Hurdle models of repayment behaviour in personal loan contracts

Abstract

This paper proposes a hurdle model of repayment behaviour in loans with fixed instalments. Using information on previous and current contracts, the approach yields a model of customer behaviour, useful, for example, in assessing the impact of determinants of default, a natural concern for credit and behavioural scoring. Under plausible assumptions, a debtor in each period faces a number of missed payments, which depends on his previous repayment decisions; meanwhile, as most debtors are expected to meet financial obligations, the number of missed payments is bound to display excess zeros, with reference to a single-part law. Each sequence of missed payments is modelled by using the binomial thinning, a conceptual tool that allows for dependence between integers by defining the support of consecutive counts. Under suitable assumptions on heterogeneity, the model can be produced under a random effects approach, leading to a two-part panel data model, estimable by quasi-maximum likelihood. The proposed approach is illustrated using a panel data set on personal loans granted by a Portuguese bank.

Appendix

This Appendix presents algebraic derivations of expressions for relevant moments of the marginal, joint and conditional distributions involved in the sequence $\left( {y_t ,t=1,\ldots ,T} \right)$.

Section 2.2 Expressions of the conditional moment $E( {y_t^k |y_{t-1} }),k\in {\varvec{\mathcal{N}}}$, and the first unconditional moment $E({y_t })$.

Proof

Consider the sequence of conditional moment generating functions (mgf), of $y_t$, given $y_{t-1}$, which, under the model’s assumptions and the definition of the binomial thinning operator can be written as

$$\begin{aligned} M_{\left. t \right| t-1} (s)\equiv & {} E_d \left( {E\left( {\exp \left( {sy_t } \right) |y_{t-1} ,d_t } \right) } \right) \nonumber \\= & {} p\exp \left( {s\left( {y_{t-1} +1} \right) } \right) +\left( {1-p} \right) \left( {1-p_1 +p_1 \exp (s)} \right) ^{y_{t-1} }. \end{aligned}$$

(17)

By evaluating derivatives of $M_{\left. t \right| t-1} \left( s \right)$ at $s=0$ one can produce conditional moments of any order: $M_{\left. t \right| t-1}^{(k)} \left( 0 \right) =E\left( {y_t^k |y_{t-1} } \right) ,k\in {\varvec{\mathcal{N}}}$. For $k=1$ and $k=2$, this yields, respectively, (4) and

$$\begin{aligned} E\left( {y_t^2 |y_{t-1} } \right)= & {} p+\left( {2p+p_1 \left( {1-p_1 } \right) \left( {1-p} \right) } \right) y_{t-1} \nonumber \\&+\left( {p+p_1^2 \left( {1-p} \right) } \right) y_{t-1}^2 . \end{aligned}$$

(18)

The unconditional first moment of $y_t , E({y_t})$, can then be obtained by successively applying the law of iterated expectations to $E\left( {y_t |y_{t-1} } \right)$; with the initial condition $y_0 \equiv 0$,

$$\begin{aligned} E\left( {y_t } \right)= & {} E\left( {E\left( {y_t |y_{t-1} } \right) } \right) =p+rE\left( {y_{t-1} } \right) \\= & {} p+rE\left( {E\left( {y_{t-1} |y_{t-2} } \right) } \right) =p\left( {1+r} \right) +r^{2}E\left( {y_{t-2} } \right) =\ldots \\= & {} p\left( {1+r+r^{2}+\ldots +r^{t-1}} \right) +r^{t}E\left( {y_0 } \right) =p\left( {1+r+r^{2}+\ldots +r^{t-1}} \right) . \end{aligned}$$

Trivially, if $p=1 (\Leftrightarrow r=1)$, then $E\left( {y_t } \right) =t$, and if $p=0$ then $E({y_t })=0$. In both cases $y_t$ is degenerate so $V({y_t })=0$. If p and $p_1$ are both less than 1 (so $r<1), E\left( {y_t } \right) =p\left( {1-r^{t}} \right) /\left( {1-r} \right)$. Aside from the trivial case $p=0, E\left( {y_t } \right)$ varies with t, so the sequence ${\varvec{y}}$ is not stationary. $\square$

Section 2.2 Expression of COV$({y_t ,y_{t-k} })$; autocorrelation function.

Proof

Through successive application of the law of iterated expectations to (4), one can write the first conditional moment of $y_t$, given $y_{t-k}$, as

$$\begin{aligned} E( {y_t |y_{t-k} } )=p( {1+r+r^{2}+\ldots +r^{k-1}} )+r^{k}y_{t-k} . \end{aligned}$$

(19)

From

$$\begin{aligned} {\hbox {COV}}( {y_t ,y_{t-k} } )= & {} E( {y_t ( {y_{t-k} -E( {y_{t-k} } )} )} )\\= & {} E( {E( {y_t |y_{t-k} } )( {y_{t-k} -E( {y_{t-k} } )} )} ) \end{aligned}$$

and from (19) it follows that

$$\begin{aligned} {\hbox {COV}}( {y_t ,y_{t-k} } )= & {} E( {( {p( {1+r+r^{2}+\ldots +r^{k-1}} )+r^{k}y_{t-k} } )( {y_{t-k} -E( {y_{t-k} } )} )} )\nonumber \\= & {} r^{k}E( {y_{t-k} ( {y_{t-k} -E( {y_{t-k} } )} )} )=r^{k}V( {y_{t-k} } ). \end{aligned}$$

(20)

In this expression $V({y_t})=E( {y_t^2 } )-E( {y_t } )^{2}$ with $E( {y_t^2 } )$ expressed as the solution to the difference equation (obtained from (18))

$$\begin{aligned} E( {y_t^2 } )= & {} E( {E( {y_t^2 |y_{t-1} } )} )\nonumber \\= & {} p+( {p+r( {1-p_1 } )} )E( {y_{t-1} } )+( {p+rp_1 } )E( {y_{t-1}^2 } ), \end{aligned}$$

with $E( {y_{t-1} } )$ given in (5) and initial condition $E( {y_1^2 } )=p$. The variance $V( {y_t } )$ involves t so the autocorrelation function, CORR$( {y_t ,y_{t-k} } )=r^{k}\sqrt{V( {y_{t-k} } )/V( {y_t } )}$, depends not only on only on lag (k) but also on t. $\square$

Section 2.2 General expression of $E({y_t|{\varvec{x}}})$ with time-varying covariates—${\varvec{x}}\equiv ( {{\varvec{x_1}} ,\ldots ,{\varvec{x_T}} } )$ does not include lags of the dependent variable.

Proof

Suppose that neither p or $p_1$ involve lags of the dependent variable. Then,

$$\begin{aligned} E( {y_1 |{\varvec{x}}_{\varvec{1}} } )= & {} p( {{\varvec{x_1}} } ),\\ E( {y_2 |{\varvec{x}}_{( \mathbf{2} )} } )= & {} E( {E( {y_2 |y_1 ,{\varvec{x}}_{( \mathbf{2} )} } )} )\\= & {} p( {{\varvec{x}}_\mathbf{2} } )+r( {{\varvec{x}}_\mathbf{2} } )E( {y_1 |{\varvec{x}}_\mathbf{1} } )=p( {{\varvec{x_2}} } )+r( {{\varvec{x_2}} } )p( {{\varvec{x_1}} } ),\\ E( {y_3 |{\varvec{x}}_{( \mathbf{3} )} } )= & {} E( {E( {y_3 |y_2 ,{\varvec{x}}_{( \mathbf{3} )} } )} )\\= & {} p({{\varvec{x}}_\mathbf{3} } )+ r( {{\varvec{x}}_\mathbf{3}} )E( {y_2 |{\varvec{x}}_{(\mathbf{2})} } )\\= & {} p( {{\varvec{x_3}} } )+r( {{\varvec{x_3}} } )( {p( {{\varvec{x_2}}} )+r( {{\varvec{x_2}} } )p( {{\varvec{x_1}} } )})\\= & {} p( {{\varvec{x}}_{\varvec{3}} } )+p( {{\varvec{x}}_\mathbf{2} } )r( {{\varvec{x}}_\mathbf{3} } )+ p( {{\varvec{x}}_\mathbf{1} } )r( {{\varvec{x}}_\mathbf{2} } )r( {{\varvec{x_3}} } ). \end{aligned}$$

A mathematical induction argument ensures the general result

$$\begin{aligned} E\left( {y_t |{\varvec{x}}} \right) =\left\{ {{\begin{array}{ll} {p\left( {{\varvec{x_1}}} \right) ,} &{} {t=1,} \\ {p_t +\mathop \sum \nolimits _{j=1}^{t-1} p_j \mathop \prod \nolimits _{k=j+1}^t r_k ,} &{} {t\ge 2,} \\ \end{array} }} \right. \end{aligned}$$

which reduces to (5) when all covariates (therefore p and $p_1 )$ are time invariant. $\square$

Section 3.4 Multiplicative unobserved effects.

Let ${\varvec{e}}_i \equiv \left( {e_{i1} ,\ldots ,e_{iT_i } } \right)$ and suppose that the mean of $y_{it}$ is affected by a multiplicative time-invariant effect, that is, $E\left( {y_{it} |{\varvec{e}}_{\varvec{i}} } \right) =e_i E\left( {y_{it} } \right)$ (for simplicity, time-invariant observable covariates are assumed so they are omitted). Hence, for $t=1$,

$$\begin{aligned} E\left( {y_{i1} |{\varvec{e}}_{\varvec{i}}} \right) =e_i E\left( {y_{i1} } \right) =e_i\,Pr \left( {d_{i1} =1} \right) =e_i p. \end{aligned}$$

For $t\ge 2$, conditionally on $d_{it} =0$ and $y_{i,t-1} , y_{it}$ follows a binomial p.f. with number of Bernoulli trials given by $y_{i,t-1}$, so any unobserved effect will intervene in the probability of success $p_1$: denote this as $p_1 \left( {e_{1it} } \right)$. Recalling (1), one can check that, for $t=2$, the term $e_{1i2}$ must verify the equation

$$\begin{aligned} E\left( {y_{i2} |e_i } \right)= & {} e_i E\left( {y_{i2} } \right) \Leftrightarrow \\ e_i p\left( {1+e_i p+p_1 \left( {e_{1i2} } \right) \left( {1-e_i p} \right) } \right)= & {} e_i p\left( {1+p+p_1 \left( {1-p} \right) } \right) \Leftrightarrow \\ e_i p+p_1 \left( {e_{1i2} } \right) \left( {1-e_i p} \right)= & {} p+p_1 \left( {1-p} \right) . \end{aligned}$$

This equation is formally different from those obtained for $t>2$, so, for each t, the roots $e_{1it}$ of the corresponding implicit equations vary with t. In words, an assumption of time-invariant heterogeneity affecting the conditional expectation of $y_{it}$ rests on a time-varying unobservable affecting $p_1$.