I worked for a few years as life actuary. They study syllabus was a bit different then vs. as I vaguely understand it now. In that era, one studied math courses to become an associate of the Society of Actuaries. Then you studied the actual machinations of the insurance business to become a Fellow.
Passing exams was a key factor in your pay scale which was a plus or a minus depending on results. But for sure, one's career track is heavily dependent on these results.
But, in short, I can tell you that I never used one iota of the math covered on the exams. They are more of a training ground to come up with fast ways to solve problems. The higher level exams require a good deal or practice and are high-pressure considering what is at stake. As an example of the intensity, I got a 10 (out of 10) on the risk theory exam (considered a tough one) answering 7 out of 15 questions - and I yet may have gotten some wrong.
The major function of actuaries was to calculate and substantiate the adequacy of reserves. This relied heavily on mortality tables - an easy math problem. A significant aspect of the endeavor entailed accounting regulations both for GAAP and state of domicile statutory requirements.
Most of the actual work in this regard was modeling of various policy lines. And each company has it's own set of well- worn models.
A latter consideration was the requirement of cash flow testing. This again was a modeling problem as the investment assets had to be tested for their performance under various, usually extreme, interest rate scenarios. This was intended to make sure there is money available to fund the reserves.
So while passing exams entailed cubic splines, the Poisson distribution, the Black-Scholes model, Ito's Lemma, the real work required familiarity with insurance regulations and their related accounting treatment.
For more information on the study materials, esp. books, you can go to the websites of the Society of Actuaries and the Casualty Actuarial Society.
Group theory may be viewed roughly as a general study of symmetry. In chemistry this applies to crystals via the study of crystallographic groups, and in art via wallpaper groups. For an example in physics, the Lie symmetry groups of partial differential equations play fundamental roles, e.g, governing conservation laws and separation of variables. See for example Weyl's Symmetry and Budden's Fascination of Groups.
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