Actuarium

Formula sheets

Every formula on the syllabus, rendered natively, with the traps examiners set for each. Print one per exam and keep it by your desk.

FAM — Fundamentals of Actuarial Mathematics

SOA

Constant force of mortality   tpx=e−μt\;{}_tp_x=e^{-\mu t}, Aˉx=μμ+δ\bar A_x=\dfrac{\mu}{\mu+\delta}, aˉx=1μ+δ\bar a_x=\dfrac{1}{\mu+\delta}

Equivalence-principle net premium (whole life, annual):   P=Axa¨x=1a¨x−d\;P=\dfrac{A_x}{\ddot a_x}=\dfrac{1}{\ddot a_x}-d

Policy value recursion   (tV+P)(1+i)=qx+t bt+1+px+t t+1V\;({}_tV+P)(1+i)=q_{x+t}\,b_{t+1}+p_{x+t}\,{}_{t+1}V

Compound Poisson aggregate   E[S]=λE[X]\;E[S]=\lambda E[X], Var⁡(S)=λE[X2]\operatorname{Var}(S)=\lambda E[X^2]

Limited expected value, Pareto(α,θ\alpha,\theta):   E[X∧u]=θα−1[1−(θu+θ)α−1]\;E[X\wedge u]=\dfrac{\theta}{\alpha-1}\left[1-\left(\dfrac{\theta}{u+\theta}\right)^{\alpha-1}\right]

Bühlmann   Z=nn+EPV/VHM\;Z=\dfrac{n}{n+EPV/VHM}

Traps to remember

  • Confusing continuous (Aˉ\bar A), discrete (AA) and mthly (A(m)A^{(m)}) benefit timing.

  • Applying the deductible after the policy limit in a per-loss calculation.

  • Using Var⁡(S)=λVar⁡(X)\operatorname{Var}(S)=\lambda\operatorname{Var}(X) instead of λE[X2]\lambda E[X^2] for compound Poisson.

  • Forgetting that per-payment quantities condition on the loss exceeding the deductible.

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