## 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 |

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.