Financial correlations measure the relationship between the changes of two or more financial variables over time. For example, the prices of
equity stocks and fixed interest bonds often move in opposite directions: when investors sell stocks, they often use the proceeds to buy bonds and vice versa. In this case, stock and bond prices are negatively correlated.
Financial correlations play a key role in modern
finance. Under the
capital asset pricing model
In finance, the capital asset pricing model (CAPM) is a model used to determine a theoretically appropriate required rate of return of an asset, to make decisions about adding assets to a well-diversified portfolio.
The model takes into accou ...
(CAPM; a model recognised by a
Nobel prize), an increase in diversification increases the return/risk ratio. Measures of risk include
value at risk
Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by ...
,
expected shortfall, and portfolio return
variance.
Financial correlation and the Pearson product-moment correlation coefficient
There are several statistical measures of the degree of financial correlations. The
Pearson product-moment correlation coefficient is sometimes applied to finance correlations. However, the limitations of Pearson correlation approach in finance are evident. First, linear dependencies as assessed by the Pearson correlation coefficient do not appear often in finance. Second, linear correlation measures are only natural dependence measures if the joint distribution of the variables is
elliptical. However, only few financial distributions such as the multivariate normal distribution and the multivariate student-t distribution are special cases of elliptical distributions, for which the linear correlation measure can be meaningfully interpreted. Third, a zero Pearson product-moment correlation coefficient does not necessarily mean independence, because only the two first moments are considered. For example,
(''y'' ≠ 0) will lead to Pearson correlation coefficient of zero, which is arguably misleading. Since the Pearson approach is unsatisfactory to model financial correlations,
quantitative analysts have developed specific financial correlation measures. Accurately estimating correlations requires the modeling process of marginals to incorporate characteristics such as
skewness
In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive, zero, negative, or undefined.
For a unimodal ...
and
kurtosis
In probability theory and statistics, kurtosis (from el, κυρτός, ''kyrtos'' or ''kurtos'', meaning "curved, arching") is a measure of the "tailedness" of the probability distribution of a real-valued random variable. Like skewness, kurtos ...
. Not accounting for these attributes can lead to severe estimation error in the correlations and covariances that have negative biases (as much as 70% of the true values). In a practical application in portfolio optimization, accurate estimation of the
variance-covariance matrix is paramount. Thus, forecasting with Monte-Carlo simulation with the Gaussian copula and well-specified marginal distributions are effective.
Financial correlation measures
Correlation Brownian motions
Steven Heston applied a correlation approach to negatively correlate stochastic stock returns
and stochastic volatility
. The core equations of the original
Heston model are the two
stochastic differential equations, SDEs
:
(1)
and
:
(2)
where S is the underlying stock,
is the expected growth rate of
, and
is the stochastic volatility of
at time t. In equation (2), g is the mean reversion rate (gravity), which pulls the variance
to its long term mean
, and
is the volatility of the volatility σ(t). dz(t) is the standard
Brownian motion
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position ins ...
, i.e.
,
is
i.i.d., in particular
is a random drawing from a standardized normal distribution n~(0,1). In equation (1), the underlying
follows the standard geometric Brownian motion, which is also applied in
Black–Scholes–Merton model, which however assumes constant volatility.
The correlation between the stochastic processes (1) and (2) is introduced by correlating the two Brownian motions
and
. The instantaneous correlation
between the Brownian motions is
:
(3).
The definition (3) can be conveniently modeled with the identity
:
(4)
where
and
are independent, and
and
are independent, t ≠ t’.
The Cointelation SDE
connects the SDE's above to the concept of mean reversion and drift which are usually concepts that are misunderstood
by practitioners.
The binomial correlation coefficient
A further financial correlation measure, is the binomial correlation approach of Lucas (1995). We define the binomial events
and
where
is the default time of entity
and
is the default time of entity
. Hence if entity
defaults before or at time
, the random indicator variable
will take the value in 1, and 0 otherwise. The same applies to
. Furthermore,
and
is the default probability of
and
respectively, and
is the joint
probability of default
Probability of default (PD) is a financial term describing the likelihood of a default over a particular time horizon. It provides an estimate of the likelihood that a borrower will be unable to meet its debt obligations.
PD is used in a variety ...
. The standard deviation of a one-trial binomial event is
, where P is the probability of outcome X. Hence, we derive the joint default dependence coefficient of the binomial events
and
as
:
(5).
By construction, equation (5) can only model binomial events, for example default and no default. The binomial correlation approach of equation (5) is a limiting case of the Pearson correlation approach discussed in section 1. As a consequence, the significant shortcomings of the Pearson correlation approach for financial modeling apply also to the binomial correlation model.
Copula correlations
A fairly recent, famous as well as infamous correlation approach applied in finance is the
copula approach. Copulas go back to
Sklar (1959). Copulas were introduced to finance by Vasicek (1987)
and Li (2000).
Copulas simplify statistical problems. They allow the joining of multiple univariate distributions to a single multivariate distribution. Formally, a copula function C transforms an n-dimensional function on the interval
,1into a unit-dimensional one:
: