XVIIIth Century Math, Chuck Norris and Updating Risks of Default

We have seen previously how machine learning and statistics could help predicting how much money a loan would pay back. Here we’ll use another branch of statistics, probabilities, to determine how likely a loan is to default knowing it has already made some payments.

Very Basic Notions

In probabilities, the chance or risk of an event A to occur is written \(P(A) \). \(P(A) \) is always between 0 an 1, where 0 means it will never occur and 1 that it is absolutely certain, and can be calculated as:

\[ P(A) = \frac{\text{Number of ways ‘A’ can happen}}{\text{Total number of possible outcomes}} \].

For instance, the probability to get a ‘Head’ when flipping a coin (provided the coin is fair) is:

\[ P(H) = \frac{\text{Head}}{\text{Head or Tail}} = \frac{1}{2} = 0.5 \].

Things get more interesting when we combine the probabilities of multiple events. For instance, the probability of drawing two heads in a row is:

\[ P(H) \times P(H) = 0.25 \].

While the probability of tossing Head at least once when doing 2 throws is equal to the probability of tossing Head at the first throw, plus the probability of tossing head at the second throw when the first one was Tail, or:

\[ P(H) + P(H) \times P(T) = 0.5 + 0.25 = 0.75 \].

But this is only valid if the two events are independent. The probability of a fair coin to fall on ‘head’ is not influenced by the previous toss (which is somehow counter-intuitive: if you toss a fair coin and get ‘Head’ 99 times in a row, most would believe that is time for ‘Tail’. It’s not, at least not more that during the very first throw). In real life, many events are dependent upon each others.

Chuck Who?

Let’s consider \(P(A) \) your probability to be in awe of someone your encounter in general to be 0.05, and\( P(C) \) your probability of bumping into Chuck Norris to be .000000003 (1 out of 320 million people in the U.S).

Your probability to bump into Chuck Norris and to be in awe is NOT the probability of bumping into Chuck Norris times the probability of being in awe in general because… well, he’s Chuck Norris, so anybody would be utterly impressed.

In probabilities, \(P(A|C) \) means the probability of event A knowing that C happens. Here, \(P(A|C) \) means the probability to be in awe knowing you bumped into Chuck Norris, which we’ll put very conservatively at only 0.99 [1].

Therefore your probability of bumping into Chuck Norris and being in awe is:

\[ P(A|C) \times P(C) = 0.99 \times 0.000000003 \]

England a long time ago

The probability for being in front of Chuck Norris and in awe being the same that the probability of being in awe and in front of Chuck Norris, we can write:

\[P(A|C) \times P(C) = P(C|A) \times P(A) \]

And therefore, the probability of being in front of Chuck Norris knowing that you’re in awe can be written:

\[ P(C|A) = \frac{P(A|C) \times P(C)}{P(A)} \]

This is known as Bayes’ Theorem, from Thomas Bayes , an English Presbyterian minister who discovered it in the XVIIIth century[2].

Here, we have \(P(C|A) = 0.99 \times 0.000000003 \div 0.05 = 0.000000059 \). Which makes even sense intuitively: as you’re difficult to impress (only 5% chance), the fact that you’re in awe right now significantly increases the probability that you’re in front of Chuck Norris himself.

Baye’s Theorem has countless uses in medicine, finance or physics. Its main interest is that it allows to update the probability of an event as new evidence is acquired, a process named Bayesian inference.

Back to the XXIst Century

A central issue when lending money is to estimate the probability that a borrower will fail to make all the due payments, an event known as a loan defaulting. As more and more payments are made, that probability shall decrease. After all, when the very last payment is made, the default probability decreases to 0.

If D is the event of a loan defaulting, and N the event of a loan to have made at least N payments already, then Bayesian inference allows use to calculate \( P(D|N) \) the updated probability that it will default knowing it has already made N payments, as:

\[ P(D|N) = \frac{P(N|D) \times P(D)}{P(N)} \]

Luckily, all the terms on the right side of the equation are relatively easy to measure.
For instance, we analyze the historical data of 218,480 Prosper loans. Amongst those, 31,924 are 36-months loans that were issued more than 3 years ago, and are therefore old enough to have reached maturity.

Let’s consider the loans with the grade ‘D’. 1,551 of them have defaulted, out of a total of 6,754, which gives an overall probability of default of:

\[ P(D) = 1551 \div 6754 = 0.2296 \]

Nota Bene: This average default rate may seem high, but one must consider that ‘D’ is a risky grade, and that defaulting loans usually make many payments before defaulting, therefore reducing the losses. Thanks to interest rates around 22% on average, even with such a high probability of default, net annual returns for investors exceeded 7.17%.

Prosper’s data do not tell how many payments each of the loans has made. But the data show for each loan how much was paid back. Knowing the loan amount \( A \), yearly interest rate \( r \) and term in months \( T \), we can calculate the monthly installment \( p \) for each loan, using the formula:

\[ p = A \times \frac{\frac{r}{12}}{1 – (1 + \frac{r}{12)})^-T} \]

Once we calculated the monthly installment, we can estimate the number of monthly payments made by a loan by dividing the total amount paid by the monthly installment.

By counting the proportion of loans that made at least N payments, both for all the 6,754 grade D loans, and for the 1,551 loans of them that we know will default, we obtain respectively \( P(N) \) and \( P(N|D) \):

Month P(N) P(N|D)
1 0.990820254664 0.960025789813
2 0.982973053006 0.925854287556
3 0.975273911756 0.892327530625
4 0.966834468463 0.855577047066
5 0.957506662718 0.814958091554
15 0.873852531833 0.450676982592
16 0.867337873853 0.422308188266
17 0.860823215872 0.393939393939
18 0.85667752443 0.375886524823
34 0.780574474386 0.0522243713733
35 0.773319514362 0.0348162475822

This kind of information is enough to update the probability of default of a loan knowing it has made N payments already. For instance, the probability of default for a 36-months, grade D loan that has made already 17 payments is:

\[ P(D|17) = \frac{P(17|D) \times P(D)}{P(17)} = \frac{0.3939\times 0.2296}{0.8608} = 10.50\% \]

This simple formula allows to make two interesting calculations. First, to estimate the probability of default of ongoing loans in an investor’s portfolio. Second, by comparing historical default rates and updated rates, it shows how healthy a marketplace is. For instance, by applying this Bayesian inference on Prosper’s 12,270 non-mature loans with the same grade ‘D’, we can calculate than on average, the default rate is now down to 15.9%, indicating significant improvements from the platform; all the more impressive since, as mentioned previously, it was already return over 7% per year for that grade.

Thomas Bayes and Chuck Norris should invest in Marketplace Lending. Although it’s probably too late for one of them.

  1. Because it’s Chuck Norris, you would expect that one probability to greater than 1, but we discounted it because he’s more than 75 years old already.  ↩
  2. He also wrote a book titled ‘An Introduction to the Doctrine of Fluxions’, which by itself brings him at a Chuck Norris level of awesomeness.  ↩


  1. Andy Hodges says:

    Very entertaining article. Nice mix of math and Chuck Norris.

  2. will says:

    great info, keep it coming

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