"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven Investment Management

Correlated Volatility Shocks
Xiao Qiao
SummerHaven Investment Management, LLC 1
1
The views expressed are those of the authors and do not necessarily reﬂect oﬃcial
positions of SummerHaven Investment Management, LLC.

Background
Volatility research is important for both academics and practitioners
Academics: Implications for volatility modeling and asset pricing
Practitioners: Risk management and pricing derivatives
Large literature on modeling volatility
Seminal contribution by Engle (1982), Bollerslev (1986)
Multivariate: Bollerslev (1990), Engle (2002), Engle and Kelly (2012)
X. Qiao, SummerHaven Correlated Vol Shocks 1

Motivation
Recent focus on the behavior of idiosyncratic volatility
Ang et al. (2006) show stocks with high idiosyncratic vol earns lower
average returns
Herskovic et al. (2016) demonstrate idiosyncratic volatilities contain a
common component, priced in the cross section of stock returns
The relationship between idiosyncratic volatility and average returns
Markowitz (1952) states investors only get compensated for
systematic risk
Merton (1987): idiosyncratic risk should be priced if investors are
under-diversiﬁed

Research Question
What accounts for the common factor in idiosyncratic
volatility?
Comovement in time-varying idiosyncratic vol (TVV)
Correlated volatility innovations (VIN)
Are both components important in describing the data?
Which component is priced in the cross-section of equity returns?

This Paper
1. Document correlated volatility shocks as a source of comovement
2. A statistical model for commonality in idiosyncratic volatility
Dynamic Factor Correlation (DFC)
Fits characteristic-sorted portfolios better than the DCC model
(Engle, 2002) and DECO (Engle and Kelly, 2012)
Closed-form likelihood, easy to estimate even for large cross sections
DFC nests the models of Bollerslev (1990) and Engle and Kelly (2012)

This Paper - Continued
3. Portfolio optimization using DFC
Mean-variance portfolios with diﬀerent covariance estimates
DFC outperforms unconditional and DECO covariance estimates
Higher Sharpe ratio and more stable mean-variance portfolios
4. Asset pricing implications of components of idiosyncratic volatility
VIN is priced in the cross section of stock returns, TVV is weakly
priced
Univariate sorts for loadings on TVV and VIN show average return
spreads of 1.62% and 3.31% per year
TVV and VIN carry distinct premiums from each other, not subsumed
by market equity (ME)

Data
Daily and monthly data from Ken French’s website
Fama and French (1992, 2015) factors
Portfolios formed on market equity (ME), book-to-market (BE/ME),
long-term reversal (LT Rev), operating proﬁtability (OP), investment
(Inv), momentum (Mom), and short-term reversal (ST Rev)
Bivariate portfolios formed on the above characteristics
Daily and monthly stock returns, prices, and shares outstanding are
from CRSP
Chicago Fed National Activities Index (CFNAI) and subindexes are
from Federal Reserve Bank of Chicago
Aruoba-Diebold-Scotti Business Conditions Index (ADS Index) from
Federal Reserve Bank of Philadelphia

Idiosyncratic Volatility Shocks are Correlated
Factor models for mean returns, univariate GARCH models to
idiosyncratic returns
H0: Cross-sectionally uncorrelated GARCH innovations
GARCH residuals are not autocorrelated, but cross-sectionally
correlated
Correlations of GARCH residuals on idiosyncratic returns:
Univariate Portfolios
ME BE/ME LT Rev OP Inv Mom ST Rev
Max 0.52 0.22 0.54 0.28 0.26 0.57 0.53
Min 0.05 0.02 0.01 -0.02 -0.01 0.08 0.03
Avg 0.19 0.12 0.24 0.09 0.09 0.28 0.18
Bivariate Portfolios
ME, BE/ME ME, OP ME, Inv ME, Mom ME, ST Rev ME, LT Rev
Max 0.75 0.49 0.39 0.71 0.63 0.50
Min -0.04 -0.03 -0.05 0.02 0.01 0.01
Avg 0.12 0.13 0.10 0.22 0.19 0.14

A Model of Correlated Volatility Shocks

Modeling Correlated Volatility Shocks
Stylized fact: idiosyncratic volatility innovations are correlated
Existing multivariate GARCH models do not account for correlated
volatility shocks
Models such Dynamic Conditional Correlation (DCC) of Engle (2002)
model time-varying correlations
We enrich the multivariate GARCH toolbox

Dynamic Factor Correlation (DFC)
Start with standard factor model with GARCH volatility
ri,t = ftβi + ai,t, hi,t = E[a2
i,t|Ft−1]
ei,t = ai,t/ hi,t
Impose factor structure on standardized residuals
ei,t =
qi,t
si,t
Where qi,t = vtξi + σi i,t, s2
i,t = Et−1[q2
i,t]
vt|t−1 ∼ N(0, hv,t)
Empirically, use hv,t = 1
N
N
i=1 e2
i,t−1 = e2
t−1

DFC Correlations
Qt = vart−1(qt),
Qt = Λ + hv,t · ξξξξξξ ,
where Λ is a N × N diagonal matrix with Λ(i, i) = 1 − ξ2
i , and
ξξξ = (ξ1, · · · , ξN)
The correlation matrix of et is given by
Rt = cort(et)
= ˜Q
−1
2
t Qt
˜Q
−1
2
t
Where ˜Qt(i, i) = 1 + (hv,t − 1)ξ2
i , and ˜Qt(i, j) = 0 for i = j

DFC is Related to Other Multivariate GARCH Models
Bollerslev’s (1990) Constant Conditional Correlation:
∀t, hv,t = 1, Rt(i, i) = R(i, i) = 1 and Rt(i, j) = R(i, j) = ξi ξj .
vart−1(at) = DtRDt Dt = diag( h1,t... hN,t)
R(i, j) = ξi ξj
Engle and Kelly’s (2012) Dynamic Equicorrelation:
Equal loadings, ξi = ξj ≡ ¯ξ,
Rt = (1 − ρt)IN + ρtJN×N
ρt =
hv,t
¯ξ2
1+(hv,t −1)¯ξ2 and JN×N denotes the N × N matrix of ones

DFC Estimation
Univariate volatility parameters θθθ, correlation parameters φφφ
L ∝ − 1
T t log |DtRtDt| + atD−1
t R−1
t D−1
t at
= −
1
T t
2 log |Dt| + atD−2
t at − etet
LV (θθθ)
−
1
T t
log |Rt| + etR−1
t et
LC (θθθ,φφφ)
Quasi-maximum likelihood approach:
1. Find volatility parameters to maximize LV (θθθ):
ˆθθθ = arg max
θθθ
{LV (θθθ)}
2. Plug in ˆθθθ into LC (θθθ,φφφ):
ˆφφφ = arg max
φφφ
{LC (ˆθθθ,φφφ)}

The Need for a New Model
Monte Carlo simulations of the DFC model
N = 3 T = 1000 T = 5000 N = 10 T = 1000 T = 5000
DFC DCC DFC DCC DFC DCC DFC DCC
Panel A: ξ = (0.1, 0.1, 0.1) Panel D: ξ = 0.1 · 110
RMSE 0.030 0.032 0.011 0.014 RMSE 0.019 0.029 0.009 0.016
MAE 0.023 0.026 0.009 0.010 MAE 0.014 0.023 0.007 0.013
Panel B: ξ = (0.2, 0.3, 0.5) Panel E: ξ = (0.2, 0.3, 0.4, 0.5, 0.6) ⊗ 12
RMSE 0.029 0.035 0.017 0.022 RMSE 0.022 0.028 0.016 0.020
MAE 0.023 0.027 0.013 0.017 MAE 0.017 0.022 0.012 0.015
Panel C: ξ = (0.5, 0.5, 0.5) Panel F: ξ = 0.5 · 110
RMSE 0.039 0.053 0.029 0.045 RMSE 0.037 0.042 0.032 0.037
MAE 0.032 0.042 0.025 0.037 MAE 0.031 0.037 0.028 0.031
Engle’s (2002) DCC cannot capture the common component in
idiosyncratic volatility shocks

Empirical Performance: DFC vs. DCC vs. DECO
ME BE/ME LTRev OP Inv Mom STRev
L(DFC) 31813.6 27972.5 26164.0 17596.8 17535.7 26146.3 26032.7
L(DCC) 31535.3 27602.1 25918.7 17430.4 17474.6 25746.7 25771.5
L(DECO) 31080.7 27432.0 25524.4 17465.2 17405.3 24164.4 25463.5
For the most part, DCC does better than DECO because it is more
general and the cross section is small
DFC outperforms both DCC and DECO

Group DFC - An Extension of DFC
Consider group structure to be more parsimonious
Reduce the number of estimated parameters for more eﬃcient
estimation and better out-of-sample properties
Equal factor loadings ξi within groups, diﬀerent loadings across
groups. Suppose K groups among N assets, within each group k all
the factor loadings are ¯ξk
In the extreme case of one group, ξi = ξj = ¯ξ and we are back to
Engle and Kelly (2012)

Group DFC on Characteristic-Based Deciles
ME BE/ME LTRev OP Inv Mom STRev
G1 1,2 1 1 1,2,3 1,2,3,4 1 1,3
G2 3 2 2,3,4 4,5,6 5 2,3 2
G3 4 3 5 7 6 4 4
G4 5 4,5,6 6 8 7 5 5
G5 6 7,8 7 9,10 8,9,10 6 6
G6 7,8 9 8 7 7,8,9
G7 9 10 9,10 8 10
G8 10 9,10
BIC(Group DFC) -25766.5 -27711.3 -25887.5 -17364.4 -17297.8 -25876.1 -25775.3
BIC(DFC) -25756.3 -27696.1 -25890.1 -17339.7 -17278.7 -25869.9 -25756.3
Increased parsimony of Group DFC leads to lower BIC
What are these groups?

Portfolio Optimization using DFC

Mean-Variance Portfolio Optimization
Suppose N securities with expected return vector µt and covariance matrix
ΣΣΣt, and target return µ0, Markowitz (1952) mean-variance portfolio:
min
wt
wtΣΣΣtwt s.t. wt1N = 1
wtµt ≥ µ0
Define At = 1NΣΣΣ−1
t 1N, Bt = 1NΣΣΣ−1
t µt, and Ct = µtΣΣΣ−1
t µt, the
mean-variance (MV) efficient portfolio is given as follows:
wMV
t (µ0) =
Ct − µ0Bt
AtCt − B2
t
ΣΣΣ−1
t 1N +
µ0At − Bt
AtCt − B2
t
ΣΣΣ−1
t µt
Ignore the constraint on target expected returns µ0, we have the Global
Minimum Variance (GMV) portfolio:
wGMV
t =
1
At
ΣΣΣ−1
t 1N
In effect comparing the covariance estimates across volatility models

Models for the Covariance Matrix
ΣΣΣt from diﬀerent volatility models, same µt
Use 49 industry portfolios as basis assets
For MV portfolios, consider µ0 = 5%, 7.5%, 10%
1 Model 1 Unconditional covariance: Historical covariance matrix of raw returns.
2 Model 2 Non-Factor DFC: DFC on raw returns, forecast ˆDt and ˆRt , ΣΣΣt = ˆDt
ˆRt
ˆDt
3 Model 3 CAPM DFC: Fit CAPM to raw returns, GARCH on rmt , DFC on CAPM
residual returns. ΣΣΣt = hmt
ˆβββt
ˆβββt + ˆDt
ˆRt
ˆDt
4 Model 4 FF3 DFC: Fit Fama and French (1992) model to raw returns, GARCH on
factors, and DFC on residual returns.
ΣΣΣt = ˆBt diag{hmt , hSMB,t , hHML,t }ˆBt + ˆDt
ˆRt
ˆDt
5 Models 5-7 30-Group, 10-Group, and 5-Group FF3 DFCs: Similar to Model 4,
but instead of DFC model, use 30-, 10-, or 5-Group DFCs based on Ken French’s
industry classiﬁcation
6 Model 8 FF3 DECO: Similar to Models 5-7, but impose one group for DECO
model on residual returns

Out-of-Sample Portfolio Volatility
Model GMV MV
µ0 - 10% 7.5% 5%
1946 - 2015
Unconditional covariance 0.694 27.814 20.800 13.789
Non-Factor DFC 0.489 19.616 14.671 9.729
CAPM DFC 0.431 17.819 13.330 8.842
FF3 DFC 0.407 17.449 13.056 8.670
FF3 30-Group DFC 0.405 17.425 13.039 8.655
FF3 10-Group DFC 0.406 17.554 13.134 8.716
FF3 5-Group DFC 0.408 17.608 13.175 8.745
FF3 DECO 0.412 17.802 13.322 8.843
FF3 30-Group DFC gives lowest out-of-sample portfolio volatility,
which implies highest OOS Sharpe ratio
All DFC models outperform unconditional covariance
DECO is beaten by several DFC models

Stability of Portfolio Weights
How much do weights change? 1
N(T−1)
N
n=1
T
t=2 |wn,t − wn,t−1|
Model GMV MV
µ0 - 10% 7.5% 5%
1946 - 2015
Unconditional covariance 5.82 259.49 194.11 157.81
Non-Factor DFC 3.29 131.71 98.57 65.46
CAPM DFC 1.56 90.57 67.76 44.96
FF3 DFC 1.54 88.95 66.55 44.16
FF3 30-Group DFC 1.47 87.90 65.77 43.63
FF3 10-Group DFC 1.37 82.89 62.00 41.12
FF3 5-Group DFC 1.34 81.22 60.76 40.30
FF3 DECO 1.37 82.41 61.66 40.92
DFC models are much more stable than unconditional covariance

Correlated Volatility Shocks and Asset Pricing

Decomposing the Common Factor in Idio Vol
Herskovic et al. (2016): CIVt is priced in the cross section
σ2
i,t − Et−1[σ2
i,t]
Unexpected Idiosyncratic Volatility
=
σ2
i,t − Et−1[σ2
i,t]
Et−1[σ2
i,t]
· Et−1[σ2
i,t]
= ˜νi,t
Volatility Innovation
· Et−1[σ2
i,t]
Time-Varying Volatility
CIVt =
1
N
N
i=1
σ2
i,t − Et−1[σ2
i,t ] ≈ −
¯˜ν · ¯σ2
N
+ ¯σ2 1
N
N
i=1
wi,t ˜νi,t
VINt
+¯˜ν
1
N
N
i=1
wi,t Et−1[σ2
i,t ]
TVVt
CIVt quintiles show average return spread of -5.4% per year

Univariate Quintiles on VIN and TVV Betas
CIVt ≈ −
¯˜ν·¯σ2
N + ¯σ2
VINt + ¯˜νTVVt
VIN beta 1 (Low) 2 3 4 5 (High) 5-1 t(5-1)
Panel A: One-way sorts on VIN-beta
E[R] − rf 9.72% 8.70% 8.75% 7.63% 6.41% -3.31% -1.72
αCAPM 2.64% 1.74% 1.42% -0.34% -3.12% -5.75% -3.11
αFF 1.63% 0.51% 0.47% -0.45% -2.35% -3.98% -2.22
TVV beta 1 (Low) 2 3 4 5 (High) 5-1 t(5-1)
Panel B: One-way sorts on TVV-beta
E[R] − rf 9.36% 8.79% 8.11% 8.72% 7.74% -1.62% -0.91
αCAPM 1.14% 1.97% 1.29% 1.14% -1.30% -2.44% -1.36
αFF 1.14% 1.33% 0.34% 0.41% -1.52% -2.67% -1.49

5 × 5 Portfolios on VIN and TVV Betas
Long position in portfolio with largest VIN and TVV betas
Short position in portfolio with smallest betas
Model Intercept Mkt.RF SMB HML WML R2
Panel A: Raw Return -7.99%
(-2.72)
Panel B: CAPM -10.13% 0.29 3.11%
(-3.47) (5.15)
Panel C: FF 3-factor -8.51% 0.14 0.57 -0.34 10.40%
(-2.98) (2.42) (6.62) (-3.83)
Panel D: Carhart 4-factor -13.18% 0.19 0.59 -0.23 0.43 15.66%
(-4.63) (3.37) (7.08) (-2.58) 7.16
Similar eﬀect as the CIV factor of Herskovic et al. (2016)

5 × 5 Portfolios on VIN Beta and ME
Low VIN β 2 3 4 High VIN β 5-1 t(5-1)
Average Excess Returns
1 (Small) 15.95% 13.60% 13.29% 11.65% 10.62% -5.33% -2.66
2 16.45% 12.71% 11.91% 11.32% 9.31% -3.10% -1.71
3 11.57% 10.97% 11.05% 10.62% 8.34% -3.23% -1.69
4 10.51% 10.52% 9.67% 10.33% 8.95% -1.56% -0.88
5 (Big) 8.52% 8.54% 8.54% 6.70% 7.26% -1.26% -0.71
5-1 -7.43% -5.06% -4.75% -4.95% -3.36% - -
t(5-1) -3.54 -2.71 -2.35 -2.28 -1.35 - -
FF3F α’s
1 (Small) 5.97% 3.47% 2.61% 0.45% -1.06% -7.03% -3.58
2 1.83% 0.98% 0.39% -1.99% -1.61% -3.44% -2.00
3 0.50% 0.67% 0.64% -0.04% -2.10% -2.60% -1.55
4 0.44% 1.01% 0.07% 0.74% -0.88% -1.32% -0.80
5 (Big) 1.15% 0.86% 1.19% -0.66% 0.20% -1.35% -0.80
5-1 -4.82% -2.61% -1.42% -1.10% 0.86% - -
t(5-1) -2.90 -1.94 -1.03 -0.74 0.45 - -

5 × 5 Portfolios on TVV Beta and ME
Low TVV β 2 3 4 High TVV β 5-1 t(5-1)
Average Excess Returns
1 (Small) 15.37% 13.61% 12.07% 13.65% 10.15% -5.22% -2.82
2 12.58% 11.38% 9.83% 11.79% 9.11% -3.47% -2.21
3 10.54% 10.53% 10.98% 10.36% 10.31% -0.23% -0.14
4 10.00% 10.81% 10.35% 9.58% 9.18% -0.81% -0.58
5 (Big) 8.48% 7.75% 8.21% 7.83% 7.94% -0.54% -0.35
5-1 -6.88% -5.86% -3.85% -5.81% -2.21% - -
t(5-1) -2.87 -3.03 -1.96 -2.93 -0.94 - -
FF3F α’s
1 (Small) 3.81% 3.41% 2.02% 3.21% -1.25% -5.06% -2.70
2 0.52% 0.89% -0.34% 0.96% -2.42% -2.94% -1.86
3 -0.76% 0.22% 0.98% -0.20% -0.43% 0.33% 0.20
4 -0.32% 1.36% 1.20% -0.08% -0.85% -0.53% -0.38
5 (Big) 1.04% 0.82% 1.08% 0.32% 0.07% -0.97% -0.62
5-1 -2.77% -2.59% -0.95% -2.89% 1.32% - -
t(5-1) -1.53 -1.85 -0.68 -2.03 0.73 - -

Conclusion
Correlated volatility shocks contribute to comovement in
idiosyncratic volatility
Dynamic Factor Correlation (DFC) directly models correlated vol
shocks and ﬁts the data better than DCC and DECO
DFC reduces to well-known multivariate volatility models under
certain restrictions
DFC can be used to build more stable mean-variance portfolios
with higher Sharpe ratios
Common factor in idio vol can be decomposed into VIN and TVV
VIN is priced in the cross section, TVV to a lesser extent

"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven Investment Management

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