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.

Exam 9 — Financial Risk & Rate of Return

CAS

CAPM   E[ri]=rf+βi(E[rM]−rf)\;E[r_i]=r_f+\beta_i\left(E[r_M]-r_f\right)

Ferrari total return   TS=IA(1+RS)+UP⋅PS\;\dfrac{T}{S}=\dfrac{I}{A}\left(1+\dfrac{R}{S}\right)+\dfrac{U}{P}\cdot\dfrac{P}{S}

Normal VaR / TVaR   VaRα=μ+zασ\;VaR_\alpha=\mu+z_\alpha\sigma,   TVaRα=μ+σϕ(zα)1−α\;TVaR_\alpha=\mu+\sigma\dfrac{\phi(z_\alpha)}{1-\alpha}

Coherence   \; monotonic, subadditive, positively homogeneous, translation invariant. VaR fails subadditivity in general; TVaR is coherent.

Modified duration   Dmod=Dmac/(1+y)\;D_{mod}=D_{mac}/(1+y); surplus duration   DS=ADA−LDLS\;D_S=\dfrac{A D_A-L D_L}{S}.

Kreps' reinsurer risk load   \; risk load ∝\propto marginal standard deviation or marginal variance of the combined portfolio.

Traps to remember

  • Beta from the regression slope of excess returns — not raw returns — when risk-free rates vary.

  • Confusing VaR (a quantile) with TVaR (a conditional mean) and their coherence properties.

  • Applying duration to surplus without leveraging by A/SA/S and L/SL/S.

  • Sign errors in Ferrari: underwriting loss reduces total return but the leverage still magnifies investment income.

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