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Comprehensive Review of Insurance Concepts and Calculations

This review consolidates key concepts from Ins. 410/FM, focusing on interest functions, annuities, random variables, and insurance types. It covers essential formulas for both discrete and continuous scenarios, alongside critical topics such as whole life, term, deferred, and endowment insurance. Key relationships are established to understand benefit premiums and second moments, with an emphasis on recursive formulas and normal approximations. Ideal for students needing to reinforce their grasp of insurance principles and calculations before exams.

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Comprehensive Review of Insurance Concepts and Calculations

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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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