be poor.

### Ensuring monotonicity of Cornish-Fisher VaR
- Cornish-Fisher expansions may produce non-monotonic VaR as a function of percentile when truncated.
- Two methods discussed:
  - Maillard (2012) analytical condition (applied as main solution in source because of intuitiveness and tractability). For truncation up to fourth order, VaR is cubic in percentile and monotonicity ensured if two inequalities hold:
    - |s_t| < √2 − 1  (equation (20) in source)
    - 9 k_t^2 − (3 + 33 s_t^2) k_t + 30 s_t^4 + 7 s_t^2 ≤ 0  (equation (21) in source)
    - where s_t = E_t / 6 and k_t = K_t / 24 (E_t and K_t are skewness and excess kurtosis respectively).
  - Rearrangement method (Chernozhukov et al. 2016) — changes the shape of entire distribution and is applicable to arbitrary order; not the main choice in source.
- Implementation detail: if predicted skewness and kurtosis fall outside the Maillard region, adjust them to the closest point within the admissible region before computing VaR.

### Conventional benchmark models used for comparison
- EGARCH-t model (used as benchmark):
  - r_t = σ_t ⋅ ((ν−2)/ν)^{1/2} ε_t  (equation (22))
  - log σ_t^2 − log σ̄^2 = ρ (log σ_{t−1}^2 − log σ̄^2) + β (|ε_{t−1}| − E[|ε_{t−1}|]) + γ ε_{t−1}  (equation (23))
  - ε_t sampled from Student t with degree of freedom ν.
  - EGARCH-t chosen because: (1) no parameter restriction, (2) captures fat-tails and volatility clustering, (3) used as benchmark in previous ML VaR studies.
- CAViaR (Conditional Autoregressive VaR):
  - Dynamic VaR autoregression: VaR_{α,t} − VaR_{α}̄ = ρ (VaR_{α,t−1} − VaR_{α}̄) + β (α − f_CCCIC(r_{t−1}, VaR_{α,t−1}))  (equation (24))
  - f_CCCIC defined as (1 + exp(G (r_{t−1} − VaR_{α,t−1})))^{−1}  (equation (25))
  - Parameters estimated by minimizing the objective min_{G,ρ,β} (1/T) ∑ (I(r_t < VaR_{α,t}) − α) [r_t − VaR_{α,t}]  (equation (26))
  - CAViaR directly models VaR percentile (does not model volatility or other moments).
- EVT (Extreme Value Theory) with generalized Pareto distribution:
  - Combine EVT with GARCH-type volatility: r_t = σ_t ε_t, and full shock CDF F_CCCIC(ε_t) piecewise combining Gaussian bulk and generalized Pareto tails with thresholds θ_u and θ_d (equation (27)).
  - Generalized Pareto CDF: F_GGPC(ε, γ, θ, β) = 1 − (1 + γ (ε − θ)/β)^{−1/γ}  (equation (28))
  - For the empirical application: θ_u = Φ(0.05) and θ_d = φ(0) = −∞ (focus on large positive shocks).

### Data and sample periods (empirical setup)
- Assets: time series of 30 assets across FX, commodity, interest rates, equity:
  - 20 currencies: EUR, JPY, CNY, KRW, GBP, MXN, INR, CAD, BRL, AUD, CHF, THB, MYR, ZAR, TWD, SGD, NOK, SEK, NZD, DKK.
  - 2 stock indices: NASDAQ 100 and Nikkei 225.
  - 3 commodities: WTI oil price, Brent oil price, Henry Hub natural gas.
  - 2 interest rates: two-year and five-year US Treasury yields.
  - Additional 3 currencies (HUF, PLN, CZK) downloaded against EUR and converted to USD.
- Four sample-period settings (daily frequency) with training and test ranges exactly as in source:
  1. Pre-Global-Financial-Crisis (Pre-GFC): training 2004/7/2 to 2005/6/30; test 2005/7/1 to 2007/6/29.
  2. Global-Financial-Crisis (GFC): training 2003/7/1 to 2007/6/29; test 2007/7/2 to 2009/12/31.
  3. pre-COVID-19 Crisis (Pre-COVID): training 2015/1/6 to 2015/12/31; test 2016//1/4 to 2019/12/31.
  4. COVID-19 Crisis (COVID): training 2018/3/1 to 2020/2/28; test 2020/3/2 to 2023/7/27.
- Descriptive observations from Table 2 (as summarized in source):
  - The three commodities show extremely large kurtosis; several currencies (e.g., Swiss Franc, Korean Won) also show large kurtosis.
  - Skewness varies: Swiss Franc shows negative skewness, Mexican Peso shows positive skewness — FX returns are not well described by normal distributions.

### Testing VaR forecasting performance (metrics used)
- Kupiec (1995) unconditional coverage test:
  - LR_uncond = 2 log [ ((1−α)^(T−I(α)) α^(I(α))) / ((1−α̂)^(T−I(α)) α̂^(I(α))) ]  (equation (29))
  - α̂ = (1/T) I(α)  (equation (30)); I(α) = ∑ I_t(α) / T  (equation (31))
  - Approximation: z = √T (α̂ − α)/√(α(1−α))  (equation (32))
- Christoffersen (1998) independence test:
  - Likelihood components L_0 and L_1 as in equations (33)-(34); LR_ind = −2 log(L_1/L_0)  (equation (35))
- Joint test: LR_joint = LR_uncond + LR_ind (chi-square with 2 degrees of freedom).
- Lopez (1999) quadratic loss function:
  - QLF = (1/r_t > C_C_C_Cα,t) ∑ [1 + (r_t − VaR_{α,t})^2]  (equation (36))
  - Designed to capture both frequency and magnitude of VaR breaches.

### Simulation testing (Monte Carlo) — setup and parameters
- Data generated by EGARCH-t model with stochastic parameters (time-varying γ_t and ν_t) — equations (37)-(40) in source.
- Parameter values used in simulation (exactly as in source):
  - σ̄ = 0.02
  - ρ_σ = 0
  - β_σ = 0
  - γ̄ = −0.2
  - ρ_γ = 0.99
  - β_γ = 0.05
  - ν̄ = 10
  - ρ_ν = 0.99
  - β_ν = 0.2
- Training sample sizes considered: T = 250, T = 500, T = 1000, T = 2000.
- Summary of simulation outcomes (as reported in source):
  - CoFiE-NN with FFNN (denoted FFNN) outperforms EGARCH-t in all three criteria across all T settings except one case (T=500) in Lopez loss.
  - FFNN outperforms CAViaR in loss function but underperforms it in Kupiec test and joint test.
  - FFNN outperforms EVT in all criteria.
  - CoFiE-NN with LSTM (denoted LSTM) outperforms EGARCH-t in Kupiec and joint tests when training size is small (T=250, T=500) but underperforms when T is large (T=1000, T=2000).
  - LSTM outperforms EGARCH-t in loss function except one case (T=250).
  - LSTM outperforms CAViaR in loss function but underperforms it in Kupiec and joint tests.
  - LSTM outperforms EVT in all criteria.
- Interpretation from source:
  - CoFiE-NN can work successfully even with small training samples.
  - CoFiE-NN tends to outperform EGARCH-t even when data-generating process resembles EGARCH-t with stochastic moments.

### Empirical results across 30 assets (summary from source)
- Accumulated breach comparisons (COVID-19 period test: training 2018/3/1–2020/2/28; test 2020/3/2–2023/7/27) visualized in Figure 2 (CoFiE-NN vs EGARCH-t vs CAViaR).
- Aggregate findings across four sample-period cases (Tables 4 summary):
  - CoFiE-NN with FFNN:
    - Outperforms EVT in most cases.
    - Generally underperforms EGARCH-t and CAViaR — FFNN may lack power compared to conventional models in some settings.
  - CoFiE-NN with LSTM:
    - Outperforms EGARCH-t in Kupiec (1995) test and joint test in most sample periods (except COVID case) but tends to underperform EGARCH-t on Lopez (1999) loss function.
    - Underperforms CAViaR in general.
    - Outperforms EVT in almost all cases (except Pre-GFC).
  - Overall summary in source: CoFiE-NN with LSTM tends to outperform EGARCH-t and EVT but tends to underperform CAViaR.

### Tail Risk Ratio (new empirical proxy introduced)
- Definition:
  - TRR_{α,i,t} = (VaR_{α,i,t} − μ_{i,t}) / σ_{i,t}  (equation (41))
  - Under CoFiE-NN the tail risk ratio equals the volatility-standardized percentile of the distribution:
    - TRR_{i,t} = w_{α,t} = x_α + (κ_{3,t}/κ_{2,t}^{1.5}) ⋅ (1/6) H_2(x_α) + (κ_{4,t}/κ_{2,t}^2) ⋅ (1/24) H_3(x_α) − [ (κ_{3,t}/κ_{2,t}^{1.5})^2 ] ⋅ (1/36) (2 H_3(x_α) + H_1(x_α)) + ⋯  (equation (42))
- Properties and remarks:
  - If skewness and kurtosis are zero, TRR reduces to x_α (the standard normal percentile).
  - TRR is time-varying when skewness and kurtosis are time-varying even if volatility is constant.
  - Under EGARCH-t with constant ν, TRR is constant and equals ((ν−2)/ν)^{1/2} F_ν^{−1}(α)  (equation (43)).
  - TRR is not defined under CAViaR unless volatility is imported from another model (which creates inconsistency).
- Empirical use: PCA on TRRs for 22 currencies (2019/8–2023/7), training 2014/8–2019/7:
  - Principal component decomposition results (Table 5 summary in source):
    - First PCA factor explains 20 percent of variance in tail risk ratios across 22 currencies (interpreted as global tail-risk factor).
    - First PCA factor explains 60 percent of variance in volatilities across same currencies (global volatility factor).
    - Interpretation: tail risks are more country-specific than volatilities.
  - PCA factor coefficient interpretations (Figures 3 and 4 summary):
    - First PCA factor (tail risk): almost all coefficients positive and similar magnitude → global factor.
    - Second PCA factor (tail risk): dominated by Chinese Yuan → China-specific tail-risk factor.
    - Third PCA factor (tail risk): related to Eastern European currencies (CZK, PLN, HUF).
    - Time evolution (Figures 4): global volatility factor spikes at onset of COVID-19 and rises after Russian invasion of Ukraine; global tail-risk factor shows more frequent spikes and a peak in June 2021 (after updated Fed dot plot).
  - Source interpretation: global tail risk captures market stress episodes that global volatility factor may not.

### Portfolio optimization under VaR constraint (link to TRR)
- VaR-constrained mean maximization problem (source notation):
  - max_{ω_t} ω_t' r_t^e  (equation (44))
  - subject to VaR_α(−ω_t' r_t') ≤ L̄  (equation (45))
  - Under multivariate normal returns, VaR constraint reduces to x_α ⋅ ω_t' Σ ω_t ≤ L̄  (equation (46)) and classic solution ω_t^* = (1/(2 λ κ_α)) Σ^{−1} r_t^e  (equation (47)).
- General (non-normal) case using TRR:
  - VaR constraint becomes w_{p,α,t} ⋅ ω_t' Σ ω_t ≤ L̄  (equation (48)), where w_{p,α,t} (portfolio TRR) depends nonlinearly on asset allocation ω_t and higher moments.
  - Implication: the VaR-constrained investor may reduce exposure when higher moments increase even if portfolio volatility is unchanged.

### Conclusion and suggested future research (as stated in source)
- CoFiE-NN summary:
  - Combines neural network (NN) forecasting of cumulants with Cornish-Fisher expansions to produce VaR forecasts.
  - Advantages: captures nonlinear dynamics of high-order moments (via NN) while retaining interpretability (via Cornish-Fisher).
  - LSTM adopted as main NN specification, though any NN can be used.
- Empirical findings:
  - CoFiE-NN with LSTM tends to outperform EGARCH-t in Kupiec (1995) test (frequency of breaches) but not in Lopez (1999) quadratic loss (magnitude of breaches).
  - CoFiE-NN underperforms CAViaR but outperforms EVT on average.
  - TRR analysis: only 20 percent of tail risk dynamics across 22 currencies is explained by a single common factor versus 60 percent for volatility.
- Future research directions outlined in source:
  - Extend CoFiE-NN to multivariate and multi-period forecasting.
  - Explore inclusion of fifth or higher order moments in CoFiE-NN.
  - Conduct more comprehensive performance analyses across more assets and settings.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024099-print-pdf.pdf*

### Annex I. Tables

### Annex I. Tables

### LSTM: Calibration and Hyperparameters
- Dimension: 6
- Number of units: 2
- Calibration method: Adams
- Maximum epochs: 250
- Gradient thresholds: 1
- Learn rate drop factor: 0.2

### Descriptive Statistics (Mean, Stdev, Skewness, Kurtosis)
- EUR: 0.00%, 0.6%, 0.1, 2.5
- JPY: 0.01%, 0.6%, -0.4, 4.4
- CNY: 0.00%, 0.2%, 0.0, 15.8
- KRW: 0.00%, 0.7%, -0.6, 46.9
- GBP: 0.00%, 0.6%, -0.6, 9.5
- MXN: 0.01%, 0.7%, 0.7, 10.4
- INR: 0.00%, 0.6%, -0.6, 9.5
- CAD: 0.00%, 0.6%, -0.1, 5.3
- BRL: 0.02%, 1.0%, 0.0, 7.7
- AUD: 0.00%, 0.8%, -0.6, 10.8
- CHF: -0.01%, 0.7%, -1.2, 34.1
- THB: 0.00%, 0.4%, 0.1, 9.2
- MYR: 0.00%, 0.4%, -0.4, 8.5
- ZAR: 0.02%, 1.1%, 0.2, 4.2
- TWD: 0.00%, 0.3%, -0.3, 14.5
- SGD: 0.00%, 0.3%, 0.0, 4.7
- NOK: 0.00%, 0.8%, 0.2, 4.1
- SEK: 0.00%, 0.7%, -0.1, 3.6
- NZD: 0.00%, 0.8%, -0.4, 4.5
- DKK: 0.00%, 0.6%, -0.2, 4.5
- CZK: -0.01%, 0.7%, 0.3, 3.8
- HUF: 0.01%, 0.9%, 0.3, 4.0
- PLN: 0.00%, 0.8%, 0.5, 5.9
- NASDAQ: 0.03%, 1.8%, 0.0, 6.6
- US 2Y: 0.00%, 4.8%, 0.0, 6.5
- US 10Y: -0.01%, 2.5%, 0.0, 25.2
- WTI: 0.02%, 5.6%, -14.2, 1718.1
- Brent: 0.02%, 2.7%, -2.0, 75.0
- Nikkei: 0.01%, 1.5%, -0.4, 6.2
- Henry Hub: 0.00%, 5.4%, -0.2, 48.8

### VaR Forecasting: Simulated Data and Tests (selected entries)
- Table 3 (a), (b), (c) present performance and winning rates of VaR forecasting using simulated data, comparing CoFiE-NN with FFNN and LSTM across tests including Kupiec (1995), Joint test, and Lopez (1999) loss function.
- Excerpted simulation results (selected T sample sizes and test statistics appear in the source; raw tabular blocks shown in source are preserved there).

### VaR Forecasting: Real Data — Performance Periods
- Tables 4 (a)–(d) report VaR forecasting performance for real data across sample periods:
  - Pre-Global Financial Crisis (Pre-GFC)
  - Global Financial Crisis (GFC)
  - Pre COVID-19 Crisis (Pre-Covid)
  - COVID-19 Crisis (Covid)
- Tables 4 (e) and 4 (f) report winning rates of CoFiE-NN against benchmarks (EGARCH-t, CAViaR, REVT) for FFNN and LSTM implementations.

### Winning Rates: CoFiE-NN (FFNN) — Winning rate (Kupiec test), Winning rate (Joint test), Winning rate (Lopez function)
- Pre-GFC: 47%, 17%, 63%; 47%, 17%, 63%; 20%, 77%, 40% (as presented in the source table format)
- GFC: 10%, 3%, 73%; 7%, 3%, 77%; 3%, 7%, 70%
- Pre-Covid: 27%, 13%, 63%; 27%, 20%, 67%; 47%, 77%, 80%
- Covid: 23%, 3%, 70%; 20%, 0%, 77%; 10%, 0%, 63%

### Winning Rates: CoFiE-NN (LSTM) — Winning rate (Kupiec test), Winning rate (Joint test), Winning rate (Lopez function)
- Pre-GFC: 70%, 17%, 80%; 67%, 47%, 80%; 20%, 77%, 43%
- GFC: 53%, 7%, 93%; 50%, 33%, 93%; 23%, 37%, 80%
- Pre-Covid: 63%, 17%, 67%; 57%, 27%, 70%; 30%, 70%, 77%
- Covid: 53%, 3%, 93%; 47%, 27%, 97%; 33%, 27%, 73%

### PCA: Proportion of Variance Explained by Each PCA Factor
- Tail risk ratio (Variance explained | Accumulation)
  - Factor 1: 19.9% | 19.9%
  - Factor 2: 9.7% | 29.6%
  - Factor 3: 7.8% | 37.4%
  - Factor 4: 6.6% | 44.0%
  - Factor 5: 6.4% | 50.4%
  - Factor 6: 5.3% | 55.6%
  - Factor 7: 5.0% | 60.7%
  - Factor 8: 4.5% | 65.2%
  - Factor 9: 4.3% | 69.5%
  - Factor 10: 3.9% | 73.3%
  - Factor 11: 3.4% | 76.7%
  - Factor 12: 3.2% | 80.0%
  - Factor 13: 3.0% | 83.0%
  - Factor 14: 2.6% | 85.6%
  - Factor 15: 2.4% | 88.0%
  - Factor 16: 2.3% | 90.3%
  - Factor 17: 2.2% | 92.5%
  - Factor 18: 2.1% | 94.5%
  - Factor 19: 1.9% | 96.5%
  - Factor 20: 1.7% | 98.1%
  - Factor 21: 1.5% | 99.6%
  - Factor 22: 0.4% | 100.0%

- Volatility (Variance explained | Accumulation)
  - Factor 1: 60.7% | 60.7%
  - Factor 2: 13.6% | 74.3%
  - Factor 3: 6.3% | 80.6%
  - Factor 4: 4.9% | 85.5%
  - Factor 5: 3.7% | 89.1%
  - Factor 6: 2.0% | 91.1%
  - Factor 7: 1.9% | 93.0%
  - Factor 8: 1.6% | 94.6%
  - Factor 9: 0.9% | 95.5%
  - Factor 10: 0.8% | 96.4%
  - Factor 11: 0.7% | 97.1%
  - Factor 12: 0.6% | 97.6%
  - Factor 13: 0.5% | 98.1%
  - Factor 14: 0.4% | 98.5%
  - Factor 15: 0.4% | 98.9%
  - Factor 16: 0.3% | 99.2%
  - Factor 17: 0.3% | 99.5%
  - Factor 18: 0.2% | 99.7%
  - Factor 19: 0.2% | 99.8%
  - Factor 20: 0.1% | 99.9%
  - Factor 21: 0.1% | 100.0%
  - Factor 22: 0.0% | 100.0%

### Figures and Additional Annex Items (referenced)
- Annex II contains figures including:
  - Figure 1. Overview of CoFiE-NN
  - Figure 2. Accumulation of VaR Breaches for 30 Assets (EUR/USD through Henry Hub and Nikkei 225)
  - Figure 3. Coefficients of PCA Factors (multiple panels for PCA factors of tail risk ratios and volatilities)
  - Figure 4. Evolution of Global Factors: Tail risk ratio and Volatility (1st PCA factor panels)

*Source: Annex I and Annex II (Tables and Figures) from the PDF content unit "wpiea2024099-print-pdf - Annex I. Tables".*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024099-print-pdf.pdf_