Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Formula Sheet (2026)

Every Exam ALTAM formula you need on the test, grouped by topic, rendered with full math notation. 16 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

16 Formulas

6 Topics

2026 Syllabus

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All Exam ALTAM Formulas

Survival Models for Contingent Cash Flows 4 items

Force of mortality — definition

μ_{x + t} = - \frac{d}{d t} ln_{t} p_{x} = \frac{f _{T} ( t )}{_{t} p _{x}}

Makeham force of mortality

μ_{x} = A + B c^{x}, A \geq 0, B > 0, c > 1

A

= accident hazard,

B c^{x}

= aging hazard

Survival function from force of mortality

_{t} p_{x} = exp (- \int_{0}^{t} μ_{x + s} d s)

Complete future lifetime — mean

e ˙_{x} = \int_{0}^{\infty}_{t} p_{x} d t

Premium and Policy Valuation for Long-Term Coverages 4 items

Endowment insurance APV (discrete)

A_{x : n ∣} = A_{x : n ∣}^{1} + v^{n}_{n} p_{x}

= term + pure endowment

Recursive policy value

(_{t} V + P) (1 + i) = q_{x + t} \cdot b_{t + 1} + p_{x + t} \cdot_{t + 1} V

b_{t + 1}

=death benefit in year

t + 1

Thiele's differential equation

\frac{d}{d t} (_{t} V ˉ) = δ_{t} V ˉ + P_{t} - μ_{x + t} (b_{t} -_{t} V ˉ)

P_{t}

=premium rate,

b_{t}

=death benefit at time

t

Insurance-annuity relationship

A_{x} = 1 - d a ¨_{x}

A ˉ_{x} = 1 - δ a ˉ_{x}

(continuous)

Joint Life Insurance and Annuities 3 items

Joint life survival function

_{t} p_{x y} =_{t} p_{x} \cdot_{t} p_{y}

(if independent)

Joint life status fails at first death

Last survivor survival function

_{t} p_{x y} =_{t} p_{x} +_{t} p_{y} -_{t} p_{x y}

Joint life annuity-due APV

a ¨_{x y} = \sum_{k = 0}^{\infty} v^{k}_{k} p_{x y}

Profit Analysis 2 items

NPV of profits

N P V = \sum_{t = 0}^{n} v_{h}^{t}_{t} p_{x} Π_{t}

v_{h}

=discount at hurdle rate;

Π_{0}

=issue expense

Profit signature

Π_{t} = (_{t - 1} V + P_{t} - e_{t}) (1 + i_{t}) - q_{x + t - 1} \cdot b_{t} - p_{x + t - 1} \cdot_{t} V

t = 1, 2, \dots

Pension Plans and Retirement Benefits 1 item

DB pension — projected unit credit (PUC) liability

A L_{t} = b_{t} \cdot a ¨_{x + t}^{ret} \cdot p_{x}^{retire} \cdot v^{r - t}

b_{t}

=accrued benefit at time

t

Universal Life Insurance 2 items

UL account value recursion

A V_{t} = (A V_{t - 1} + P_{t} - e_{t} - C O I_{t}) (1 + i_{t}^{credit})

C O I_{t}

=cost of insurance charge

UL cost of insurance

C O I_{t} = v q_{x + t - 1} \cdot (b_{t} - A V_{t})

(net amount at risk

\times

discounted

q

)

Frequently Asked Questions

Is the Exam ALTAM formula sheet free?

Yes. The full Exam ALTAM formula sheet is free, with no signup, no email, and no credit card required. 16 formulas across 6 topics, all rendered with the same KaTeX math notation used in the FreeFellow study app.

Can I download the Exam ALTAM formula sheet as a printable PDF?

Yes. A 1080x1350 portrait PDF (Instagram and LinkedIn carousel native size, also great for tablet study) is linked at the top of this page. The PDF is fully self-contained: math is pre-rendered, fonts are embedded, no internet connection needed once downloaded.

What's covered on the Exam ALTAM formula sheet?

Every formula is grouped by official syllabus topic, with the formula in math notation plus a one-line note on when to use it (or a watch-out from CAIA, CFA, or other prep-provider commentary). Coverage is calibrated to the 2026 syllabus and refreshed when the corpus changes.

What is FreeFellow's relationship with SOA?

No. FreeFellow is not affiliated with the SOA or any examination body. This is an independent study aid covering the published syllabus.

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The full question bank with detailed solutions, mixed practice, readiness tracking, lessons (where available), and the formula sheet are all free forever. Fellow ($79/quarter or $199/year per track) unlocks timed mock exams, spaced-repetition flashcards, performance analytics, AI essay grading, and a personalized study plan.

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