A Cash Flow-Based Approach to Estimate Default Probabilities

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2 A Cash Flow-Based Approach to Estimate Default Probabilities Francisco Hawas Faculty of Physical Sciences and Mathematics Mathematical Modeling Center University of Chile Santiago, CHILE Arturo Cifuentes Financial Regulation and Stability Center, CREM Faculty of Economics and Business University of Chile Santiago, CHILE October, 2013

3 Executive Summary This paper introduces a practical and flexible approach to estimate the default probability of a company. Since a company s default is actually triggered by lack of sufficient funds, namely cash flows, we base our method on modeling the cash flows. In this sense, our approach departs radically from that of other researchers who base their predictions on financial ratios or naïve representation of asset values. We model the cash flows assuming a fairly general stochastic characterization that can be easily accommodated to handle multiple cash sources. Then, we rely on a Monte Carlo simulation technique. An important advantage of this approach is that it permits not only to estimate the default probability of the company under study but also a number of figures of merit, as well as their distributions. Additionally, from a regulator s viewpoint, this method is particularly insightful as it permits to assess in the case of systemically important entities not only their likelihood of default but also the feasibility of rescuing them. A simple example demonstrates the usefulness of the technique.

4 Introduction Estimating the likelihood that a company might default on its debt is an important consideration for regulators, especially if such default might entail the risk of triggering a cascade of failures as is the case with systemically important institutions. Additionally, having reliable tools to estimate the risk of default is also important for financial managers (they need to assess how much debt is reasonable to take) as well as creditors and investors (they need to assess the risk of not being repaid). Unfortunately, despite almost fifty years of efforts since Altman (1968) and Beaver (1966) got started on this topic one fact remains: the overall record of predictive models is poor and no method has yet gained widespread acceptance. It is not the aim of this report to review past efforts regarding this topic as other authors have dealt with this issue already (see, for example, Mansi et al. 2010; Blochlinger 2013; Bismark and Pasaribu 2011; Bielecki et al. 2013; and Frunza 2013). Our goal is simply to propose a method which departs from previous efforts in the sense that it is based on modeling the cash flows generated by the entity under analysis. We think this approach is promising for a number of reasons. First, it deals with the problem at its root: the cash flows (a default is in essence a failure to generate sufficient cash flows). Second, it provides enough flexibility to accommodate several cash sources with different probabilistic features (as it happens oftentimes in most companies). And third, it is easy to implement since it does not rely on obscure mathematical jargon or information that is difficult to obtain. The next section formulates the problem in a formal fashion. Then, we discuss a numerical algorithm to tackle the relevant computations, we demonstrate the usefulness of the method with an example, and we finish with a discussion of possible extensions plus a brief set of conclusions. Problem Statement Conceptually, a company is an engine that produces cash flows over a certain period of time and often from multiple sources. These cash flows are uncertain, that is, stochastic in nature. Typically, a company has also debt which is deterministic and whose payments are spread out and well defined over a certain time span. The problem consists of estimating the likelihood that the aggregate cash flows might not be sufficient to make the debt payments.

5 More formally, we assume that we have M cash sources and N time periods (for convenience equally spaced). Since we have M cash sources (or assets) we denote as X i (i=1,, M) the vector associated with the cash flows generated by source i at times t 1,, t N. Thus, X i = (x i1,, x in ) t. To clarify, x ij refers to the cash flow generated by source i at time t j. Moreover, we assume that the vectors X i follow multi-normal distributions, X i MN(μ i, C i ), where μ i represents the vector of expected values and C i denotes the corresponding correlation matrix. In short, μ i = (E(x i1 ),, E(x in )) t where E refers to the expected value operator and i=1,, M. Also for convenience we will use the notation μ ij to designate E(x ij ). Notice that there are no cash flows associated with t 0. In principle, there is no reason to assume that the cash flows generated by asset i, at different points in time, are uncorrelated. As a practical matter, however, we will make the simplifying assumption that the cash flow corresponding to time t j, is only correlated to the two neighboring cash flows, that is, those corresponding to times t j-1 and t j+1. This leads to a correlation matrix (C i ) having a tri-diagonal structure. We designate the value of this correlation factor, which we take it to be constant for each cash source, as ρ i (i=1,, M). It might appear that the proposed correlation structure is overly simplistic and that one should employ a more general correlation matrix structure (for example, using a fully populated matrix). However, recent studies by Hawas and Cifuentes (2013a) have indicated that with the exception of some extreme cases this assumption renders good accuracy. Finally, we also assume that the analyst has been able to estimate the standard deviation of the cash flows which, again for convenience, we express in terms of the coefficient of variation (λ). In short, σ ij = λ ij μ ij for i=1,, M and j=1,, N. This completes the characterization of the cash flows associated with each cash source. Additionally, the different vectors of cash sources (X i with i=1,, M) might be themselves correlated. For simplicity, we will assume that this correlation takes place only at the present time, that is, x pj and x rs are supposed to be independent if j s; but if j = s then the correlation between x pj and x rj, (which we denote as ρ pr ) is taken to be, in general, different than zero and the same for all j s. Finally, the debt payments are specified by the vector D=(d 1,, d N ) t which is known and deterministic. The vector Z= (z 1,, z N ) t where z i = x 1i +x 2i + +

6 x Mi (i=1,, N) represents the total cash flow (from all sources) that the company generates at each time. If for any j (j=1,, N) occurs that d j > z j then the company defaults as it does not have enough cash to cover the debt payment. This, of course, under the assumption that the company has to service the debt payment at time t j with cash generated at the same time period. In other words, we are assuming that the company has no reserves (all excess cash generated in previous periods was either paid as dividends or used to cover capital expenditures). How to relax this assumption (a rather straightforward matter) is dealt with in the final section. This description completes the specification of the problem. In essence, we know D and the probabilistic characterization of the cash flow vectors X 1,, X M and the issue reduces to estimate the likelihood that Z might have a component less than its corresponding D counterparty (z j < d j ) for some j (j=1,, N). For the avoidance of doubt, it is helpful to clarify graphically the structure of the aggregate correlation matrix, a MN x MN matrix which we call C*. Just for illustration purposes let us assume that we have three sources of cash (M=3) and four time periods (N=4). Exhibit 1. Structure of the aggregate correlation matrix (C*) for the case M=3 and N=4.

7 The preceding diagram (Exhibit 1) shows the structure of such matrix. The entries not shown correspond to zeroes. Furthermore, in the context of this example, the vector Z can be written as Z=(z 1, z 2, z 3, z 4 ) t = (x 11 +x 21 + x 31, x 12 +x 22 + x 32, x 13 +x 23 + x 33, x 14 +x 24 + x 34 ) t. And finally, we refer to the vector that includes all the components of the cash flows, as X*. That is (noting that MN=12), X*= (x 1 *,, x 12 * ) t = (x 11, x 12, x 13, x 14, x 21, x 22, x 23, x 24, x 31, x 32, x 33, x 34 ) t with, of course, the corresponding vector of expected values designated as μ*. Thus, in brief, X* MN(μ*, C*). Simulation Technique An efficient technique to tackle the problem at hand is via a Monte Carlo simulation approach. This technique reduces to generating a family of X* vectors satisfying the condition X* MN(μ*, C*). The algorithm we describe is based on a paper by Hawas and Cifuentes (2013a) and can be summarized as follows: [0] Find the Cholesky decomposition of C* (the aggregate correlation matrix). This can be accomplished using standard commercial software packages such as MATLAB or Mathematica. Hence, C*can be expressed as C*= L L t in which L is a lower triangular matrix; [1] Generate U= (u 1,, u MN ) t where u i (i=1,, MN) are random draws from iid N(0, 1); [2] Compute V= LU, with V=(v 1,, v MN ) t ; [3] Determine W* (the desired sample vector) using the expression w* i = E(x* i ) + σ i v i (for i=1,, MN) where σ i is obtained multiplying E(x* i ) by the corresponding coefficient of variation; [4] Determine the sample vector Z by adding the appropriate components of vector W*; that is = for values of i=1, 2,, N;

8 [5] Determine for such vector Z (a) if a default has occurred (namely, if z j < d j for some j between 0 and N), and (b) if such default has occurred record the period (j) when the default occurs. Repeating steps [1, 2, 3, 4 and 5] many times we can estimate: (i) the expected value the default probability; and (ii) the expected value of the time to default (assuming a default takes place) plus other relevant figures of merit. This is accomplished by averaging the appropriate quantities across all samples. Example of Application The purpose of this example is to showcase the benefits of the method rather than obscuring the computations with unnecessary cash flow complexity. To this end we use a simple situation involving only three sources of cash (M=3) and ten time periods (N=10). The cash flows are specified as follows: Source 1. For i=1,, 10; E(x 1i ) = μ 1i = 25; λ 1i =0.3; and ρ 1 = 0.3 Source 2. For i=1,, 10; E(x 2i ) = μ 2i = 10; λ 2i =0.5; and ρ 2 = 0.3 Source 3. For i=1,, 10; E(x 3i ) = μ 3i = 60; λ 3i =0.05; and ρ 3 = 0.3 In addition we assume that ρ 13 = ρ 12 = ρ 23 = 0.2. Debt Payments The debt (D) vector is defined as d i = 100 β (i=1,, 10) where 100 is a basic reference value and β represents a scaling factor. The goal is to examine for values of β between 0 and 1 the default likelihood of a company that relies on the three above-mentioned sources to meet its obligations.

9 Exhibit 2. Default probabilities estimated with a Monte Carlo simulation. The table shows, for different values of β, the likelihood that the company might default on each period. P-of-Def is the overall default probability (the sum of the period-by-period default probabilities). Exhibit 3. Default probability as a function of the period, for two values of β (75% and 100%). The results of the Monte Carlo simulation (with 300,000 random samples of the vector X*) are shown in Exhibit 2. This table is self-explanatory: higher

10 levels of debt are associated with higher default probabilities. It is also clear from the table (and from Exhibit 3) that for higher values of β the defaults are frontloaded and for lower β s they tend to be evenly distributed over time. Exhibit 4. Time to defaults statistics (units in periods), for different values of β, assuming that a default has actually occurred. Exhibit 4 displays the expected time-to-default expressed in units of periods based on those cases in which a default actually occurred plus other relevant metrics. It is interesting to notice that for high levels of debt (presumably the situation for which the type of analysis discussed herein is more relevant) the time-to-default follows a markedly non-normal distribution. This should act as a warning against the validity of predictions based on assumptions of normality. Additionally although this issue is beyond the scope of this article the results presented here, or more precisely, the framework we have outlined here, should be considered as a useful tool to assess the soundness of some intensity-based default models. (This issue is discussed in more detail in the Conclusions section at the end.) Finally, in most realistic situations it is likely that the analyst will possess reliable information regarding the nature of the cash flows (namely, the value of the corresponding expected values) but less reliable information about the magnitude of their standard deviations. Furthermore, the correlation values (both, inter-temporal as well as between different cash sources) are clearly the

11 most challenging values to estimate. With that as background, it would be prudent to perform several sensitivity analyses to explore the influence of the standard deviations and correlations on the results. To demonstrate the usefulness of this type of analysis we consider two situations: a high-debt level (β = 85%) case and a low-debt level (β = 65%) case. The corresponding default probabilities, as indicated before in Exhibit 2, are 83.2% and 3.0% respectively. The idea is to see how much these values would change if we change the assumptions made for the coefficients of variation and correlations. Exhibit 5 summarizes the results which were obtained by perturbing 10% the base values, one at a time, while keeping the other variables constant. Exhibit 5. Sensitivity analyses for two cases (β=85% and β=65%). The base values for the coefficients of variation and correlations are shown on the top left panel; and the corresponding results obtained with those values (Reference Values) are shown on the top-right panel. The bottom-right panel shows the value of the probability of default and the time to default, assuming we increase in 10% (one at a time), the value of the variables indicated on the bottom-left panel.

12 We notice that for the high-debt level case (β=85%) the results are insensitive to variations in the values assigned to the λ s and ρ s. This is somewhat expected since a company with such a high default probability (83.2%) is already at the brink of collapse no matter what. The opposite occurs for the case of low-debt level (β=65%). The initial estimate of the default probability (3%) is extremely sensitive to changes in the coefficients of variations and a bit less so in terms of the correlations (with changes in the inter-temporal correlations being less influential than changes in the correlation between the different cash sources). This highlights the importance of doing extensive sensitivity analyses before awarding a high investment-grade rating (AAA or AA) to a company s debt. Also, and more important, these findings should serve as a warning whenever we are presented with a situation in which the default probabilities fluctuate around small values (maybe 5% or less, but definitively for estimates of the order of 1%). And considering the inherent uncertainty in some of the factors explored, the case for a detailed sensitivity analysis is even more compelling. The fact that the time to default is more stable should not be surprising since this is a second-order variable. Extensions and Further Applications The previous example has demonstrated the feasibility and usefulness of the framework we have outlined. This framework, with slight modifications, can be adapted to treat more general cases if needed. For example: [1] Suppose we wish to describe the cash flows using a more elaborated correlation matrix structure for either the inter-temporal dependence or the dependence between the different cash sources. The algorithm described to generate the random vectors, which are the basis for the Monte Carlo simulation, can still be applied. In fact, this algorithm does not impose any conditions on the correlation matrices other than being positive definite. Even fully populated correlation matrices can be handled with this method. [2] We have assumed here that the cash flows, from each cash source, follow a normal distribution. In the event that the analyst wants to characterize some cash flow using a different distribution (for instance, a uniform or boundednormal distribution) the approach outlined here can still be applied with the caveat that the algorithm used to generate the vectors X* s has to be modified a bit. This topic is treated in detail elsewhere (Hawas and Cifuentes; 2013b).

13 [3] In the example we assumed that the company pays the debt with cash flows generated in the same period and the excess cash is paid out (no reserves are built). If we wish to introduce the possibility that the company can build a reserve fund (for instance, by keeping a certain fraction of the excess cash if there is excess cash on a given period) this feature can be easily managed outside the X*-generating algorithm. It would just involve a minor modification to the software engine that determines if there has been a default. It just involves creating a variable to track the cash in the reserve amount to see if using it might prevent a default. [4] Even though the example described involved a straightforward debt structure, it should be clear that it can accommodate much more general debt profiles (namely, time-dependent payments, time-dependent amortization profiles, or multiple debt obligations with different priorities). [5] Finally, an important consideration in the case of systemically important institutions when they are in a weak position at least from a regulatory viewpoint is the ability to determine if they can be rescued (if so desired). That is, to estimate the level of support they might need to survive, and ultimately distinguish between what can be a liquidity or solvency problem. The technique we have presented here is a suitable tool to address all these issues. The reason is that any rescue package, no matter how complex, it can always be modeled in terms of its fundamentals, that is, in terms of cash flows. Therefore, our approach is ideally suited to investigate what level of cash flow support might be needed to prevent a corporation from defaulting (and whether that cash flow injection is technically or politically feasible). Conclusions We have introduced a fairly simple yet flexible technique to estimate the default probability of a general corporate entity. Our approach departs radically from previous attempts at dealing with this problem since it is based on modeling the cash flows the company relies on. Unlike options-based methods we do not make simplistic assumptions regarding the time-dependent behavior of the cash flows (Brownian motion, constant volatility and the like). Nor do we rely on ratios that are supposed to be constant when in fact in any real situation are highly time-dependent. Therefore, at least from a phenomenological viewpoint, our approach is more sound since it is based on modeling realistically the random variable that ultimately determines whether a company can service its debt or no: the cash flows. We do not rely on modeling variables such as asset prices, leverage ratios, and the like which are

14 only indirectly related to the debt-paying capacity of a company. Furthermore, our method can capture real-life situations such as multiple cash sources, different levels of inter-dependency among them, and different levels of precision in their specifications all features that the standard models cannot even attempt to grasp. Additionally (even though this feature is not shown here) the present method lends itself naturally to estimate confidence intervals for all the relevant metrics (default probabilities, time to default, etc.) In summary, this method offers important improvements compared to the current state-of-the-arts techniques. Finally, given the flexibility that the present method affords in terms of modeling the cash flows, it would be useful to explore the limitations that the current intensity-based models have. We suspect that they might only be able to capture the features of simplistic cash flows patterns. This topic we leave it for future research. We also hope that this approach will help to refocus future efforts. We think that by paying more attention to the key variable (cash flows) and less attention to secondary variables, it will be possible to make substantial advances towards having better predictive tools. Not an outlandish idea after fifty years of failures

15 References Altman, E.I. (1968) Financial Ratios, Discriminant Analysis and the Prediction of Corporate Bankruptcy, Journal of Finance, 23 (4): Beaver, W. (1966) Financial Ratios as Predictors of Failure, Journal of Accounting Research, 5: Bielecki, T. and Cousin, A. (2013) A Bottom-Up Dynamic Model of Portfolio Credit Risk. Available from SSRN, paper number= Bismark, R. and Pasaribu, F. (2011) Capital Structure and Corporate Failure Prediction, Available from SSRN, paper number= Blochlinger, A. (2013) The Next Generation of Default Prediction Models: Incorporating Signal Strength and Dependency. Available from SSRN, paper number= Frunza, M-C. (2013) Are Default Probability Models Relevant for Low Default Portfolios? Available from SSRN, paper number= Hawas, F. and Cifuentes, A. (2013a) Stochastic Cash Flows with Inter- Temporal Correlations, submitted for publication. Hawas, F. and Cifuentes, A. (2013b) A Gaussian Copula-Based Simulation Approach to Valuation Problems with Stochastic Cash Flows, submitted for publication. Mansi, S.A., Maxwell, W.F. and Zhang, A. (2010) Bankruptcy Prediction Models and the Cost of Debt, Available from SSRN, paper number=

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