Stressed Correlations and Volatilities –
How to Fulfill Requirements of the Basel
Committee
Christoph Becker∗
Wolfgang M. Schmidt†
Preliminary version, March 24, 2011
Abstract
We propose a new approach to the definition of stress scenarios for
volatilities and correlations which fulfills the requirements of the Basel Committee on Banking Supervision for the quantification of market risk. Correlations and volatilities are functions of one and the same market factor, which
is the key to stressing them in a consistent and intuitive way. Our approach is
based on a new asset price model where correlations and volatilities depend
on the current state of the market. The state of the market captures marketwide movements in equity-prices and thereby fulfills minimum requirements
for risk factors stated by the Basel Committee. For sample portfolios we
compare correlations and volatilities in a normal market and under stress and
explore consequences for value-at-risk. Stressed value-at-risk exceeds the
standard value-at-risk by a factor of 3 to 4, confirming estimates from the
Basel Committee.
We finally compare our modeling approach with multivariate GARCH
models. For all data analyzed our model proved to be superior in capturing
the dynamics of volatilities and correlations.
Keywords: correlation, volatility, Basel III, GARCH models
JEL classification codes: C13, C32, C58, G11, G12
We thank Darrell Duffie, participants of EFA 2010 conference, Peter Ruckdeschel, Alexander
Szimayer, Uwe Kuechler, Natalie Packham, Robert Tompkins, Thomas Heidorn, Matthias Fengler,
and Marlene Mueller for helpful suggestions and discussions. This paper is part of a research project
that is funded by the Frankfurt Institute for Risk Management and Regulation (FIRM). Special thanks
to Keith Jarrett for his music.
∗ Frankfurt School of Finance & Management, Sonnemannstr. 9-11, 60314 Frankfurt am Main,
e-mail: [email protected]
† Frankfurt School of Finance & Management, Sonnemannstr. 9-11, 60314 Frankfurt am Main,
e-mail: [email protected]
1
1
Introduction
We suggest a model that provides a natural setting to define consistent stress scenarios for volatilities and correlations. The stress scenarios are based on historical
experience and correspond to pre-specified probabilities. In our model the vector
of volatilities σ and the correlation matrix ρ depend on one and the same market
state F. The market state F is generic; to comply with suggestions of the Basel
Committee we define the market state as the realized drift of a market index. We
estimate the dependence structure of volatilities σ (F) and correlations ρ(F) on
the market state F from daily stock prices. In other words, we propose and estimate a nonlinear one-factor model for the dynamics of volatilities and correlations.
Stressed volatilities and correlations are then defined by
σ ( fα ),
ρ( fα ),
where fα is the α-quantile of the empirical distribution of observed market states
F. The concept is transparent and, by construction, volatilities and correlations are
stressed in a consistent manner. Furthermore, the choice of the quantile probability α relates to the probability of the stress scenario. We examine consequences
of stressed volatilities and correlations on portfolio value-at-risk. We find that the
stressed value-at-risk exceeds a standard value-at-risk by a factor of 3 − 4, confirming results in BCBS [2009a].
Our approach is motivated by the requirements of the Basel Committee on
Banking Supervision (BCBS) on how to determine capital requirements for the
market risk of a stock portfolio. If a bank uses internal models to determine capital requirements, the committee demands that the “bank must calculate a ‘stressed
value-at-risk’ measure” where “the relevant market factors were experiencing a period of stress”, see BCBS [2009b] and the updated document BCBS [2011b]. One
type of stress tests suggested by the committee is to “evaluate the sensitivity of the
bank’s market risk exposure to changes in the assumptions about volatilities and
correlations”. However, in the document BCBS [2011a] the committee acknowledges that banks that wish to fulfill this requirement face difficult technical questions. For risk factors the minimum requirement stated in BCBS [2011b] is that
“there should be a risk factor that is designed to capture market-wide movements
in equity-prices (e.g. a market index)”. One question is how to use such a factor to
stress volatilities and correlations in a consistent manner. Another question is how
to stress volatilities and correlations in a way that is justified by historical experience. Furthermore, stressing correlations in a portfolio with more than two assets
is technically difficult because one needs to maintain the positive semi-definiteness
of the correlation matrix. By increasing individual correlations by a fixed quantity
the positive semi-definiteness of the correlation matrix is lost in general. Therefore
2
a naive approach that bases stressed correlations on estimates of bivariate models
is infeasible.
Popular models like multivariate GARCH models do not seem to be able to
fulfill all the requirements posed by the Basel Committee. Our approach for modeling the dynamics of volatilities and correlations is related to existing GARCH
approaches. In comparison, out of all the time series we considered, our model
proves to capture the behavior of correlations and volatilities better than the Dynamic Conditional Correlation GARCH model by Engle [2002].
The paper is organized as follows: In Section 2 we outline our model and
present the main results. We investigate the dependence structure of volatilities
and correlation on the market state F and consider examples of stressed volatilities and correlations. Furthermore, we explore consequences for value-at-risk. In
Section 3 we describe the model in more detail. The relationship between our
model and multivariate GARCH models is discussed in Section 4. In Section 5 we
compare our model with the Dynamic Conditional Correlation GARCH model, a
model with constant volatilities and correlations, and two moving averages based
on the past 30 and 90 days, respectively. We analyze how the models capture the
dynamics of volatilities and correlations in a given time series and find that our
model outperforms.
2
Stressed volatilities and correlations and their impact
on VaR
To describe the main results we sketch our model and provide details in Section
3. We develop a diffusion model for the dynamics of assets S1 , . . . , Sn , where
volatilities and correlations depend on a market state F. The market state F is
the realized drift of a market index over a fixed number of past observations. We
estimate our model for stocks from the S&P 500 using daily data for the period
Jan 1990 − Nov 2010. Here, the market state F is defined as the realized drift of
the S&P 500. We estimate the model by applying a maximum likelihood estimator.
The dependence of volatilities and correlations on the market state is first analyzed
by considering pairs of stocks. The estimates in Figure 1 show that correlations
are increased in bear markets and stable in normal and bull markets. Furthermore,
correlations and volatilities seem to be co-moving. The horizontal lines in Figure
1 are estimated constant correlations and volatilities in a corresponding model that
assumes only constant volatilities and correlations. The difference between market
state dependent volatilities as well as correlations and their constant counterparts,
respectively, indicates how strongly these quantities change in a crisis.
However, we cannot determine stressed correlations for a larger portfolio by
3
(a) Bank of America - Citigroup
(b) Colgate - Exxon
(c) Microsoft - Halliburton
(d) Walt Disney - Pfizer
(e) Walmart - Johnson & Johnson
(f) Merck - Pfizer
Figure 1: Typical dependency structures of correlation ρ and volatilities σ1 , σ2 of
daily stock returns on the market state, that is, the realized drift of the S&P 500
over a rolling windows of nF = 75 business days. The realized drift is annualized.
Data from 1990 − 2010.
computing stressed correlations in a bivariate manner because we may obtain a
non-positive-semidefinite matrix. Therefore, for a portfolio consisting of n stocks
we simultaneously estimate the vector of volatilities σ (F) and the n × n - correlation matrix ρ(F).
Recall that the market state is computed as a realized drift over a fixed number
of past daily observations. This number of past observations has a convenient
interpretation as the memory of the market. We estimate our model for different
market memories and find that the optimal memory is about 75 business days, see
Figure 6 in Appendix B.
As a result, the vector of volatilities and the correlation matrix are known functions σ (F), ρ(F) of the market state F. How can we define stressed volatilities and
correlations that comply with the requirements in BCBS [2009b]? We propose to
define risk scenarios by shifting the market state to predefined quantiles of its em-
4
pirical distribution. That is, we shift the market state F to α-quantiles fα with, for
example, α ∈ {0.1%, 1%, 5%} and compute the corresponding correlation matrices
ρ( fα ) = (ρi, j ( fα ))i, j=1,...,n
(1)
σ ( fα ) = σ1 ( fα ), . . . , σn ( fα ) .
(2)
and vectors of volatilities
Since the market state is defined as the realized drift of an appropriate index,
it captures the systemic market risk component of the stock portfolio. By design
the proposed stress scenarios for volatilities and correlations fulfill the minimum
requirements posed by BCBS [2011b]. In particular, the choice of the quantile
probability α relates to the probability of the stress scenario. Moreover, volatilities
and correlations are stressed in a consistent way because they are stressed simultaneously and based on one and the same market factor F.
We illustrate our approach by analyzing a sample portfolio consisting of six
stocks: Bank of America (BoA), Exxon, General Electric (GE), Microsoft (MS),
Pfizer and Walmart. Figures 7 and 8 in Appendix B show the estimated dependence
structure of correlations and volatilities on the market state. We emphasize that
these estimations must be performed for the portfolio as a whole and not on a
pairwise basis. Table 1 shows the correlation matrix and volatilities for the market
state F at the 5% - quantile of its empirical distribution, Table 2 for the market state
F at the median.
Vol.
MS
0.517512
Walmart
0.304813
GE
0.604834
Pfizer
0.345711
Exxon
0.406842
BoA
1.355577
Corr.
MS
Walmart
GE
Pfizer
Exxon
BoA
MS
1.000000
0.391786
0.445221
0.538888
0.620236
0.582697
Walmart
0.391786
1.000000
0.410489
0.529755
0.438121
0.135700
GE
0.445221
0.410489
1.000000
0.588320
0.588359
0.547035
Pfizer
0.538888
0.529755
0.588320
1.000000
0.632803
0.546533
Exxon
0.620236
0.438121
0.588359
0.632803
1.000000
0.406522
BoA
0.582697
0.135700
0.547035
0.546533
0.406522
1.000000
Table 1: Volatilities and correlations for the market state at the 5%-quantile of
its empirical distribution. The market state is computed as the realized drift of
the S&P500 over the past 75 business days. The model is estimated for period
Jan 2004 − Nov 2010.
We analyze the portfolio’s value-at-risk in our proposed stress scenarios. Let
us assume that we invest V0 = 100$ in our sample portfolio. For asset weights γi
5
Vol.
MS
0.196424
Walmart
0.168472
GE
0.200905
Pfizer
0.212423
Exxon
0.201778
BoA
0.285326
Corr
MS
Walmart
GE
Pfizer
Exxon
BaA
MS
1.000000
0.350425
0.437016
0.284452
0.332468
0.238774
Walmart
0.350425
1.000000
0.389480
0.336892
0.287761
0.373636
GE
0.437016
0.389480
1.000000
0.376709
0.384305
0.707337
Pfizer
0.284452
0.336892
0.376709
1.000000
0.299811
0.384892
Exxon
0.332468
0.287761
0.384305
0.299811
1.000000
0.323369
BoA
0.238774
0.373636
0.707337
0.384892
0.323369
1.000000
Table 2: Volatilities and correlations for the market state at the median of its
empirical distribution. The market state is computed as the realized drift of
the S&P500 over the past 75 business days. The model is estimated for period
Jan 2004 − Nov 2010.
we compute a 10-day ‘stressed value-at-risk’ for level α by
!
!
s
r
n
10
= −V0 exp
VaRstressed
Φ−1 (α) ∑ γi γ j σi ( fα )σ j ( fα )ρi, j ( fα ) − 1 ,
α
250
i, j=1
where volatilities and correlations are evaluated at the α-quantile fα of the empirical distribution of the market state F. The function Φ is the cumulative distribution
function of the standard normal distribution. Furthermore, we compute a ‘nonstressed value-at-risk’ for level α by
!
!
s
r
n
10
VaRnon-stressed
= −V0 exp
Φ−1 (α) ∑ γi γ j σiconst σ const
ρi,const
−1 ,
α
j
j
250
i, j=1
where volatilities and correlations are estimated from a corresponding model with
constant volatilities and correlations. In Figure 2a we plot the functions
α 7→ VaRstressed
,
α
α 7→ VaRnon-stressed
,
α
α ∈ (0, 0.25)
(3)
for our sample portfolio by assuming equal asset weights γi . Figure 2b shows the
ratio of stressed and non-stressed value-at-risk,
α 7→
VaRstressed
α
,
non-stressed
VaRα
α ∈ (0, 0.25).
We observe that the stressed 99%, 10-day value-at-risk VaRstressed
computed with
0.99
market state dependent volatilities and correlations is 3 times higher than the value6
(a) Value-at-Risk
(b) Ratio of Values-at-Risk
Figure 2: Estimated ten-day VaR for different probabilities α for our sample portfolio and portfolio value 100$. The model is estimated on Jan 2004 − Nov 2010.
computed with constant volatilities and correlations. For a
at-risk VaRnon-stressed
0.99
1
portfolio of 20 stocks we even observe a ratio of 4, see Figure 3.
Our results confirm the findings of BCBS [2009a] who report that the ratio of
the stressed value-at-risk and the non-stressed value-at-risk as computed by banks
is in the range of 0.68 − 7 with median 2.6.
To summarize the advantages of our approach we conclude that, firstly, by
defining stress scenarios for volatilities and correlations by (1)-(2), we fulfill the
minimum requirements posed by BCBS [2011b], that is, “there should be a risk
factor that is designed to capture market-wide movements in equity-prices (e.g. a
market index)”. Moreover, volatilities and correlations are stressed in a consistent
way because they are stressed simultaneously and based on one and the same market factor F. Secondly, our approach confirms findings of the impact study BCBS
[2009a] on how much a stressed value-at-risk should exceed a standard value-atrisk. Thirdly, the correlation matrix (1) and the vector of volatilities (2) can be used
as inputs for a market risk analysis in any model where daily returns are assumed
to be normally distributed, see the discussion in Section 4. Moreover, as will be
shown in Section 5, the model seems to perform better than the Dynamic Conditional Correlation GARCH model by Engle [2002] in capturing the dynamics of
correlations and volatilities within given samples.
1 Apple, American Express, AT&T, Bank of America, Boeing, Chevron, Citigroup, Coca Cola,
Exxon, Ford, General Electric, J.P. Morgan, Johnson & Johnson, McDonalds, Merck, Microsoft,
Pfizer, Procter & Gamble, Walmart, Walt Disney.
7
(a) Value-at-Risk
(b) Ratio of Values-at-Risk
Figure 3: Estimated ten-day VaR for different probabilities α for a portfolio of 20
stocks and portfolio value 100$. The model is estimated on Jan 2004 − Nov 2010.
3
An asset price model with state-dependent correlation
in continuous time
In Section 2 we have sketched the main elements of our model. In this section we
state its complete definition.
We consider a diffusion type asset price model driven by Brownian motion.
Our assumption is that asset volatilities and asset correlation depend on the current
state of the market, which is interpreted as a common risk factor. An example of
the state of the market is the realized drift of a market index, which is determined
on a rolling window of past and current asset price realizations. The dependency
of the asset dynamics on past asset realizations leads us to stochastic differential
equations with time delay, so called stochastic delay differential equations, see Mao
[2007] or Mohammed [1984]. In every point in time t volatilities and correlations
in our model are determined by states in the past interval [t − r,t] for some fixed
length r of the memory window.
We work on a probability space (Ω, F , P) equipped with a filtration F =
(Ft )t≥0 satisfying the usual conditions. The filtration is rich enough to carry at
least n independent Wiener processes. Before we state the dynamics of asset price
processes S = (S1, . . . , Sn ), we introduce the concept of the segment process.
Let S = S(t) t∈[−r,∞) be a continuous Rn -valued stochastic process. For every
8
t ≥ 0 we define the [t − r,t]-segment of the process S by
St (u) = S(t + u),
u ∈ [−r, 0],
that is, St is a mapping that takes values in the space C ([−r, 0], Rn ) of continuous
functions from [−r, 0] to Rn ,
St : Ω 7→ C ([−r, 0], Rn ) .
We call (St )t∈[0,∞) the segment process with time delay r. For a segment φ ∈
C ([−r, 0], Rn ) we define the norm
||φ ||∞ = sup ||φ (u)||2 ,
u∈[−r,0]
with || · ||2 the Euclidean norm on Rn . We base our analysis on the model
dSi (t) = µi (θ , St )Si (t) dt + σi θ , F(St ) Si (t) dW i (t),
dW i (t) · dW j (t) = ρi, j θ , F(St ) dt,
i, j = 1, . . . , n
n
S0 ∈ C [−r, 0], R+ ,
(4)
(5)
(6)
with Wiener processes W 1 , . . . ,W n . The drifts µi , the volatilities σi and the instantaneous correlations ρi, j are functions,
µi : Θ ×C ([−r, 0], Rn ) → R,
σi : Θ × R → [0, ∞),
ρi, j : Θ × R → [−1, 1],
i, j = 1, . . . , n,
that are parameterized with some parameter θ ∈ Θ ⊂ R p , p ∈ N. The volatilities σi
and the correlations ρi, j depend on the market state F(St ), which we describe by a
market state function F of the past segment St ,
F : C ([−r, 0], Rn ) → R.
Since the market state depends on a past window of assets S1 , . . . , Sn , the model is
a system of delay equations.
We assume that the instantaneous correlation matrix
ρ(θ , x) = ρi, j (θ , x) i, j=1,...,n
is positive definite for all (θ , x) ∈ Θ × Rn , hence there exists a unique Cholesky
decomposition. For future reference we define the Cholesky decomposition C(θ , x)
of the instantaneous covariance matrix, that is,
C(θ , x)C(θ , x)T = diag σ (θ , x) ρ(θ , x) diag σ (θ , x) , (θ , x) ∈ Θ × R, (7)
9
with diag σ (θ , x) a matrix with components σ1 (θ , x), . . . , σn (θ , x) on the diagonal and zeros otherwise.
Before we define the functional form of the market state F and introduce appropriate parameterizations for the drifts µi , volatilities σi and correlations ρi, j , we
state sufficient conditions for the existence of a unique solution of our model in its
most general form (4)-(6). Furthermore, we show that the instantaneous correlation
can be interpreted as a proxy for the correlation of daily log-returns. For proofs see
Becker and Schmidt [2010].
Proposition 1. Assume that for every θ ∈ Θ and for i = 1, . . . , n
µi ◦ exp(θ , ·) : C ([−r, 0], Rn ) → R
is locally Lipschitz-continuous and fulfills the linear growth condition
|µi ◦ exp(θ , x)|2 ≤ D 1 + ||x||2∞ ,
x ∈ C ([−r, 0], Rn ) ,
(8)
with a constant D > 0. The concatenation µi ◦ exp is defined as
µi ◦ exp( f ) = µi exp ◦ f 1 , . . . , exp ◦ f n , f = ( f 1 , . . . , f n ) ∈ C ([−r, 0], Rn ) . (9)
Furthermore, assume that for every θ ∈ Θ and i, j = 1, . . . , n
σi (θ , ·) : (R, | · |) → ([0, ∞), | · |) ,
ρi, j (θ , ·) : (R, | · |) → ([−1, 1], | · |) ,
F ◦ exp : C ([−r, 0], Rn ) , || · ||∞ → R, | · |
are locally Lipschitz continuous, where the concatenation F ◦ exp is defined analogously to µi ◦ exp. Let the volatility functions σi (θ , ·) be bounded, and the instantaneous correlation ρ(θ , x) be positive definite for all (θ , x).
Then for every θ ∈ Θ there exist F-adapted Wiener processes W 1 , . . . ,W n
and an F-adapted process S = (S1 , . . . , Sn ) with strictly positive paths that satisfy the system of delay equations (4)-(6). The distribution of S on the path space
C [−r, ∞), Rn is unique. If the drifts µi are bounded it holds that
n
E
sup
∑
2
S (t)
< ∞.
i
(10)
t∈[−r,T ] i=1
From now on we assume that the conditions of Proposition 1 hold and S denotes
the unique positive solution of the system (4)-(6).
10
The instantaneous correlation ρi, j admits a convenient interpretation as the correlation between daily log-returns of assets Si and S j given the market state F(St ),
that is
Si (t + 1 day)
S j (t + 1 day) ρi, j θ , F(St ) ≈ Corr log
, log
Ft .
Si (t)
S j (t)
The following lemma makes this more precise.
Lemma 1. Denote the components of the Cholesky-decomposition C(θ , x) introduced in (7) as
θ ∈ Θ, x ∈ C ([−r, 0], Rn ) .
C(θ , x) = (ci, j (θ , x))i, j=1,...,n ,
For every θ ∈ Θ let the mappings
x 7→ µi (θ , ·) ◦ exp(x),
i = 1, . . . , n,
and
x 7→ ci, j (θ , ·) ◦ exp(x),
x 7→ c2i, j (θ , ·) ◦ exp(x),
i, j = 1, . . . , n
be uniformly Lipschitz-continuous, with the concatenation of functions defined as
in (9). Furthermore, let the volatilities σi be bounded and condition (8) hold.
Then for every t ≥ 0 and the sequence ∆m = 1/m, m ∈ N, it holds P-almost
surely
Si (t + ∆m )
S j (t + ∆m ) lim Corr log
F
=
ρ
θ
,
F
(S
)
.
,
log
t
i,
j
t
m→∞
Si (t)
S j (t) The market state function F is a common risk factor for the dynamics of volatilities and correlations. The minimum requirement for such a risk factor stated in
BCBS [2011b] is that it shall “capture market-wide movements in equity-prices”.
Therefore we define the market state as an average realized drift of assets S1 , . . . , Sn ,
F (St ) = F S(t), S(t − ∆t), . . . , S(t − nF ∆t)
j
1 n 1 nF
S (t − (k − 1)∆t)
=
∑ nF ∆t ∑ log S j (t − k∆t)
n j=1
k=1
j
1 c2
S (t − (k − 1)∆t)
+
σ log
: k = 1, . . . , nF .
(11)
2∆t
S j (t − k∆t)
with
c2 (x1 , . . . , xn ) = 1
σ
F
nF − 1
11
nF
∑
j=1
1
xj −
nF
nF
∑ xk
k=1
!2
.
(12)
In case of constant volatilities σi ≡ σi θ , F(St ) and constant drifts µi ≡ µi (θ , St ),
the proposed function F is an unbiased estimator of ∑ni=1 µi /n. Alternatively, we
can define the market state function F as the realized drift of a stock market index.
It is straightforward to prove that the market state (11) fulfills the conditions in
Proposition 1.
For the functional form of volatilities and correlations in the continuous time
model (4) - (6) we want to achieve a maximum of flexibility. Parameterizations
for certain classes of correlation matrices have been proposed, for example, by
Ding and Engle [2001]. However, it is not clear whether these parameterizations
are suitable for our model and the problems we are targeting. Furthermore, parameterizing correlation matrices is technically difficult because of the required
positive definiteness, see, for example, Rebonato and Jackel [1999]. Therefore
we do not define separate parameterizations for pairwise correlations ρi, j (θ , ·) and
for volatilities σi (θ , ·). Instead, we introduce a parameterization for the Cholesky
decomposition
C(θ , x) = (ci, j (θ , x))i, j=1,...,n
of the instantaneous covariance matrix (7) via


ξ , θ ),
i> j
hi, j (x,
ci, j (θ , x) = α + hi, j (x, ξ , θ ), i = j


0,
i < j.
(13)
The functions hi, j (·, ξ , θ ) are cubic splines through a common set of equidistant
discretization points ξ for the values of the market state,
ξ = (ξl )l=1,...,nCov ,
nCov ∈ N,
(14)
and individual sets θli, j
of values for every entry of the Cholesky del=1,...,nCov
composition ci, j . These individual sets are collected in one vector θ that we estimate from real market data. We choose the points (ξl )l=1,...,nCov such that they, for
given asset realizations s(ti ) i=1...,n , cover the range of realized market states
n
o
F s(ti−nF ), . . . , s(ti ) , i ≥ 1 + nF .
The variable α is a small, positive number that guarantees that the covariance matrix defined via (13) and hence the correlation matrix ρ(·, ·) is positive definite for
all (θ , x) ∈ Θ × R.
For the drift functions µi we use either constant values or the weighted form
m
Si (t − j∆t)
µi (θ , St ) = ∑ β j log i
,
(15)
S (t − ( j + 1)∆t)
j=0
12
with ∑mj=0 β j = 1, m ∈ N, and β j ≥ 0 for all j. In our estimates for volatilities
and correlations we found that the estimation results are quite insensitive to the
particular choice of the drift function. Therefore, we assume constant drifts µi in
our estimates.
4
Relation to multivariate GARCH models
Multivariate GARCH models are standard models to describe the dynamics of
volatilities and correlations, see, for example, Silvennoinen and Ter¨asvirta [2009].
For a vector of returns
r(k) = (r1 (k), . . . , rn (k))
with discrete time k = 1, 2, . . . , multivariate GARCH models describe the dynamics
of r via
r(k) = C(k)η(k).
(16)
The matrix C(k) is the Cholesky decomposition of the covariance matrix H(k) ∈
Rn,n , which by assumption depends on the observations up to time k − 1. The
Rn -valued random process η has independent, identically distributed realizations
η(k), k = 1, 2, . . . . The random variable η(k) has components (η1 (k), . . . , ηn (k))
with covariances
(
1, i = j
Cov (ηi (k), η j (k)) =
.
0, i 6= j
The variable η(k) is assumed to be independent of the observations up to time
k − 1. The assumption
E (ηi (k)) = 0,
for all i, k,
implies then that the returns are centered. For normally distributed η(k) the dynamics (16) yields that
r(k) ∼ N (0, H(k)) .
(17)
The core element of multivariate GARCH models are updating rules for the covariance matrix H(k) that are based on past observations of the covariance matrix
H and past returns r. One group of models defines linear updating rules, see,
for example, the VEC-model by Bollerslev et al. [1988], while a second group
introduces nonlinearities, see, for example, the Dynamic Conditional Correlation
GARCH model by Engle [2002].
13
To compare our approach with multivariate GARCH models we translate our
continuous time dynamics into a dynamics in discrete time. Let S follow the dynamics (4)-(6) introduced in Section 3 and define the discretized asset price process
S˜ by
˜ = S(k∆t),
S(k)
k = 0, 1, . . . .
We investigate the discrete time dynamics of the returns
!
S˜i (k)
,
ri (k) = log
i = 1, . . . , n.
S˜i k − 1
(18)
For the process S˜ we denote by pk the transition density given all information up
to time k − 1,
˜ ∈ dyk log S(
˜ j) = y j : j = 1, . . . , k − 1 .
pk (yk |y1 , . . . , yk−1 , θ )dyk = Pθ log S(k)
The transition density pk is unknown in our model. Therefore we approximate pk
with the density of a n-dimensional normal distribution with distribution parameters implied by the Euler scheme. Neglecting the drift2 we obtain a distribution for
asset returns like (17),
r(k) ∼ N (0, H(k)) ,
(19)
with
˜ k−1 ) ,
H(k) = ∆t diag σ θ , F(S˜k−1 ) ρ θ , F(S˜k−1 ) diag σ θ , F(S
and diag σ (θ , F(S˜k−1 )) a matrix with diagonal
(20)
σ1 (θ , F(S˜k−1 )), . . . , σn (θ , F(S˜k−1 ))
and zeros otherwise. The market state F depends on the segment S˜k−1 of the discretized path,
˜ − 1), . . . , S(k
˜ − 1 − nF ) .
S˜k−1 = S(k
The dynamics of returns is similar to (16),
r(k) ≈ C(k)η(k),
(21)
with C(k) the Cholesky decomposition of the instantaneous covariance matrix (20),
and (η(k)) , k = 1, 2, . . . independent identically distributed random vectors with
2 Our estimates for volatilities and correlations prove to be quite insensitive to the particular choice
of the drift function.
14
n-dimensional standard normal distribution. Formula (21) explains why our approach is related to multivariate GARCH models. However, instead of updating
the covariance matrix based directly on past observations of returns and covariance
matrices, our model first updates the state of the market, which then determines the
covariance matrix. Note that the market state function F as defined in (11) can be
˜ − 1), . . . , S(i
˜ − 1 − nF ), but
expressed not only as a function of price realizations S(i
also as a function of realized returns r(i − 1), . . . , r(i − nF ). Like in the Dynamic
Conditional Correlation GARCH model volatilities and correlations are non-linear
functions of realized returns.
5
Capturing the dynamics of volatilities and correlations:
a comparison with GARCH type models
How well does our model (4)-(6) capture the real-world dynamics of volatilities
and correlations? To answer this question we follow an approach suggested by
Engle and Colacito [2006]. They compare different models by optimizing a portfolio based on volatilities and correlations that are predicted by the corresponding
model. The model that yields the smallest portfolio variance is then best able to
capture the real dynamics of volatilities and correlations.
We use the notation introduced in Section 4. For the period from time k − 1 to
k we denote the asset weights in the portfolio by
w(k) = (w1 (k), . . . , wn (k)) ∈ Rn .
If the sum ∑ni=1 wi (k) differs from one we invest the difference
n
1 − ∑ wi (k)
i=1
in a riskfree asset with return r f . The return of the asset portfolio is then
!
n
n
rportfolio (k) =
∑ wi (k)ri (k) +
i=1
n
=
1 − ∑ wi (k) r f
(22)
i=1
∑ wi (k) (ri (k) − r f ) + r f .
(23)
i=1
Denote by H(k) = (hi, j (k)) the covariance matrix of the returns ri (k), which is
predicted by the model at time k − 1. Then the predicted portfolio volatility is
s
n
∑
wi (k)w j (k)hi, j (k).
i, j=1
15
Observe that the second summand in (22) does not contribute to the predicted
volatility. Denote by µ = (µ1 , . . . , µn ) the vector of expected excess returns,
µi = E(ri (k) − r f ),
i = 1, . . . , n,
where we assume that these expectations are identical for all k. For the period
[k − 1, k], the asset weights w(k) minimizing the portfolio variance are the solution
of the problem
wT (k)H(k)w(k),
(24)
minn
w(k)∈R ,
s.t. wT (k)µ=µ0
where µ0 ∈ R is the required excess return of our portfolio. The solution to problem
(24) is
H −1 (k)µ
w(k) = T −1
µ0 .
(25)
µ H (k)µ
The conditional portfolio variance for period [k − 1, k] is
2
σ (k) = Ek−1 wT (k) (r(k) − r) ,
(26)
where r = Ek−1 r(k) is the conditional mean of returns r(k), which is assumed to
be constant. By Theorem 2 in Engle and Colacito [2006], the unknown conditional
mean r in (26) can be safely approximated by the sample mean.
The portfolio weights w(k) in (25) depend on the model predicted conditional
covariance matrix H(k) and the vector µ. The corresponding weights using the
true (however, unknown) conditional covariance matrix would lead to a different
portfolio. Comparing the variances of these two portfolios Engle and Colacito
[2006], Theorem 1, show that the latter variance is always smaller, no matter the
choice of µ. This justifies that the model generating the smallest portfolio variance
is considered to be superior.
We analyze two specifications for the vector µ, which can be interpreted as two
portfolio strategies. In the first strategy we assume that
µ = (1, . . . , 1) ∈ Rn .
Since no asset outperforms, the portfolio can be interpreted as a minimum variance
portfolio. In the second strategy the vector of expected excess returns is chosen as
µ = (1, 0, . . . , 0) ∈ Rn ,
which corresponds to a strategy where the first asset is held for return whereas the
others are hedging positions.
16
MinVar
Hedge
DCC
0.220075
0.290439
Const
0.223416
0.291045
Avg30
0.230884
0.306560
Avg90
0.221615
0.294265
StateDepen
0.219187
0.289201
Table 3: Annualized portfolio volatilities, averaged over all choices of four-asset
portfolios. Data are from 1997 − 2004.
MinVar
Hedge
DCC
0.175241
0.184381
Const
0.173437
0.182614
Avg30
0.177969
0.192667
Avg90
0.173076
0.184465
StateDepen
0.165852
0.179317
Table 4: Annualized portfolio volatilities, averaged over all choices of four-asset
portfolios. Data are from 2004 − 2010.
As an example, in the following we consider sample portfolios that consist of
four stocks out of ten given large US stocks3 . We compare five different models:
the discretized version (21) of our state-dependent model (denoted by ’StateDepen’ in the following tables); the standard mean-reverting Dynamic Conditional
Correlation GARCH model by Engle [2002] (denoted by ’DCC’); a model with
constant volatilities and correlations (’Const’); and two moving averages over the
past 30 and 90 days, respectively (’Avg30’, ’Avg90’).
Recall that we want to investigate how well the models capture the dynamics of
volatilities and correlations on a given set of data. To this end we estimate the models4 and optimize the portfolios period by period in the time frames from Jan 1997
- Jan 2004 and from Jan 2004 - Nov 2010. Tables 3 and 4 show the annualized realized portfolio return volatilities, averaged over all choices of portfolios that consist
of four stocks out of ten. The differences in portfolio return volatilities are small
but systematic. Tables 5 and 6 show the percentage of all four-asset portfolios for
which the model in the respective row yields a smaller realized portfolio volatility than the model in the respective column. We observe that the state-dependent
model (21) outperforms the other models systematically.
3 AT&T, Coca Cola, Exxon, Ford, General Electric, Johnson & Johnson, Microsoft, J.P. Morgan,
Procter & Gamble, Walmart.
4 Formula (23) for the portfolio return is only justified for simple returns. Therefore the DCCGARCH model, the moving averages and the constant model are estimated based on simple returns.
17
MinVar
DCC
Const
Avg30
Avg90
StateDepen
DCC
–
0.152381
0.000000
0.228571
0.671429
Const
0.847619
–
0.042857
0.657143
1.000000
Avg30
1.000000
0.957143
–
1.000000
0.985714
Avg90
0.771429
0.342857
0.000000
–
0.790476
StateDepen
0.328571
0.000000
0.014286
0.209524
–
Hedge
DCC
Const
Avg30
Avg90
StateDepen
DCC
–
0.409524
0.000000
0.009524
0.747619
Const
0.590476
–
0.000000
0.128571
0.985714
Avg30
1.000000
1.000000
–
1.000000
1.000000
Avg90
0.990476
0.871429
0.000000
–
0.980952
StateDepen
0.252381
0.014286
0.000000
0.019048
–
Table 5: Percentage of four-asset portfolios for which the model in the row yields
a smaller realized portfolio volatility than the model in the column. Data are from
1997 − 2004.
MinVar
DCC
Const
Avg30
Avg90
StateDepen
DCC
–
0.700000
0.376190
0.647619
0.947619
Const
0.300000
–
0.252381
0.471429
1.000000
Avg30
0.623810
0.747619
–
0.842857
1.000000
Avg90
0.352381
0.528571
0.157143
–
0.966667
StateDepen
0.052381
0.000000
0.000000
0.033333
–
Hedge
DCC
Const
Avg30
Avg90
StateDepen
DCC
–
0.823810
0.014286
0.480952
0.985714
Const
0.176190
–
0.000000
0.209524
1.000000
Avg30
0.985714
1.000000
–
1.000000
1.000000
Avg90
0.519048
0.790476
0.000000
–
1.000000
StateDepen
0.014286
0.000000
0.000000
0.000000
–
Table 6: Percentage of four-asset portfolios for which the model in the row yields
a smaller realized portfolio volatility than the model in the column. Data are from
2004 − 2010.
18
A
Estimation method
We estimate the parameter θ ∈ Θ in the continuous time model (4)-(6) from discrete time market observations. Statistical methods for stochastic delay equations
are not yet well-developed. For delay equations with affine drift a first estimation
approach is developed in K¨uchler and Sørensen [2009a] and K¨uchler and Sørensen
[2009b]. It is not obvious how their approach can be generalized to the setting
of our model5 . We propose an approximate maximum likelihood estimator that is
heuristically motivated and is shown to work well in simulation experiments. A
proof of the consistency and asymptotic distribution of this estimator is beyond the
scope of this paper and subject of future research.
Consider daily realizations of the process S,
s(tk ) k=1,...,N = s1 (tk ), . . . , sn (tk ) k=1,...,N ,
where tk+1 − tk = ∆t = 1/250. It is natural to base our estimator on log-prices.
Denote by pk the conditional density
Pθ log S(tk ) ∈ dyk log S(t j ) = y j : j = 1, . . . , k − 1 = pk (yk |y1 , . . . , yk−1 , θ )dyk ,
which is unknown in our model. The log-likelihood function is then given by
N
∑
log pk log s(tk )| log s(t1 ), . . . , log s(tk−1 ), θ .
(27)
k=1+nF
We approximate the unknown density pk with the density p˜k of a n-dimensional
normal distribution. The parameters of p˜k are motivated by the Euler scheme for
stochastic delay equations6 , cf. K¨uchler and Platen [2000]. The density p˜k has
mean
!
2
1 n
i
log s (tk−1 ) + µi (θ , stk−1 ) − ∑ σ˜ i,p θ , F(stk−1 )
∆t
,
2 p=1
i=1,...,n
and covariance matrix (cf. (7))
T
∆t diag σ θ , F(stk−1 ) ρ θ , F(stk−1 ) diag σ θ , F(stk−1 )
,
both depending on the parameter
θ . Recall that stk−1 denotes the segment of observations s(tk−1 ), . . . , s(tk−1−nF ) . As in our setup (11) and (15), we let µi (θ , stk−1 )
5 Private
communication with Uwe K¨uchler.
case of non-delay stochastic differential equations, our approach reduces to the well-known
parameter estimation technique, see, for example, Hurn et al. [2007].
6 In
19
and F(stk−1 ) depend on daily past observations s(tk−1 ), s(tk−2 ), . . . . We propose an
approximate maximum likelihood estimator
θb = argmaxθ ∈Θ L(θ ),
(28)
with likelihood function
N
L(θ ) =
∑
k=1+nF
log p˜k log s(tk ) log s(t1 ), . . . , log s(tk−1 ), θ .
(29)
For a number of n assets, the computational effort of estimator (28) grows
at the order of n3 , because the density p˜k requires us to compute the inverse of an
n × n-covariance matrix. This makes the application of estimator (28) to large portfolios practically infeasible from a computational point of view. This is a common
problem also for other models like multivariate GARCH, see, for example, BCBS
[2011a]. Inspired by methods in Engle et al. [2009], we reduce the computational
effort to order n2 . This is achieved by replacing the likelihood function (29) by
the sum over all bivariate likelihood functions Li1 ,i2 (θ ) that refers to the asset pair
Si1 , Si2 ,
Li1 ,i2 (θ )
(30)
N
=
∑
k=1+nF
log p˜k
log(si1 , si2 )(tk ) log s(t1 ), . . . , log s(tk−1 ), θ .
Here, p˜k is the density of a two-dimensional normal distribution with parameters
suggested by the Euler scheme. Observe that the covariance matrix of Si1 , Si2 depends on the whole vector S of assets. The parameter θ is now estimated from
n
θb = argmaxθ ∈Θ
Li1 ,i2 (θ ).
∑
(31)
i1 ,i2 =1,i1 <i2
The computational effort of this estimator is at the order n2 .
Still, we face a high-dimensional optimization problem. The effort there can
be reduced by subsequently applying estimator (31) to sub-portfolios with m =
2, 3, 4, . . . assets. More precisely, we estimate the parameters for the sub-portfolio
with assets S1 , . . . , Sm and then use these parameters as pre-estimated values in the
estimation of the sub-portfolio with assets S1 , . . . , Sm+1 .
To reduce the computational effort further, we would have to depart from modeling correlations individually, thereby lose too much flexibility.
Recall that the market state (11) depends on the market memory nF , which has
to be estimated from data as well. We estimate the market memory from
n
nc
F = argmaxnF ∈{2,...,250} max
∑
θ ∈Θ i ,i =1,i <i
1 2
1 2
20
Li1 ,i2 (θ , nF ).
(32)
The approximate maximum likelihood estimator (28) and thus our estimators
(31) and (32) can be severely biased, because for large Euler discretization steps
∆t the density p˜k of a normal distribution may be a poor approximation for the
transition density pk . For daily observations we justify the reliability of estimator
(31) by re-estimating a given parameterization in model (4) - (6) from simulated
discrete time asset realizations. We use n = 4 assets and nCov = 8 discretization
points (14) for every spline function of the instantaneous covariance matrix. In
the model (4) - (6) we use the market state function (11) with nF = 75 days and
constant drifts µk = 0.1 for all i = 1, . . . , 4. For the dependency of volatilities and
instantaneous correlation
on the
market state we describe the Cholesky decomposition C F(St ) = ci, j F(St ) i, j of the instantaneous covariance matrix by (13),
with

2


 π arctan (sin(i + j+p0.9x)) ,
i > j
2
ci, j (x) = 0.001 + π arctan
|2 j + 0.9x| , i = j
i, j = 1, . . . , n,


0,
i < j,
We use an Euler scheme to generate a time series of daily asset realizations over
a period of 20 years. To keep the discretization bias small in the Monte Carlo
simulation we use 2000 additional equidistant discretization steps per day. Figure
4 shows that estimator (31) yields a reliable estimate of the dependency structure
of volatilities and correlation on the market state. Figure 5 shows that estimator
(32) is able to identify the market memory.
21
Figure 4: Re-estimation of pre-specified dependency structures of volatilities and
correlations on the market state. Black lines indicate the empirical estimate (31),
red lines indicate the true model dependencies.
22
Figure 5: Re-estimation of the market memory nF of the market state function F in
the model (4) - (6). The Black line are estimated maxima of the likelihood function
in (31), the red line indicates the true market memory.
B
Results of empirical estimations
(a) Colgate - S&P 500
(b) Microsoft - S&P 500
(c) Halliburton - S&P 500
(d) Walt Disney - S&P 500
(e) Pfizer - S&P 500
(f) Walmart - S&P 500
3
Figure 6: Plots of the estimated maximum of the likelihood function (31) versus varying market memories of the market state function in the model (4) - (6).
We observe that the market memory is about 75 business days. Estimates are for
1990 − 2010.
23
(a) Microsoft-Walmart
(b) Microsoft-GE
(c) GE - Walmart
(d) Pfizer-Microsoft
(e) Pfizer-Walmart
(f) Pfizer-GE
(g) Microsoft-Exxon
(h) Walmart-Exxon
(i) GE-Exxon
(j) Pfizer-Exxon
(k) Microsoft-BoA
(l) Walmart-BoA
(m) GE-BoA
(n) Pfizer-BoA
(o) Exxon-BoA
Figure 7: Dependence structure of return correlations on the market state for a
portfolio of six stocks. The market state is defined as the realized drift of the
S&P500 over a rolling window of 75 business days. Estimates are for Jan 2004 −
Nov 2010.
24
(a) Microsoft
(b) Walmart
(c) General Electric
(d) Pfizer
(e) Exxon
(f) Bank of America
Figure 8: Dependence of return volatilities on the market state. The market state
is defined as the realized drift of the S&P500 over a rolling window of 75 business
days. Data from Jan 2004 − Nov 2010.
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26