Premiums

1. Premiums Lessons 21 - 26

2. Overview • Benefit Premium: the premium needed to cover the cost of benefits • Contract Premium: the amount actually charged, including expenses. • Annual Benefit Premium vs. Single Benefit Premium • Equivalence Principle: requires equality between the APV of the benefit premiums and the APV of the benefits.

3. Types of Insurance • Whole life: Āx = Pbar(Āx)āx • Pbar(Āx) = (1-δāx)/āx = (1/āx) – δ = δĀx/(1-Āx) • N-Year Endowment: Āx:n┐= Pbar(Āx:n┐)āx:n┐ • Pbar(Āx:n┐) = (1/āx:n┐) – δ = δĀx:n┐/ (1-Āx:n┐) • N-Year Term: Āx1:n┐= Pbar(Āx1:n┐) āx:n┐ • Pbar(Āx1:n┐) = Āx1:n┐/ āx:n┐ = δĀx1:n┐/(1-Āx:n┐) • N-Year Deferred: nPbar(n|Āx) = n|Āx/āx:n┐ • Premium payments made only for the deferral period • N-Pay Whole Life: nPbar(Āx) = Āx/ āx:n┐ • N-Year Deferred Annuity: Pbar(n|āx) = n|āx/āx:n┐

4. Loss at Issue • 0L = the PV of the amount the insurance company loses, or the excess of benefits over premiums. • 0L = vT – PāT┐= vT – P((1-vT)/δ) = vT(1+(P/δ)) – (P/δ) • E[0L] = Āx – Pāx = Āx(1 + (P/δ)) – (P/δ) • With the equivalence principle, E[0L] = 0 • Var[0L] = (2Āx – Āx2)(b + (bP/δ))2 • If discrete take bars off As and change δ to d

5. Discrete Premiums • Px = the annual benefit premium payable at the beginning of each year for whole life insurance. = (1/äx) – d = (dAx)/(1-Ax) • Px:n┐= (1/äx:n┐) – d = dAx:n┐/(1-Ax:n┐)

6. Three Premium Principle Formula • nPx - Px1:n┐= Px:n1┐Ax+n • Px:n┐- nPx = Px:n1┐(1-Ax+n)

7. Topics that we did not focus on • Percentiles