# Contents

• The motivation
• The model
• The mathematics

# Motivation

• I am a credit risk manager. At the start of the year, I have assessed the probabilities of my debtors defaulting in the course of the year (which I am rather naively assuming to be independent). At the end of the year, I have a list of those who have defaulted and those who have not and I would like to evaluate how good my probability assessments were, on average.
• I am a portfolio manager in an oil and gas exploration company. At the start of the year, my subsurface teams have assessed the probabilities that the exploration wells I drill during the year will discover oil. At the end of the year, I have a list of those that discovered oil and those that did not and I would like to evaluate how good my probability assessments were, on average.
• I am testing myself at assessing probabilities, so I have taken a calibration test where I have answered a list of questions and provided a probability that my answer to that question is correct. I would like to see whether, on average, I over- or under-estimate probability

# The model

We have a sequence of probabilities, each of which describes our belief in the likelihood that a given event will take place (default, discovery, etc.). The events are binary, so they either happen or they don’t.

1. You can hit it with a Monte Carlo sledge hammer
2. You can calculate the probabilities directly
3. You can use the Central limit theorem and approximate the distribution with a Normal distribution.

## Direct calculation

We call the outcomes that we count successes . This makes sense for oil discoveries, but is frankly a little odd for credit defaults. Nonetheless.

## Central limit theorem

If we think of each event as a random variable that takes a value 1 or 0 then the count is just the number we get adding all these random variables together.

# The tool

You can find a copy of the spreadsheet tool here.

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## More from Graeme Keith

Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.

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## Graeme Keith

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Mathematical modelling for business and the business of mathematical modelling. See stochastic.dk/articles for a categorized list of all my articles on medium.