BaR - Balance at Risk

Transcription

1 BaR - Balance at Risk Working Paper Abstract This paper introduces an approach designed to the case of personal credit risk. We define a structural model for the balance of an individual, allowing for cashflow seasonality and deterministic trends in the process. This formulation is best suited for the case of short term loans. Based on this model, we build risk measures associated with the probability of default conditional on time. We illustrate empirical applications by estimating an empirical model with simulated data and, based on it, finding the values of yield rate and maturity that maximizes the expected profit from a short-term debt contract. Keywords: balance at risk, credit risk, personal finance JEL: D1, G21, G21 1

2 1 Introduction The assessment of consumer credit quality is an important aspect in banking and finance [Thomas, 2000]. The existence of a developed financial market allows an individual to sell future cashflow in exchange for a present purchase of goods such as a TV, car or real state [Shiller, 2013]. Over the years, higher demand for consumer credit products and competitive incentives towards cost minimization of credit analysis have motivated a transition from a subjective evaluation of credit risk towards the use of quantitative models for the socalled 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 a wide range of techniques such as artificial neural networks [Khashman, 2010, Oreski et al., 2012], support vector machine [Huang et al., 2007], logistic regression [Wiginton, 1980, Crook and Bellotti, 2010], decision tree [Matuszyk et al., 2010], mathematical programming [Crook et al., 2007], just to cite some examples 1. Another, less explored, approach to consumer credit risk assessment is the structural models which are more oriented on obtaining the probability of default (PD) of a loan. The first structural model for credit risk was an option-based approach proposed by Merton [1974] focusing on modeling the stochastic dynamics of firm 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 the equity holders do not exercise their call option on the company s assets - the option expires out of the money. Thus, the probability of default on the loan equals to the probability of the option being out of the money at expiration date [Allen et al., 2004]. Since Merton s seminal work, other corporate structural 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, not much attention has been given to structural models for consumer credit risks. Perli and Nayda [2004] discuss an option-based models in the same direction as corporate credit risk: if the consumer assets is below a threshold then he/she will default on the loan. De Andrade and Thomas [2007] propose a structural model in which the default probability is based on the reputation of the consumer. In their model, the consumer has a call option on the value of his/her reputation and the strike price of this option is the debt value. The 1 Reviews of other techniques can be found on Baesens et al. [2003] and Lessmanna et al. [2013]. 2 Reduced-form models, in which default is modeled exogenously, would also be alternative to structural models, see Jarrow et al. [2012] 2

3 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 Andrade and Thomas [2007]. We model directly the balance of the individual client by explicitly defining a stochastic process for his/her income and expenses. This approach allows for a larger flexibility for the simulation of scenarios and a more realistic assessment of the probability of default since it does not need any characteristic information from the applicant. The method is best suited for the case of short term loans, where the distribution of the cashflows over time can significantly impact the default rate of the debt. The model is easily justified since a better assessment of the default rate of the applicants is very important as short term loans can improve the long 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 assessment is still a challenge. Thus, we also contribute to the literature by taking into account the conditions of the economy directly, such as unemployment and wages level. We name the model as Balance at Risk - BaR. The theory that supports this proposal along with derivations for risk measures are provided in the paper. With an artificial dataset, we illustrate the usefulness of the model by first estimating an empirical model and then using it for calculating forward probabilities of a default that are conditional on time. We also show how the model can be used in a loan application by finding the parameters of a debt contract that maximizes the expected return of the financial transaction. The paper is organized as follows. First, we discuss the theory supporting the empirical BaR model. Then, an application model is examined. In the last 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 manage credit risk into a mathematical and theoretical framework. Unless otherwise stated, the content is based on the following notation. Consider the filtered atom-less probability space X T := X (Ω, F, (F T ) T T, P) of monetary values, where Ω is the sample space, F is the set of possible events in Ω, F T is a filtration with F 0 = {, Ω}, F = σ ( T 0 F T ), T := R +, with the usual assumptions, and P is a probability measure that is defined in Ω of the events contained in F. 3

4 We consider adapted random processes X T : X T R to represent the variables in our approach. Thus, E[X T ] is the expected value of X T under P. All equalities and inequalities are considered to be almost surely in P. F XT := F X FT is the probability function of X T, with inverse F 1 X T and density f XT. B T, R T, C T, I T, E T, L T represent at time T, respectively, balance, risk free rate, net cash flow, income, expense and loan to be paid. We assume that R T and L T are F T dt measurable, since their values are known by contract in the previous period. The main idea is to consider the balance of an agent as a stochastic process composed by his/her net cash flows, in order to measure credit risk incurred for some financial institution when it loans money for this agent. The BaR approach is, basically, the analysis of risk measures through this stochastic process. Thus, in this section, we define and expose some properties of the stochastic process under analysis as well as the credit risk measures that are derived from it. Definition 2.1. Let B T, R T, C T, I T, E T : X T R, and C T = I T E T. The closed form for the stochastic process that represents the balance is given by: ( T B T = B 0 exp db T = 0 ([ ( T B 0 exp ) T R t dt ( T C t exp t ) R s ds dt. (1) )] ) R t dt R T + C T dt. (2) Remark 2.2. The stochastic process in Definition 2.1 depends only on cash flows, and can be considered as an information that summarizes distinct dimensions (economic, social, geographic, etc.) of the agent behavior. Equations (1) and (2) can be recursively generalized for any time T T, respectively conform: ( T ) T ( T B T = B T exp R t dt + C t exp T T t ([ ( T )] ) db T = B T exp R t dt R T + C T dt. T ) R s ds dt, Moreover, it is possible to obtain simpler formulations under null initial balance, i.e. B 0 = 0. Obviously, the properties of B T as a stochastic process are directly dependent of those possessed by C T, and consequently by I T and E T. At this point, we do not make any assumption about the stochastic process that governs such random variables. Our goal is to keep the model so general as possible. 4

5 Nonetheless, for a special case, it is possible to derive F BT analytically given F IT and F ET. To that, we use the concept of probability functions convolution. Thus, we present the definition of this concept and two very known lemmas in probability theory. With this background, we prove a theorem. Definition 2.3. The convolution between two probability densities f X is given by: f Y and f X f Y (x) = f X (x u)f Y (u)du. (3) Lemma 2.4. Let X and Y be two independent random variables. f X+Y = f X f Y (x). Lemma 2.5. Let k R. Then f(x) = 1 ( x ) k f X. k Then Proposition 2.6. Let B T, R T, C T, I T, E T : X T R, C T = I T E T and Γ t = T R t s ds. If I T, E T are independent T T and C T, C S are independent T, S T, T S, then: F BT (x) = f CtΓ t (x) = x [ ] lim f B 0 Γ 0 (s) f CtΓt (s) T 0 ds (4) dt 0 [ ( )] [ ( )] 1 x 1 x Γ t f I t Γ t f E t, t [0, T ]. (5) Γ t f CtΓ t (x) T 0 = f C0 Γ 0 (x) f C0+dt Γ 0+dt (x) f CT Γ T (x). (6) Proof. Because R T is F T dt measurable by assumption, Γ t R t [0, T ]. Since C t Γ t = I t Γ t E t Γ[ t, t [0, ( T ], )] we have [ by Lemmas ( )] 2.4 and 2.5 that: 1 x 1 x f CtΓt (x) = f ItΓt E tγ t = Γ t f I t Γ t Γ t f E t, t [0, T ]. Γ t Repeating the argument for C t Γ t, t [0, T ], which are independent by assumption, we obtain that f T 0 CtΓtdt(x) = lim f C tγ t (x) T 0. Since f B0 Γ 0 is dt 0 a Dirac measure, which is independent to any probability function, and F X (x) = x f X(s)ds, we have that F BT (x) = x This concludes the proof. Γ t [ lim dt 0 f B 0 Γ 0 (s) f CtΓ t (s) T 0 Remark 2.7. Naturally, such assumptions of independence in Proposition 2.6 are too restrictive, and can be questioned in real life. However, even with such constraints the complexity degree becomes huge and it leads to an analytical expression that is hard to be computed. Moreover, it is at least an alternative to obtain F BT from distributions of the cash flows. One can drop ] ds. 5

6 the assumption that I T and E T are independent if the distribution for C T is directly known. Nonetheless, in practical situations, it is also possible to directly estimate F BT from data of the balance using numerical simulations, for instance. Having exposed the stochastic process, we turn our focus to the next step on the BaR approach, which is the measurement of credit risk. We concentrate into two risk measures, the Probability of Default (PD) and Loss Given Default (LGD). Since both PD and LGD are very well known risk measures for credit risk, we do not pursue to debate their theoretical properties in here. Nonetheless, we formally define them, and briefly discuss their financial meaning. Definition 2.8. Let B T, L T : X T R. The Probability of Default, P D T : X T [0, 1], and Loss Given Default, LGD T : X T R, risk measures are defined, respectively as follows: P D T := P(B T L T ) = F BT (L T ). (7) LGD T := E [ B T L T B T L T = F 1 B T (P D T ) ]. (8) Remark 2.9. PD is the probability of an agent not having enough balance to pay his/her loan at time T. It is a very simple and intuitive credit risk measure. However, PD alone does not give the whole picture of the situation since it does not consider the loss magnitudes in case of a default. This shortcoming is handled by the LGD, which indicates how much money is necessary to fulfill the loss in case of a default. This measure has similarities to the well known market risk measure Expected Shortfall (ES), but for LGD the expectation is truncated by L T instead of some quantile of interest. Moreover, because formulations (7) and (8) are both directly dependent of F BT, under the hypothesis of Proposition 2.6, one can obtain analytical formulations 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 is represented by t = 1...T, which is the usual notation in empirical time series models. We use a discrete version of the model by setting the income and the expense equation of an individual as separate stochastic processes. We expect the income to be far more predictable than the expenses since most of the 6

7 society is bounded by work contracts that explicitly define the amount of financial reward one receives per unit of time. When looking at expenses, however, we can expect a higher quantity of noise as the individual decision to purchase goods and services is a function of diverse economic, social and personal factors. The heterogeneous stochastic properties of inflow and outflow of cash motivates separating the process of income and expenses. Another important part in empirically modeling the inflow and outflow of financial resources is recognizing the existence of seasonalities. An individual might receive more cash in particular periods of time than others. For instance, a worker in Brazil is entitled to an extra salary and an additional holiday premium throughout the year. These are usually paid in June and December. Identifying these particular months is essential towards a realistic model for credit risk, specially in the case of short term loans. The expectation of receiving more cash in particular months will affect the balance, adding a time dependency on the dynamic of cashflows, and therefore the forward probability of default. A realistic empirical model should also consider the effect of unemployment. If an individual loses its work contract by any reason, the income of cash will cease and this will impact the probability of default. When the individual loses his/her job, all that is left to support his expenses is the current value of the balance. Considering these effects, we propose a discrete version of Equations 1 and 2 for our empirical example: B t = B t 1 (1 + r t ) + I i,t E i,t (9) { 0 if S t = 1 (unnemployed) I i,t = (10) F I + MI i,t if S t = 2 (employed) E i,t = F E + ME i,t (11) 7

8 B t = Balance at time t (t = 1..T ) r t = Risk free yeild (monthly), assumed constant (r t = r) I i,t = Total income at month i, time t E i,t = Total expense at motnh i, time t F I = Fixed monthly income MI = Monthly extra income at month i F E = 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 (S t ) will follow a discrete markov chain given by transition matrix P : [ ] p11 1 p P = 22 1 p 11 p 22 (12) Since there is only two states and two transition probabilities, we can estimate the transition probabilities by inverting the expected duration of the states E(Dur(S t = k)) = 1 1 p kk. p 11 = 1 1/E(T imeemployed) (13) p 22 = 1 1/E(T imeunemployed) (14) Thus, based on how much time it takes for a worker in the same profession as the applicant to get a job or lose his/her job, it is easy to transform this information into transition probabilities with Equations 13 and 14. With this setup, we allow for economic conditions in the job market to affect the probability of default of the applicant. If the duration of unemployment increases, so does increase the likelihood of lower income and therefore we can expect an increase of the probability of default on a loan. Also, notice that as the balance in Equation 9 will change over time. Meaning that, in similar fashion as behavioral models [Thomas, 2000], the probability of default should be recalculated as time moves forwards. If the effective balance level increases with the good financial behavior of the applicant, the likelihood of future default decreases and the individual should receive a reward by paying 8

9 lower interest rates. 3.1 Estimating and simulating the parameters As an example of an empirical application, we are going to assume the existence of a history of inflows and outflows of a balance account for five years. The individual banking statements should be easily available within a commercial bank. In commercial applications of the model, it might be interesting to consider the creation of a central organization with the aggregate banking statements from the applicants. On the one hand, banks would benefit from the higher amount of information from the applications, On the other 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 and outflows of money as random Normal variables with different means and standard deviations according to the month of the year. This should emulate the 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 and expense data. Notice that the behavior of the income is far more stable then 9

10 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 usually higher than the expenditures. We further assume that the applicant generally stays three years employed and, when unemployed, he/she can acquire a new job in three months. This information could be retrieved from the applicant s job records or from other sources. As an example, the Bureau of labor statistics reports 3 that the average duration of employment in the US for march 2016 is 29 weeks (7.25 months). Given the general mathematical formulation discussed in the previous section, the empirical model can be formulated in different ways. For simplicity we are going to assume that the income and expenses will follow a linear model conditional on time. However, it is important to point out that more sophisticated methods such as cubic splines can be used to model the seasonality of the banking data. Such an approach has already been used with success in Finance for the case of modeling volatility and term structure [Audrino and Bühlmann, 2009, Engle and Rangel, 2008, Jarrow et al., 2012]. The estimation of the model is straightforward. We define an statistical model for income and expenses as: 12 I i,t = α I + β i D i + ɛ i,t (15) i=2 12 E i,t = α E + φ i D i + η i,t (16) where D i is a dummy variable that takes value 1 if the current month is i and zero otherwise. We exclude the dummy for the first month, i = 1 to avoid identification issues in the regression model. Using as input the simulated dataset, we estimate by least squares the empirical model defined in Equations 15 and 16. The resulting coefficients from the estimation are omitted. Notice that they do capture the seasonality of the data. With the information regarding the regression models and the initial balance set to 1000, we simulate the future balance of the individual by sampling the estimated residuals from the regression, ɛ i,t and η i,t, conditional on the month of the year. This allowed for a seasonality not only on the expected income and expense but also on their corresponding volatility. We perform simulations with a time horizon of 24 months. Once the balance is simulated, it is straightforward to calculate the PD i=2 3 See 10

11 of the individual by simply looking at the number of simulated scenarios in 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 figure represents the probability of the applicant falling short on his expenses. We call 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 in month 12, which is when we defined a larger income of cash. We also see that the probabilities of default generally increases with time, meaning that this applicant is more likely to run out of cash as time go by. Now, we illustrate a common practical case in which the applicant is asking for a loan that can be paid with monthly installments of 400 for two years or monthly installments of 200 for four years. Given the previous regression for the expenses, it is easy to implement this information in the model by defining E i,t = α E i=2 φ id i + η i,t, where is the expected change in the expense represented as the monthly loan payment value. Figure 3 presents the impact in the PD. Based on Figure 3, the payment of the loan will have an explicit impact in the forward PD. For the case of monthly payments of 400, the applicant 11

12 Figure 3: Default probabilities given a new loan Table 1: Risk metrics for empirical BAR model Case Loss Given default Installments of 400 per month Installments of 200 per month is more likely to default on its loan in month 23. However, when setting a monthly payment of 200, the PD decreases significantly, indicating a better financial contract for both sides. This illustration shows how the empirical model is flexible and could accommodate different, practical scenarios in the evaluation of credit risk. Next, in Table 1, we present the calculation of the Loss Given Default (LGD) for each scenario. As we can see, the first scenario leads to a LGD of , which is comparatively higher than the second case with longer maturity and lower monthly installments. As suspected, the second setup presents a lower expected cost for the bank. 12

13 3.2 Optimizing parameters of a loan based on a BaR model In practice, the bank receives data from the applicant and must decide on the structure of the offered debt contract including its maturity and rate of return. If the contract is perceived as risky, the bank is likely to charge a higher yield rate and set a lower maturity range. In this section, we are going to present an illustration of how the BaR formulation can be used to optimize the parameters from the point of view of the bank. We consider the case where the applicant is requesting a loan for 10, 000 units 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 monthly installment that is defined for a given maturity and an annual yield rate. This is equivalent to a fixed rate bond with monthly coupons [Fabozzi and Mann, 2012]. Using an annual yield rate, we define the installment value as: Where: r m = (1 + r a ) (17) r m I = 5000 (1 (1 + r m ) T ) (18) r m = Monthly yield of debt contract r y = 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 the decrease of the maturity of the contract will increase the value of the installments, but also the risk of the contract. In the BaR model, a higher installment will lead to a higher level of uncertainty and, consequently, a higher probability of default. From the bank side, a natural question to investigate is: Given the expected cashflow dynamic of the applicant, which values of yield and maturity will maximize the profit of the contract? We analyze this question by defining the objective function as the expected return from the loan, which is calculated based on the simulation procedure defined in the previous section. Notice that the risk is also considered, as higher values of installments will increase the likelihood of a default and therefore decrease the expected profit. 13

14 Figure 4: Result of grid search procedure We first create a vector of values for maturity (T = 1, 2,..., 23, 24) and yields (r y = 0.025, 0.05,..., 0.275, 0.5). For each pair of maturity and yield we simulated the BaR model and calculated the expected return from the contract. We performed a grid search procedure in order to find the pair of maturity and yield that maximizes the value of E(R), the expected cashflow divided 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 maximizes the expected profit of the contract is 22 months and a 37.5% rate of return on the debt. The white spots in the figures shows that a low maturity and a higher yield leads to a null return of the contract. This is explained by the fact that the higher installment (see Equation 18) will drain the balance of the applicant quickly, leading to a default on the contract. We also point out that the risk of the contract, LGD (Equation 8), is minimized with a maturity of 16 months and a 5% yield. This exercise clearly shows how the BaR model can be used to better design short-term financial contracts by taking into account the applicant s banking history. 14

15 4 Conclusions In this paper, we propose an alternative model for calculating personal financial risk by defining a stochastic process for the income and expenditure of cash of an individual. Formal theoretical definitions for this approach along with usual risk measures are presented. Based on an artificial dataset, we illustrate the usage of the model by estimating an empirical version of the general formulation and calculating the forward probabilities of default on different loan scenarios. We also present an example of an optimization scenario, where we find the optimal maturity and yield rate of a short-term loan. The main advantage in using our approach is its flexibility. By directly using a stochastic process for the income and expenses, we allow for seasonality to take its role in the analysis of short-term credit risk. Using this model one can verify the impact of several economic factors in the personal credit risk of an individual, whether it be the increase of a loan payment, its maturity date, specific payment schedules or changes in the applicant s job market. Empirical applications of the model are straightforward. Based on banking data and information from the job market, one can estimate and simulate the future income and expenses of a person. As argued in the paper, the creation of a central organization for storing and analyzing banking data could facilitate the process and benefit for both parties in the transaction. As banks deals with a portfolio of individual loans, a suggestion for a future study is to investigate a multivariate version of the model. It would be interesting to formulate a model that incorporates systematic dependencies within a pool of applicant s cashflows, where economic shocks such as the increase of general unemployment will affect the overall probability of default on loans. Future investigations with real banking data are also suggested. Based on individual records, one could study the dynamics of cashflows and test for the structure of an empirical model that efficiently represents the dataset and, therefore, provides realistic values of default probabilities. Understanding and modeling how applicants react when faced with a low balance in this account is also suggested. A person is likely to hold its expenses once his/her balance passes a particular threshold. An investigation with real data could shed light regarding this effect. These and other ideas are left for future development of the BaR model. 15

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