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

n
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 )1/ n
i 1
Variance and Standard Deviation
• Variance (2)is a measure of dispersion
• Standard deviation  is square root of
variance
var( x)   prob( x  xi )( xi  E( x)) 2
i 1

n
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
5
0
-10
-5
0
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)
5
10
15
Normal Versus Fat-Tailed
Norm al Versus Fat Tailed Distributions
0.45
0.4
0.35
0.3
0.25
f(x)
Normal Distribution
Cauchy Distribution
0.2
0.15
0.1
0.05
0
-15
-10
-5
0
Return x
5
10
15
Expected Utility
• Pascal’s 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
11/(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 Company’s portfolio faces risk