# Games of Chance

I worked for a Property & Casualty insurer in Seattle. It was my first year out of college. My desk covered with exam problems. My stress level low, still 7 weeks before my exam. My boss walks up. He asks what I'm planning for vacation? That morning I emailed him asking for next Thursday and Friday off. I replied - "flying to Vegas for a golf trip. Do you ever play 21?" I'll never forget his response, very actuary of him, "I don't play games of chance."

His response immediately made me feel self-conscious. His language, so elegant, so professional. It gave me the impression that just asking was a mistake. As if I should already known his answer. Of course, "real actuaries" don't gamble. I remember thinking, I know the expectation is negative, from the exams of course, but alcoholic beverages were free if I was gambling. I could keep my losses to the low negative expectation I figured. Probably break even compared to a night out with friends (my numeraire at the time).

Why did I ask in the first place? My reason was simple. Pure curiosity of how a probability professional gambles. At the time, our job was setting rates and reserves for auto, home, etc... based on frequency and severity. In other words, weren't we the preeminent experts in probability? Than why the cold response? Gambling games were a serious specialization in mathematics, right? Wouldn't actuaries be interested in the probability-based logic? Answers to these questions didn't come for many years.

Years later, as an investment actuary in California, I was sitting in a large conference room. Bright fluorescent lights reflecting off a long shiny conference table. On each side of me were black empty chairs. The chairs across the table were filled. Insurance executives, division heads, corporate risk managers, all the top people, actuaries included. It was my turn to explain the new Product Approval Guidelines I’d been working on for months. Our Chief Actuary had been fired following the financial crisis of 2007-2008 and the company was sitting on billions of dollars of toxic liabilities that had been grossly miss-priced. From what I could tell, prior the crisis, the actuaries had priced the VA Equity Guarantees using average/expected future equity market scenarios. Since their models hadn't generated any claims after adding the guarantees (i.e. they appeared to cost the company nothing), someone determined that 0.10% was an appropriate charge. In the conference room that day, my results indicated the cost was at least 10 times that amount. It gets better though, the cost wasn't even stable, it would vary over time and depend on prevailing implied volatility and interest rates.

What I presented that day is commonly referred to as risk-neutral pricing. Common knowledge for practicing actuaries since the financial crisis. My objective here isn't to explain risk-neutral methodologies. Although I believe many actuaries, even today, would better understand Arabic. The objective is to explore exactly what "probability" all of us are using daily in our jobs. More precisely, confusing conflicting definitions of probability is problematic for actuaries and leads to confusion of risk-neutral in the real-world. And maybe, the confusion results from the history of the profession. Lastly, I hope to begin a dialogue that improves the evolutionary process of how actuaries manage financial risk going forward.

Probability, by definition, is the measure of an event's likelihood. As a mathematical concept, and on actuarial exams, the measurement is defined in two distinct and sometimes inconsistent ways. There is the frequency ("Frequency") definition based on how frequently something occurs, while the other is based on one’s degree of belief ("Belief"). The syllabus contains both conflicting definitions. Frequency for insurance claims and belief (or Bayesian) for credibility analysis. More recently, belief as a measure of probability is being increasing used along with the rise of mathematical finance in the actuarial curriculum since the financial crisis.

It's important to note that whether frequency or belief is used to measure probability, both can be consistent with the following three axiom's:

1. P(AllEvents) =1

2. 0<= P(Event_i) <=1 for all i

3. P(any union of independent events) =P(Event_1) + P(Event_2) + ... + P(Event_n) if mutually exclusive

Since the real-world is far more complex than flipping “fair coins” and rolling “fair dice”, actuaries need to grasp the meaning, applications, and limitations of each measure to become better risk managers. Some may consider these distinctions trivial. However, I know from practical experience that misunderstanding these can lead to financial disasters. I remember an actuary once telling me the belief measure was kind of flimsy, and that frequency as a measure was more legitimate, or at least he said something like this. I replied, "you mean like a 50/50-coin flip, right?" Consider this, coins aren’t even 50/50. Text books always have the adjective "fair" before coin for a reason. The frequency probability of flipping a coin is never truly 50/50. The “fair coin flip” is only theoretical.

discuss in "Probability Interpretations" is {Frequency & Not-Belief}, {Not-Frequency & Belief}, {Frequency & Belief}, and {Not-Frequency & Not-Belief}.