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Probability Insurance

This document covers key concepts in probability and risk management. It discusses how probability theory originated and developed over time. Key concepts discussed include random variables, independent events, expected value, variance, standard deviation, covariance, correlation, and different probability distributions like normal and fat-tailed. It also covers topics like present discounted value, annuities, and problems faced by insurance companies in managing risk.

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Kairu Hiroshima
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240 views18 pages

Probability Insurance

This document covers key concepts in probability and risk management. It discusses how probability theory originated and developed over time. Key concepts discussed include random variables, independent events, expected value, variance, standard deviation, covariance, correlation, and different probability distributions like normal and fat-tailed. It also covers topics like present discounted value, annuities, and problems faced by insurance companies in managing risk.

Uploaded by

Kairu Hiroshima
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPT, PDF, TXT or read online on Scribd
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Lecture 2

The Universal Principle of Risk


Management
Pooling and Hedging of Risk

Probability and Insurance


Concept of probability began in 1660s
Concept of probability grew from interest in
gambling.
Mahabarata story (ca. 400 AD) of Nala and
Rtuparna, suggests some probability theory
was understood in India then.
Fire of London 1666 and Insurance

Probability and Its Rules


Random variable: A quantity determined by
the outcome of an experiment
Discrete and continuous random variables
Independent trials
Probability P, 0<P<1
Multiplication rule for independent events:
Prob(A and B) = Prob(A)Prob(B)

Insurance and Multiplication


Rule
Probability of n independent accidents = Pn
Probability of x accidents in n policies
(Binomial Distributon):

f ( x) P (1 P)
x

( n x)

n! /( x!(n x)!)

Expected Value, Mean, Average


E ( x) x i 1 prob ( x xi ) xi

E ( x ) x

f ( x) xdx

x xi / n
i 1

Geometric Mean
For positive numbers only
Better than arithmetic mean when used for
(gross) returns
Geometric Arithmetic
n

G( x) ( xi )
i 1

1/ n

Variance and Standard Deviation


Variance (2)is a measure of dispersion
Standard deviation is square root of
variance
i 1

var( x) prob( x xi )( xi E( x)) 2

s ( xi x ) 2 / n
2
x

i 1

Covariance
A Measure of how much two variables
move together
n

cov( x, y ) ( x x)( y y ) / n
i 1

Correlation
A scaled measure of how much two
variables move together
-1 1

cov( x, y ) /( s x s y )

Regression, Beta=.5, corr=.93


Return XYZ Corporation against Market 1990-2001
25

Return on XYZ Corporation

20

15
Each point represents a year.
Linear (Each point represents a year.)
10

0
-10

-5

10

Return on the Market

15

20

25

Distributions
Normal distribution (Gaussian) (bell-shaped
curve)
Fat-tailed distribution common in finance

Normal Distribution
Norm al Distribution w ith Zero Mean
0.45
0.4
0.35
0.3
0.25
f(x)

Standard Dev. = 3
Standard Dev. = 1

0.2
0.15
0.1
0.05
0
-15

-10

-5

0
Return (x)

10

15

Normal Versus Fat-Tailed


Normal Versus Fat Tailed Distributions
0.45
0.4
0.35
0.3

f(x)

0.25

Normal Distribution
Cauchy Distribution

0.2
0.15
0.1
0.05
0
-15

-10

-5

0
Return x

10

15

Expected Utility
Pascals Conjecture
St. Petersburg Paradox, Bernoulli: Toss coin
until you get a head, k tosses, win 2(k-1)
coins.
With log utility, a win after k periods is
worth ln(2k-1)

E(U ) prob( x xi )U ( xi )
i 1

Present Discounted Value (PDV)


PDV of a dollar in one year = 1/(1+r)
PDV of a dollar in n years = 1/(1+r)n
PDV of a stream of payments x1,..,xn
T

PDV xt /(1 r ) t
t 1

Consol and Annuity Formulas


Consol pays constant quantity x forever
Growing consol pays x(1+g)^t in t years.
Annuity pays x from time 1 to T

Consol PDV x / r

Growing Consol PDV x /( r g )


Annuity PDV x

11 /(1 r )T
r

Insurance Annuities
Life annuities: Pay a stream of income until a
person dies.
Uncertainty faced by insurer is termination
date T

Problems Faced by Insurance


Companies
Probabilities may change through time
Policy holders may alter probabilities
(moral hazard)
Policy holders may not be representative of
population from which probabilities were
derived
Insurance Companys portfolio faces risk

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