Writing Practice Problems for Self-Study
How creating your own practice problems deepens understanding of actuarial exam material.
Why Write Your Own Problems
Creating practice problems requires a deeper understanding than solving them. You must identify what concepts to test, construct a solvable scenario, calculate the correct answer, and anticipate plausible wrong answers. This process activates a completely different mode of thinking than solving problems written by someone else. When you solve an existing problem, you are working within a framework someone else designed. When you write a problem, you must understand the material well enough to design that framework yourself.
This process reveals gaps in your understanding that passive studying misses. You might think you understand annuity-due calculations until you try to write a problem that tests them and realize you are unsure how to set up a scenario where the payment timing genuinely matters. It also develops the ability to recognize problem structures, which helps on exam day when you need to quickly identify which technique to apply.
Starting with Modifications
Start by modifying existing problems: change the numbers, add a twist, or combine elements from two different problems. This is the easiest entry point because the problem structure is already provided. Take a problem from your study manual that involves finding the present value of an annuity with level payments, and modify it so that payments increase by a fixed amount each period, turning it into an increasing annuity problem.
Other effective modifications include changing the interest rate type (from an annual effective rate to a nominal rate compounded monthly), adding an additional condition (such as a deferred start or a period certain), or switching the unknown variable (if the original problem solves for the present value, rewrite it to solve for the payment amount or the interest rate instead). Each modification forces you to rethink the solution approach and deepens your understanding of how the variables interact.
Creating Original Scenarios
Progress to creating original scenarios based on real-world situations. For Exam FM, you might design a problem around a car loan with a balloon payment, a retirement savings plan with increasing contributions, or a bond with an unusual call feature. For Exam P, you could create a scenario involving conditional probability in a medical testing context, or a problem about the distribution of insurance claims in a given month.
Drawing from real-world contexts makes your problems more interesting and helps you see how abstract mathematical concepts apply to practical situations. This perspective is valuable not only for the exam but also for your future actuarial career, where you will constantly translate real-world situations into mathematical models.
Designing Problems at the Right Difficulty Level
Write problems at the difficulty level you want to practice, not just easy ones. Exam problems typically require combining two or three concepts within a single question, so your practice problems should do the same. A good exam-level problem for Exam FM might combine time value of money with loan amortization and ask for the outstanding balance after a specific payment, where the interest rate changes partway through the loan.
To calibrate difficulty, study the structure of actual SOA or CAS sample problems. Notice how they typically present a scenario with several given pieces of information, require you to identify the relevant technique, set up equations, and solve for a specific quantity. Your problems should follow a similar pattern: enough information to determine a unique answer, a clear question, and a solution that requires more than a single formula application.
Writing Complete Solutions
Include your solution and explain the reasoning behind each step. A complete solution should walk through the problem as if you were teaching someone else. State which formula or technique you are using and why, show all algebraic steps, and verify your answer using an alternative method when possible. For example, if you solve for a present value using the annuity formula, verify it by building a payment schedule and summing the discounted payments individually.
Writing detailed solutions also serves as a form of the Feynman technique. If you cannot explain each step clearly and logically, you do not fully understand the material. The places where your explanation becomes vague or hand-wavy are exactly the concepts you need to study further.
Exchanging Problems with Study Partners
Exchange problems with study partners for mutual benefit. Having someone else attempt your problem provides valuable feedback: if they misinterpret the question, it may be poorly worded; if they get stuck at a specific step, that step may require additional information or context that you forgot to include. Solving problems written by your study partners exposes you to different perspectives and problem structures that you would not have created yourself.
If you do not have a local study partner, online actuarial communities and forums are excellent places to share and exchange practice problems. Posting a well-crafted original problem and its solution contributes to the community while also giving you feedback on your problem-writing skills. Some candidates organize weekly problem exchanges where each member contributes one original problem to a shared set.
How Many Problems to Write
Even writing five original problems per topic forces a level of engagement that solving 20 existing problems may not achieve, making it a high-value study activity. A realistic goal is to write two to three original problems per week, focusing on the topics you are currently studying. By the end of a three-month study period, you will have accumulated 25 to 35 original problems that you can use as a personalized practice exam.
Quality matters more than quantity. A single well-designed problem that combines multiple concepts and has a thorough solution is more valuable than five trivial problems that each test only one formula. Invest your problem-writing time in creating the kind of challenging, multi-step problems that will actually appear on the exam.