The core mathematics of the actuarial profession: interest theory, life contingencies, loss models, credibility and ratemaking, plus how to plan the SOA and CAS exams.

Actuarial Mathematics

Choose your first exam (P or FM). Write one sentence on why: for example, which one is closer to what you studied most recently.

Open the official exam page on soa.org and find the current syllabus or learning objectives. List the main topic areas and the weight or share of the exam each one has, if the page gives it.

On the same page, note the exam format and length, the approved calculators and how to register. Write down the date you checked.

Rate yourself 1 to 5 on each topic area (5 = could teach it). Mark the two weakest areas.

Pick a target exam window at least 10 to 14 weeks away. Count the weeks and plan about 200 to 300 study hours in total; work out the hours per week this needs and check it fits your life.

Make a weekly calendar or spreadsheet: one or two topic areas per week, weakest areas early, and the last three to four weeks reserved for full timed practice exams and review.

Choose your practice material: the official sample or practice questions linked from the SOA exam page if available, plus any study manual or free resource you can access. Record which ones.

Do one timed set of 10 problems now, before you study, and record your score and time. You will compare against it later.

Look up the SOA VEE page and note which VEE subjects you could earn through courses you have taken or will take.

## Why it matters

In the US and Canada, "actuary" is a career you qualify into by passing a long series of professional exams while you work. Employers hire entry-level analysts who have passed one or two exams, pay for further study time, and raise salaries as exams are passed. The exams take most people several years to finish, so planning the first two well matters more than any single lesson in this course. This lesson explains the structure and has you write a study plan.

**Requirements change often.** The organisations below update exam names, syllabi, fees and pathways regularly. Read the official pages for the current rules; this lesson only gives the overall shape. Links checked on 2026-10-23.

## The two US actuarial societies

- **Society of Actuaries (SOA)**: life insurance, health, retirement and pensions, investments, and general corporate finance. Credentials: **ASA** (Associate) and then **FSA** (Fellow); also CERA for enterprise risk management. Official requirements: [SOA exam requirements](https://www.soa.org/education/exam-req/default/) and [ASA requirements](https://www.soa.org/education/exam-req/edu-asa-req/). Checked on 2026-10-23.
- **Casualty Actuarial Society (CAS)**: property and casualty insurance (auto, home, workers' compensation, liability, reinsurance). Credentials: **ACAS** and then **FCAS**. Official information: [CAS exams](https://www.casact.org/exams-admissions/exams). Checked on 2026-10-23.

In Canada the CIA (Canadian Institute of Actuaries) grants FCIA, largely recognising SOA and CAS exams. In the UK and many Commonwealth countries the IFoA (Institute and Faculty of Actuaries) runs its own exams.

## The first exams: P and FM

Nearly everyone starts with two multiple-choice, computer-based exams that the SOA and CAS share:

- **Exam P (Probability)**: probability rules, discrete and continuous distributions, multivariate distributions, and insurance applications. It uses the content of the Probability & Statistical Inference course. [Exam P page](https://www.soa.org/education/exam-req/edu-exam-p-detail/). Checked on 2026-10-23.
- **Exam FM (Financial Mathematics)**: interest theory, annuities, loans, bonds, duration and immunization: module 1 of this course, in more depth. [Exam FM page](https://www.soa.org/education/exam-req/edu-exam-fm-detail/). Checked on 2026-10-23.

Both rely heavily on calculus (integration for continuous distributions, derivatives for duration) and on fast, accurate arithmetic with an approved calculator. Read the exam page for the approved calculator list.

## After P and FM

The later path depends on the society and changes over time. In broad terms:

- More exams on actuarial models (long-term and short-term), statistics for risk modelling and predictive analytics.
- **VEE** (Validation by Educational Experience) credits, earned through approved university courses or other approved options in subjects such as economics, accounting and finance, and mathematical statistics. See [SOA VEE](https://www.soa.org/education/exam-req/edu-vee/). Checked on 2026-10-23.
- E-learning modules and a professionalism course (the CAS runs a [Course on Professionalism](https://www.casact.org/exams-admissions/course-professionalism); the SOA has its own). Checked on 2026-10-23.
- Fellowship: specialised exams and modules in your practice area.

## How people actually prepare

- **Study hours.** Candidates commonly report studying a few hundred hours for each preliminary exam, spread over two to four months. Plan for that, not for a weekend of cramming.
- **Practice problems are the method.** Read the theory once, then spend most of your time on timed problem sets. The SOA publishes the syllabus and learning objectives; third-party study manuals and online seminars (paid) add large problem banks.
- **Calculator fluency** saves minutes per question: practise with the exact model you will use in the exam.
- **Pass rates** are published by the societies. They show that many people fail at least once; treat a fail as information, not a verdict.
- **Employer support.** Actuarial employers usually provide paid study hours and cover exam fees for analysts. Internships are a common way in; ask about the study programme in interviews.

## Common mistakes

- **Starting without the official syllabus.** Third-party material may be out of date; the syllabus defines what is tested.
- **Too little time.** Booking an exam date first and studying for whatever time remains rarely works.
- **Passive study.** Rereading notes feels productive but does not build speed. Timed practice does.
- **Ignoring VEE and other requirements** until late, when they could have been earned in a degree.

## Quick recap

- SOA (life, health, retirement, finance) leads to ASA/FSA; CAS (property and casualty) to ACAS/FCAS.
- Exams P and FM come first and are shared; then more exams, VEE credits, modules and professionalism.
- Requirements change: always confirm on soa.org and casact.org.

## Try it

The **Do** tab walks you through writing a dated study plan for Exam P or FM using the official pages.


Understand how actuaries qualify through SOA or CAS exams and build a realistic study plan for Exam P and FM.

SOA and CAS exam paths

First Steps to Become an Actuary

Exam Study Tips

Create a sheet with three tabs: Inputs, Calc and Notes. On Inputs, enter the triangle from the lesson: accident years 2022 to 2025 down column A, ages 12, 24, 36, 48 across row 1, cumulative paid amounts in B2:E5 (leave the unknown cells empty). Add a labelled cell for the tail factor and set it to 1.

On Calc, compute the three volume-weighted age-to-age factors with SUM formulas that refer to the Inputs tab. Check that you get 1.5000, 1.1000 and 1.0303.

Compute the cumulative development factors with PRODUCT and the tail factor cell. Check CDF at 12 months = 1.7000.

For each accident year, pick up the latest diagonal value (paid to date), multiply by its CDF to get the ultimate, and subtract to get the unpaid amount.

Add a total row and a check cell: =IF(ABS(total_ultimate - total_paid - total_unpaid) < 0.01, "OK", "ERROR"). Confirm the total unpaid is 1,200 and the check says OK.

Sensitivity test: change the tail factor to 1.02 and note the new total unpaid. Then set it back to 1.

Make sure no formula contains a typed-in factor (search the Calc tab for numbers inside formulas), shade the input cells, and protect or lock the formula cells.

On Notes, write the data source ("teaching example"), the date, the method (volume-weighted chain ladder, tail 1.0) and one limitation of the method.

## Why it matters

Actuaries use Python, R and SQL more every year, but spreadsheets are still everywhere: rate indications, reserve reviews, pension valuations and quick checks are often built, reviewed and handed over in Excel. A reviewer must be able to follow every number. This lesson uses the most famous actuarial spreadsheet of all, the **loss development triangle** and the **chain ladder** method, to show how to build a clear, checkable model. It connects directly to the P&C reserves from module 2.

## The loss development triangle

Claims take time to settle. A car accident in 2025 might be paid partly in 2025, more in 2026 and the last of it in 2028. Arrange cumulative paid losses by **accident year** (rows) and **development age** (columns, in months since the start of the accident year):

| accident year | 12 | 24 | 36 | 48 |
|---|---|---|---|---|
| 2022 | 1,000 | 1,500 | 1,650 | 1,700 |
| 2023 | 1,100 | 1,700 | 1,870 | |
| 2024 | 1,200 | 1,750 | | |
| 2025 | 1,300 | | | |

(Amounts in thousands.) Each diagonal is a calendar year. The bottom-right is empty because those ages have not happened yet. The reserving question is: what will each row grow to (its **ultimate** loss)?

## The chain ladder method

**1. Age-to-age factors.** For each pair of ages, divide the sum of the later column by the sum of the earlier column, using only rows that have both (the **volume-weighted** average):

```
12-24: (1,500 + 1,700 + 1,750) / (1,000 + 1,100 + 1,200) = 4,950 / 3,300 = 1.5000
24-36: (1,650 + 1,870) / (1,500 + 1,700)                 = 3,520 / 3,200 = 1.1000
36-48: 1,700 / 1,650                                      =                 1.0303
```

**2. Cumulative development factors (CDFs).** Multiply the factors from each age to the end. We assume no development after 48 months (a **tail factor** of 1.0; real reviews estimate one):

```
CDF 48 = 1.0000
CDF 36 = 1.0303
CDF 24 = 1.1000 * 1.0303 = 1.1333
CDF 12 = 1.5000 * 1.1333 = 1.7000
```

**3. Ultimates and reserves.** Multiply each row's latest value by its CDF. The **unpaid** (reserve) is ultimate minus paid to date:

| accident year | paid to date | CDF | ultimate | unpaid |
|---|---|---|---|---|
| 2022 | 1,700 | 1.0000 | 1,700.00 | 0.00 |
| 2023 | 1,870 | 1.0303 | 1,926.67 | 56.67 |
| 2024 | 1,750 | 1.1333 | 1,983.33 | 233.33 |
| 2025 | 1,300 | 1.7000 | 2,210.00 | 910.00 |
| total | 6,620 | | 7,820.00 | 1,200.00 |

The insurer needs about 1.2 million held for these claims. Most of it is for the most recent year, whose estimate is also the least certain: a small change in the 12-24 factor moves it a lot. Methods such as **Bornhuetter-Ferguson** (which blends the chain ladder with an expected loss ratio, a credibility idea) are used to stabilise recent years.

## Spreadsheet formulas

With the triangle in `B2:E5` (rows = years, columns = ages):

```
12-24 factor (B8):   =SUM(C2:C4)/SUM(B2:B4)
24-36 factor (C8):   =SUM(D2:D3)/SUM(C2:C3)
36-48 factor (D8):   =E2/D2
CDF at 12 (B9):      =PRODUCT(B8:D8)*tail
```

Put every assumption (tail factor, any selected factor that overrides the average) in its own clearly labelled **input cell**, never typed inside a formula.

## Good spreadsheet practice

- **Separate inputs, calculations and outputs**, ideally on different tabs, with inputs shaded one colour.
- **No hard-coded numbers in formulas.** `=C2*1.0303` hides an assumption; `=C2*$D$9` points to it.
- **Check totals.** Add a reconciliation: total ultimate minus total paid must equal total unpaid. Show a check cell that says OK or ERROR.
- **Document.** A notes tab with the data source, date, method and who reviewed it. ASOP 41 (actuarial communications) expects that another actuary could follow your work.
- **Version and lock.** Save dated versions; protect formula cells so they are not overwritten.
- **Know when to move to code.** Large triangles, many segments or stochastic methods (bootstrapping, Mack's model) are easier to reproduce in Python or R, such as the open-source `chainladder` Python package.

## Common mistakes

- **Averaging the individual ratios** with a simple mean when the volume-weighted average was intended (or the reverse), without saying which.
- **Using rows that lack the later age** in a factor's denominator.
- **Forgetting the tail.** Long-tailed lines (liability, workers' compensation) develop for many years.
- **Mixing paid and incurred** data in one triangle.

## Quick recap

- Triangles arrange cumulative losses by accident year and development age.
- Chain ladder: age-to-age factors, then CDFs, then ultimate = latest x CDF, unpaid = ultimate - paid.
- Build spreadsheets with separate inputs, no hard-coded numbers, check cells and notes.

## Try it

The **Do** tab walks you through building this triangle and reserve estimate in a free spreadsheet, with a check that the total unpaid is 1,200.


Build a loss development triangle and a chain ladder reserve estimate in a spreadsheet.

Actuarial work in spreadsheets

How to Read a Loss Triangle - P&C Insurance Basics - Loss Reserving - Actuarial 101

Actuarial Reserving 1 | Development Method

## Why it matters

Actuaries' numbers decide whether pensions are funded, whether insurers can pay claims and what millions of people pay for cover. Users of that work usually cannot check it themselves, so they rely on the actuary's integrity and competence. That trust is protected by a professional framework: a code of conduct, standards of practice, qualification requirements and a discipline process. Every credentialed actuary is bound by it, and the SOA and CAS both require professionalism education before associateship.

This lesson describes the US framework. Other countries have their own (for example the IFoA's Actuaries' Code in the UK and the CIA's rules in Canada). Read the official documents for exact wording; links checked on 2026-10-23.

## The pieces of the US framework

- **Code of Professional Conduct.** Adopted by the US actuarial organisations, including the SOA, the CAS and the American Academy of Actuaries. It is a set of **precepts** with annotations. See the [SOA's Code of Professional Conduct page](https://www.soa.org/about/governance/about-code-of-professional-conduct/). Checked on 2026-10-23.
- **Actuarial Standards of Practice (ASOPs).** Written by the Actuarial Standards Board (ASB), they describe what an actuary should consider, document and disclose when doing particular kinds of work. See the [ASB's standards of practice](https://www.actuarialstandardsboard.org/standards-of-practice/). Checked on 2026-10-23.
- **Qualification standards.** Rules on the education and experience needed before you issue a statement of actuarial opinion in a given area, plus continuing education.
- **Counseling and discipline.** A body (the Actuarial Board for Counseling and Discipline) handles questions and complaints, and the organisations can discipline members.

## Key precepts in plain English

The first precepts set the tone:

1. **Integrity and competence.** Act honestly, with integrity and competence, and in a way that upholds the profession's reputation. Your duty is to the public as well as to your employer or client.
2. **Qualification.** Do actuarial work only when you are qualified to do it, through education and experience.
3. **Standards of practice.** Make sure your work meets the applicable standards of practice.
4. **Communication.** Communications must be clear and appropriate and must identify the actuary responsible.

Other precepts cover conflicts of interest, confidentiality, courtesy toward other actuaries, and what to do when you learn that another actuary may have broken the Code. Read the full text; the wording matters.

## ASOPs you will meet early

Some standards apply to almost all work:

- **ASOP No. 1, Introductory Actuarial Standard of Practice**: how the ASOPs work and key terms, such as "should" (a requirement unless deviation is justified and disclosed).
- **ASOP No. 23, Data Quality**: review data for reasonableness and consistency, and disclose limitations.
- **ASOP No. 41, Actuarial Communications**: what reports must contain so that another qualified actuary could assess the work.
- **ASOP No. 56, Modeling**: understanding, testing and documenting models, including ones built by others.

Others are specific to an area, such as pensions, health, ratemaking or reserving.

## Case 1: pressure on an assumption

You are a reserving analyst. Your manager asks you to lower a loss development factor "because the CFO wants a better quarter", without any new data. The Code requires honesty and integrity, and the ASOPs require assumptions to be reasonable and supported. The professional response is to explain, in writing if needed, why the current assumption is supported; consider whether there *is* evidence for a change; and escalate to the responsible (signing) actuary if the pressure continues. Changing it silently is not acceptable.

## Case 2: outside your competence

A friend's small company asks you, a pricing actuary in auto insurance, to sign a pension plan valuation. Precept 2 and the qualification standards apply: pension work has its own requirements (in the US, many pension opinions require an Enrolled Actuary). Decline, or work under a qualified actuary.

## Case 3: bad data

The claims extract you received has duplicated rows for one month. ASOP 23 does not require you to audit the data, but you must review it for reasonableness, fix or adjust what you can, and **disclose** the issue and its likely effect in your communication.

## Case 4: a confusing report

Your summary to the board shows a reserve number without stating the method, the key assumptions or the uncertainty. ASOP 41 expects the communication to identify the responsible actuary, the methods and assumptions, and material limitations, at a level suited to the audience.

## Common mistakes

- **Thinking professionalism is only for fellows.** Students and analysts are expected to learn and follow the framework from the start.
- **Confusing "my employer wants it" with "it is reasonable".** The duty to the public does not disappear.
- **Undocumented judgement.** Judgement is part of the job, but it must be explained and disclosed.
- **Relying on summaries.** Read the Code and ASOPs themselves; they are free online.

## Quick recap

- US framework: Code of Professional Conduct, ASOPs (from the ASB), qualification standards, counseling and discipline.
- Act with integrity and competence; work only where qualified; follow the standards; communicate clearly.
- ASOP 23 (data), 41 (communications) and 56 (modeling) apply to almost every task.

This lesson has no activity: work through the cases in the quiz on the **Do** tab.


Apply the Code of Professional Conduct and the Actuarial Standards of Practice to realistic workplace cases.

Actuarial professionalism

Actuarial Professionalism and the Academy

Professionalism and Code of Conduct for Actuaries: Interview with Kevin Allport

Plan the SOA and CAS exam path, build a reserving triangle in a spreadsheet, and apply the profession's code and standards.