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Insurance

Insurance. Lessons 10 - 15. Review. Review concepts from Ins 410/FM Interest functions: i , d, v, δ Annuities certain: a n ┐(immediate, due, present value, future value, etc.). Overview. Random variable: Z (decorated or not decorated)

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Insurance

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  1. Insurance Lessons 10 - 15

  2. Review • Review concepts from Ins 410/FM • Interest functions: i, d, v, δ • Annuities certain: an┐(immediate, due, present value, future value, etc.)

  3. Overview • Random variable: Z (decorated or not decorated) • Most formulas are the same for discrete and continuous – continuous A’s and Z’s have bars over them • DeMoivre and Constant μ simplifications are different • Āx= ∫ benefit ∙ interest ∙ probability • Exam questions often call Āx/Ax the single benefit premium

  4. Expectation • Discrete: E[Z] = Σ vk+1 ∙ pdf • pdf = kǀqx • Continuous: E[Z bar]= 0∫∞ e-δt ∙ pdfdt • pdf = tpxμx+t

  5. Variance • Variance: (Second moment) – (first moment)2 • Second moment = 2Ax • 2Ax-Ax2only works for fully continuous and fully discrete whole life and endowment • Working with second moments is the same as first moments except use: • 2i = 2i + i2 • 2v=v2 • 2δ=2δ

  6. Types of Insurance • Whole life (Āx)– receive benefit at time of death • Term (Āx1:n┐) – receive benefit at time of death if it is before term is up; otherwise no benefit • Deferred (n|Āx) – receive benefit at time of death if death occurs after deferral period; no benefit within period • Pure endowment (Āx:n┐1 or nEx)– receive benefit at time n if still alive; otherwise no benefit • Endowment (Āx:n┐) - receive benefit at time of death if it is before term is up or at time n if still alive

  7. Relationships • The life table has whole life values and nExvalues so rewriting the equations using these terms is important: • Whole life = term + deferred • Deferred (n year) = nEx ∙ Ax+n • nEx= npx ∙ vn • Endowment = term + pure endowment

  8. Constant μ • Most calculations will either involve the life table or say that μ is constant • Know whole life formulas; others can be derived using relationships on previous slide • Continuous: Āx= μ/(μ+δ) • Discrete Ax = q/(q+i)

  9. Discrete to Continuous • Under UDD assumption, multiply discrete by i/δ to get continuous • For endowment insurance, only multiply term component by i/δ • Second moment: multiply by (2i + i2)/2δ • Multiply by i/i(m) to get m-thly payable

  10. Recursive formulas • Whole life: Ax = v ∙ qx + v ∙ px∙Ax+t • At ω-1, qx = 1 and px=0

  11. Other formulas to memorize • Table 10.1 – summary of random variables and symbols for each type of insurance • DeMoivre (table 12.1 will be helpful) • Variance of endowment insurance (section 11.3) • Normal approximation

  12. Topics that we did not focus on • Percentiles • Increasing/decreasing insurance • Gamma integrands

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