The Easter Bunny: A Diversifier of Risk

The Easter bunny named Alun is busy preparing for the upcoming holiday. It’s been a rough year for Alun due to inflation, prompting Alun to seek ways to save money while hiding eggs. He finds a great deal online for a wicker basket, ideal for distributing the eggs. However, the bargain seemed too good to be true, as the basket arrives in poor condition, potentially leading to a heartbreaking scenario of broken eggs and disappointed children. If all the eggs break, children will be left with none to discover, turning Easter into a catastrophe. Our resourceful bunny must devise a strategy to ensure some eggs remain intact. Luckily he is skilled with the art of mathematics.

Initially, Alun considers placing all eggs in a single basket, but realizes the risk of losing everything if the basket breaks. Understanding the probability of basket breakage (denoted as 'p'), he decides to diversify his approach. He wants to order multiple baskets of the same quality from the same website, intending to spread the risk across multiple baskets.

It's important to note that the analysis solely focuses on independent breaking chances. If for example the chance of breaking was dependent on a different factor, say whether it rains or not, this strategy does not work. Rainfall could potentially ruin all or none of the baskets, so we would need to diversify in another direction to mitigate these risks. This could be done by getting different kinds of baskets protected against the weather or diversify the risk of rain by placing all the eggs on different days. Unfortunately for Alun placing eggs throughout the year is not possible as easter is only celebrated on a particular day (see article RiskQuest - Easter article).

As mentioned, the probability of one basket breaking is p and therefore the chance of not all baskets breaking is 1-p. When considering 'n' independent baskets, the probability of all of them breaking is the product of the individual probabilities, which results in:

To get the probability of not all breaking, this is subtracted from one obtaining:

Let’s examine what happens when the amount of baskets heads towards infinity using the limit laws of calculus:

As a probability is by construction between 0 and 1, this leads to:

This was a pleasant surprise for the Easter bunny as this meant that with enough baskets, the probability of a catastrophe would become 0 and easter would be saved. However, he can’t buy unlimited wicker baskets, therefore he should allow some potential risk. He remembers that just as for banks, the Capital Requirements Regulation (CRR) of the European Union mandates the Easter bunny solvency for 999 out of 1000 years. In IRB modelling this is known as a Probability of Default (PD) of 0.01%, in Easter terminology it is referred to as the Probability of Broken Eggs (PBE). Thus, he accepts the risk of a disaster occurring once every 1000 years, equating to a PBE of 0.01% and ensuring 99.9% certainty that not all eggs will break. To determine how many baskets he would need for different probabilities of breakage, he plots the following graph.

As our Easter bunny expects the probability of breaking to be somewhere above 70%, he sees that he would have to buy at least 20 baskets to mitigate the risk of a catastrophic egg loss. However, as he thinks about the necessity of purchasing such a large quantity of baskets, he begins to consider the environmental implications of his decision. Each basket represents not only a financial cost but also an environmental cost, from the resources required to manufacture them to the energy consumed in their production and transportation. The prospect of purchasing 20 or more baskets raises concerns about the ecological footprint associated with his Easter preparations. Even though the Easter bunny is tempted to save money with a ‘good’ deal he found online, he's worried about choosing short-term financial gains over long-term environmental sustainability. By disregarding the environmental costs (even those in scope 3. See RiskQuest - ESG article) as a negative externality to the rest of the world, he contributes to environmental degradation and climate change.

Fulfilling his standing as a role model, Alun decides to invest a little extra to purchase a basket with only a 0.1% chance of breaking, embodying the wisdom of the Dutch phrase "goedkoop is duurkoop" (translation: Cheap is expensive). In doing so, he contributes to safeguarding an environment where future generations can continue to enjoy egg hunts with friends and family. The Easter bunny sends wishes for an eggcellent Easter egg hunt this year and for all the years ahead. Well...with no egg hunt for just 1 in 1000 years.