A debt behaviour model

The model concerns the following random variables:

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  A discrete Markov process $B_{t}$ which records the behavioural state of the debtor during the time period $t$ - measured in months. The state is measured in the middle of each month.

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  A discrete-valued process $T_{t}$ which records the strongest debt management intervention that was applied to the debtor during the time period $t$.

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  $R$ an entity-specific variable, $R$ gives the final result of the debtor’s most immediate previous debt case - NA, paid in full, liquidation/bankrupty, full write-off, partial write-off.

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  $X_{t}$ is the economic state at time period $t$. This measure is obtained through clustering a pertinent collection of economic variables: change in CPI, change in unemployment, change in the average weekly wage, etc. The underlying variables for $X_{t}$ are varying quarterly, so $X_{t}$ will be constant in blocks of three months.

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  $S_{t}$ is a latent discrete Markov process which categorizes debtors in a time period into the behavioural scheme that governs the generation of $B_{t}$. The model supposes that $T_{t-1}$ influences $S_{t}$, and hence influences $B_{t}$ indirectly.

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  $D_{t}$ is a positive real-valued variable, given by

  $$D_{t}=\frac{\text{Debt amount at time $t$, including penalties and interest}}{\text{Largest amount of debt owed up to time $t$, excluding penalties and interest}}$$

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  $Y_{t}$ is a categorization of $D_{t}$ into $\{0,1\}$ - this is governed by a parameter $\alpha$ that needs to be inferred. the notion is that as a debtor gets closer to being paid in full, its probability of making a large lump-sum payment to clear its debt may change.

We introduce a set of parameters as follows:

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  $\alpha$: defined by $Y_{t}:=0$ if and only if $D_{t}\leq\alpha$.

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  $Q_{S}$: a list of transition matrices, one for each combination of values of $R,X_{t},T_{t-1}$.

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  $\pi_{S}$: a list of initial probabilities, one for each combination of values of $R,X_{t}$.

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  $Q_{B}$: a list of transition matrices, one for each combination of values of $Y_{t-1}$ and $S_{t}$.

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  $\pi_{B}$: a list of initial probabilities, one for each value of $S_{1}$.

Figure 2 depicts the causal structure of the variables and the parameters - we have now expressed each of the variables as a vector of length as long as the number of observation periods.

Every debt case begins at a time period $u$ and ends at a time period $l$. If the debt case is indexed by $i$, the the beginning is $u_{i}$ and the end is $l_{i}$. There will be observations of $T_{t}$, $B_{t}$, $D_{t}$, and $X_{t}$ from $u_{i}$ through to $l_{i}$.

The log-likelihood of observing a single debt case is maximized when we maximize:

$$l_{0}=\sum_{t=u+1}^{t=l}(\ln(Q_{B}^{Y_{t-1},S_{t}}(B_{t-1},B_{t}))+\ln(Q_{S}^{X_{t},R,T_{t-1}}(S_{t-1},S_{t})))+\ln(\pi^{S_{u}}_{B}(B_{u}))+\ln(\pi_{S}^{X_{u},R}(S_{u}))$$

We apply the EM algorithm to $l_{0}$, taking the expected value of $l_{0}$ conditional on $\{B_{t},X_{t},D_{t},T_{t},R\}$ and the $k$-th iteration of the parameters $\{\alpha,Q_{B},Q_{S},\pi_{B},\pi_{S}\}$, $\Theta^{k}$.

For this we define the responsibilities for each debt case, $i$, and time $t$, $t=u_{i},\ldots,l_{i}$:

$$\gamma_{i,t}(s):=p(S_{t}=s|T_{u_{i}}^{l_{i}-1},X_{u_{i}}^{l_{i}},B_{u_{i}}^{l_{i}},R_{i},D_{u_{i}}^{l_{i}-1})$$

for $t\geq u_{i}$; and for $t>u_{i}$,

$$\Gamma_{i,t}(p,q):=p(S_{t}=q,S_{t-1}=p|T_{u_{i}}^{l_{i}-1},B_{u_{i}}^{l_{i}},R_{i},D_{u_{i}}^{l_{i}-1})$$

It is clear that $\gamma_{i,t}(s)=\sum_{p}\Gamma_{i,t}(p,s)$, or if $t=u_{i}$, $\gamma_{i,u_{i}}(s)=\sum_{q}\Gamma_{i,u_{i}+1}(s,q)$ - hence we need only compute $\Gamma_{i,t}$.

This is done using the Forward-Backward algorithm: