BaR - Balance at Risk

Working Paper

Abstract

This paper introduces an approach designed to the case of personalcredit risk. We define a structural model for the balance of an indi-vidual, allowing for cashflow seasonality and deterministic trends inthe process. This formulation is best suited for the case of short termloans. Based on this model, we build risk measures associated withthe probability of default conditional on time. We illustrate empiricalapplications by estimating an empirical model with simulated dataand, based on it, finding the values of yield rate and maturity thatmaximizes the expected profit from a short-term debt contract.

Keywords: balance at risk, credit risk, personal financeJEL: D1, G21, G21

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1 Introduction

The assessment of consumer credit quality is an important aspect in bankingand finance [Thomas, 2000]. The existence of a developed financial marketallows an individual to sell future cashflow in exchange for a present purchaseof goods such as a TV, car or real state [Shiller, 2013]. Over the years, higherdemand for consumer credit products and competitive incentives towards costminimization of credit analysis have motivated a transition from a subjectiveevaluation of credit risk towards the use of quantitative models for the so-called credit and behavior scoring [Hand and Henley, 1997, Thomas, 2000,Crook et al., 2007].

Traditional models for consumer credit risk assessment tend to rely in awide range of techniques such as artificial neural networks [Khashman, 2010,Oreski et al., 2012], support vector machine [Huang et al., 2007], logistic re-gression [Wiginton, 1980, Crook and Bellotti, 2010], decision tree [Matuszyket al., 2010], mathematical programming [Crook et al., 2007], just to citesome examples 1. Another, less explored, approach to consumer credit riskassessment is the structural models which are more oriented on obtaining theprobability of default (PD) of a loan.

The first structural model for credit risk was an option-based approachproposed by Merton [1974] focusing on modeling the stochastic dynamics offirm’s value. If the firm’s value falls bellow its debt value on its maturity date,then, the firm is on a default situation. This is the same of saying that theequity holders do not exercise their call option on the company’s assets - theoption expires out of the money. Thus, the probability of default on the loanequals to the probability of the option being out of the money at expirationdate [Allen et al., 2004]. Since Merton’s seminal work, other corporate struc-tural credit risk models have appeared, for instance, Longstaff and Schwartz[1995], Leland and Toft [1996] and Collin-Dufresne and Goldstein [2001] 2.

Despite the developments on structural corporate credit risk models, notmuch attention has been given to structural models for consumer credit risks.Perli and Nayda [2004] discuss an option-based models in the same directionas corporate credit risk: if the consumer assets is below a threshold thenhe/she will default on the loan. De Andrade and Thomas [2007] propose astructural model in which the default probability is based on the reputationof the consumer. In their model, the consumer has a call option on the valueof his/her reputation and the strike price of this option is the debt value. The

1Reviews of other techniques can be found on Baesens et al. [2003] and Lessmannaet al. [2013].

2Reduced-form models, in which default is modeled exogenously, would also be alter-native to structural models, see Jarrow et al. [2012]

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credit/behavior score is the proxy for the value of reputation. Thomas [2009]classifies this model as a consumer structural model based on reputation.

Our approach is a consumer structural model based on affordability [Thomas,2009] which differs not only from Perli and Nayda [2004] but also from De An-drade and Thomas [2007]. We model directly the balance of the individualclient by explicitly defining a stochastic process for his/her income and ex-penses. This approach allows for a larger flexibility for the simulation ofscenarios and a more realistic assessment of the probability of default sinceit does not need any characteristic information from the applicant. Themethod is best suited for the case of short term loans, where the distributionof the cashflows over time can significantly impact the default rate of thedebt. The model is easily justified since a better assessment of the defaultrate of the applicants is very important as short term loans can improve thelong term relationship between the bank and its clients [Bodenhorn, 2001].

Furthermore, as discussed by Thomas [2010] and Crook and Bellotti[2010], introducing economics and market conditions into consumer risk as-sessment is still a challenge. Thus, we also contribute to the literature bytaking into account the conditions of the economy directly, such as unem-ployment and wages level. We name the model as Balance at Risk - BaR.

The theory that supports this proposal along with derivations for riskmeasures are provided in the paper. With an artificial dataset, we illustratethe usefulness of the model by first estimating an empirical model and thenusing it for calculating forward probabilities of a default that are conditionalon time. We also show how the model can be used in a loan applicationby finding the parameters of a debt contract that maximizes the expectedreturn of the financial transaction.

The paper is organized as follows. First, we discuss the theory supportingthe empirical BaR model. Then, an application model is examined. In thelast section, we finish the article with the usual concluding remarks.

2 Balance at Risk - BaR

In this section, we present our proposed approach BaR to measure and man-age credit risk into a mathematical and theoretical framework. Unless oth-erwise stated, the content is based on the following notation. Consider thefiltered atom-less probability space XT := X (Ω,F , (FT )T∈T,P) of monetaryvalues, where Ω is the sample space, F is the set of possible events in Ω, FT

is a filtration with F0 = ∅,Ω, F = σ (∪T≥0FT ), T := R+ ∪ ∞, with theusual assumptions, and P is a probability measure that is defined in Ω of theevents contained in F .

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We consider adapted random processes XT : XT → R to represent thevariables in our approach. Thus, E[XT ] is the expected value of XT underP. All equalities and inequalities are considered to be almost surely in P.FXT

:= FX|FTis the probability function of XT , with inverse F−1

XTand density

fXT. BT , RT , CT , IT , ET , LT represent at time T , respectively, balance, risk

free rate, net cash flow, income, expense and loan to be paid. We assume thatRT and LT are FT−dt measurable, since their values are known by contractin the previous period.

The main idea is to consider the balance of an agent as a stochastic processcomposed by his/her net cash flows, in order to measure credit risk incurredfor some financial institution when it loans money for this agent. The BaRapproach is, basically, the analysis of risk measures through this stochasticprocess. Thus, in this section, we define and expose some properties of thestochastic process under analysis as well as the credit risk measures that arederived from it.

Definition 2.1. Let BT , RT , CT , IT , ET : XT → R, and CT = IT − ET . Theclosed form for the stochastic process that represents the balance is given by:

BT = B0 exp

(∫ T

0

Rtdt

)+

∫ T

0

Ct exp

(∫ T

t

Rsds

)dt. (1)

dBT =

([B0 exp

(∫ T

0

Rtdt

)]RT + CT

)dt. (2)

Remark 2.2. The stochastic process in Definition 2.1 depends only on cashflows, and can be considered as an information that summarizes distinctdimensions (economic, social, geographic, etc.) of the agent behavior. Equa-tions (1) and (2) can be recursively generalized for any time T ∗ ≤ T , respec-tively conform:

BT = BT ∗ exp

(∫ T

T ∗Rtdt

)+

∫ T

T ∗Ct exp

(∫ T

t

Rsds

)dt,

dBT =

([BT ∗ exp

(∫ T

T ∗Rtdt

)]RT + CT

)dt.

Moreover, it is possible to obtain simpler formulations under null initial bal-ance, i.e. B0 = 0.

Obviously, the properties of BT as a stochastic process are directly depen-dent of those possessed by CT , and consequently by IT and ET . At this point,we do not make any assumption about the stochastic process that governssuch random variables. Our goal is to keep the model so general as possible.

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Nonetheless, for a special case, it is possible to derive FBTanalytically given

FIT and FET. To that, we use the concept of probability functions convo-

lution. Thus, we present the definition of this concept and two very knownlemmas in probability theory. With this background, we prove a theorem.

Definition 2.3. The convolution between two probability densities fX andfY is given by:

fXfY (x) =

∫ ∞−∞

fX(x− u)fY (u)du. (3)

Lemma 2.4. Let X and Y be two independent random variables. ThenfX+Y = fXfY (x).

Lemma 2.5. Let k ∈ R. Then f(x) =1

|k|fX

(xk

).

Proposition 2.6. Let BT , RT , CT , IT , ET : XT → R, CT = IT −ET and Γt =∫ T

tRsds. If IT ,−ET are independent ∀T ∈ T and CT , CS are independent

∀ T, S ∈ T, T 6= S, then:

FBT(x) =

∫ x

−∞

[limdt→0

fB0Γ0(s)fCtΓt(s)|T0]ds (4)

fCtΓt(x) =

[1

|Γt|fIt

(x

Γt

)]

[1

|Γt|f−Et

(x

Γt

)],∀t ∈ [0, T ]. (5)

fCtΓt(x)|T0 = fC0Γ0(x)fC0+dtΓ0+dt(x) · · ·fCT ΓT

(x). (6)

Proof. Because RT is FT−dt measurable by assumption, Γt ∈ R∀t ∈ [0, T ].Since CtΓt = ItΓt − EtΓt,∀t ∈ [0, T ], we have by Lemmas 2.4 and 2.5 that:

fCtΓt(x) = fItΓt−EtΓt =

[1

|Γt|fIt

(x

Γt

)]

[1

|Γt|f−Et

(x

Γt

)],∀t ∈ [0, T ].

Repeating the argument for CtΓt, t ∈ [0, T ], which are independent byassumption, we obtain that f∫ T

0 CtΓtdt(x) = lim

dt→0fCtΓt(x)|T0 . Since fB0Γ0 is

a Dirac measure, which is independent to any probability function, and

FX(x) =∫ x

−∞ fX(s)ds, we have that FBT(x) =

∫ x

−∞

[limdt→0

fB0Γ0(s)fCtΓt(s)|T0]ds.

This concludes the proof.

Remark 2.7. Naturally, such assumptions of independence in Proposition2.6 are too restrictive, and can be questioned in real life. However, evenwith such constraints the complexity degree becomes huge and it leads to ananalytical expression that is hard to be computed. Moreover, it is at least analternative to obtain FBT

from distributions of the cash flows. One can drop

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the assumption that IT and −ET are independent if the distribution for CT

is directly known. Nonetheless, in practical situations, it is also possible todirectly estimate FBT

from data of the balance using numerical simulations,for instance.

Having exposed the stochastic process, we turn our focus to the nextstep on the BaR approach, which is the measurement of credit risk. Weconcentrate into two risk measures, the Probability of Default (PD) andLoss Given Default (LGD). Since both PD and LGD are very well knownrisk measures for credit risk, we do not pursue to debate their theoreticalproperties in here. Nonetheless, we formally define them, and briefly discusstheir financial meaning.

Definition 2.8. Let BT , LT : XT → R. The Probability of Default, PDT :XT → [0, 1], and Loss Given Default, LGDT : XT → R, risk measures aredefined, respectively as follows:

PDT := P(BT ≤ LT ) = FBT(LT ) . (7)

LGDT := −E[BT − LT |BT ≤ LT = F−1

BT(PDT )

]. (8)

Remark 2.9. PD is the probability of an agent not having enough balanceto pay his/her loan at time T . It is a very simple and intuitive credit riskmeasure. However, PD alone does not give the whole picture of the situa-tion since it does not consider the loss magnitudes in case of a default. Thisshortcoming is handled by the LGD, which indicates how much money isnecessary to fulfill the loss in case of a default. This measure has similari-ties to the well known market risk measure Expected Shortfall (ES), but forLGD the expectation is truncated by LT instead of some quantile of inter-est. Moreover, because formulations (7) and (8) are both directly dependentof FBT

, under the hypothesis of Proposition 2.6, one can obtain analyticalformulations for these two risk measures.

3 An empirical BaR model

Now, we consider a specific BaR model nested in the previous formulations.From this point forward, we change the notation so that the time index isrepresented by t = 1...T , which is the usual notation in empirical time seriesmodels.

We use a discrete version of the model by setting the income and the ex-pense equation of an individual as separate stochastic processes. We expectthe income to be far more predictable than the expenses since most of the

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society is bounded by work contracts that explicitly define the amount offinancial reward one receives per unit of time. When looking at expenses,however, we can expect a higher quantity of noise as the individual deci-sion to purchase goods and services is a function of diverse economic, socialand personal factors. The heterogeneous stochastic properties of inflow andoutflow of cash motivates separating the process of income and expenses.

Another important part in empirically modeling the inflow and outflowof financial resources is recognizing the existence of seasonalities. An indi-vidual might receive more cash in particular periods of time than others. Forinstance, a worker in Brazil is entitled to an extra salary and an additionalholiday premium throughout the year. These are usually paid in June andDecember. Identifying these particular months is essential towards a realisticmodel for credit risk, specially in the case of short term loans. The expec-tation of receiving more cash in particular months will affect the balance,adding a time dependency on the dynamic of cashflows, and therefore theforward probability of default.

A realistic empirical model should also consider the effect of unemploy-ment. If an individual loses its work contract by any reason, the income ofcash will cease and this will impact the probability of default. When theindividual loses his/her job, all that is left to support his expenses is thecurrent value of the balance.

Considering these effects, we propose a discrete version of Equations 1and 2 for our empirical example:

Bt = Bt−1(1 + rt) + Ii,t − Ei,t (9)

Ii,t =

0 if St = 1 (unnemployed)

FI +MIi,t if St = 2 (employed)(10)

Ei,t = FE +MEi,t (11)

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Bt = Balance at time t (t = 1..T )

rt = Risk free yeild (monthly), assumed constant (rt = r)

Ii,t = Total income at month i, time t

Ei,t = Total expense at motnh i, time t

FI = Fixed monthly income

MI = Monthly extra income at month i

FE = Fixed monthly expense

ME = Monthly extra expense at month i

Following the ideas described in Huh et al. [2010] and Malik and Thomas[2012], the states of employment and unemployment (St) will follow a discretemarkov chain given by transition matrix P :

P =

[p11 1− p22

1− p11 p22

](12)

Since there is only two states and two transition probabilities, we canestimate the transition probabilities by inverting the expected duration ofthe states E(Dur(St = k)) = 1

1−pkk.

p11 = 1− 1/E(TimeEmployed) (13)

p22 = 1− 1/E(TimeUnemployed) (14)

Thus, based on how much time it takes for a worker in the same profes-sion as the applicant to get a job or lose his/her job, it is easy to transformthis information into transition probabilities with Equations 13 and 14. Withthis setup, we allow for economic conditions in the job market to affect theprobability of default of the applicant. If the duration of unemployment in-creases, so does increase the likelihood of lower income and therefore we canexpect an increase of the probability of default on a loan. Also, notice thatas the balance in Equation 9 will change over time. Meaning that, in sim-ilar fashion as behavioral models [Thomas, 2000], the probability of defaultshould be recalculated as time moves forwards. If the effective balance levelincreases with the good financial behavior of the applicant, the likelihood offuture default decreases and the individual should receive a reward by paying

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lower interest rates.

3.1 Estimating and simulating the parameters

As an example of an empirical application, we are going to assume the ex-istence of a history of inflows and outflows of a balance account for fiveyears. The individual banking statements should be easily available withina commercial bank. In commercial applications of the model, it might beinteresting to consider the creation of a central organization with the aggre-gate banking statements from the applicants. On the one hand, banks wouldbenefit from the higher amount of information from the applications, On theother hand, the clients would benefit from the competition within banks,which should create a downwards pressure in the interest rates.

In order to illustrate the use of the model, we simulate the inflows andoutflows of money as random Normal variables with different means andstandard deviations according to the month of the year. This should emulatethe expected noise from an empirical banking data.

Figure 1: Simulated data for income and expenses

In Figure 1, we present the time series plot of the simulated income andexpense data. Notice that the behavior of the income is far more stable then

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the expenses, with a clear seasonality in the months of june and december.The balance of this individual has a clearly upward trends as income is usuallyhigher than the expenditures. We further assume that the applicant generallystays three years employed and, when unemployed, he/she can acquire a newjob in three months. This information could be retrieved from the applicant’sjob records or from other sources. As an example, the Bureau of laborstatistics reports3 that the average duration of employment in the US formarch 2016 is 29 weeks (7.25 months).

Given the general mathematical formulation discussed in the previoussection, the empirical model can be formulated in different ways. For sim-plicity we are going to assume that the income and expenses will follow alinear model conditional on time. However, it is important to point out thatmore sophisticated methods such as cubic splines can be used to model theseasonality of the banking data. Such an approach has already been usedwith success in Finance for the case of modeling volatility and term structure[Audrino and Buhlmann, 2009, Engle and Rangel, 2008, Jarrow et al., 2012].

The estimation of the model is straightforward. We define an statisticalmodel for income and expenses as:

Ii,t = αI +12∑i=2

βiDi + εi,t (15)

Ei,t = αE +12∑i=2

φiDi + ηi,t (16)

where Di is a dummy variable that takes value 1 if the current month isi and zero otherwise. We exclude the dummy for the first month, i = 1to avoid identification issues in the regression model. Using as input thesimulated dataset, we estimate by least squares the empirical model definedin Equations 15 and 16. The resulting coefficients from the estimation areomitted. Notice that they do capture the seasonality of the data.

With the information regarding the regression models and the initial bal-ance set to 1000, we simulate the future balance of the individual by samplingthe estimated residuals from the regression, εi,t and ηi,t, conditional on themonth of the year. This allowed for a seasonality not only on the expectedincome and expense but also on their corresponding volatility. We perform10000 simulations with a time horizon of 24 months.

Once the balance is simulated, it is straightforward to calculate the PD

3See http://www.bls.gov/news.release/empsit.t12.htm

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of the individual by simply looking at the number of simulated scenariosin which the resulting balance was negative for each forward point in time(see Equation 7). Since there are no loans in this first example, this figurerepresents the probability of the applicant falling short on his expenses. Wecall this figure the benchmark default rate curve.

Figure 2: Default probability

From Figure 2, we notice that the PD is also seasonal, with a drop inmonth 12, which is when we defined a larger income of cash. We also seethat the probabilities of default generally increases with time, meaning thatthis applicant is more likely to run out of cash as time go by.

Now, we illustrate a common practical case in which the applicant isasking for a loan that can be paid with monthly installments of 400 fortwo years or monthly installments of 200 for four years. Given the previousregression for the expenses, it is easy to implement this information in themodel by defining Ei,t = αE + ∆ +

∑12i=2 φiDi + ηi,t, where ∆ is the expected

change in the expense represented as the monthly loan payment value. Figure3 presents the impact in the PD.

Based on Figure 3, the payment of the loan will have an explicit impactin the forward PD. For the case of monthly payments of 400, the applicant

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Figure 3: Default probabilities given a new loan

Table 1: Risk metrics for empirical BAR model

Case Loss Given defaultInstallments of 400 per month 1620.76Installments of 200 per month 1108.28

is more likely to default on its loan in month 23. However, when setting amonthly payment of 200, the PD decreases significantly, indicating a betterfinancial contract for both sides. This illustration shows how the empiricalmodel is flexible and could accommodate different, practical scenarios in theevaluation of credit risk. Next, in Table 1, we present the calculation of theLoss Given Default (LGD) for each scenario.

As we can see, the first scenario leads to a LGD of 1620.76, which iscomparatively higher than the second case with longer maturity and lowermonthly installments. As suspected, the second setup presents a lower ex-pected cost for the bank.

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3.2 Optimizing parameters of a loan based on a BaRmodel

In practice, the bank receives data from the applicant and must decide onthe structure of the offered debt contract including its maturity and rate ofreturn. If the contract is perceived as risky, the bank is likely to charge ahigher yield rate and set a lower maturity range. In this section, we aregoing to present an illustration of how the BaR formulation can be used tooptimize the parameters from the point of view of the bank.

We consider the case where the applicant is requesting a loan for 10, 000units of cash and has the same banking records used in the previous example(see Figure 1). The debt is payed back by the applicant in a fixed monthlyinstallment that is defined for a given maturity and an annual yield rate.This is equivalent to a fixed rate bond with monthly coupons [Fabozzi andMann, 2012]. Using an annual yield rate, we define the installment value as:

rm = (1 + ra)112 − 1 (17)

I = 5000rm

(1− (1 + rm)−T )(18)

Where:

rm = Monthly yield of debt contract

ry = Yearly yield of debt contract

T = Maturity of contract (in months)

I = Fixed value of monthly installments

From Equation 18, we can see that the increase of yield rate and thedecrease of the maturity of the contract will increase the value of the in-stallments, but also the risk of the contract. In the BaR model, a higherinstallment will lead to a higher level of uncertainty and, consequently, ahigher probability of default.

From the bank side, a natural question to investigate is: Given the ex-pected cashflow dynamic of the applicant, which values of yield and maturitywill maximize the profit of the contract? We analyze this question by defin-ing the objective function as the expected return from the loan, which iscalculated based on the simulation procedure defined in the previous section.Notice that the risk is also considered, as higher values of installments willincrease the likelihood of a default and therefore decrease the expected profit.

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Figure 4: Result of grid search procedure

We first create a vector of values for maturity (T = 1, 2, ..., 23, 24) andyields (ry = 0.025, 0.05, ..., 0.275, 0.5). For each pair of maturity and yieldwe simulated the BaR model and calculated the expected return from thecontract. We performed a grid search procedure in order to find the pair ofmaturity and yield that maximizes the value of E(R), the expected cashflowdivided by the total value of the loan. The result is presented in Figure 4.

From Figure 4 we can see that the pair of maturity and yield that max-imizes the expected profit of the contract is 22 months and a 37.5% rate ofreturn on the debt. The white spots in the figures shows that a low maturityand a higher yield leads to a null return of the contract. This is explained bythe fact that the higher installment (see Equation 18) will drain the balanceof the applicant quickly, leading to a default on the contract. We also pointout that the risk of the contract, LGD (Equation 8), is minimized with amaturity of 16 months and a 5% yield. This exercise clearly shows how theBaR model can be used to better design short-term financial contracts bytaking into account the applicant’s banking history.

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4 Conclusions

In this paper, we propose an alternative model for calculating personal fi-nancial risk by defining a stochastic process for the income and expenditureof cash of an individual. Formal theoretical definitions for this approachalong with usual risk measures are presented. Based on an artificial dataset,we illustrate the usage of the model by estimating an empirical version ofthe general formulation and calculating the forward probabilities of defaulton different loan scenarios. We also present an example of an optimizationscenario, where we find the optimal maturity and yield rate of a short-termloan.

The main advantage in using our approach is its flexibility. By directlyusing a stochastic process for the income and expenses, we allow for sea-sonality to take its role in the analysis of short-term credit risk. Using thismodel one can verify the impact of several economic factors in the personalcredit risk of an individual, whether it be the increase of a loan payment, itsmaturity date, specific payment schedules or changes in the applicant’s jobmarket.

Empirical applications of the model are straightforward. Based on bank-ing data and information from the job market, one can estimate and simulatethe future income and expenses of a person. As argued in the paper, the cre-ation of a central organization for storing and analyzing banking data couldfacilitate the process and benefit for both parties in the transaction.

As banks deals with a portfolio of individual loans, a suggestion for a fu-ture study is to investigate a multivariate version of the model. It would beinteresting to formulate a model that incorporates systematic dependencieswithin a pool of applicant’s cashflows, where economic shocks such as the in-crease of general unemployment will affect the overall probability of defaulton loans. Future investigations with real banking data are also suggested.Based on individual records, one could study the dynamics of cashflows andtest for the structure of an empirical model that efficiently represents thedataset and, therefore, provides realistic values of default probabilities. Un-derstanding and modeling how applicants react when faced with a low bal-ance in this account is also suggested. A person is likely to hold its expensesonce his/her balance passes a particular threshold. An investigation withreal data could shed light regarding this effect. These and other ideas areleft for future development of the BaR model.

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