Credit risk

Following Merton [13], we assume a company’s asset value V_i(t) to be the sum of time-independent liabilities F_i and equity g_i(t), V_i(t) = F_i + g_i(t), and model its dynamics by (1) with μ_i and σ_i denoting drift and volatility and dW_i being a random process to be specified. Further, we assume publicly listed companies and thus trace back changes in asset value to stock price returns. This allows us to determine the parameters μ_i and σ_i from stock return time series.

This deduction has a remarkable weakness though: stock returns do not exclusively reflect the actual changes in the company’s equity at any time, as could be assumed. Instead, they are subject to speculation and hence might be an inadequate approximation. However, this drawback is especially important on short return time scales Δt—days or weeks, say—and is becoming less and less substantial as Δt approaches longer time scales. Hence, calculations on annual horizons which are considered hereafter are in line with the reasoning set out above.

Credit risk can be easily introduced in this framework: let us consider a company with V_0i > F_i and liabilities, F_i, that mature after one year (without any coupon payments in between). If V_i(T) > F_i holds, the company is able to make the required payments and thus fulfills its obligations. Contrarily, if V_i(T) < F_i holds, the company fails to make the payments, ie it defaults. The normalized loss l_i the creditors are suffering can be expressed as (2) with Θ(x) denoting the Heaviside step function.

In the following, all loss simulations refer to a one-year time horizon and are carried out using the time parameter T = 252 days = 12 × 21 days. This is due to the described correspondence between asset value and stocks and the fact that a calender year has about 252 trading days, ie the stock markets are open 252 days a year. Moreover, it is useful to introduce the ratio of liabilities and asset value, F_i/V_i(t), the so-called leverage. The default criterion then reads F_i/V_i(T) < 1 (no default) and F_i/V_i(T) > 1 (default), respectively.

Next, we pass on from single credit contracts to credit portfolios. The importance of a single contract in a portfolio is determined by comparing the face value of this specific contract with the sum of all face values. To this end we use the fractional face values (3) Then we can write the normalized portfolio loss L as (4) Here, i enumerates the K credit contracts. K is hereafter often referred to as portfolio size. Due to normalization, L ranges from zero to one. Further note that all credit contracts are supposed to mature at time T and are to be coupon-free, ie the total face value must be payed at maturity.

According to the explanations above, the dynamics of such a portfolio of correlated credit contracts is well described by correlated stock price returns, which are frequently modeled using a multivariate normal distribution (5) Here, r = (r₁, r₂, …, r_K) denotes the K-dimensional return vector and r′ its transpose. The components of r are coupled via a K × K covariance matrix Σ, which can be expressed as Σ = σ C σ with C being the K × K correlation matrix and σ = diag(σ₁, σ₂, …, σ_K) a diagonal matrix containing the volatilities σ_i of the different stock returns r_i. Assuming returns according to Eq (5) results in a multivariate log-normal distribution for the stock prices.

More realistic, however, is the aforementioned random matrix concept of [17], an extension of [21], which accounts for non-stationary covariances. The key idea is a new application of the Wishart random matrix model (see eg [18]). In the original setting, generic spectral properties of correlation and covariance matrices were modeled for the case of large dimensions by using a statistical ensemble of random matrices. Hence, a certain ergodicity reasoning is underlying, as spectral averages have to equal ensemble averages—in the limit of large matrix dimensions. In this sense, the ensemble is fictitious. In the above references we advertise another interpretation of the Wishart model: Due to non-stationarity the covariances and correlations fluctuate in time. We show that, when looking at larger time intervals, the resulting fluctuating correlation and covariance matrices form a truly existing, non-fictitious ensemble which can very well be modeled by a random matrix ensemble. Ergodicity is not an issue. Importantly, the matrix dimension is in this interpretation given by the number of considered assets, it can have any value and it is not to be taken to infinity. Finite size effects are thus not artefacts, they are real effects which are automatically contained in our model. In the different context of random matrix theory for dynamical systems, there is recent progress in studying finite size effects [22], [23]. In our modeling, Eq (5) with the stationary covariance matrix Σ serves as a starting point. Averaging over covariance matrices fluctuating around Σ in the course of time finally yields the multivariate ensemble averaged distribution (6) Here, the parameter N controls the strength of fluctuations around the mean Σ. Compared to the multivariate normal distribution with a fixed covariance matrix, the ensemble averaged distribution exhibits fat tails for any finite N, the more pronounced the smaller N. N is determined by fitting and depends strongly on the considered return interval Δt. In [17] it was shown particularly that daily returns are matched best by N = 5, while annual returns behave normally (N ≫ 1). Yet, the ensemble approach remains useful if simplified correlation matrices with homogeneous off-diagonal elements C_ij = Corr(V_i, V_j) = c_a, i ≠ j, are used. (The subscript “a” stands for “asset” and is used to distinguish asset correlations c_a from other correlation coefficients.) In [24, 25] it was shown that distributions of annual stock returns are fitted best by N = 5 in this case, as opposed to N ≫ 1 in the case of full correlation matrices.

We obtain the asset values at maturity V_i(T) from the stock price returns r_i via (7) according to Itō’s lemma [26].

Copulas

Let us assume a bivariate random variable X = (X₁, X₂) that is comprehensively described by means of its joint probability density, f(x₁, x₂), say. The main idea of copulas, which were introduced in [27], is to separate the statistical dependence of X₁ and X₂ from the two marginal distributions. This is achieved by the copula (8) which is the cumulative joint distribution of the quantiles u and v with (u, v) ∈ [0, 1] × [0, 1]. denotes the inverse marginal cdfs. This construction—the copula as a composition of inverse marginal cdfs and the joint cdf—allows us to analyze statistical dependences regardless of the marginal distributions. Instead of the copula itself, we will work with the copula density cop(u, v), which is given by (9) In particular, we are interested in estimating and analyzing copula densities obtained from empirical or simulation data. The construction of such an empirical copula density from a bivariate data set of length n takes the following steps. First, we need to replace the actual values (x_i, y_i), i ∈ {1, …, n} with their ranks (rank(x_i), rank(y_i)); this is conducted separately for x_i- and y_i-values. Next, the ranked values, 1 ≤ rank(x_i) ≤ n are scaled to the interval [0, 1]. These normalized ranks produce the empirical cdf-values of the initial data points in the data set.

Binning these new data points yields a two-dimensional histogram which is—if normalized appropriately—an empirical approximation of the copula density. The number of bins is chosen to be b = 20 in each direction.

Simulation setup

Here and in all simulations hereafter, we consider two credit portfolios which are set up according to Fig 1. On a given financial market, illustrated by means of its correlation matrix, we choose these portfolios to be non-overlapping, ie no credit contract is part of both portfolios. The number of contracts in each portfolio is K, ie they are of equal size. We mark the two portfolios in Fig 1 as black-rimmed squares. We note that the off-diagonal squares are important as well. They illustrate the asset correlations of companies which are part of different portfolios and thus cause interdependences between the two portfolio losses.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. Exemplary correlation matrix.

This example illustrates a financial market. The black-rimmed squares (solid and dashed) correspond to two disjoint portfolios.

https://doi.org/10.1371/journal.pone.0190263.g001

The portfolio loss copula of these portfolios is obtained as follows: We simulate the correlated asset values, V_i(T) with T = 252, of all companies within the financial market. For both portfolios, we choose the affiliated V_i(T) and calculate the corresponding portfolio losses L₁ and L₂ according to Eq (4). Repeating this portfolio loss calculation a few thousand times yields enough data to estimate the copula histogram of the two portfolio losses.

We repeat these steps several thousand times for different pairs of portfolios in order to avoid results which are due to the specific features of two particular portfolios. All these pairs are non-overlapping and operate on the same financial market; each portfolio is of size K. We determine the copula histogram in each case and average over them. The resulting mean copula histogram provides information about the average portfolio interdependence. However, it still depends on the portfolio size K as well as on the considered market.

As mentioned above, statistical dependences are often reduced to correlation coefficients or—at the level of copulas—to Gaussian copulas. Here, we would like to study the deviations of the averaged empirical copula from the related Gaussian copula. In order to achieve this Gaussian copula, we need to estimate its parameter c in the first place. Here, c equals the averaged correlation coefficient of portfolio losses C_L1L2 = Corr(L₁, L₂), which is determined—similarly to the averaged copula histogram—by averaging over the portfolio loss correlation for every portfolio pair.

Moreover, we employ leverages, which are drawn from a uniform distribution on the interval [0.6, 0.9], (10) for every non-homogeneous, ie heterogeneous, portfolio hereafter.

Impact of asset correlations on portfolio loss copulas

For the simulation of asset value processes and loss calculation, we specify μ = 10⁻³ day⁻¹, σ = 0.03 day^−1/2 and leverages F/V₀ = 0.75. Our choice of rather small portfolios, K = 50, will be discussed at the end of this section. Furthermore, we assume a market with vanishing asset correlation, c_a = 0. The simulation of asset values V_i(T), which underlies the portfolio loss calculation and thus the averaged copula histogram, is run in two different ways. On the one hand, we assume Gaussian dynamics, ie we use—in the framework of the ensemble averaged distribution—the “fat-tail” parameter N → ∞. On the other hand, we choose N = 5—in accordance with the findings of [25] for homogeneous average correlation matrices and annual time horizons. The resulting averaged copula histograms for 10000 portfolio loss simulations and 1000 portfolio pairs are shown in Fig 2. (The choice of different portfolio pairs is unnecessary for homogeneous portfolios, as they are all equivalent. However, it matters for heterogeneous portfolios and we employ—except for the input parameters—the same simulation in either case.)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. Averaged copula histograms for vanishing mean correlations.

The copula histograms of losses L₁ and L₂ of homogeneous portfolios with vanishing (mean) asset correlations, ie c_a = 0, averaged. The asset values are multivariate normal with N → ∞ (top) and multivariate heavy-tailed with N = 5 (bottom). The coloring indicates the local deviations from the Gaussian copula with c = C_L1L2 = 0.752.

https://doi.org/10.1371/journal.pone.0190263.g002

For N → ∞ (top panel), we find an independence copula, which exhibits a constant level of one due to normalization. We observe only minimal deviations due to the finite simulation length. Measuring the average portfolio loss correlation yields C_L1L2 = 0 and thus the related Gaussian copula is an independence copula as well. The reason for this result is quite obvious: the simulation of asset values is based on uncorrelated Gaussian random numbers, ie they are statistically independent. Unsurprisingly, quantities derived from these random numbers do not exhibit dependences either.

The combination of c_a = 0 and N = 5 yields a different result (bottom panel of Fig 2); its deviations from the independence copula are striking. Ignoring the coloring for a moment and considering solely the bar heights is sufficient to recognize that this copula is not Gaussian: the bar in the (0, 0)-corner is twice as high as its equivalent in the (1, 1)-corner. Contrarily, Gaussian copulas are perfectly symmetric regarding the line spanned by (0, 1) and (1, 0). Nevertheless, we calculate the correlation coefficient c = C_L1L2 = 0.752 of the simulated portfolio losses and determine the corresponding Gaussian copula. The difference between the actual copula and the Gaussian copula within each bin is illustrated by means of coloring. The color bar (on the right in Fig 2) provides the translation between color and value. In general, the colors yellow, orange and red indicate that the actual copula exhibits a stronger dependence within the given (u, v)-interval than predicted by a Gaussian copula. Contrarily, turquoise and blue are illustrations of local dependences weaker than Gaussian.

The differences from the independence copula (N → ∞) are due to the fact that finite values of N, eg N = 5, cause fluctuating correlations (cf section Model and methods). Here, they are centered at c_a = 0, ie positive and negative fluctuations are equally likely. This raises the question why the portfolio losses exhibit such a strong positive correlation, c = C_L1L2 = 0.752. The answer to this question is twofold: First, a large correlation matrix with strong positive and negative correlations implies, roughly speaking, a devision of the companies into two blocks with positive correlations within the blocks and negative correlations between them. Second, credit risk is—as discussed before—highly asymmetric. There is no positive impact of prospering and thus non-defaulting companies on portfolio losses. The Heaviside functions in Eq (4) cut off all these non-defaults and project them onto zero. This way, anti-correlations of asset values contribute to the portfolio loss correlation only in a rather limited fashion. Even worse, due to the aforementioned block-structure, the negative correlation between the blocks makes it more likely that one of the blocks is adversely affected. And the positive correlations within the blocks imply a high risk of concurrent defaults.

Drift dependence of portfolio loss copulas

Defaulting portfolios (L > 0) contribute to the portfolio loss copula as well as non-defaulting portfolios (L = 0). We would like to study the impact of each on the portfolio loss copula in greater detail. First, we vary the asset value drifts μ in order to influence the default-non-default ratio. Deviating from the specifications above, c_a = 0.3, instead of c_a = 0, and N → ∞ are used. Moreover, we choose the slightly lower volatility σ = 0.02 day^−1/2. The resulting averaged copulas for three different drift parameters are shown in Fig 3. For μ = 10⁻³ day⁻¹ (top panel), non-defaults occur with a probability of 39.1%. The “plateau” within [0, 0.3] × [0, 0.3] is due to simultaneous non-defaults of both portfolios. More striking, however, is the strong tail dependence in the (1,1)-corner. Even though we observe such an obviously non-Gaussian averaged copula, we estimate the average portfolio loss correlation and obtain C_L1L2 = 0.851.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Averaged copula histograms for non-vanishing mean correlations.

The copula histograms of losses L₁ and L₂ of homogeneous portfolios with asset correlations c_a = 0.3. The drifts are μ = 10⁻³ day⁻¹ (top), 3 × 10⁻⁴ day⁻¹ (middle) and −3 × 10⁻³ day⁻¹ (bottom). The coloring refers to the local deviations from the Gaussian copula with C_L1L2. The asset values are multivariate normal (N → ∞).

https://doi.org/10.1371/journal.pone.0190263.g003

Choosing the smaller drift μ = 3 × 10⁻⁴ day⁻¹ (middle panel) yields—unsurprisingly—a lower probability of non-defaults, namely 12.8%, and a copula which is far more Gaussian compared to the previous one. Nevertheless, there are several deviations from an ideal Gaussian copula. The (1, 1)-tail is narrower and more pointed. And even though we have not changed the dependences of asset values, the resulting average portfolio loss correlation is C_L1L2 = 0.904 and thus approximately 5% higher compared to the result in the top panel.

Finally, the drift μ = −3 × 10⁻³ day⁻¹ leaves no non-default events at all. We observe an ideal Gaussian copula (no coloring except for white) with an average portfolio loss correlation of C_L1L2 = 0.954. Again, this portfolio loss correlation is 5% higher compared to the previous result. We have seen that portfolio loss correlations increase and that the averaged copula turns ever more Gaussian if the percentage of default events increases. In order to explain these findings, let us analyze the according marginal distributions of the portfolio losses (Fig 4). For μ = 10⁻³ day⁻¹ and μ = 3 × 10⁻⁴ day⁻¹, ie for portfolio default probabilities of 60.9% and 87.2%, respectively, we find extremely asymmetric portfolio loss pdfs. Both exhibit a fat tail for high portfolio losses and a delta peak at zero due to non-defaults. In contrast, for μ = −3 × 10⁻³ day⁻¹, ie for a default probability of 100%, the pdf has no delta peak at zero—as the chance of non-defaults is 0%—and moreover is almost symmetric. Nevertheless, it shows some deviations from a normal distribution, even though we observe an ideally Gaussian dependence of the portfolio losses (see bottom panel of Fig 3).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Portfolio loss distributions.

Semi-log scaled portfolio loss pdfs for homogeneous portfolios (asset correlations c_a = 0.3) are shown with drift parameters μ = 10⁻³ day⁻¹ (blue), 3 × 10⁻⁴ day⁻¹ (red) and −3 × 10⁻³ day⁻¹ (green). The asset values are multivariate normal (N → ∞).

https://doi.org/10.1371/journal.pone.0190263.g004

Moreover, let us consider enlarged versions (K ≫ 50) of these three portfolios for a moment. It can be easily shown that such scalings lead to a decreasing number of non-default events and thus to smaller delta peaks of the portfolio loss pdfs at zero, which vanish entirely if K is large enough. Despite this change, however, the shapes of the pdfs remain almost the same for K ≫ 50. In contrast to this virtual K-independence of the marginal distributions, the statistical dependences turn ever more Gaussian as K increases and the number of non-default events decreases.

Taking all these findings into consideration, we infer that the loss of information—resulting from the projections of non-default events onto zero—is responsible for the decrease in portfolio loss correlation with decreasing likeliness of default events. Furthermore and even more importantly, we deduce that these projections onto zero cause the observed deviations of the statistical dependences from Gaussian copulas.

Portfolio loss correlation

Even though non-Gaussian portfolio loss copulas cannot be described comprehensively by means of correlation coefficients, correlations are frequently applied in credit risk, eg in [28] and [20]. It is worth pointing out that our concept of loss or default correlations differs from theirs, as we account for the actual loss values. In contrast, Lucas applies the frequently used “binary” approach, ie he only distinguishes default and non-default, while Li’s concept is based on so-called “survival times”.

Here, we examine portfolio loss correlations more closely, because the reduction of an entire statistical dependence to a single number simplifies the analyses of further parameter dependences considerably. Especially, we are interested in the dependence of portfolio loss correlations on the underlying asset correlations and on the portfolio size K. They are shown in Fig 5 for a homogeneous portfolio—as described above—with drifts μ = 2 × 10⁻³ day⁻¹ (top row) and μ = −3 × 10⁻³ day⁻¹ (bottom row) and “fat-tail” parameters N → ∞ (left column) and N = 5 (right column). Portfolio size K serves as a curve parameter and takes on values between K = 1 and K = 150. (For a list of all portfolio sizes K, cf caption of Fig 5.)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Portfolio loss correlation.

As a function of asset correlation c_a, the portfolio loss correlation C_L1L2 is shown. The size K of the underlying homogeneous portfolio ranges from 1 (blue) over 2, 3, 4, 7, 10, 15, 25, 50, 100 to 150 (green). The daily drifts are set to μ = 2 × 10⁻³ day⁻¹ (top row) and μ = −3 × 10⁻³ day⁻¹ (bottom row). All asset values are multivariate normal with N → ∞ (left column) and multivariate heavy-tailed with N = 5 (right column). The bisecting line is shown in red.

https://doi.org/10.1371/journal.pone.0190263.g005

First, we find the portfolio loss correlation to be a monotonic function of asset correlation as well as of portfolio size—regardless of drift μ and “fat-tail” parameter N. Keeping in mind the results of the previous subsection, we already know that correlation coefficients capture the entire dependence of portfolio losses for μ = −3 × 10⁻³ day⁻¹ and N → ∞ (bottom left panel). Obviously, asset and loss correlation equal each other for K = 1. As K increases, portfolio loss correlation is a concave function of asset correlation and increases much more steeply. An asset correlation differing only slightly from zero is sufficient to make it take on values close to one. This behavior is related to credit portfolio granularity, which increases with portfolio size K and causes ever more deterministic results. In the limit K → ∞ and for every c_a > 0, all portfolio losses are the same and thus are perfectly correlated. In order to preserve some randomness to achieve non-trivial results, we decided to choose the rather small portfolio size of K = 50 in the previous subsections.

For μ = 2 × 10⁻³ day⁻¹ and N → ∞ (top left panel of Fig 5), portfolio loss correlations are only a first approximation of the actual statistical dependence of portfolio losses. Here, concavity holds only for large K (namely, K = 50, 100, 150), while portfolio loss correlation is a convex function of asset correlation for small K (namely, K = 1, 2, 3, 4). The functional dependences for all K in between are neither concave nor convex. If convexity holds, we find the portfolio loss correlations to be always smaller than the corresponding asset correlations.

Comparing the results for N → ∞ in the top and bottom left panel reveals that the strength of portfolio loss correlation is not only a function of asset correlation, but moreover a function of the ratio of non-default and default events. This result is of particular importance for stress testing, as it implies that the correlations of single credit losses as well as of credit portfolio losses are stronger during financial crises than in calm market periods!

Contrasting these results with the results for N = 5 (right column), we find remarkable deviations for μ = 2 × 10⁻³ day⁻¹. We observe large portfolio loss correlations for c_a = 0 and c_a close to zero, which are in line with the portfolio loss correlation we determined in the context of Fig 2. There, we discussed fluctuating asset correlations that occur for N = 5 and argued that the contributions of negative asset correlations to the portfolio loss are negligible. Hence, positive asset correlations dominate and cause the large portfolio loss correlations we just mentioned. For μ = −3 × 10⁻³ day⁻¹, however, the portfolio loss correlations depend only slightly on the “fat-tail” parameter N, as the impact of the strongly negative drift on the asset values and thus on the portfolio loss is much more important compared to the contribution of the involved random process.

Impact of parameter heterogeneity

First, we consider a rather simple heterogeneous portfolio. It equals the homogeneous portfolio we have discussed so far, except for the fact that the daily volatilities σ_i are not set to σ_i = σ = 0.02 day^−1/2, but their values are drawn from the uniform distribution . Moreover, it is important to mention that we choose the daily drift μ = −3 × 10⁻³ day⁻¹ and the Gaussian limit (N → ∞) of the ensemble averaged random process. The resulting portfolio loss copula for such portfolios is shown in Fig 6. Due to the strongly negative drift, the probability of non-default events is zero. Nevertheless, we observe deviations from the Gaussian copula with parameter C_L1L2. This result is in contrast to the Gaussian copula displayed in the bottom panel of Fig 3, where a homogeneous σ = 0.02 day^−1/2 is assumed. For such a perfectly homogeneous portfolio, the Gaussian dependence of asset values is preserved at the level of portfolio losses. However, the Gaussian loss dependence is altered by the heterogeneous choice of one or more parameters; here, volatility. Thus, we have identified two causes of non-Gaussian copulas: parameter heterogeneity and the aforementioned projections of non-defaults onto zero.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. Averaged copula histogram for heterogeneous volatilities.

Copula histogram of portfolio losses L₁ and L₂ of two portfolios with heterogeneous volatilities, , averaged. The coloring indicates the local deviations from the Gaussian copula with C_L1L2.

https://doi.org/10.1371/journal.pone.0190263.g006

Simulation setup and results

In order to set up realistic, fully heterogeneous portfolios we employ empirical data for drifts, volatilities and correlations. They are based on stock return data over the period 01/1993–04/2014 from S&P 500 at New York Stock Exchange (all 272 companies available over the whole period) and from Nikkei 225 at Tokio Stock Exchange (all 179 companies available over the whole period). All data are freely avaliable at Yahoo! finance [29], see S1 Appendix, entitled Download of financial data from Yahoo! finance. We want to to obtain an average portfolio loss copula of empirical portfolios which is first averaged over many portfolio pairs and then averaged over the 21-year interval. To achieve this goal, we run the following procedure 20000 times: First, an annual time interval within the 21-year period is randomly chosen. We determine the drifts, volatilities and correlations of all companies on this interval and draw two portfolios of size K = 50. Furthermore, the leverages F_i/V_0i are drawn from the uniform distribution in Eq (10). Finally, 10000 portfolio loss simulations (N → ∞) are run and the portfolio loss copula is estimated. This Monte-Carlo simulation enables us to determine the desired time-averaged portfolio loss copula of empirical portfolios. We consider three different cases: First, portfolio 1 is always drawn from S&P 500, while portfolio 2 is always based on the Nikkei 225 (top panel of Fig 7). Next, both portfolios are drawn from the S&P 500 (middle panel), and finally, from Nikkei 225 (bottom panel). In the latter two cases we divided these stock markets into two “sub-markets” before drawing any portfolios. Thus, we avoid a given company to be part of portfolio 1 in one iteration and part of portfolio 2 in another iteration. Due to this division into “sub-markets”, the almost perfect symmetry of all three copulas regarding the line spanned by (0, 0) and (1, 1) is not trivial, but an important feature. We will return to this point.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. Time-averaged copula histogram.

Copula of losses L₁ and L₂ of two empirical portfolios (K = 50), time-averaged. Top: portfolio 1 is always drawn from S&P 500 and portfolio 2 from Nikkei 225, middle: both portfolios are drawn from S&P 500, bottom: both portfolios are drawn from Nikkei 225. The coloring indicates the local deviations from the Gaussian copula with C_L1L2. The asset values are multivariate normal (N → ∞).

https://doi.org/10.1371/journal.pone.0190263.g007

In the S&P-Nikkei case, the deviations of the copula from the Gaussian copula are obvious, as we observe dependences of extreme events in the (1, 1)-corner that are more pronounced than the Gaussian prediction. Further deviations are the narrower and more pointed (1, 1)-tail as well as the flatter (0, 0)-tail, which renders it quite asymmetric regarding the line spanned by (0, 1) and (1, 0). We interpret this result as follows. We find extreme portfolio losses to occur more often simultaneously than it is the case for small portfolio losses. Thus, a modeling of portfolio loss dependences by means of Gaussian copulas is highly erroneous and might cause severe underestimations of the actual risks.

In the S&P-S&P as well as in the Nikkei-Nikkei case, the time-averaged copulas bear a certain resemblance to the result in the top panel. However, we find higher coupling strengths, as can be seen from Table 1 which lists the average asset correlation and the portfolio loss correlation for all three cases.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 1. Asset correlation and portfolio loss correlation.

https://doi.org/10.1371/journal.pone.0190263.t001

For K = 50, the averaged marginal distributions of empirical portfolio losses are depicted in Fig 8. The S&P 500-related pdf is shown in red, the Nikkei 225-related result in green. They are highly asymmetric and exhibit distinct tails as well as delta peaks at zero. These features are in accordance with the blue and red curve in Fig 4, which illustrate the portfolio loss distributions of a homogeneous portfolio with drifts μ = 10⁻³ day⁻¹ (blue) and μ = 3 × 10⁻⁴ day⁻¹ (red), respectively. In contrast, the averaged empirical portfolio loss pdfs have much heavier tails. While the risk of extreme portfolio losses (L > 0.7) is equally high for S&P- and Nikkei-based portfolios, the Nikkei-based portfolio loss pdf is almost half a magnitude higher for large portfolio losses (0.15 < L < 0.45). Considering the delta peak more closely, we find the time-averaged probabilities of portfolio non-default events to be 11.7% (S&P 500) and 12.2% (Nikkei 225), respectively.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8. Time-averaged portfolio loss distributions.

Semi-log scaled time-averaged portfolio loss pdfs of empirical portfolios of size K = 50. The portfolios are drawn from S&P 500 (red) and Nikkei 225 (green), respectively. The asset values are multivariate normal (N → ∞).

https://doi.org/10.1371/journal.pone.0190263.g008

Portfolio size and portfolio loss correlation

Although we have seen that a Gaussian modeling of loss dependences falls short in most cases, it can be seen as a rough but useful approximation of the actual dependence. Especially, as such a reduction of the full statistical dependence to a single number simplifies the study of further parameter dependences. Here, we consider the average loss correlation for empirical portfolios as a function of portfolio size K. So far, we have discussed K = 50 and have found portfolio loss correlations which were much smaller than the ones for homogeneous portfolios of the same size. This is unsurprising, because empirical portfolios are heterogeneous and thus exhibit higher idiosyncrasies for a given portfolio size. Once again, we distinguish three cases: Both portfolios are S&P 500-based, both are Nikkei 225-based, one is S&P 500-based and the other one Nikkei 225-based. Taking a look at Fig 9, we observe steep increases in portfolio loss correlation with increasing K, which is in line with our findings for homogeneous portfolios. This behavior is due to the decreased idiosyncrasy of large portfolios, which causes very similar portfolio losses and, thus, high portfolio loss correlations. In all three cases, idiosyncrasy is almost negligible for K ≥ 80. The above mentioned almost perfect symmetry of all three copulas regarding the line spanned by (0, 0) and (1, 1) as observed in Fig 7 can now be understood from Fig 9. This behavior is due to the almost negligible idiosyncrasy for credit portfolios of size K = 50.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. Time-averaged portfolio loss correlation.

As a function of portfolio size K for empirical portfolios, the time-averaged portfolio loss correlation is shown. Red: both portfolios are drawn from S&P 500, green: both portfolios are drawn from Nikkei 225, blue: portfolio 1 is always drawn from S&P 500 and portfolio 2 always from Nikkei 225.

https://doi.org/10.1371/journal.pone.0190263.g009

The results in Fig 9 are of high practical relevance, as the underlying portfolios are based on realistic, empirically determined parameter sets. For S&P 500-based portfolios we find, in particular, loss correlations C_L1L2 > 0.5 for K ≥ 14, C_L1L2 > 0.7 for K ≥ 40 and even C_L1L2 > 0.85 for K ≥ 150. This means that losses of medium-sized disjoint portfolios (K = 150) are almost perfectly correlated and that even small (K = 40) and very small (K = 14) portfolios exhibit rather strong coupling. Thus, high dependences among banks are not limited to “big players”, which hold portfolios of several thousand credit contracts, but affect small financial institutions in a very similar fashion.

In the Japanese stock market we observe even higher portfolio loss correlations, as the underlying asset correlations are on average higher compared to the US market (see above). The opposite is the case if one Japanese and one US-American portfolio is considered. Here, we find considerably weaker asset correlations and thus loss correlation coefficients C_L1L2 ≤ 0.35.