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

Every Exam ALTAM formula you need on the test, grouped by topic and 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 Free forever
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All Exam ALTAM Formulas

Survival Models for Contingent Cash Flows 4 items
Force of mortality — definition
μx+t=−ddtln⁡tpx=fT(t)tpx\mu_{x+t}=-\dfrac{d}{dt}\ln {}_t p_x = \dfrac{f_T(t)}{{}_t p_x}
Makeham force of mortality
μx=A+Bcx,A≥0, B>0, c>1\mu_x = A + Bc^x,\quad A\ge0,\,B>0,\,c>1
AA = accident hazard, BcxBc^x = aging hazard
Survival function from force of mortality
tpx=exp⁡ ⁣(−∫0tμx+s ds){}_t p_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)
Complete future lifetime — mean
e˙x=∫0∞tpx dt\dot{e}_x=\int_0^{\infty}{}_t p_x\,dt
Premium and Policy Valuation for Long-Term Coverages 4 items
Endowment insurance APV (discrete)
Ax:n‾∣=Ax:n‾∣1+vn npxA_{x:\overline{n}|}=A^1_{x:\overline{n}|}+v^n\,{}_n p_x
= term + pure endowment
Recursive policy value
(tV+P)(1+i)=qx+t⋅bt+1+px+t⋅t+1V({}_t V + P)(1+i) = q_{x+t}\cdot b_{t+1} + p_{x+t}\cdot {}_{t+1}V
bt+1b_{t+1}=death benefit in year t+1t+1
Thiele's differential equation
ddt(tVˉ)=δ tVˉ+Pt−μx+t(bt−tVˉ)\dfrac{d}{dt}({}_t\bar{V}) = \delta\,{}_t\bar{V} + P_t - \mu_{x+t}(b_t - {}_t\bar{V})
PtP_t=premium rate, btb_t=death benefit at time tt
Insurance-annuity relationship
Ax=1−d a¨xA_x=1-d\,\ddot{a}_x
Aˉx=1−δ aˉx\bar{A}_x=1-\delta\,\bar{a}_x (continuous)
Joint Life Insurance and Annuities 3 items
Joint life survival function
tpxy=tpx⋅tpy{}_t p_{xy}={}_t p_x\cdot{}_t p_y (if independent)
Joint life status fails at first death
Last survivor survival function
tpxy‾=tpx+tpy−tpxy{}_t p_{\overline{xy}}={}_t p_x+{}_t p_y-{}_t p_{xy}
Joint life annuity-due APV
a¨xy=∑k=0∞vk kpxy\ddot{a}_{xy}=\sum_{k=0}^{\infty}v^k\,{}_k p_{xy}
Profit Analysis 2 items
NPV of profits
NPV=∑t=0nvht tpx ΠtNPV=\sum_{t=0}^{n}v_h^t\,{}_t p_x\,\Pi_t
vhv_h=discount at hurdle rate; Π0\Pi_0=issue expense
Profit signature
Πt=(t−1V+Pt−et)(1+it)−qx+t−1⋅bt−px+t−1⋅tV\Pi_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,…t=1,2,\ldots
Pension Plans and Retirement Benefits 1 item
DB pension — projected unit credit (PUC) liability
ALt=bt⋅a¨x+tret⋅pxretire⋅vr−tAL_t = b_t \cdot \ddot{a}_{x+t}^{\text{ret}} \cdot {}_{}p_x^{\text{retire}}\cdot v^{r-t}
btb_t=accrued benefit at time tt
Universal Life Insurance 2 items
UL account value recursion
AVt=(AVt−1+Pt−et−COIt)(1+itcredit)AV_{t} = (AV_{t-1}+P_t-e_t-COI_t)(1+i_t^{\text{credit}})
COItCOI_t=cost of insurance charge
UL cost of insurance
COIt=v qx+t−1⋅(bt−AVt)COI_t = v\,q_{x+t-1}\cdot(b_t - AV_t)
(net amount at risk ×\times discounted qq)
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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.
What else is free at FreeFellow for Exam ALTAM candidates?
The full original question bank is free with an account, subject to usage limits. Worked solutions, written lessons, mixed practice, and your readiness score stay free. The formula sheet is free too. Fellow is $39 per month or $79 per quarter, per exam family (USD). Fellow Plus is $49 per month, $99 per quarter, or $199 per year, per exam family (USD). Every annual plan is Fellow Plus. Fellow adds timed mock exams, spaced-repetition flashcards, performance analytics, and a personalized study plan. AI grading: 5 attempts a day on Fellow; Fellow Plus removes that allowance, subject to grading rate and usage limits.

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