insurance.pptx

8/14/2019 Insurance.pptx

1/12

InsuranceLessons 10 - 15


2/12

Review

Review concepts from Ins 410/FM

Interest functions: i, d, v,

Annuities certain: an(immediate, due, presentvalue, future value, etc.)


3/12

Overview

Random variable: Z (decorated or not decorated)

Most formulas are the same for discrete and

continuouscontinuous As and Zs have bars

over them

DeMoivre and Constant simplifications are different

x= benefit interest probability

Exam questions often call x/Axthe single benefitpremium


4/12

Expectation

Discrete: E[Z] = vk+1 pdf

pdf = kqx

Continuous: E[Z bar]=0e-t pdf dt

pdf = tpx x+t


5/12

Variance

Variance: (Second moment)(first moment)2

Second moment = 2Ax

2Ax-Ax

2only 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

2=2


6/12

Types of Insurance

Whole life (x)receive benefit at time of death Term (x

1:n) receive benefit at time of death if

it is before term is up; otherwise no benefit

Deferred (n|x)receive benefit at time of deathif death occurs after deferral period; no benefitwithin period

Pure endowment (x:n1or nEx)receive benefit

at time n if still alive; otherwise no benefit

Endowment (x:n) - receive benefit at time ofdeath if it is before term is up or at time n if stillalive


7/12

Relationships

The life table has whole life values and nEx

values 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


8/12

Constant

Most calculations will either involve the life

table or say that is constant

Know whole life formulas; others can be

derived using relationships on previous slide

Continuous: x = /(+)

Discrete Ax= q/(q+i)


9/12

Discrete to Continuous

Under UDD assumption, multiply discrete by

i/to get continuous

For endowment insurance, only multiply term

component by i/

Second moment: multiply by (2i + i2)/2

Multiply by i/i(m) to get m-thly payable


10/12

Recursive formulas

Whole life: Ax= v qx+ v px Ax+t At -1, qx= 1 and px =0


11/12

Other formulas to memorize

Table 10.1summary of random variablesand symbols for each type of insurance

DeMoivre (table 12.1 will be helpful)

Variance of endowment insurance (section

11.3)

Normal approximation


12/12

Topics that we did not focus on

Percentiles

Increasing/decreasing insurance

Gamma integrands

insurance.pptx

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