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

fhawas@dim.uchile.cl

Arturo Cifuentes

Financial Regulation and Stability Center, CREM

Faculty of Economics and Business

University of Chile

Santiago, CHILE

arturo.cifuentes@fen.uchile.cl

October, 2013

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.

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.

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 Xi (i=1, …, M) the vector associated with the cash flows

generated by source i at times t1, …, tN. Thus, Xi = (xi1, …, xiN)

t. To clarify, xij

refers to the cash flow generated by source i at time tj. Moreover, we assume

that the vectors Xi follow multi-normal distributions, X

i ∼MN(μi, Ci), where μi

represents the vector of expected values and Ci denotes the corresponding

correlation matrix. In short, μi = (E(xi1), …, E(xiN))t where E refers to the

expected value operator and i=1, …, M. Also for convenience we will use the

notation μij to designate E(xij). Notice that there are no cash flows associated

with t0.

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 tj, is only correlated to the two neighboring cash flows,

that is, those corresponding to times tj-1 and tj+1. This leads to a correlation

matrix (Ci) 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 ρi

(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, σij = λij μij 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 (Xi 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, xpj and xrs are supposed to be

independent if j ≠ s; but if j = s then the correlation between xpj and xrj, (which

we denote as ρpr) 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=(d1, …, dN)t which is

known and deterministic. The vector Z= (z1, …, zN)t

where zi= x1i +x2i+ … +

xMi (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 dj > zj

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 tj 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 X1, …,

XM

and the issue reduces to estimate the likelihood that Z might have a

component less than its corresponding D counterparty (zj < dj) 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.

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=(z1, z2, z3 ,z4)t

= (x11+x21+ x31, x12+x22+ x32, x13+x23+ x33, x14+x24+ x34)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*= (x1*, … , x12

*)

t = (x11, x12, x13, x14, x21, x22, x23, x24, x31, x32, x33, x34)

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 Lt in

which L is a lower triangular matrix;

[1] Generate U= (u1, …, uMN)t where ui (i=1, …, MN) are random draws from

iid N(0, 1);

[2] Compute V= LU, with V=(v1, …, vMN)t;

[3] Determine W* (the desired sample vector) using the expression

w*i = E(x*i ) + σi vi (for i=1, …, MN) where σi 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;

[5] Determine for such vector Z (a) if a default has occurred (namely, if zj < dj

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(x1i) = μ1i = 25; λ1i=0.3; and ρ1= 0.3

Source 2.

For i=1, …, 10;

E(x2i) = μ2i = 10; λ2i=0.5; and ρ2= 0.3

Source 3.

For i=1, …, 10;

E(x3i) = μ3i = 60; λ3i=0.05; and ρ3= 0.3

In addition we assume that ρ13 = ρ12 = ρ23 = 0.2.

Debt Payments

The debt (D) vector is defined as di = 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.

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

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

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.

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 bounded-

normal 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).

[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

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…

References

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Prediction of Corporate Bankruptcy, Journal of Finance, 23 (4): 589-609.

Beaver, W. (1966) Financial Ratios as Predictors of Failure, Journal of

Accounting Research, 5: 71-111.

Bielecki, T. and Cousin, A. (2013) A Bottom-Up Dynamic Model of

Portfolio Credit Risk. Available from SSRN, paper number=1844574.

Bismark, R. and Pasaribu, F. (2011) Capital Structure and Corporate Failure

Prediction, Available from SSRN, paper number=1978628.

Blochlinger, A. (2013) The Next Generation of Default Prediction Models:

Incorporating Signal Strength and Dependency. Available from SSRN, paper

number=10 0231 .

Frunza, M-C. (2013) Are Default Probability Models Relevant for Low

Default Portfolios? Available from SSRN, paper number=2282675.

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

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