State dependent correlations in the Vasicek default model

© 2020 A. Metzler, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Depend. Model. 2020; 8:298–329 Research Article Open Access A. Metzler* State dependent correlations in the Vasicek default model https://doi.org/10.1515/demo-2020-0017 Received April 23, 2020; accepted October 8, 2020 Abstract: This paper incorporates state dependent correlations (those that vary systematically with the state of the economy) into the Vasicek default model. Other approaches to randomizing correlation in the Vasicek model have either assumed that correlation is independent of the systematic risk factor (zero state dependence) or is an explicit function of the systematic risk factor (perfect state dependence). By contrast, our approach allows for an arbitrary degree of state dependence and includes both zero and perfect state dependence as special cases. This is accomplished by expressing the factor loading as a function of an auxiliary (Gaussian) variable that is correlated with the systematic risk factor. Using Federal Reserve data on delinquency rates we use maximum likelihood to estimate the parameters of the model, and nd the empirical degree of state dependence to be quite high (but generally not perfect). We also nd that randomizing correlation, without allowing for state dependence, does not improve the empirical performance of the Vasicek model.


Introduction
This paper incorporates state dependent correlations (those that vary systematically with the state of the economy) into the Vasicek default model. Other approaches to randomizing correlation in related models, such as [2,7,8,15,25,28], have either assumed that correlation is independent of the systematic risk factor (zero state dependence) or is an explicit function of the systematic risk factor (perfect state dependence). By contrast, our approach allows for an arbitrary degree of state dependence and includes both zero and perfect state dependence as special cases. This is accomplished by expressing the factor loading as a function of an auxiliary (Gaussian) variable that is correlated with the systematic risk factor; the degree to which the two are correlated can be interpreted as the degree of state dependence.
We t several di erent models to Federal Reserve data on delinquency rates, and compare their performance according to the Akaike Information Criteria (AIC). We nd that a state-dependent model with two correlation regimes outperforms the traditional Vasicek model, and that the estimated degree of state dependence is very high across all loan types. Importantly, we also nd the traditional Vasicek model outperforms a model with stochastic, but state-independent, correlations. In other words, randomizing correlation without allowing for state dependence does not improve the statistical performance of the Vasicek model. . Gaussian copula model The Vasicek model is a one-period (i.e. static) model used to construct default indicators of correlated exposures over a given time horizon. Closely related is the Gaussian copula model, a dynamic model used to construct the default times of correlated exposures. Like the Vasicek model, the Gaussian copula model is quite in uential, in particular it played a major role in the development of credit derivative markets; see [22] for an engaging description of that process.
Although the Vasicek and Gaussian copula models do share the same dependence structure, there are subtle but important di erence in the way that they are applied in practice. Speci cally, the Gaussian copula model is typically used to price and hedge credit derivatives (which would be found on the bank's so-called "trading book"), whereas the Vasicek model is typically used to compute risk measures associated with portfolios of loans that the bank itself has extended (i.e. risk management of the so-called "banking book"). In the former case the relevant probability measure is risk-neutral (i.e. Q), in the latter case it is historical (i.e. P). Another important di erence lies in the number of exposures. The number of exposures underlying a typical credit derivative pales in comparison to the number of loans in a typical banking book; in applications of the Vasicek model the number of exposures is typically large enough that so-called large portfolio approximation (described in the introduction to Section 2) is justi ed.
Because the Vasicek and Gaussian copula models share the same underlying dependence structure, it is important to consider the literature on the latter when proposing extensions to the former. Randomizing correlation has received considerable attention in the context of the Gaussian copula model; see [2,7,8,15,25,28] for example. Each of these extensions assumes either zero state dependence, in which case correlation is independent of the systematic risk factor, or perfect state dependence, in which case correlation is a function of the systematic risk factor. Our approach is more general in the sense that it allows for an arbitrary degree of state dependence, including both extremes as special cases.
Because our interest in this paper lies with the Vasicek model, we work almost exclusively with the large portfolio approximation. That being said, Section 6 does consider the impact of the proposed extension on the dependence structure (conditional tail probabilities and hazard rates, speci cally) in the case of two exposures. While this does give the reader some idea of how the proposed extension might a ect the pricing and hedging of credit derivatives, a full treatment of that impact is beyond the scope of the this paper.

. Regulatory capital
Regulatory capital is the capital that banks are legally compelled to raise as a cushion against the most adverse economic scenarios. Formally, the Basel Committee on Banking Supervision (BCBS) de nes regulatory capital on a given portfolio as the di erence between the (i) . th percentile and (ii) mean, of the portfolio's loss distribution. The BCBS further speci es a particular formula, often called the risk-weight function, that banks must use in order to compute regulatory capital. See [5] for a detailed discussion.
The statistical model underpinning the risk-weight function is the Vasicek model. Speci cally, the riskweight function is an approximation to the exact value of regulatory capital in the Vasicek model. The approximation was originally introduced and justi ed in [11] in so-called Asymptotic Single Risk Factor (ASRF) framework, which includes a very wide range of models. See [24] for a more recent discussion of the ASRF framework.
For n ≥ we let Φn(x; µ, Σ) denote the multivariate normal cdf with mean vector µ and covariance matrix Σ, evaluated at the point x ∈ R n . Note that where s×t is the s × t matrix with every entry equal to zero. Suppose that Z is normally distributed with zero mean and unit variance, and let a and b be n × vectors. Then In order to see that this is true, note that the rst expectation can be written as The display above is demonstrably equal to P(Z ≤ c, U − b Z ≤ a , . . . , Un − bn Z ≤ an), where (U , . . . , Un) is multivariate normal with zero mean and covariance matrix Σ, and Z is a standard normal independent of the vector (U , U , . . . , Un). The result follows upon checking the covariance matrix. By symmetry the second expectation can be obtained from the rst by replacing c with −c and b with −b. Finally, the third equation is obtained by summing the rst two.

Vasicek model
In order to model correlation between the default status of distinct exposures, the Vasicek model associates a "credit quality" variable X i with every exposure (here i indexes exposures). Credit quality is decomposed into systematic and idiosyncratic risk sources as follows: . is a sequence of i.i.d. N( , ) random variables and a , a , . . . is a sequence of constants such that a i ∈ ( , ) for every i. M is referred to as the systematic risk factor, and is interpreted as representing the overall state of the economy. For instance if M = − . then the economy is 1.2 standard deviations below average (whatever that means). The variable Y i is an idiosyncratic risk factor that is speci c to exposure i. The parameter a i is a factor loading that dictates how sensitive exposure i is to the overall economy. Alternatively, the factor loading governs the relative importance of systematic risk, as compared to idiosyncratic risk. It is clear that the correlation between distinct exposures i ≠ j is a i a j . In the model, exposure i defaults if the realized value of credit quality is su ciently low. Speci cally, default occurs if and only if X i ≤ Φ − (q i ), where q i ∈ ( , ) is a constant. Since X i is standard normal (recall that M and Y i are independent standard normal variables), it follows that q i is the default probability associated with exposure i. It is important to note that credit quality is latent, i.e. it is never observed directly. We only observe default indicator (X i ≤ Φ − (q i )), and not the credit quality variable itself. Here (A) denotes the indicator of event A.
The default rate among the rst N exposures is . The probability distribution of the default rate for large values of N is a fundamental object in quantitative risk management. The key insight in [29] is that, while the exact distribution might be complicated for nite N, the limiting distribution as N → ∞ is eminently tractable. Indeed, under very mild conditions on the model parameters (to be discussed more formally in the next paragraph) the almost sure limit D := lim N→∞ D N is well-de ned, depending only the realized value of the systematic risk factor (and not the realized values of the idiosyncratic risk factors). In other words, in essentially any case of practical interest we can write D = v(M) for some tractable function v. As a tractable function of a single Gaussian variable, the probability distribution of D is easy to derive. In the remainder of this paper we will refer to the variable D is often called the large portfolio default rate.
We conclude this section with a more formal version of the discussion in the previous paragraph. To begin, Theorem A2 in [20] ensures that D N − E[D N |M] converges to zero almost surely as M → ∞, for all values of the model parameters. If the model parameters are such that E[D N |M] converges to some limiting variable, then (i) that limiting variable is necessarily a (measurable) function of M and (ii) D N converges to the same limit. In other words, if the model parameters are such that E[D N |M] converges almost surely as N → ∞, then (i) the variable D is well-de ned and (ii) D = v(M) for some function v. As discussed in [26], in practice it is common to place each exposure into one of G groups and to assume that exposures within a given group are homogeneous (i.e. all exposures in group g would have the common default probability qg and factor loading ag). Under these assumptions, and as discussed in [26] we have E[D N |M = m] = G g= w g,N · Φ(Φ − (qg); ag m, − a g ), where w g,N is the proportion of the rst N exposures that fall into group g. If, for each g, w g,N converges to some constant wg as N → ∞, then E[D N |M = m] converges to G g= wg · Φ(Φ − (qg); ag m, − a g ) and we have D = G g= wg · Φ(Φ − (qg); ag M, − a g ). In the special case that G = (homogeneous exposures) we have D = Φ(Φ − (q); aM, − a ), where q is the common default probability and a is the common factor loading. Note that in general (i.e. for arbitrary G), D is a monotone function of M, whence the probability distribution of D is straightforward to obtain.

. Homogeneous Vasicek model
In the case of homogeneous exposures, neither default probabilities nor factor loadings depend on i. If a ∈ ( , ) denotes the common factor loading and ρ := a , then ρ is the correlation between distinct credit qualities. And if q denotes the common default probability, then x := Φ − (q) denotes the common default threshold. Mathematically it is more convenient to work with the variables x and a, economically the variables q and ρ have more meaning.
Given that M = m, credit qualities are i.i.d. Gaussian variables with mean am and variance − a , and the default indicators are i.i.d. Bernoulli variables with success probability P( is the default threshold. By the strong law of large numbers, then, we have that D N → Φ(x; am, − a ) as N → ∞. The implication is that the large portfolio default rate is given by For xed x ∈ R and a ∈ ( , ) we de ne the function vx,a : R → ( , ) as follows: We call this function a Vasicek default curve. Given a homogenous portfolio with correlation ρ and default probability q, the relation between the state of the economy and the large portfolio default rate is described by the function vx,a(·) with x = Φ − (q) and a = ρ / . The relationship between the systematic risk factor and the large portfolio default rate (i.e. the graph of the function vx,a) is illustrated in Figure 2.1 below. Important points are that (i) higher correlation exacerbates both good times and bad (i.e. higher correlation makes good times better and bad times worse) and (ii) higher correlation makes the default rate more sensitive to the state of the economy. For later use, it is worth noting that the inverse and derivative of a Vasicek default curve are given by and respectively.
Since D is a monotone (decreasing) function of M and M is standard normal, it is easy to obtain the probability distribution of D. Indeed For xed x ∈ R and a ∈ ( , ) we de ne the function sx,a : ( , ) → ( , ) as follows: (2.5) We call sx,a(·) a Vasicek survival function. Given a homogenous portfolio with correlation ρ and default probability q, the probability that the default rate exceeds a given threshold d is given by sx,a(d), with x = Φ − (q) and a = ρ / . Di erentiating − sx,a(d) we get the probability density function of the large portfolio default rate, which we denote by fx,a(d) and call a Vasicek density. It is readily veri ed that the Vasicek density is given by The implication is that higher correlation makes extreme default rates (be they large or small) much more likely.

Incorporating state dependence in correlations
In order to introduce state dependent correlations into the homogeneous Vasicek model, we begin by writing credit qualities in the form . . is an i.i.d. sequence of Gaussian variables with zero mean and unit variance.
• A is a random variable supported on some subset of the unit interval [ , ], and independent of the sequence Y , Y , . . .. Note, importantly, that we have not insisted that A and M are independent. Indeed we have only insisted that the marginal of M is Gaussian with mean zero and unit variance, and that the marginal of A is supported on some subset of [ , ]. Beyond this we have (intentionally) said nothing about the joint distribution of A and M. There are myriad ways to construct the joint distribution of A and M. For instance one could specify the marginal distribution of A and the copula of the pair (A, M), in which case the joint distribution of the pair would be fully and uniquely speci ed. The construction that we use in this paper is described more fully in Section 3.1, for now we simply assume that the joint distribution is essentially arbitrary.
The marginal and joint distributions of credit qualities will depend on the joint distribution of A and M, which we have yet to specify. That being said, certain insights are possible based on the limited assumptions we have made to this point. If A and M are independent then µ(a) ≡ and σ (a) ≡ , otherwise the exact forms of µ(·) and σ (·) will depend on the joint distribution of the pair (which we have yet to specify). That being said, it is not hard to show that, regardless of the exact form of the joint distribution, we have and Cov(X i , X j |A = a) = a σ (a) . We can therefore think of the random variable ρ(A) as a stochastic correlation between credit qualities. If A and M are independent then ρ(a) = a , otherwise the form of ρ(·) will depend on the form of the joint distribution of A and M. To complete the model, we assume that exposure i defaults if and only if the realized value of credit quality is su ciently low. In particular we assume that default occurs if and only if X i ≤ xq, where xq is the q th percentile of X i and q is the marginal default probability.

. Joint distribution of A and M
In order for correlation to vary systematically with the state of the economy, we must allow for dependence between A and M. Indeed if A and M are independent then correlation is stochastic but not state dependent. In order to create dependence between A and M we rst introduce an auxiliary standard normal variable T, which we permit to be correlated with M, and then write the factor loading as a function of T. More specically: The role of the function g is to govern the marginal behaviour of the factor loading A (equivalently, the correlation ρ(A)). By choosing g appropriately we can ensure that A (or ρ(A)) has any desired probability distribution. The role of the parameter β is to govern the degree of state dependence, by which we mean the tendency for correlation to vary systematically with the state of the economy. If β = then correlation is stochastic but independent of the state of the economy, and we obtain a model where credit qualities follow a Gaussian mixture. This is the approach taken in the stochastic correlation models proposed by [7] and [8] (see [15] for further discussion of these models). Although those approaches do allow for heterogenous factor loadings, the basic idea that factor loadings are randomized independently of the systematic risk factor is fundamentally the same. If β = ± then correlation is determined completely by the state of the economy, and we obtain both the random factor loading model proposed by [2] or the local correlation models proposed by [7] and [28] (see [15] for further discussion of these models). Intermediate values of β interpolate between these two extremes, and allow us to consider the more realistic case of a non-trivial, but imperfect, relationship between correlations and the overall state of the economy.

Remark 3.2.
If g is a monotone function, then this approach e ectively assumes a Gaussian copula between the systematic risk factor M and the factor loading A. The marginal behaviour of the factor loading is governed by the function g, and the dependence between M and A is governed by the parameter β.

. . Discrete factor loading
With a view to developing intuition, we focus on the case where the factor loading is discrete. A discrete loading can be obtained by taking g to be a simple function of the form where K ≥ is an integer, a , a , . . . , a K are numbers in the unit interval and −∞ = t < t < . . . < t K− < t K = ∞. With this speci cation the factor loading A is a discrete random variable taking on the value a k with probability Given that A = a k (equivalently, T ∈ (t k− , t k ]), the correlation between credit qualities is where we have used (3.3), and where σ k := Var(M|A = a k ) = Var(M|t k− < T ≤ t k ). Note that σ k depends on β (the correlation between M and T), as well as the threshold values t k− and t k . An explicit formula for σ k will be given in Section 4.1.3. Given the value of σ k , the relation (3.6) is easily inverted to express a k in terms of ρ k , indeed The implication is that the model can either be parametrized in terms of factor loadings or correlations. The former are easier to work with mathematically, the latter are (perhaps arguably) more natural at an intuitive level. In order to facilitate either parametrization, we introduce the variable and call the event {R = k} = {t k− < T ≤ t k } the k th (correlation) regime. Note that in the k th regime, the value of the factor loading is a k and the correlation between distinct credit qualities is ρ k , and that p k = P(R = k) is the probability that the k th regime is realized. Finally, note that the threshold values t k can be recovered from regime probabilities p k via the relation t k = Φ − ( k j= p j ). The implication is that the model can be parametrized in terms of regime probabilities as opposed to threshold values, the former being far more natural to work with at an intuitive level.
In order to parametrize the version of the model with K regimes, we must rst specify (i) the marginal default probability q, (ii) the degree of state dependence β and (iii) the probability mass function of the regime variable R (i.e. any K − of the probabilities p , p , . . . , p K ). Having xed the values of these quantities, we must then either specify (i) regime correlations ρ , ρ , . . . , ρ K , in which case factor loadings a k are then implied, or (ii) regime factor loadings a , a , . . . , a K , in which case correlations ρ k are then implied. In total, the K-regime version of the model has K + parameters.
It is not di cult to show that the joint distribution of the factor loading A and the systematic risk factor M is the same under the speci cation with parameters as it is under the speci cation with parameters where a k = a K−k+ and p k = p K−k+ . The implication is that there is a one-to-one correspondence between speci cations with strictly negative state dependence, and those with strictly positive state dependence. For this reason we may, without loss of generality, restrict attention to either non-negative or non-positive values of β.

Remark 3.3.
In the remainder of this paper we will, without loss of generality, assume that the degree of state dependence is non-negative. In other words, we assume that β ≥ in all instances.

Implementing the model
This section explains how to implement the discrete-loading version of the proposed model, and builds intuition for the impact of state dependence on important model-implied quantities. It culminates with closed form expressions for the probability density and survival functions of the large-portfolio default rate, and a discussion of important features of its distribution such as quantiles and tail expectations (which must in general be computed numerically).
In order to build intuition we consider a numerical example with two regimes. Parameter values are obtained by tting the two-regime version of the proposed model to the All Loans data set described in Section 5. We compare important model outputs with those obtained from the Vasicek model t to the same data set. For the two-regime model factor loadings are (a , a ) = ( . , . ), regime probabilities are (p , p ) = ( . , . ), the degree of state dependence is β = . and the marginal default probability is q = .
. For the Vasicek model, the factor loading is a = . and the marginal default probability is q = .
. Note that p a + p a ≈ a, so that the mean factor loading in the state dependent model is approximately the same as the factor loading in the Vasicek model.

. Correlation regime (R) and economic environment (M)
This section considers the relationship between the regime variable R (a discrete variable) and the systematic risk factor M (a continuous variable).

. . Correlation regime conditional on economic environment
Recall that p k = P(R = k) is the unconditional probability of regime k, and let p k (m) := P(R = k|M = m) be the conditional probability that regime k is realized, given the realized value of the systematic risk factor. Then where we have used the facts that P(R = k|M = m) = P(t k− < T ≤ T k |M = m) and that the conditional distribution of T, given that M = m, is Gaussian with mean βm and variance − β . If β = then p k (m) ≡ p k for all m. This re ects the fact that in the absence of state dependence, correlation is stochastic but unrelated to the state of the economy. If β = then p k (m) = (t k− < m ≤ t k ), which is either zero or one depending on the value of m. This re ects the fact that under perfect state dependence, correlation is completely determined by the state of the economy. If < β < then the behaviour of p k (·) depends on k. In particular: • p (·) is a decreasing function such that p (−∞) = and p (∞) = . The more adverse the economic scenario, the more likely it is that the rst regime is realized; during the most adverse scenarios it is virtually certain that the rst regime prevails. • p K (·) is an increasing function such that p K (−∞) = and p K (∞) = . The more favourable the economic scenario, the more likely it is that regime K is realized; during the most favourable scenarios it is virtually certain that regime K prevails. • For < k < K − , p k (·) is a unimodal function (increasing and then decreasing) such that p k (−∞) = p k (∞) = . Middle regimes are most likely to be observed during moderate economic scenarios, and it is extremely unlikely that middle regimes prevail during extreme economic scenarios (be they favourable or adverse). Figure 4.1(a) illustrates the graphs of p (·) and p (·) in our numerical example. We see that the rst regime (where the factor loading is higher) is almost certain to prevail during below average economic scenarios, in the sense that p (m) ≈ for m < . We also see that the second regime (where the factor loading is lower) is certain to prevail during very good economic scenarios, speci cally those that are two standard deviations above average (i.e. p (m) ≈ whenever m > ).

. . Economic environment given correlation regime
Let ϕ k (m) denote the density of the systematic risk factor in regime k, i.e. the conditional density of M given where we have used Bayes' rule and the fact that the unconditional distribution of the systematic risk factor is Gaussian with zero mean and unit variance. If β = then ϕ k (m) = ϕ(m) for all k and m. In the absence of state dependence, learning the correlation regime does not in uence the likelihood of any economic scenario. If β = then ϕ k (m) = ϕ(m) · (t k− < m ≤ t k )/p k , which is a truncated normal distribution concentrated on those values of m that correspond to regime k. If < β < then ϕ k (m) > ϕ(m) if and only if p k (m) > p k . If scenario m increases the likelihood of regime k, then ϕ k assigns relatively more weight to scenario m (as compared to the unconditional density ϕ). illustrates the conditional density of the systematic risk factor in each regime, in our numerical example. We see that the conditional density in the rst regime concentrates its mass on adverse scenarios. The implication is that if the (random) factor loading takes on its higher value, then it is likely that the economic scenario will be below average. We also see that the conditional density in the second regime concentrates its mass on favourable scenarios, the implication being that if the factor loading takes on its lower value then it is likely that the economy is doing well.  More precisely, it depicts the graphs of p (·) (solid) and p (·) (dashed) in our numerical example. The horizontal axis represents the realized value of the systematic risk factor (a value of, say, m = − . corresponds to an economy that is 1.2 standard deviations below average) and the vertical axis represents the conditional probability that a particular regime prevails, given the realized value of the systematic risk factor. Panel (b) depicts the conditional density of the systematic risk factor, given the prevailing regime. More precisely, it depicts the graphs of ϕ (·) (solid) and ϕ (·) (dashed) in our numerical example.
We conclude this section with the observation that if h is some function, then The identity (4.3) is easily veri ed by writing out and comparing the de ning integrals, and will be used repeatedly in what follows.

. . Conditional mean and variance of M
The mean and variance of the conditional density ϕ k are straightforward to compute. Using the tower property (recall that R is a function of T) and the fact that the conditional distribution of M, given that T = t, is Gaussian with mean βt and variance − β , we get that and Thus and Now, the conditional distribution of T, given that R = k, is a truncated Gaussian -speci cally, the standard Gaussian truncated to the interval [t k− , t k ]. The moments of the truncated Gaussian are well known, and we get that and where tϕ(t) = whenever t = ±∞. Inserting (4.6) into (4.4) and (4.7) into (4.6) we nally get that and In our numerical example we have (µ , µ ) = (− . , . ) and (σ , σ ) = ( . , . ).

Remark 4.1.
Although it is not obvious from (4.7), well-known properties of the truncated Gaussian distribution ensure that Var(T|R = k) < . It follows from (4.5), then, that the conditional variance σ k is a strictly decreasing function of β. In particular, we have that σ k = whenever β = and σ k < whenever β > .

. Factor loadings and correlations
Recall that the correlation between distinct credit qualities in regime k is . (4.10) In the absence of state dependence we have σ k = and ρ k = a k . In the presence of state dependence we have σ k < , and it is readily veri ed that this implies ρ k < a k . In the presence of state dependence, then, the relationship between the factor loading and correlation is not the same as it is in the Vasicek model. In our numerical example we have (ρ , ρ ) = ( . , . ). It is readily veri ed that ρ k is increasing in σ k , for xed a k . And since σ k is decreasing in β (recall Remark 4.1), it follows that ρ k is a decreasing function of β, for xed a k . In other words, if the factor loading is held xed then the correlation between distinct credit qualities decreases as the degree of state dependence becomes stronger. Although this may seem counter-intuitive at rst, the intuition is straightforward. If factor loadings are held xed but the degree of state dependence increases, then the variability of the idiosyncratic component of credit qualities ( − a k ) stays constant while the variability of the systematic component (a k σ k ) decreases. The systematic component therefore becomes less important, relative to the idiosyncratic component, and since correlation here measures the relative importance of the systematic component, correlation will decrease. The upper half of Table 4.1 illustrates this phenomenon in the context of our numerical example. We see that the degree of state dependence has a non-linear impact on correlations. For example in the absence of state dependence a factor loading of a = . would produce a rst-regime correlation of 6.3%, in the presence of moderate state dependence (β = . ) the same factor loading would produce a rst-regime correlation of 5.7%, and in the presence of perfect state dependence it would produce a correlation of only 4.1%.  The upper half of this table illustrates the impact of state dependence on correlations when factor loadings are held xed, in the context of our numerical example. Recall that factor loadings are (a , a ) = ( . , . ), regime probabilities are (p , p ) = ( . , . ) and the marginal default probability is q = .
. For each of the indicated values of β, we compute σ k using (4.9) and then compute ρ k using (4.10), all other parameter values are held xed. The lower half of the table illustrates the factor loading required to maintain correlations of (ρ , ρ ) = ( . , . ) as we vary the degree of state dependence.
The relationship (4.10) is easily inverted to express factor loadings in terms of correlations.
It is of passing interest to consider how a k varies with β when ρ k is held xed, i.e. the factor loading that would be required in order to maintain a given level of correlation as state dependence becomes stronger.
If correlation is to be held xed as the degree of state dependence increases, the factor loading must also increase. Indeed, if the factor loading were to stay constant then the variability of the systematic component (a k σ k ) would decrease but the variability of the idiosyncratic component ( −a k ) would not change, leading to a decrease in correlation. In order to maintain a xed level of correlation as the variability of the systematic risk factor (σ k ) decreases, then, we must increase its relative importance (a k ) in order to maintain a given level of correlation. The lower half of Table 4.1 illustrates this phenomenon in the context of our numerical example. In order to ensure a rst-regime correlation of 4.4% in the absence of state dependence (β = ) a factor loading of a = . , in the presence of moderate state dependence (β = . ) a factor loading of a = . is required and in the presence of perfect state dependence (β = . ) a factor loading of a = .
is required.

. Credit quality distributions
In this section we derive the marginal distribution of an individual credit quality, as well as the joint distribution of an arbitrary number of credit qualities. In preparation we note that if M = m and R = k, then Since the Y i are independent (and independent of R and M), it follows that the X i are i.i.d. Gaussian variables with mean a k m and variance − a k . Thus where a k = (a k , a k , . . . , a k ) T is n × and In is the identity matrix of size n.

. . Marginal distribution of credit quality
Using (4.12) we get the marginal cdf of X i in regime k is given by (4.14) In order to evaluate (4.14) we use (4.3), (1.1) and (1.5) to get that where Di erentiating (4.15) we nd that the conditional density of X i in regime k is where The density appearing in (4.17) is a member of the parametric family considered by [3], which contains the skew-normal family introduced by [4] as a special case and is discussed in more detail in Appendix B. We will henceforth refer to this parametric family as the ABGM family, after the authors of [3].
In the absence of state dependence (i.e. if β = ) then the density (4.17) reduces to ϕ(x), whence X i is standard normal in any regime. In the presence of state dependence (i.e. if β > ) the distribution of X i in regime k is a member of the ABGM family (with parameter values depending on k), and based on the discussion in Appendix B we can make the following general observations. In the rst regime (k = ) the density of X i is skewed to left, its left tail is proportional to ϕ(x) and its right tail is much thinner than ϕ(x). The exact opposite is true in the last regime (k = K), where the density is skewed to the right, the left tail is much thinner than ϕ(x) and the right tail is proportional to ϕ(x). For intermediate regimes ( < k < K) the density of X i can be skewed in either direction and both tails are much thinner than ϕ(x). Panel (a) in Figure  4.2 illustrates the conditional densities (4.17) in our numerical example. In the rst regime (where the factor loading is higher) the credit quality density is similar to a skew-normal density with negative skew, and for the second regime (where the factor loading is lower) it resembles a skew-normal with positive skew.  Recall that the default threshold, xq, is the q th percentile of the unconditional distribution of X i . In order to compute the default threshold xq, we therefore require an expression for the unconditional cdf. Using (4.15) and the fact that P( where we recall that x k and Σ k are given by (4.16).

Remark 4.2.
In the absence of state dependence (4.19) reduces to Φ(x) and the default threshold is xq = Φ − (q), which only depends on the default probability q and none of the other model parameters. In the presence of state dependence the default threshold depends on all model parameters, and must be determined numerically (Φ − (q) is typically a good initial guess).
In our numerical example the Vasicek threshold is xq = − . and the state dependent threshold is xq = − . . Since P(X i ∈ dx) = K k= P(X i ∈ dx|R = k) · P(R = k), an expression for the unconditional density of X i can be obtained by multiplying (4.17) by p k and summing over k. The end result is where r(x) = K k= r k (x). Note that (4.20) is a convex combination of ABGM densities (i.e. the unconditional distribution of X i is a mixture of ABGM distributions); since the ABGM family of densities is not closed under convex combinations, (4.20) is not a member of that family in general. In the absence of state dependence we have r(x) ≡ for all x, whence (4.20) reduces to ϕ(x) and X i is standard normal. Otherwise the function r is not identically equal to one and X i is not standard normal. That being said it will always be the case that r(x) → as x → ±∞, whence the density of X i is asymptotically identical to that of a standard normal variable. Figure 4.2 illustrates the density of X i in our numerical example and, for a point of reference, compares it to the standard normal density. The two densities are eminently similar, which suggests that the degree of state dependence does not have a material impact on the marginal behaviour of credit qualities.

. . Joint distribution of credit qualities
Using (4.13) we get the joint cdf of X = (X , X , . . . , Xn) T in regime k is given by where we recall that x = (x , x , . . . , xn) T , a k = (a k , a k , . . . , a k ) T is n × and In is the identity matrix of size n. In order to evaluate (4.21) we use (4.3), (1.2) and (1.5) to get that where the n × vector b k and n × n matrix Σ k are given by Di erentiating (4.22) we get that the conditional density of X in regime k is given by where Multiplying (4.22) and (4.23) by p k and summing over k we get that the unconditional cdf and pdf of X are given by and respectively. Note that in the absence of state dependence (i.e. if β = ) then s k (x) = p k for all x and (4.25) reduces to a mixture of n-dimensional Gaussian densities.

. Large portfolio default rate
Suppose that the realized value of the systematic risk factor is m, and that regime k prevails. Then the default indicators (X ≤ xq), (X ≤ xq), . . . , are i.i.d. Bernoulli variables with success probability where v k (·) := vx q ,a k (·) (4.26) is a Vasicek curve. The implication is that the large-portfolio default rate D is given by Note that the large portfolio default rate is a function of both M (a continuous variable) and R (a discrete variable). Note also that the unconditional (i.e. long-run, or through-the-cycle) default rate is which is as expected.

. . Default rate conditional on systematic risk factor
Given the realized value of the systematic risk factor (m, say), the large-portfolio default rate is a discrete variable taking one of the values v (m), v (m), . . . , v K (m) with respective probabilities p (m), p (m), . . . , p K (m). The mean default rate is therefore The relationship between the systematic risk factor and the large-portfolio default rate is summarized by the function m → E[D|M = m], and it is interesting to note that this function is a weighted average of Vasicek curves. Recall that our numerical example consists of (i) the two-regime state dependent model and (ii) the Vasicek model, both tted to the same set of data. Conditional on the realized value of the systematic risk factor, the large-portfolio default rate in (i) is a random variable taking on one of two possible values, whereas the large-portfolio default rate in (ii) is known with certainty. More precisely, if the realized value of the systematic risk factor is m then the large-portfolio default rate in the state dependent model is either v (m) or v (m), whereas it is certain to be v(m) in the state dependent model. Here v and v are Vasicek curves with factor loadings a = . and a = . , respectively, and a common default threshold of x = − . , while v is a Vasicek curve with factor loading a = . and default threshold x = − . .  is at least one standard deviation below average. Similarly, the state dependent model is virtually certain to produce a higher default rate if the economy is at least two standard deviations above average. In other words, the default rate will be higher in the state dependent model during extreme economic scenarios, be they good or bad. This is re ected in  In "typical" economic environments (those that are no more than one standard deviation away from average), the average default rate is lower in the state dependent model than it is in the Vasicek model. Otherwise, if the economic environment is at least one standard deviation above or below average, the default rate is higher in the state dependent model. Figure 4.3(c) illustrates how the two curves v (the high-correlation curve) and v (the low-correlation curve) are spliced together to produce the mean default rate E[D|M = m]. The average default rate is indistinguishable from the high-correlation curve for m < . , and indistinguishable from the low-correlation curve for m > . . In between, the average default rate is relatively constant at approximately 2%.

. . Default rate conditional on correlation regime
Given that regime k prevails, the large-portfolio default rate is D = v k (M). Conditional on the correlation regime, then, the default rate is a continuous random variable and its probability density function in regime is a Vasicek density. Figure 4.4 illustrates the conditional density of the large-portfolio default rate in each regime, in the context of our numerical example. We see that the distribution of the default rate in the rst regime is more concentrated on large default rates, whereas whereas it is more concentrated on lower default rates in the second regime. The reason is that the systematic risk factor tends to take on more adverse values during the rst regime (recall Figure 4.1(b)), and that the large-portfolio default rate tends to be higher when the systematic risk factor takes on adverse values (recall Figure 4.3(c)). This gure illustrates the conditional density of the large-portfolio default rate in each regime, in the context of our numerical example. Speci cally, it graphs g (·) (the conditional density in the rst regime, solid line) and g (·) (the conditional density in the second regime, dashed-line), where the expression for g k (·) is given by (4.29). We see that the rst-regime density is relatively more concentrated on large default rates.
The mean of the default rate in regime k can be computed in closed form. Indeed, we have where we have used the identity 1 Recall that if X is a random variable with density f X and Y = h(X) for some decreasing function h, then the density of Y is f Y (y) = −f X (h − (y))/h (h − (y)).
In the absence of state dependence the mean default rate does not depend on the correlation regime, in particular E[D|R = k] = q for every k. By contrast, in the presence of state dependence mean default rates can vary widely across correlation regimes. In our numerical example the average default rate in the rst regime is 4% (slightly higher than the long-run average default rate of 3.7%) whereas the default rate in the second regime is 2% (half the long-run rate). The default rate tends to be higher in the rst regime, since the systematic risk factor tends to take on more adverse values there.

. Default rate distribution
The unconditional density of the large-portfolio default rate is easily recovered from the conditional densities g k given in (4.29). Indeed, if g(d) denotes the unconditional density of D, then Important features of this distribution, such as tail probabilities and tail expectations, can be computed in closed form.
In order to compute tail probabilities, we use the fact that D = v k (M) in regime k and the identity (4.3) to write can be evaluated using (1.3), and the end result is Finally, using the fact that P(D ≥ d) = K k= P(D ≥ d|R = k) · P(R = k), we get Note that if β = , then (4.32) reduces to In order to compute tail expectations, it remains to compute the quantity E[D · (D ≥ d)], and we do use using an approach that is similar to the one used in the previous paragraph. We begin with which can be evaluated in closed form using (1.1) and then (1.3). The end result is Thus, (4.33) Combining (4.32) and (4.33), we nally get that tail expectations are given by . (4.34) .

Quantiles and tail expectations
Let dp denote the p th percentile of the default rate, i.e. dp satis es P(D ≥ dp) = − p. It does not appear possible to invert (4.32) in closed form, so dp must in general be computed numerically. Since dp is, for xed p, the unique root of the function d → P(D ≥ d) − ( − p), numerical computation of dp is straightforward. Figure 4.5 illustrates high percentiles of the default rate in our numerical example. As expected, percentiles in the state dependent model are considerably larger than those in the Vasicek model. For instance at the 99.9% level, the state dependent percentile is 14.4% as compared to a Vasicek percentile of 12.8%. In relative terms, di erence is nearly 13%. At the 99.99% level the state dependent and Vasicek percentiles are . % and . %, respectively, a di erence of more than 15% in relative terms.  The p th percentile is denoted dp and satis es P(D ≥ dp) = − p. In the Vasicek model dp = vx,a(z −p ), where zp is the p th percentile of the standard Gaussian distribution. In the state dependent model dp is computed by numerically nding the root of the function

Empirical analysis
In this section we estimate the parameters of the proposed model, using publicly available Federal Reserve data on delinquency rates for various types of loans². There are eleven categories of loans for which data is available: all loans (AL), business loans (BL), loans secured by real estate (SRE), lease nancing receivables (LFR), other consumer loans (OCL), commercial real estate (excluding farmland) loans (CRE), farmland loans (F), loans to nance agricultural production (AP), single-family residential mortgages (RM), consumer loans (CL) and credit card loans (CCL). Loans are considered delinquent if they are 30 more days over due, and delinquency rates are aggregated across the largest 100 nancial institutions. The data is seasonally adjusted, and consists of quarterly delinquency rates expressed on an annual basis. One series (AL) ranges from the rst quarter of 1985 to the third quarter of 2019 (139 observations), ve series (AP, BL, SRE, CL, LFR) range from the rst quarter of 1987 to the third quarter of 2019 (131 observations), and the remaining series (RM, CCL, OCL, CRE, F) range from the rst quarter of 1991 to the third quarter of 2019 (115 observations).
For each time series we use maximum likelihood to estimate the parameters of each of the following models.
(i) The Vasicek model. This is a two-parameter model with parameter vector θ = (x, a).
(ii) A two-regime model with stochastic, but state-independent, correlation. This is a four-parameter model with parameter vector θ = (x, a , a , p ). Note that β = here. (iii) A two-regime model with state dependent correlation. This is a ve-parameter model with parameter vector is θ = (x, a , a , p , β). (iv) A three-regime model with state dependent correlation. This is a seven-parameter model with parameter vector θ = (x, a , a , a , p , p , β).

. Estimation procedure
We consider the observed sequence of delinquency rates d , d , . . . , dn as i.i.d. drawings from the largeportfolio default rate density g(d) given in (4.31). The assumption of temporal independence is admittedly strong, but temporal dynamics are well beyond the scope of this paper. where g θ is that version of the density (4.31) with parameter vector θ. The function is simply n − times the negative of the log-likelihood function (the normalization n − is introduced to ensure numerical stability with respect to the number of observations). In an e ort to identify the global minimum of (5.1) we run the algorithm 500 di erent times, using 500 randomly generated initial points. This produces a set of candidate estimatesθ ,θ , . . . ,θ , whereθ k is the point to which the algorithm converged (or otherwise terminated) after being initialized at the k th initial point. The maximum likelihood estimate is then taken to be that point θ k that produces the smallest value of (θ k ).
When tting model (ii) with fmincon we constrain a , a and p to lie in the unit interval [ , ] but do not impose any constraints on the default threshold x. For model (iii) we impose the additional constraint that β lie in the unit interval but we do not impose any restriction on the relationship between a and a (e.g. we do not impose the restriction a > a ). Finally, for model (iv) we impose the additional constraints that p , a and p + p lie in the unit interval, and we do not impose any restrictions on the relationship between p , p or a , a , a . To protect against over tting we also impose a non-linear constraint designed to ensure that the function m → E[D|M = m] is continuous when β = . Speci cally, we insist that v k (t k ) = v k+ (t k ) for ≤ k ≤ K − .
In order to understand why this might protect against over tting, consider the two-regime case and let d = v (t ) and d = v (t ). If d > d then according to the model it is never possible to observe default rates in the interval [d , d ]. And if d > d then the default rate is not a monotone function of the systematic risk factor (which seems counterintuitive), and the e ect is that "extra" mass is placed on the interval [d , d ]. In either case, this strikes us as an opportunity for the algorithm to over t by placing too much or too little mass on ranges that are either over-or under-represented in the data.

. Model performance
The Aikaike Information Criteria (AIC) can be used to gauge the empirical performance of each model. AIC values for each model and time series are reported in Table 5.1. The two most striking features of the data are that (i) the model with stochastic, but state independent, correlation is the worst-performing model in every case except for one (credit card loans) and (ii) the state dependent model with two regimes outperforms the Vasicek model in every case.
Remark 5.1. Together, these observations suggest that simply randomizing correlation, without allowing for state dependence, does not improve the performance of the Vasicek model.

Series
Vasicek Mixture Two Regime Three Regime Vasicek model provides a superior t to the data) would be rejected for every series, at the 5% level it would be rejected for all but one of the series and at the 1% level it would be rejected for all but three of the series. Overall the results strongly suggest that di erences observed in Table 5.1 are statistically signi cant. In particular, they support the observation that the state dependent model, with either two or three states, provides a superior t to the data as compared to the Vasicek model. The rst group accounts for seven of the eleven series, and is characterized by the facts that correlation is higher in the rst regime than it is in the second (ρ > ρ ) and that the rst regime is (considerably) more likely than the second (p > p ). The second group (two series) is characterized by the facts that correlation is higher in the rst regime (ρ > ρ ) and that the rst regime is (considerably) less likely than the second (p < p ). The third group (two series) is characterized by the fact that correlation is lower in the rst regime (ρ < ρ ) and that the two regimes are equally likely (p ≈ p ).  The two most striking features of the data in Table 5.3 are that (i) the estimated degree of state dependence is very high in every case (the median estimate is 93%, the smallest estimate is 72%) and that (ii) correlation tends to be higher during adverse scenarios for a strong majority of the data (together the rst two groups account for over 80% of the data). In the introduction we noted that correlations between many asset classes tend to rise during adverse economic scenarios, and wondered whether the same was true of the late credit quality variables that underpin the Vasicek model. The results presented in Table 5.3 answer this question in the a rmative.

. Implications for risk management
Recall from Section 1.3 that the regulatory capital on a given portfolio is de ned as the di erence between the (i) th percentile and (ii) mean, of the portfolio loss distribution. If we make the (admittedly strong) assumptions that all exposures are the same size and that recovery rates are non-random and homogeneous across exposures then portfolio loss is proportional to the portfolio's default rate, in which case regulatory capital on the portfolio is proportional to the di erence between the (i) th percentile and (ii) mean, of the default rate. Now, the mean default rate predicted by both the state dependent and Vasicek models are nearly identical, so a comparison of model-implied high percentiles can give us a sense of how model-implied regulatory capital might compare.  Table 5.3 and d sd (p) is the corresponding percentile in the state dependent model with two regimes (also with parameters as reported in Table 5.3). Note that if exposures were equally-sized and recovery rates were non-random and homogeneous across exposures, then the percentage di erence in . th percentiles would correspond exactly to the percentage di erence in model-implied regulatory capital. -. -.
The same three groups that emerged in Table 5.3 also emerge in Table 5.4. For the rst group, the Vasicek model underestimates high percentiles (relative to the state dependent model) to a moderate degree. For the second group, Vasicek underestimates high percentiles to a high degree, and for the third group it actually overestimates high percentiles.
Remark 5.2. The most striking feature of the data in Table 5.4 is that Vasicek underestimates high percentiles whenever correlation tends to be higher during adverse scenarios (i.e. for every series for which a > a ), and that the underestimate is severe if the high-correlation regime is relatively unlikely (i.e. if p < p ). The clear implication is that if the state dependent model is a more faithful representation of the data (and Table 5.1 suggests that it is), then the Vasicek model will tend to underestimate regulatory capital, possibly to a very high degree.

Implied copula
Recall (Section 4.3.2) that the joint cdf of X = (X , X , . . . , Xn) T is Note that b k and Σ k are n × and n × n, respectively. The copula of X is therefore Cn(u , u , . . . , un) := Fn(F − (u ), . . . , F − (un)) , (6.1) where F = F is the marginal cdf of X i . Recall (Section 4.3.1) that in the presence of state dependence (i.e. if β > ), F − (u) must be computed numerically and that Φ − (u) typically provides a good initial guess.
Of particular interest in risk management and credit derivative pricing are the coe cients of tail dependence. Recall that in the bivariate case the coe cient of lower tail dependence is de ned as where i ≠ j. Figure 6.1 illustrates the small-u behaviour of in the presence of state dependence. Numerical evidence suggests that (i) the coe cient of lower tail dependence is zero, but that (ii) the quantities C (u, u)/u decay at a much slower rate in the presence of state dependence, than they do in the case of the Gaussian copula.

. Conditional hazard function
Now let h(t) be some pdf and let H(t) be the corresponding cdf, and suppose we wish to construct a vector T = (T , T , . . . , Tn) T such that (i) the marginal distribution of T i is H and (ii) the copula of T is Cn. Such an exercise is common in the credit derivatives literature, where the components of T represent the default times of n exposures and the user is interested in pricing or hedging some derivative whose payo is a function of T. The desired vector T can be constructed by setting The conditional hazard function of T , given T , is of particular interest in credit derivatives modelling. It is de ned as λ(t |t ) := − d dt log P(T > t |T = t ) , (6.3) and we note that conditional survival probabilities can be recovered via It is not hard to show that in the present context, we have where f (x , x ) is the joint density of (X , X ). Figure 6.2 illustrates the conditional hazard function (6.3) in both the proposed model and the Gaussian copula model, assuming the marginal distribution of T i is exponential. In panel (a) the second name defaults much earlier than expected, in the sense that the realized value of T is equal to the th percentile of its distribution. We see (unsurprisingly) that in both models, an early default of one name causes the entire hazard curve for the other name to shift upwards. At the short end of the term structure (i.e. for su ciently small values of t ) the impact of the early default on the conditional hazard rate is larger in the proposed model, than it is in the Gaussian copula model. The implication is that the conditional survival probability (6.4) is smaller in the proposed model than it is in the Gaussian copula model, for su ciently small values of t . Numerical integration of the conditional hazard curves appearing in the gure reveals that "su ciently small" here means t ≤ . . . ., at which point the conditional survival probability in both models is approximately 1.14%; i.e. su ciently small here is actually quite deep in the right tail of the conditional distribution of T . We conclude that the impact of the early default in the proposed model is more pronounced than it is in the Gaussian copula model.
In panel (b) of Figure 6.2 the second name defaults much later than expected, in the sense that the realized value of T is equal to the th percentile of its distribution. Unsurprisingly, the late default of the second name causes the entire hazard curve of the rst name to shift downwards in both models. As in panel (a) the impact at the short end of the term structure is larger in the proposed model, as compared to the Gaussian copula model. The implication is that conditional survival probabilities (6.4) are higher in the proposed model, than they are in the Gaussian model, for su ciently small values of t . Numerical integration reveals that "su ciently small" corresponds to t ≤ . . . ., at which point the conditional survival probability in both models is approximately 16.2%. As in the case of the early default, we conclude that the impact of the early default int he proposed model is more pronounced than it is in the Gaussian copula model. (other parameter values are as in Figure 6.1). The conditional hazard function is de ned in (6.3), and we compute it using (6.5). The solid line corresponds to the proposed model, the dashed line corresponds to the Gaussian copula model and the dash-dot line corresponds to the unconditional hazard rate (which is the same in both models). Panel (a) corresponds to the case where the second name defaults much earlier than expected, in the sense that the realized value of T is equal to the th percentile of its marginal distribution (i.e. it depicts λ(t | . )). Panel (b) corresponds to the case where the second name defaults much later than expected, in the sense that the realized value of T is equal to the th percentile of its marginal distribution (i.e. it depicts λ(t | . )).

Conclusion
This paper incorporates state dependent correlations (those that vary systematically with the state of the economy) into the Vasicek default model. This is accomplished by expressing the factor loading as a function of an auxiliary (Gaussian) variable that is correlated with the systematic risk factor; the degree to which the two are correlated can be interpreted as the degree of state dependence. We show how to implement the model in the special case where the factor loading is a discrete random variable, and illustrate that ignoring state dependence when it is in fact present will tend to underestimate important risk measures such as high percentiles and tail expectations. We t several di erent models to Federal Reserve data on delinquency rates, and compare their performance according to the Akaike Information Criteria (AIC). We nd that a state-dependent model with two correlation regimes outperforms the traditional Vasicek model, and that the estimated degree of state dependence is very high across all loan types. Importantly, we also nd the traditional Vasicek model outperforms a model with stochastic, but state-independent, correlations. In other words, randomizing correlation without allowing for state dependence does not improve the statistical performance of the Vasicek model. We also nd that failure to incorporate state dependence, when it is in fact present, can lead one to underestimate important risk measures. The most pressing avenue for future research is to incorporate realistic time dynamics for the systematic risk factor.

Appendix A
In this appendix we derive the properties (P1), (P2) and (P3) that appear in the introductory paragraphs of Section 3. • If both v and v are nite, then r is a unimodal function such that r(−∞) = r(∞) = . In this case both tails are thinner than the standard normal, in the sense that limu→±∞ g(u)/ϕ(u) = .
• If exactly one of v or v is in nite, then r is a monotone function from R to [ , ] and either (i) r(−∞) = and r(∞) = or (ii) r(−∞) = and r(∞) = . In the former case, the right tail of the density is proportional to a Gaussian and the left tail is much thinner than a Gaussian. In the latter case, the left tail is proportional to a Gaussian and the right tail is much thinner. If v = −∞ and v = or v = and v = −∞ then g(u) a skew-normal density (the skew-normal family is discussed in [4]). Finally, moments of the ABGM can be determined via an appeal to the truncated normal distribution.