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. 21 formulas across 6 topics, calibrated to the 2026 syllabus. Free forever, no signup required.

21 Formulas
6 Topics
2026 Syllabus
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All Exam ALTAM Formulas

Survival Models for Contingent Cash Flows 5 items
Survival function from force of mortality
tpx=exp⁡ ⁣(−∫0tμx+s ds){}_t p_x = \exp\!\left(-\int_0^t \mu_{x+s}\,ds\right)
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
Complete future lifetime — mean
e˙x=∫0∞tpx dt\dot{e}_x=\int_0^{\infty}{}_t p_x\,dt
Curtate future lifetime — mean
ex=∑k=1∞kpxe_x=\sum_{k=1}^{\infty}{}_k p_x
Premium and Policy Valuation for Long-Term Coverages 8 items
Whole life insurance APV (continuous)
Aˉx=∫0∞e−δt tpx μx+t dt\bar{A}_x=\int_0^{\infty}e^{-\delta t}\,{}_t p_x\,\mu_{x+t}\,dt
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
Life annuity-due APV
a¨x=∑k=0∞vk kpx=1−Axd\ddot{a}_x=\sum_{k=0}^{\infty}v^k\,{}_k p_x=\dfrac{1-A_x}{d}
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)
Net premium (benefit equivalence)
P=Axa¨x(whole life, discrete)P=\dfrac{A_x}{\ddot{a}_x}\quad\text{(whole life, discrete)}
Prospective policy value
tV=Ax+t−P a¨x+t{}_t V=A_{x+t}-P\,\ddot{a}_{x+t}
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
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
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
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
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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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.
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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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