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Reserves

Reserves. Lessons 27 to 33. Concept. Insurance is funded by level premiums, but mortality is increasing with age Insured overpays in early years and underpays in later years Deficit is made up through funds already received (reserve). Random Variable.

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Reserves

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  1. Reserves Lessons 27 to 33

  2. Concept • Insurance is funded by level premiums, but mortality is increasing with age • Insured overpays in early years and underpays in later years • Deficit is made up through funds already received (reserve)

  3. Random Variable • Company’s prospective loss at time t = tL • PV of benefit at time t minus PV of premiums received • Reserve = expected value of loss • tV = E[tL ǀ T(x) > t]

  4. Notation • tV is the benefit reserve at time t • V has a bar if the premium is continuous • V is followed by parentheses with the type of insurance/annuity that it is for • Limited payment of the premium is denoted by a superscript to the left of the V • If both the premium and the insurance are discrete, the parentheses are dropped.

  5. Formulas • Prospective: kVx= Ax+k– Pxäx+k • Premium difference: kVx= äx+k(Px+k– Px) • The fund needed at time t is the amount needed to pay for a life annuity of the difference between the premium needed at time t and the premium actually being collected • Paid up insurance: kVx= Ax+k(1 – Px/Px+k) • Px/Px+kis the proportion of Ax+kthat is covered by future premiums, so the complement is the ratio needed to be held in reserves

  6. Formulas (continued) • Retrospective: kVx=Px(s double dot x angle k)-kKx • Accumulated value of premiums minus accumulated value of insurance • Annuity ratio: kVx:n┐=1-(äx+k:n-k┐ / äx:n┐) • Insurance ratio: kVx:n┐=(Ax+k:n-k┐- Ax:n)/(1- Ax:n┐) • Premium ratio: kVx:n┐= (Px+k:n-k┐- Px:n)/(Px+k:n-k┐+d)

  7. Formulas (continued) • Three premium principle • Given two plans with: • Identical benefits • Either both discrete or both continuous • Different premiums • Because there is no difference in AV of the insurance, the difference in reserves (by retrospective formula) must be the AV of the difference in premiums

  8. Variance of Loss • Var[kLǀ K(x) > k] = Var(Z) [1+ (P/δ)]2 • If equivalence principle is used then the variance formula is the same as in premium lessons, but now the numerator is evaluated at x+t and denominator is evaluated at x

  9. Recursive Formula • (k-1V + Pk-1)(1+i) = bk∙ qx+k-1 + kV ∙ px+k-1 • Start with reserve at k-1 and add the premium received at beginning of year • Accumulate one year of interest • This must equal the sum of: • The benefit for those who die • The next year’s reserve for those who live

  10. Other topics • Refund of reserve • Memorize formula if possible (derivation on page 666) • Fractional durations • Hattendorf

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