## Modelling Cashflow and Bad Debt

A trader extends credit to his customers. The credit extended falls into four categories - paid, bad debt, owing - if within a thirty day period - or overdue - if outside the period, up to thirty more days. Any money owing after this period becomes a bad dent. At some point there is £4,500 owing and £2,000 overdue. How much of this will turn into real. paid cash for the business?
The table of transition probabilities is - from experience:
 From\To Paid Bad Debt Owing Overdue Paid 1 0 0 0 Bad Debt 0 1 0 0 Owing 0.5 0 0.3 0.2 Overdue 0.4 0.3 0.2 0.1
This means for example that there is a probability of 0.5 that a debt owing will be paid before it is overdue.
The transition matrix is
$A= \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0.5 & 0 & 0.3 & 0.2 \\ 0.4 & 0.3 & 0.2 & 0.1 \end{array} \right)$

We can write this as
$A= \left( \begin{array}{cc} I & O \\ K & M \end{array} \right)$
where
$I= \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right), \; O= \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right), \; K= \left( \begin{array}{cc} 0.5 & 0 \\ 0.4 & 0.3 \end{array} \right), \; M= \left( \begin{array}{cc} 0.3 & 0.2 \\ 0.2 & 0.1 \end{array} \right)$

Then
$(I-M)^{-1}K= \left( \begin{array}{cc} 0.8983 & 0.1017 \\ 0.6441 & 0.3559 \end{array} \right)$
.
This is a matrix of transition probabilities from
$(Owing, Overdue)$
to
$(Paid, Bad \; Debts)$
.
$(4500 \; 2000) \left( \begin{array}{cc} 0.8983 & 0.1017 \\ 0.6441 & 0.3559 \end{array} \right)=(5330.55 \; 1169.45)$
.
It is predicted that £5330.55 will be paid and £1169.45 will turn into bad debts.