# Annuities

Study Annuities for SOA Exam FM. Topic weight 15–20%. Per-objective worked-example approaches, exam traps, vendor chapter map (ASM, ACTEX, Coaching Actuaries, TIA), and a 5–7 day micro study plan.

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[Actuarial](https://exclam.ai/actuarial/index.md)›[Exam FM](https://exclam.ai/actuarial/exam-fm/index.md)›Annuities

Exam FM topic · 15–20% of exam

Level and varying annuities paid annually or more frequently, annuities due and immediate, and deferred annuities.

[Back to Exam FM](https://exclam.ai/actuarial/exam-fm/index.md)

## Per-objective worked-example outlines

For each learning objective on Annuities, here is the approach an exam item would test — the setup, the ordering of your reasoning, and the formula or identity you need to bring to the page. Approaches, not full solutions, by design. Verify against the current [soa.org](https://www.soa.org/) syllabus before your sitting.

### Compute present and accumulated values for level, arithmetic, and geometric annuities

Setup

You are given a series of payments — level, arithmetically increasing, or geometrically growing — and a valuation date.

Approach

Identify whether the annuity is due or immediate, then write the basic level-annuity PV, a-angle-n or ä-angle-n. For arithmetic increases, use (Ia)-angle-n or its decreasing counterpart; for geometric growth, divide payments by the growth-adjusted rate (1 + i) / (1 + g) - 1 and value as a level annuity. Always re-anchor the cash flow timeline to the valuation date.

Key identity

a-angle-n = (1 - v^n) / i; ä-angle-n = a-angle-n · (1 + i).

### Work with annuities payable more frequently than interest is convertible

Setup

Payments occur m times per year but interest is quoted at an annual effective rate (or vice versa).

Approach

Either convert the rate to a per-period effective rate matching the payment frequency, or use the m-thly annuity formulas a^{(m)}-angle-n in terms of i and i^{(m)}. The two approaches give identical answers — pick whichever the calculator handles cleanly. Double-check whether the annuity is "due" (start of period) or "immediate" (end of period).

Key identity

a^{(m)}-angle-n = (1 - v^n) / i^{(m)}.

### Value annuities with level and varying payments including perpetuities

Setup

A perpetuity or a deferred annuity is mixed with another stream and you must price the combination as of a given date.

Approach

Decompose the cash flows into pieces you can value with closed forms: perpetuity-immediate is 1/i, perpetuity-due is 1/d, and a deferred annuity is v^k times the standard annuity. Add the present values; do not try to handle the combined stream as a single formula.

Key identity

Perpetuity-immediate PV = 1/i; Perpetuity-due PV = 1/d = (1 + i)/i.

## Common exam traps on Annuities

Recurring patterns where candidates lose points on Annuities-style items. Each entry pairs the trap with the fix.

Trap

Mixing up annuity-due and annuity-immediate when reading the timing of the first payment.

Fix

Draw a payment timeline and circle the first payment date; multiply by (1 + i) to convert immediate to due.

Trap

Using i when the payments are m-thly without converting to i^{(m)}.

Fix

Match the payment period to the rate period before using any annuity formula.

Trap

Forgetting that geometric growth requires a modified interest rate.

Fix

Use the adjusted rate (1 + i)/(1 + g) - 1 and treat the stream as level under that rate.

Trap

Mistreating a deferred perpetuity as a finite annuity.

Fix

Recognize "forever after period k" patterns and use v^k · (1/i).

## Where to find Annuities in popular manuals

Pointers to where each major vendor covers this topic, so you can grab the right chapter without combing the full manual. We do not reproduce vendor content — just the location. Chapter and lesson numbers shift between editions; use these as a guide, not as a citation.

ASM

Annuity chapters covering basic, m-thly, increasing/decreasing, and continuous

ACTEX

Annuity chapter and varying annuity follow-up chapter

Coaching Actuaries

Learn modules on Annuities; Adapt category "Annuities"

The Infinite Actuary

Annuity video block including arithmetic and geometric variations

## 7-day Annuities micro plan

A focused 7-day sub-schedule for Annuities specifically, at roughly 1.5–2.5 hours per day. Drop it inside your full Exam FM plan as a single coverage module.

Day 1

Read the level annuity chapter; build flashcards for a-angle-n, ä-angle-n, s-angle-n, s̈-angle-n.

Day 2

Drill 20 basic level annuity problems on the calculator, alternating between PV and accumulated value modes.

Day 3

Arithmetic annuities (Ia, Da, increasing perpetuity) — 10 problems; build a small reference card with the formulas.

Day 4

Geometric annuities and adjusted-rate problems — 10 problems including a couple with negative growth rates.

Day 5

m-thly annuities — 15 problems requiring rate conversion or the i^{(m)} form.

Day 6

Mixed annuity drill including perpetuities and deferred annuities; 25 problems with a 90-minute timer.

Day 7

Re-do flagged problems and rebuild your annuity cheat sheet from memory; compare to vendor formula sheet.

## How exclam.ai helps you master Annuities

### Flashcards from your manual

Upload your ACTEX Exam FM digital edition, scanned ASM pages, TIA handouts, or your own notes. exclam.ai extracts the Annuities sections and generates flashcards automatically, tuned to the exam traps above.

### Worked-example drilling

Each per-objective approach above maps to a quiz template. exclam.ai re-surfaces missed items until you can recall both the setup and the key identity from cold.

### FSRS spaced repetition

Because Annuities is 15–20% of your exam, losing it during review costs you. FSRS brings it back at the optimal moment.

## Annuities in the Exam FM context

SOA Exam FM has 8 topic areas. Annuities is weighted at approximately 15–20% of the exam, here is where it sits relative to the other topics.

| Topic area | Weight |
| --- | --- |
| Time Value of Money | 10–15% |
| → Annuities | 15–20% |
| Loans | 10–15% |
| Bonds | 10–15% |
| General Cash Flows and Portfolios | 15–20% |
| Immunization | 10–15% |
| Interest Rate Swaps | 0–5% |
| Determinants of Interest Rates | 0–10% |

## Other Exam FM topics

[Time Value of Money 10–15% Interest rates, discount rates, present and future value, and the relationships between different interest measurement conventions. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/time-value-of-money/index.md)

[Loans 10–15% Loan repayment, amortization schedules, sinking funds, and varying loan payments. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/loans/index.md)

[Bonds 10–15% Bond pricing, yield to maturity, premium and discount, and book value schedules. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/bonds/index.md)

[General Cash Flows and Portfolios 15–20% Valuing arbitrary cash flow streams, net present value, internal rate of return, dollar-weighted and time-weighted returns. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/general-cash-flows-and-portfolios/index.md)

[Immunization 10–15% Portfolio immunization strategies, Redington immunization, full immunization, and asset-liability matching. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/immunization/index.md)

[Interest Rate Swaps 0–5% Basic mechanics of interest rate swaps, including fixed and floating legs and swap rate determination. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/interest-rate-swaps/index.md)

[Determinants of Interest Rates 0–10% Economic and market factors that shape interest rates, including inflation, risk premiums, and the yield curve. Study this topic](https://exclam.ai/actuarial/exam-fm/topics/determinants-of-interest-rates/index.md)