Financial correlation
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 (CAPM; a model recognised by a Nobel prize), an increase in diversification increases the return/risk ratio. Measures of risk include value at risk, 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, formula_1 ("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 and kurtosis. 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 formula_2 and stochastic volatility formula_3. The core equations of the original Heston model are the two stochastic differential equations, SDEs and where S is the underlying stock, formula_6 is the expected growth rate  of formula_7, and formula_3 is the stochastic volatility of formula_7 at time t. In equation (2), g is the mean reversion rate (gravity), which pulls the variance formula_10 to its long term mean formula_11, and formula_12 is the volatility of the volatility σ(t). dz(t) is the standard Brownian motion, i.e. formula_13, formula_14 is i.i.d., in particular formula_15 is a random drawing from a standardized normal distribution n~(0,1). In equation (1), the underlying formula_7 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 formula_17 and formula_18. The instantaneous correlation formula_19 between the Brownian motions is The definition (3) can be conveniently modeled with the identity where formula_22 and formula_23 are independent, and formula_24 and formula_25 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 formula_26 and formula_27 where formula_28 is  the default time of entity formula_29 and formula_30 is the default time of entity formula_31. Hence if entity formula_29 defaults before or at time formula_33, the random indicator variable formula_34 will take the value in 1, and 0 otherwise. The same applies to formula_31. Furthermore, formula_36 and formula_37 is the default probability of formula_29 and formula_31 respectively, and formula_40 is the joint probability of default. The standard deviation of a one-trial binomial event is formula_41, where P is the probability of outcome X. Hence, we derive the joint default dependence coefficient of the binomial events formula_42and formula_43 as 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 [0,1] into a unit-dimensional one: More explicitly, let formula_46 be a uniform random vector with formula_47 and formula_48. Then there exists a copula function formula_49 such that where F is the joint cumulative distribution function and formula_51, "i" = 1, ..., "n" are the univariate marginal distributions. formula_52 is the inverse of formula_51. If the marginal distributions formula_54 are continuous, it follows that C is unique. For properties and proofs of equation (11), see Sklar (1959) and Nelsen (2006 ). Numerous types of copula functions exist. They can be broadly categorized in one-parameter copulas as the Gaussian copula, and the Archimedean copula, which comprise Gumbel, Clayton and Frank copulas. Often cited two-parameter copulas are student-t, Fréchet, and Marshall-Olkin. For an overview of these copulas, see Nelsen (2006). Hence, from equations (7) and (8) we derive the Gaussian default time copula CGD, In equation (9) the terms formula_59 map the cumulative default probabilities Q of asset i for time t, formula_56, percentile to percentile to standard normal. The mapped standard normal marginal distributions formula_61 are then joined  to a single n-variate distribution formula_62 by applying the correlation structure of the multivariate normal distribution with correlation matrix R. The probability of n correlated defaults at time t is given by formula_62. Copulae and the 2007–08 financial crisis. Numerous non-academic articles have been written demonizing the copula approach and blaming it for the 2007/2008 global financial crisis, see for example Salmon 2009, Jones 2009, and Lohr 2009. . (a) Tail dependence In a crisis, financial correlations typically increase, see studies by Das, Duffie, Kapadia, and Saita (2007) and Duffie, Eckner, Horel and Saita (2009) and references therein. Hence  it would be desirable to apply a correlation model with high co-movements in the lower tail of the joint distribution. It can be mathematically shown that the Gaussian copula has relative low tail dependence, as seen in the following scatter plots. Figure 1: Scatter plots of different copula models As seen in Figure 1b, the student-t copula exhibits higher tail dependence and might be better suited to model financial correlations. Also, as seen in Figure 1(c), the Gumbel copula exhibits high tail dependence especially for negative co-movements. Assuming that correlations increase when asset prices decrease, the Gumbel  copula might also be a good correlation approach for financial modeling. (b) Calibration A further criticism of the Gaussian copula is the difficulty to calibrate it to market prices. In practice, typically a single correlation parameter (not a correlation matrix) is used to model the default correlation between any two entities in a collateralized debt obligation, CDO. Conceptually this correlation parameter should be the same for the entire CDO portfolio. However, traders randomly alter the correlation parameter for different tranches, in order to derive desired tranche spreads. Traders increase the correlation for ‘extreme’ tranches as the equity tranche or senior  tranches, referred to as the correlation smile. This is similar to the often cited implied volatility smile in the Black–Scholes–Merton model. Here traders increase the implied volatility especially for out-of-the money puts, but also for out-of-the money calls to increase the option price.. In a mean-variance optimization framework, 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. Allowing the modeling process to allow for empirical characteristics in stock returns such as auto-regression, asymmetric volatility, skewness , and kurtosis is important. Not accounting for these attributes lead to severe estimation error in the correlations and variances that have negative biases (as much as 70% of the true values). (c) Risk management A further criticism of the Copula approach is that the copula model is static and consequently allows only limited risk management, see Finger (2009) or Donnelly and Embrechts (2010). The original copulas models of Vasicek (1987) and Li (2000) and several extensions of the model as Hull and White (2004) or Gregory and Laurent (2004) do have a one period time horizon, i.e. are static. In  particular, there is no stochastic process for the critical underlying variables default intensity and default correlation. However, even in these early copula formulations, back testing and stress testing the variables for different time horizons can give valuable sensitivities, see Whetten and Adelson (2004) and Meissner, Hector, and. Rasmussen (2008). In addition, the copula variables can be made a function of time as in Hull, Predescu, and White (2005). This still does not create a fully dynamic stochastic process with drift and noise, which allows flexible hedging and risk management. The best solutions are truly dynamic copula frameworks, see section ‘Dynamic  Copulas’ below. Irrational complacency. Before the global 2007–08 financial crisis, numerous market participants trusted the copula model uncritically and naively. However, the 2007–08 crisis was less a matter of a particular correlation model, but rather an issue of "irrational complacency". In the extremely benign time period from 2003 to 2006, proper hedging, proper risk management and stress test results were largely ignored. The prime example is AIG's London subsidiary, which had sold credit default swaps and collateralized debt obligations in an amount of close to $500 billion without conducting any major hedging. For an insightful paper on inadequate  risk management leading up to the crisis, see “A personal view of the crisis – Confessions of a Risk Manager” (The Economist 2008). In particular, if any credit correlation model is fed with benign input data as low default intensities and low default correlation, the risk output figures will be benign, ‘garbage in garbage out’ in modeling terminology. Dynamic copulas. A core enhancement of copula models are dynamic copulas, introduced by Albanese et al. (2005) and (2007). The "dynamic conditioning" approach models the evolution of multi-factor super-lattices, which correlate the return processes of each entity at each time step . Binomial dynamic copulas apply combinatorial methods to avoid Monte Carlo simulations. Richer dynamic Gaussian copulas apply Monte Carlo simulation and come at the cost of requiring powerful computer technology. Conditionally independent default (CID) correlation modeling. In order to avoid specifying the default correlation between each entity pair in a portfolio a factorization is often applied. This leads to conditionally independent default (CID) modeling. The most widely applied CID model is the one-factor Gaussian copula (OFGC) model. It was the de facto market model for pricing CDOs before the 2007/2008 global financial crisis. The core equation of the OFGC model  where formula_65 and formula_66 are random drawings from formula_67 and formula_68. As a result, the latent variable formula_69, sometimes interpreted as the asset value of i, see Turc, Very, Benhamou and Alvarez et al. (2005), is also n~(0,1). The common factor formula_65 can be interpreted as the economic environment, possibly represented by the return of the S&P 500. formula_66 is the idiosyncratic component, the ‘strength’ of entity i, possibly measured by entity i's stock price return. From equation (10) we see, that the correlation between entities i is modeled indirectly by conditioning the latent variable formula_69 on the common  factor formula_65. For example, for p =1, the latent variables of all entities formula_74, so the formula_69 are identical in each simulation. For p = 0, all latent variable for all entities formula_76, hence the formula_69 are independent. Importantly, once we fix the value of M, the defaults of the n entities are (conditionally on M) mutually independent. As of 2010, the OFGC is the basis for credit risk management in Basel II. The benefits of the model are simplicity and intuition. One of the main shortcomings of the model is that traders when pricing CDOs randomly alter the  correlation parameter for different CDO tranches to achieve desired tranche spreads. However conceptually, the correlation parameter should be identical for the whole portfolio. Contagion default modeling. Contagion default modeling can be viewed as a variation of CID modeling. As discussed in section 2.3, in the CID framework, correlation is modeled by conditioning on a common market factor M, which impacts all entities to the same degree. The lower the random drawing for M, the higher is the default intensity of all entities (unless ρ = 0). Hence CID modeling can elucidate default clustering. In contrast, contagion approaches model the default  intensity of an entity as a function of the default of another entity. Hence contagion default modeling incorporates counterparty risk, i.e. the direct impact of a defaulting entity on the default intensity of another entity. In particular, after a default of a particular entity, the default intensity of all assets in the portfolio increases. This default contagion then typically fades exponentially to non-contagious default intensity levels. See the papers of Davis and Lo (2001) and Jarrow and Yu (2001), who pioneered contagion default modeling. Top–down correlation approaches. Within the credit correlation modeling framework, a fairly new correlation approach  is top–down modeling. Here the evolution of the portfolio intensity distribution is derived directly, i.e. abstracting from the default intensities of individual entities. Top-down models are typically applied in practice if: Top–down models are typically more parsimonious, computationally efficient and can often be calibrated better to market prices than bottom–up models. Although seemingly important information such as the default intensities of individual entities is disregarded, a top-down model can typically better capture properties of the portfolio such as volatility or correlation smiles. In addition, the default information of individual entities can often be inferred by  random thinning techniques, see Giesecke, Goldberg and Ding (2007) for details. Within the top-down framework, Schönbucher (2006) creates a time-inhomogeneous Markov-chain of transition rates. Default correlation is introduced by changes in the volatility of transition rates. For certain parameter constellations, higher volatility means faster transition to lower states as default, and as a consequence implies higher default correlation, and vice versa. Similarly, Hurd and Kuznetsov (2006a) and (2006b) induce correlation by a random change in the speed of time. A faster speed of time means faster transition to a lower state, possibly default, and as a result  increases default correlation, and vice versa. For a comparative analysis of correlation approaches in finance, see Albanese, Li, Lobachevskiy, and Meissner (2010).