Bayes' Theorem is a proven application of conditional probability rule which has been well received in the line of corporate finance and management of companies in evaluating operating and financial business decisions. The theorem provides the mechanics to allow the incorporation of a new piece of information or observation into the probability equation that derives the Posterior (or “updated”) Probability of occurrence of an event.
Bayes’ formula is expressed below.
(Exhibit A)
Through a simple example, this newsletter offers clarification on how the probability of occurrence of an event (previously calculated) can be modified, to incorporate a new information using Bayes’ Theorem.
Typically, it is not a pre-requisite for the occurrence of an event to be strongly correlated to the new information/observation in order to apply Bayes' Theorem. Assuming this – based on the historical records, Company A estimated the probability of default in payments by its customers (also known as prior probability) for next year as follows.
10% of its customers will likely default in payment next year.
90% of its customers will not default in in payment next year.
(Management has defined default as unsettled debt balance which is > 270 days past due)
Suppose: a month later, the internal credit control report revealed following information:
85% of these defaulted customers has overdue debt balance, x > 180 days
15% of them has debt balance, x ≤180 days
On the other hand, the prompt-paymasters have following debt profiles:
20% of them has overdue debt balance, x > 180 days
80% of them has debt balance, x ≤180 days
To visualize the permutations of scenarios, all relevant new information are presented in the respective branches in the Probability Tree diagram.
(Exhibit B)
Based on the formula in Exhibit A, the Posterior Probability of occurrence of Default Events conditional on customers having debt balances overdue > 180 day, can be calculated as follows.
(Exhibit C)
There, the probability of occurrence of an event of default, conditional upon on the new information that is customers having overdue debt balance beyond 180 days is 32% (as opposed to the earlier probability of default of 10% estimated using the historical data).
Let's breakdown the calculation into two permutations:
Permutation 1 - Yellow Line. (Exhibit D)
Permutation 2 - Blue line. (Exhibit E)
Bayes’ Theorem allows refinement & modification of management’s earlier estimate (ie prior probability of default) to reflect the Posterior Probability of default. The posterior probabilities resulting from Bayes’ Theorem are summarized in tables on the right in Exhibits D & E below:
(Exhibit D)
(Exhibit E)
The new piece of information on overdue debts is insightful as it presents the severity of the potential default situations which is now adjusted to 32% probability of occurrence (from prior probability of 10%, basing on historical actual default cases). One observation here is - in the case of prompt-paymasters, there still stands a 2% exposure whereby event of default will occur, given that 20% of these prompt-paying customers are likely to continue to record overdue debts > 180 days in next year. This could be an area where management would want to delve in and consider whether the current credit term would still be acceptable to this group of customers knowing that there is a 2% potential risk of default in payment. With the knowledge of customers' payment habits, management can then decide on whether or not to tighten Company A' s credit control procedures and monitor more closely, on the collectability of debts particularly of those customers with debts overdue > 180 days; and thereby more effectively manage the Company’s operating cash liquidity.
To sum up – Based on the historical data, management has already had in hands, the actual probability of default of customers (deemed as Event A) . Management may be interested to find out whether the new information on overdue debts (deemed as Event B) which comes in later, has a bearing with the existing default rate.
By applying Bayes' theorem, P ( Event A | Event B) is calculated as the posterior probability for its variable dependency on the new information, and in the example above - the Overdue Debt > 180 days. This probability notation, P ( Event A | Event B), assumes that the occurrence of Event A may likely be caused by the delay in payment by customers which presumably results in unsettled debt balances crossing 180 days threshold. Omission of Event B in the probability analysis will result in wrong decision being made such as: granting certain customers extended credit terms as opposed to imposing a more stringent one which may result in financial loss to Company A next year.
One may appreciate that it is with Bayesian Theorem which makes further examination and analysis of the potential impact of a new event possible. By revising previous probabilities using a pertinent forward-looking information, a substantially different perspective about a financial condition is presented for management's evaluation. This enables management to make a more comprehensive assessment and hence their action plans and resources can be redirected to areas where preventive measures are effectively implemented before a potentially serious operating/ financial issue crops up.
We assist business owners to review critical business processes of their companies with the objective to help improve the financial health of companies.
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