Factor Copula Application - S&P 100 and Systemic Risk

Summary

The finite-sample simulation study (up to $N=100$) confirms the SMM estimator and its asymptotic theory work well. The empirical application to all 100 S&P 100 constituents (2008-2010) finds a fat-tailed, left-skewed common factor: significant tail dependence, heterogeneous (industry-driven) dependence, and asymmetric dependence — crashes more correlated than booms. The Normal copula is rejected; the skew $t$-$t$ factor copula fits best and yields superior estimates of systemic-risk measures (marginal expected shortfall and a multi-stock variant).

Overview

This note collects the empirical results: the Monte Carlo validation, the equidependence and block-equidependence estimates, the asymmetric/tail-dependence findings, and the systemic-risk application. The methods are in SMM Estimation of Factor Copulas; the models in Multi-Factor and Block Dependence Structures; the tail-dependence theory in Tail Dependence in Factor Copulas.

Main Content

Simulation design and results (Sec. 3.3, Tables 1-5)

Three factor copulas of form $X_i=Z+{\epsilon }_i$ with $Z\sim \text{Skew }t({\sigma }_z^2,\nu ,\lambda )$, ${\epsilon }_i\sim \text{iid }t(\nu )$, ${\sigma }_z^2=1$ (rank correlation $\approx 0.5$): (1) Normal ($\nu \to \infty ,\lambda =0$), (2) symmetric $t$-$t$ ($\nu =4,\lambda =0$, tail dependence), (3) skew $t$-$t$ ($\nu =4,\lambda =-0.5$, asymmetric + tail dependence). Estimate ${\nu }^{-1}\in [0,0.5)$. Dimensions $N\in \{3,10,100\}$; for $N=100$ a block-equidependence model (10 groups). Marginals: iid Normal or AR(1)-GARCH(1,1). $T=1000$ ($\approx$4 yrs daily), $S=25T$ simulations, 100 replications, identity weight matrix.

• Table 1: estimates centered on true values; small bias; precision improves with $N$ (equidependence). MLE→SMM efficiency loss 25% ($N=3$) to 10% ($N=100$); GMM→SMM loss ≤3%.
• Table 2: $N=100$ flexible-loadings block model well estimated; shape params (${\nu }^{-1},\lambda$) slightly less precise than ${\sigma }_z^2$.
• Tables 3-4: 95% CI coverage near nominal for step size ${\epsilon }_T\in \{0.01,0.03,0.1\}$; collapses if step too small.
• Table 5: $J$-test rejection rates near nominal, best for ${\epsilon }_T\ge 0.01$.

Data and marginal models (Sec. 4, Table 7)

All 100 S&P 100 constituents as of Dec 2010; April 2008-Dec 2010, $T=696$ trade days (window set by Philip Morris’s April-2008 addition). Marginals filtered by AR(1)-GJR-GARCH with lagged market return:

$$r_{it}={\varphi }_{0i}+{\varphi }_{1i}{r}_{i,t-1}+{\varphi }_{mi}{r}_{m,t-1}+{\epsilon }_{it}$$ $${\sigma }_{it}^2={\omega }_i+{\beta }_i{\sigma }_{i,t-1}^2+{\alpha }_i{\epsilon }_{i,t-1}^2+{\gamma }_i{\epsilon }_{i,t-1}^{2}1\{{\epsilon }_{i,t-1}\le 0\}+{\alpha }_{mi}{\epsilon }_{m,t-1}^2+{\gamma }_{mi}{\epsilon }_{m,t-1}^{2}1\{{\epsilon }_{m,t-1}\le 0\}$$

GJR-GARCH preferred by BIC; leverage parameter ${\gamma }_i>0$ for 97/100 stocks. Marginals estimated nonparametrically (EDF) given heterogeneous skewness/kurtosis. Summary dependence over 4950 pairs: linear and rank correlation both $\approx 0.42$-$0.44$; rank correlation IQR 0.37-0.50 (mild heterogeneity); 1% tail dependence $\approx 0.06$; the ${\tau }_{0.90}-{\tau }_{0.10}$ difference is negative for >75% of pairs — strong evidence of asymmetric dependence.

Systemic-risk measures (Sec. 4.3)

Marginal Expected Shortfall (Brownlees & Engle 2011): expected return on stock $i$ given the market return is below a low threshold,

$$ME{S}_{it}=-E_{t-1}[r_{it}\mid r_{mt}<C]$$

The factor copula (full model for all 100 stocks) also enables a multi-stock variant $kES$ — expected return on $i$ given that more than $k$ stocks have crashed:

$$kE{S}_{it}=-E_{t-1}\left[r_{it}\left|\right(\sum _{j=1}^{N}1\{r_{jt}<C\})>k\right]$$

Models are ranked by MSE/relative-MSE against realized returns on crisis days:

$$MS{E}_i=\frac{1}{T}\sum _{t=1}^T(r_{it}-ME{S}_{it}{)}^{2}1\{r_{mt}<C\},RelMS{E}_i=\frac{1}{T}\sum _{t=1}^T(\frac{r_{it}-ME{S}_{it}}{ME{S}_{it}}{)}^{2}1\{r_{mt}<C\}$$

Unlike CAPM/Brownlees-Engle (which need only a bivariate model and use the market index to flag crises), the factor copula uses crashes in individual stocks as turmoil flags.

Examples

Asymmetric, fat-tailed dependence in S&P 100 returns (Tables 8-10, Figs 4-5)

Setup: Eight copulas estimated by SMM — four existing (Clayton, Normal, $t$, skew $t$ with equicorrelation) and four factor copulas ($t$-Normal, skew $t$-Normal, $t$-$t$, skew $t$-$t$). Step size ${\epsilon }_T=0.1$, 1000 bootstraps. Result (equidependence, Table 8): common-factor variance ${\sigma }_z^2\approx 0.9$ (avg correlation $\approx 0.47$); inverse DoF ${\nu }^{-1}\approx 1/25$, significant only for asymmetric models; asymmetry $\lambda$ significantly negative in all models ($t$-stats -2.1 to -4.4) → crashes more likely than booms. The three asymmetric models ($Q_{SMM}$) outperform all others, but all models fail the $J$-test ($p\approx 0$), pointing to the equidependence assumption. Result (block, Tables 9-10): with industry factors, ${\nu }^{-1}\approx 1/14$ (stronger tail dependence) and $\lambda$ larger/more negative. Implied lower tail dependence averages 0.82 (range 0.70-0.99) vs upper tail dependence 0.07 (range 0.02-0.74) — strong asymmetry. The skew $t$-$t$ block copula is the only model passing the $J$-test ($p=0.07$). Figures 4-5: the Normal copula overestimates upper-tail and underestimates lower-tail dependence; the skew $t$-$t$ factor copula fits both tails well. Conditioning on $j$ crashes out of 100, the Normal copula is adequate for moderate tail events but the skew $t$-$t$ is needed for extreme (once-in-a-quarter, 1/66) ones. Interpretation: Risk management using a Normal copula takes too benign a view; basket/CDO securities may be mispriced; diversification benefits are lower than under Normality because large negative shocks originate from a fat-tailed common factor hitting all stocks at once.

Superior systemic-risk estimates (Table 11)

Setup: Estimate MES and $kES$ ($k=30$) at thresholds $C\in \{-2\%,-4\%\}$, comparing Brownlees-Engle, CAPM, Historical, and four block-equidependence copulas (Normal, $t$, skew $t$, skew $t$-$t$ factor). Result: For MES, Brownlees-Engle is best under MSE with the skew $t$-$t$ factor copula second; under Relative MSE the factor copula is best for both thresholds (skew $t$ second). Historical and CAPM are worst. For $kES$ (needs the full 100-stock joint distribution, so CAPM/Brownlees-Engle cannot apply), the skew $t$-$t$ factor copula performs best on both metrics and thresholds. Interpretation: The high-dimensional factor copula not only characterizes the dependence structure but delivers improved systemic-risk estimates, especially for multi-firm crash measures only a full joint model can produce.

Connections

• SMM Estimation of Factor Copulas — the estimation method whose finite-sample properties are validated here.
• Tail Dependence in Factor Copulas — Propositions 2 & 3 supply the reported tail-dependence coefficients.
• Multi-Factor and Block Dependence Structures — the block-equidependence model used empirically.
• Factor Copulas - Overview — the crisis motivation realized in these findings.

See Also

• SMM Copula Simulation and Application — companion-paper note on related copula simulation/application results.
• Econometrics