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Multiple Lives

Multiple Lives. Lessons 34 to 40. Overview. Probabilities based on a collection of lives, rather than just one life Often used for husband and wife pair, insured and beneficiary, etc. Joint Life. Joint life status: ( xy ) Fails as soon as the first life dies (both need to be alive)

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Multiple Lives

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  1. Multiple Lives Lessons 34 to 40

  2. Overview • Probabilities based on a collection of lives, rather than just one life • Often used for husband and wife pair, insured and beneficiary, etc.

  3. Joint Life • Joint life status: (xy) • Fails as soon as the first life dies (both need to be alive) • T(xy) is the time of the first death • Every equation regarding p and q from chapters 3 and 4 still apply • Independence: tpxy= tpx ∙ tpy

  4. Last Survivor • Last survivor status: (xy bar) • Fails only when every member of the status fails • T(xy bar) is the time of the last death • Independence: tqxybar = tqx ∙tqy • Relation to joint life: • T(x) + T(y) = T(xy) + T(xy bar) • Not a case of just true or false – be careful when calculating tǀuqxybar

  5. Expectation • Use the relationship equation: • ex + ey = exy + exybar • Special cases: • DeMoivre • Constant μ

  6. Variance • Var(Txy) = 2 0∫∞t ∙ tpxydt – (exy complete)2 • Cov[T(xy),T(xy bar)]= Cov[T(x),T(y)] + (ex - exy)(ey- exy) • All e complete

  7. Insurance • Use relationship equation: • Ax + Ay = Axy + Axybar *Relationship equation does not work for premiums

  8. Annuities • Relationship equation: • ax + ay = axy + axybar • Reversionary equation: • axǀy = ay– axy • A lot of these problems don’t use a specific formula, but manipulation of formulas from chapter 6

  9. Contingent Probabilities • Unlike regular joint life or last survivor, this status depends on which member dies first and which dies second • Chart of relationships on pg. 800

  10. Common Shock • Method to model dependent future lifetimes • Uses T(x), T(y), and Z • Z = time until death by common shock • Exponential

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