1) Deductible2) Policy Limit3) Combining deductible and policy limit4) Proportional Insurance5) Dealing with the conditional loss distributionMath 370X - Lecture 11Risk Management ConceptsSaumil PadhyaNovember 28, 20161) DeductibleLet X be the amount of loss incurred, and Y be the amount of payment through insurance.For a deductible amount of d, if a loss of amount X occurs, the insurer pays nothing if the loss isless than d and pays the policyholder the amount of the loss in excess of d if the loss is greaterthan d.Y =0, X ≤ dX − d, X > dE(Y ) =∞Zd(x − d) · fX(x)dx =∞Zd[1 − FX(x)]dxThe above policy is known as anordinary deductible insurance. Two variations on deductibleinsurance are described below:i) Franchise deductible:Y =0, X ≤ dX, X > dii) Disappearing deductible: (less likely to appear on exam)Y =0, X ≤ dd0·X − dd0− d, d < X ≤ d0X, X > d012) Policy LimitLet X be the amount of loss incurred, and Y be the amount of payment through insurance.A policy limit of u indicates that the insurer will pay a maximum amount of u when a loss occurs.Y =X, X ≤ uu, X > uE(Y ) =uZ0x · fX(x)dx + u · [1 − FX(u)] =uZ0[1 − FX(x)]dx3) Combining deductible and policy limitLet X be the amount of loss incurred, and Y be the amount of payment through insurance.If a policy has a deductible of d and a policy limit of u, then the claim amount paid by the insurercan be described asY =0, X ≤ dX − d, d < X ≤ u + du, X > u + dE(Y ) =u+dZd(x − d) · fX(x)dx + u · [1 − FX(u + d)] =u+dZd[1 − FX(x)]dx4) Proportional InsuranceA fraction α is specified, and the payment amount is αX.Y = αX5) Dealing with the conditional loss distributionLet X be the total loss amount, and B be the loss amountgiven that a loss has occured. Theprobability that a loss will occur is given as q.To compute Var(X), we use the conditional variance formulaV ar(X) = E[V ar(X|I)] + V ar[E(X|I)]2whereV ar(X|I) =0, w.p. 1 − qV ar(B), w.p. qE(X|I) =0, w.p. 1 − qE(B), w.p. qSimilarly,E(X) = E[ E(X|I)]where E(X|I) is defined as
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