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Multiple Lives

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This overview explores the concepts of Joint Life and Last Survivor models in the context of insurance. It delves into how these models define the probabilities based on collections of lives, emphasizing key terms like joint life status, independence, and time of death. The document also discusses expectations, relationship equations for insurance and annuities, contingent probabilities, and methods for modeling dependent future lifetimes, including the common shock scenario. Understanding these complex relationships is essential for accurate insurance calculations.

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Multiple Lives

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Presentation Transcript

  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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