1. The Theory of Choice: Utility Theory Given Uncertainty

2. Introduction
In our previous lectures we talked about individual decision making in the absence of uncertainty
In reality, we usually make decision under uncertainty
example:
1. uncertainty from product quality (second-hand vehicle)
2. uncertainty in dealing with others
-> often the outcome depends on what others do
3. purchase of financial assets (stocks and bonds) whose return is contingent on which state is realized.
This is the essence of Financial Economics

3. Expected utility theory
In economics, game theory and decision theory the expected utility theorem or expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes (gambles) is represented by
a function of the payout (whether in money or other goods),
the probability of occurrence,
risk aversion,
and the different utility
……of the same payout to people with different assets or personal preferences.

4. Founder Fathers of the Theory
Daniel Bernoulli described the complete theory in 1738.
John von Neumann and Oskar Morgenstern reinterpreted and presented an axiomatization of the same theory in 1944.
They proved that any "normal" preference relation over a finite set of states can be written as an expected utility, sometimes referred to as von Neumann-Morgenstern utility.

5. Importance of the theory
This theory has proved useful to explain some popular choices that seem to contradict the expected value criterion (which takes into account only the size of the payout and the probability of occurrence), such as gambling and insurance.

6. 2 Goals 1) Individual maximizes their expected Utility10
E(W) = 0.4(10) + 0.6(2) = 5.2
0.4
Asset i
E[U(W)] = 0.4U(10) + 0.6U(2) = ?
0.6 2
Prefer the one with higher E[U(W)] 9
0.3
Asset j
E(W) = 0.3(9) + 0.7(4) = 5.5
E[U(W)] = 0.3U(9) + 0.7U(4) = ?
0.7 4
2) Individual preferences over risk and return y C2
Return x C1
Risk

7. Probability
probability of an event occurring is the relative frequency with which the event occurs
if αi = the probability of event i occurring and there are n possible events (states) then
1. αi > 0, i = 1…n
2.  αi = 1 (summation of probabilities = 1)
Lottery (X) with prizes (outcomes, states, events) X1, X2, X3,...,Xn with corresponding probabilities α1, α2, α3,...,αn , respectively (mutually exclusive and exhaustive) then the expected value of this lottery is
E(X) = α1X1 + α2X2 + α3X αnXn
E(X) =  αiXi n n
‘i=1

8. Example 1 Gamble (X) flip of a coin if heads, you receive $1 X1 = +1
if tails, you pay $1 X2 = -1
E(X) = (0.5) (1) + (0.5) (-1) = 0
if you play this game many times, it is likely that you break-even

9. Example 2 Gamble (X) flip of a coin if heads, you receive $10 X1 = +10
if tails, you pay $1 X2 = -1
E(X) = (0.5) (10) + (0.5) (-1) = 4.50
if you play this game many times, you will be a big winner
How much would you pay to play this game:
perhaps as much as a $4.50
But of course the answer depends upon your preference to risk

10. Fair Gambles if the cost to play = expected value of
these gambles the outcome
then the gamble is said to be actuarially fair
Common empirical findings:
1. individuals may agree to flip a coin for small amounts of money, but usually refuse to bet large sums of money
2. people will pay small amounts of money to play actuarially
unfair games (Lotto 6 -49, where cost = $1, but E(X) < 1)
- but will avoid paying a lot
Why do these empirical findings occur?
Becoz’ it is not about E(W)

11. Expected Utility Theory
Objective: to develop a theory of rational decision-making under uncertainty with the minimum sets of reasonable assumptions possible
the following five axioms of cardinal utility provide the minimum set of conditions for consistent and rational behaviour
What do these axioms of expected utility mean?
  1. all individuals are assumed to make completely rational decisions (reasonable)
2. people are assumed to make these rational decisions among
thousands of alternatives (hard)

12. 5 Axioms of Choice under uncertainty
A1.Comparability (also known as completeness).
For the entire set of uncertain alternatives, an individual can say either that
either x is preferred to outcome y (x > y)
or y is preferred to x (y > x)
or indifferent between x and y (x ~ y).
A2.Transitivity (also know as consistency).
If an individual prefers x to y and y to z, then x is preferred to z. If (x > y and y > z, then x > z).
Similarly, if an individual is indifferent between x and y and is also indifferent between y and z, then the individual is indifferent between x and z. If (x ~ y and y ~ z, then x ~ z).

13. 5 Axioms of Choice under uncertainty
A3.Strong Independence.
Suppose we construct a gamble where the individual has a probability α of receiving outcome x and a probability (1-α) of receiving outcome z. This gamble is written as:
G(x,z:α)
Strong independence says that if the individual is indifferent to x and y, then he will also be indifferent as to a first gamble set up between x with probability α and a mutually exclusive outcome z, and a second gamble set up between y with probability α and the same mutually exclusive outcome z.
  If x ~ y, then G(x,z:α) ~ G(y,z:α)
NOTE: The mutual exclusiveness of the third outcome z is critical to the axiom of strong independence.

14. 5 Axioms of Choice under uncertainty
A4.Measurability. (CARDINAL UTILITY)
If outcome y is less preferred than x (y < x) but more than z (y > z),
then there is a unique probability α such that:
the individual will be indifferent between
[1] y and
[2] A gamble between x with probability α
z with probability (1-α).
In Maths,
if x > y > z or x > y > z ,
then there exists a unique α such that y ~ G(x,z:α) 

15. 5 Axioms of Choice under uncertainty
A5.Ranking. (CARDINAL UTILITY)
If alternatives y and u both lie somewhere between x and
z and we can establish gambles such that an individual is
indifferent between y and a gamble between x (with probability α1)
and z, while also indifferent between u and a second gamble, this
time between x (with probability α2) and z, then if α1 is greater
than α2, y is preferred to u.
If x > y > z and x > u > z
then if y ~ G(x,z:α1) and u ~ G(x,z:α2),
  then it follows that if α1 > α2 then y > u,
or if α1 = α2, then y ~ u

16. Other Axioms
Axiom 6 Monotonicity. If two lotteries with the same alternatives differ only in probabilities, then the lottery that gives the higher probability to obtain the most preferred alternative is preferred.
Axiom 7 Substitutability. The lottery xi can always be substituted for its certainty equivalent xi in any other lottery since the consumer is indifferent between them..

17. One more assumption People are greedy, prefer more wealth than less.
The 5 axioms and this assumption is all we need in order to develop an expected utility theorem and actually apply the rule of
max E[U(W)] = max ∑iαiU(Wi)

18. Utility Functions Utility functions must have 2 properties
1. order preserving: if U(x) > U(y) => x > y
2. Expected utility can be used to rank combinations of risky
alternatives:
U[G(x,y:α)] = αU(x) + (1-α) U(y)
Deriving Expected utility theorem, one of the most elegant derivations in Economics, is tough. Don’t worry about a formal derivation. Just apply it.
Remark:
  Utility functions are unique to individuals
- there is no way to compare one individual's utility function with
another individual's utility
- interpersonal comparisons of utility are impossible
if we give 2 people $1,000 there is no way to determine who is happier

19. One more element: Risk Aversion
Consider the following gamble:
Prospect a prob = α G(a,b:α)
prospect b prob = 1-α
Question: Will we prefer the expected value of the gamble with
certainty, or will we prefer the gamble itself?
ie. consider the gamble with
10% chance of winning $100
90% chance of winning $0 E(gamble) = $10
would you prefer the $10 for sure or would you prefer the gamble?   
if prefer the gamble, you are risk loving
if indifferent to the options, risk neutral
if prefer the expected value over the gamble, risk averse

20. Attitudes to Risk
Intuitively, whether someone accepts a gamble or not depends on his attitude to risk
The Risk Averse Person
The Risk Neutral Person
The Risk Loving Person

21. Attitude to risk
To define these attitudes, we use the concept of a fair gamble
In essence, a fair gamble allows you receive the same amount of money through two distinct ways:
Gambling or not gambling

22. Attitudes to Risk and Fair Gambles
A risk averse person will never accept a fair gamble
A risk loving person will always accept a fair gamble
A risk neutral person will be indifferent
towards a fair gamble

23. What Does This Mean?
Given the choice between earning the same amount of money through a gamble or through certainty
The risk averse person will opt for certainty
The risk loving person will opt for the gamble
The risk neutral person will be indifferent

24. Why does the risk averse person reject the fair gamble?
Answer: because her marginal utility of money diminishes

25. Example
Your wealth is $10. I toss a coin and offer you $1 if it heads and take $1 from you if it tails
This is a fair gamble: 0.5×11+0.5×9=10, but you reject it
Because, your gain in utility from another $1 is less than your loss in utility from losing $1
Your MU diminishes, you are risk averse

26. Equivalent Concepts A person is risk averse
A person’s marginal utility of money diminishes
A person’s utility function, u(c), is concave

27. Risk Averse: concave function
An economic agent is risk averse if it values an incremental increase in wealth less highly than an incremental decrease of the same magnitude and will reject a fair gamble.
As wealth increases utility increases at a decreasing rate ( marginal utility decreases)
U’’(w)<0

28. The Risk Averse Utility Function
U(W)
U(w+1)
U(w)
U'(W) > 0
U''(W) < 0
Let U(W) = ln(W)
U'(W) = 1/w
U''(W) = - 1/W2
MU positive
But diminishing
U(w-1) W
w-1
w+1 w

29. Two more concepts
The certainty equivalent of a gamble: the sum of money, X, which, if received with certainty will yield the same utility as the gamble
X is CE if u(X) = EU=pG×u(cG)+pB×u(cB)
The risk premium associated with a gamble is the maximum amount a person is prepared to pay to avoid the gamble
RP = ER - CE

31. Risk Lover/ Seeker
Values an incremental increase in wealth more highly than an incremental decrease and will accept a fair gamble
The utility function is convex

32. Utility Functions : Graphs
Utility U(W)
Risk Lover
U'(W) > 0
U''(W) > 0
U(w+1)
U(w)
U(w-1)
w-1 w
Wealth
w+1

34. Risk Neutral
Values an incremental increase in wealth the same as an incremental decrease and indifferent on a fair gamble.

35. U'(W) > 0
U''(W) = 0

36. Preferences to Risk U(W) U(W) U(W) a b W a b W a b W Risk Preferring
U(b)
U(b)
U(b)
U(a)
U(a)
U(a) a b W a b W a b W
Risk Preferring
Risk Neutral
Risk Aversion
U'(W) > 0
U''(W) = 0
U'(W) > 0
U''(W) > 0
U'(W) > 0
U''(W) < 0

37. U[E(W)] and E[U(W)]
U[(E(W)] is the utility associated with the known level of expected wealth (although there is uncertainty around what the level of wealth will be, there is no such uncertainty about its expected value)
E[U(W)] is the expected utility of wealth is utility associated with level of wealth that may obtain
The relationship between U[E(W)] and E[U(W)] is very important

38. Expected Utility
Assume that the utility function is natural logs: U(W) = ln(W)
Then MU(W) is decreasing
U(W) = ln(W)
U'(W)=1/W
MU>0
MU''(W) < => MU diminishing
Consider the following example:
  80% chance of winning $5
20% chance of winning $30
E(W) = (.80)*(5) + (0.2)*(30) = $10
U[E(W)] = U(10) = 2.30
E[U(W)] = (0.8)*[U(5)] + (0.2)*[U(30)]
= (0.8)*(1.61) + (0.2)*(3.40)
= 1.97
Therefore, U[(E(W)] > E[U(W)] -- uncertainty reduces utility

39. The Arrow-Pratt measures of Risk Aversion
Risk aversion can be measured in two ways:
Absolute risk aversion
Relative risk aversion

40. Absolute Risk Aversion: A(w)
Is the risk aversion with the actual dollar amount an individual will choose to hold in risky assets, given a certain wealth level w.
Absolute risk aversion

41. Relative Risk Aversion: R(w)
If we want to measure the percentage of wealth held in risky assets, for a given wealth level w, we simply multiply the Arrow-Pratt measure of absolute risk-aversion by the wealth w, to get a measure of relative risk-aversion, i.e.:
Relative Risk Aversion R(w) = W.A(w)
Relative Risk aversion

42. HOW ABSOLUTE RISK-AVERSION CHANGES WITH WEALTH
Type of Risk-Aversion
Description
Example of Bernoulli Function
Increasing absolute risk-aversion A’(w)>0
As wealth increases, hold fewer dollars in risky assets
w-cw2
Constant absolute risk-aversion A’(w)=0
As wealth increases, hold the same dollar amount in risky assets
-e-cw
Decreasing absolute risk-aversion A’(w)<0
As wealth increases, hold more dollars in risky assets
ln(w)

43. HOW RELATIVE RISK-AVERSION CHANGES WITH WEALTH
Type of Risk-Aversion
Description
Example of Bernoulli Function
Increasing relative risk-aversion R’(w)>0
As wealth increases, hold a smaller percentage of wealth in risky assets
w - cw2
Constant relative risk-aversion R’(w)=0
As wealth increases, hold the same percentage of wealth in risky assets
ln(w)
Decreasing relative risk-aversion R’(w)<0
As wealth increases, hold a larger percentage of wealth in risky assets
-e2w-1/2

44. Utility Function : Power
Constant Relative Risk Aversion
U(W) = W(1-g) / (1-g) g > 0, g ≠ 1
U’(W) = W-g
U’’(W) = -gW-g-1
RA(W) = g/W
RR(W) = g (a constant)

45. The Empirical Evidence
Empirical evidence (Friend and Blume 1975) indicates that individuals have decreasing ARA and constant RRA = 2
Power Utility Function
U(W) = -W-1
U'(W) = W-2 > 0
U"(W) = -2W-3 < 0
ARA = 2/W => dARA/dW < 0
RRA = 2W/W = 2 => dRRA/dW = 0
This power utility function is consistent with the empirical evidence of Friend and Blume (1975)
U(W) = -1/W

46. Utility Function : Logarithmic
As g  1, logarithmic utility is a limiting case of power utility
U(W) = ln(W)
U’(W) = 1/W
U’’(W) = -1/W2
RA(W) = 1/W
RR(W) = 1

47. Utility Function : Quadratic
U(W) = W – (b/2)W2 b > 0
U’(W) = 1 – bW
U’’ = -b
RA(W) = b/(1-bW)
RR(W) = bW / (1-bW)

48. Quadratic Utility quadratic utility exhibits increasing ARA
Quadratic Utility - widely used in the academic literature
U(W) = a W - b W2
U'(W) = a - 2bW U"(W) = -2b
-U"(W) b
ARA = =
U'(W) a -2bW
d(ARA)
> 0 dW
  b
RRA =
a/W - 2b
d(RRA)
quadratic utility exhibits
increasing ARA
and increasing RRA
ie an individual with increasing RRA would
become more averse to a given percentage
loss in W as W increases -

49. Utility Function : Negative Exponential
Constant Absolute Risk Aversion
U(W) = a – be-cW c > 0
RA(W) = c
RR(W) = cW

50. How do I decide whether an investment is “profitable”.
Decision is based on three Questions:
Is return commensurate with “risk”?
Does investment diversify my portfolio or concentrate exposure?
Is investment consistent with my preferred operating risk?

51. Efficiency Revisited
Portfolios are efficient if they are not dominated by other portfolios
Portfolios are inefficient if at least one other portfolio dominates them
Rational investors prefer efficient investments

52. Common Measures of Risk and Reward
Internal Rate of Return
Return on Equity
Net Present Value
Loss Ratio
Return on Capital
Expected Policyholder Deficit

53. Utility Theory (Please suppress groans)
What to do then?
Utility Theory (Please suppress groans)
Basic premise is “Tell me how much a return of W is worth to you...”
“…then we can see if the investment improves your expected worth.”

54. Review of Utility Theory
A utility function is a transformation that maps dollars to utility (worth).
The shape of this function reflects our investment objectives and preferred operating risks.
Common features include
Wealth Preference and Risk Aversion

55. Wealth Preference “Greed is good.”
A utility function U(w) possesses Wealth Preference if and only if U’(w)³ 0 for all w with at least one strict inequality.
In other words, my utility function is increasing (there are a lot of ways to be increasing, though).

56. Risk Aversion I hate losing more than I like winning.
A utility function U(w) possesses Risk Aversion if and only if it satisfies Wealth Preference and U’’(w)£ 0 for all w with at least one strict inequality.
In other words, my utility function is increasing at a decreasing rate (i.e. it’s curved).

57. A Less Common Feature: Ruin Aversion
Also called Decreasing Absolute Risk Aversion, Skewness Preference, etc.
Losing a little is bad, but losing everything is intolerable. Enter reinsurance...

58. Ruin Aversion
A utility function U(w) possesses Ruin Aversion if and only if it satisfies Risk Aversion and U’’’(w)³ 0 for all w with at least one strict inequality.
In other words, my utility is curved but “flattening out” as it goes.

60. These three features of utility functions are nested.
Fine Point
These three features of utility functions are nested.
Wealth Preference
Risk Aversion
Ruin Aversion

61. A Tool for Evaluating Reinsurance Alternatives
Stochastic Dominance
A Tool for Evaluating Reinsurance Alternatives

62. Introduction
Stochastic dominance is an alternative technique employed in the portfolio construction process
Stochastic “denotes the process of selecting from among a group of theoretically possible alternatives those elements or factors whose combination will most closely approximate a desired result”
Stochastic models are not always exact
Stochastic models are useful shorthand representations of complicated processes

63. Introduction to Stochastic Dominance
The approach to the maximization of expected utility discussed in previously is based on the assumption that the preferences of the decision makers are known, easily obtained or quantified.
In a number of cases one may be confronted with the necessity of making a prediction about a decision maker’s preferences between risky prospects with limited or no knowledge of the underlying utility function.
Under these conditions the decision-theoretic approach is of limited value.
SD is introduced to help solve this problem. It helps to resolve risky choices while making the weakest possible assumptions. The most general form of SD makes no assumptions about the form of the probability distribution.

64. SD is said to occur if the expected utility of one risky prospect exceeds the expected utility of another for all possible utility functions within a defined class.
Under SD approach one is interested in defining selection rules that minimize the admissible set of risk prospects by discarding those that are dominated.
The set of risky prospects that are found not to be dominated according to some rule(s) is referred to as the stochastically efficient set.
SD thus helps to isolate a smaller set of prospects by excluding those that are inferior. This is important for decisions analysts since it reduces the number of alternatives requiring explicit consideration.

65. Identification of the stochastically efficient set involves comparisons of cumulative distribution functions (CDFs) for risky outcomes.
It utilizes the property that decision makers prefer low probabilities to be associated with less preferred outcomes and high probabilities to be associated with more preferred outcomes.
SD is applicable when the probability distributions are continuous, discrete or mixed.

66. Assumptions in traditional stochastic dominance:
1. Individuals are utility maximizers. - stochastic dominance assumes expected utility maximization.
2. Two alternatives to be compared (And these are mutually exclusive alternatives - that is, the other must be chosen and not a convex combination of both).
The stochastic dominance is developed based on population distributions

67. Three Alternative Criteria
The three main SD selection rules are: First Order Stochastic Dominance, Second Order Stochastic Dominance, and Third Order Stochastic Dominance.
1. First Order Stochastic Dominance – the least restrictive of the three.
It assumes that for every investor: more is better.
2. Second Order Stochastic Dominance – it assumes that:
Investors prefers more to less
Investors are risk averse
3. Third Order Stochastic Dominance
Third derivative of the utility function is positive

70. First Order Stochastic Dominance
Given two distributions f and g, f dominates g by FOSD when the decision maker has positive marginal utility of wealth for all x.
And for all wealth levels the cumulative probabilities under the f distribution is less than or equal to the cumulative distribution under the g distribution.
This requires that for all x the cumulative probability distribution for f is always to the right of the cumulative probability of g.

71. The most general efficiency criteria relies only on the assumption that utility is nondecreasing in income, or the decision maker prefers more of at least one good to less.
FSD Rule: Given two cumulative distribution functions F and G, an option F will be preferred to the second option G by FSD independent of concavity if F(x)  G(x) for all return x with at least one strict inequality.

72. Intuitively, this rule states that one alternative F will dominate G if its cumulative distribution function always lies to the right of G’s:

74. Mathematically, FSD is dependent on the integrals of the utility function times each alternative distribution function:

75. Note that the utility function is the same for each investment alternative, but the distribution function changes. If investment F dominates investment G, then the difference, D, defined as

76. Integrating by parts

77. Cumulative distribution A will be preferred over cumulative distribution B if every value of distribution A lies below or on distribution B, provided the distributions are not identical
The distribution lines do not cross

78. Curves may never cross.

79. Second-Order Stochastic Dominance
Second Degree Stochastic Dominance
Building on FSD, second degree stochastic dominance SSD invokes risk aversion by inferring that the utility function is concave, implying that the second derivative of the utility function is negative.
SSD Rule A necessary and sufficient condition for an alternative F to be preferred to a second alternative G by all risk averse decision makers is that

81. Alternative F is preferred to Alternative G if the cumulative probability of G minus the cumulative probability of F is always non-negative
SSD can be a significant aid in reducing the security universe to a workable number of efficient alternatives

82. Graphically, another explanation of SSD can be determined by: Alternative F dominates alternative G for all risk averse individuals if the cumulative area under F exceeds the area under the cumulative distribution function G for all values x, or if the cumulative area between F and G is non-negative for all x.

83. Curves may cross but
not “too soon”.

84. Third Order Stochastic Dominance (TOSD)
For TOSD the following conditions are necessary:
Investor prefers more to less .
Investors are risk averse.
Third derivative of the utility function is positive.  
If f dominates g then we have a risk averter with diminishing absolute risk aversion.

85. Third-Order Stochastic Dominance
Under Ruin Aversion, A is uniformly preferred to B if and only if for all w with at least at least one strict inequality.
Small, probable loss is preferable to remote, possible ruin

86. Curves may cross sooner than SSD.

87. In general, the FOSD, SOSD, and TOSD orderings for a decision problem have certain properties or relations in common. These include: 
Transitivity – if prospect A dominates prospects B, and B dominates C in terms of FOSD (SOSD or TOSD), then prospect A dominates C in terms of the same degree of stochastic dominance.
Partial ordering – the dominance relations imply that the set of utility functions comprising a class for one degree of SD contains the set of utility functions comprising a class for a higher degree of SD, but not conversely. That is, if A dominates B by FSD, then A dominates B by SOSD, and A dominates B by TOSD also, but the reverse does not hold.
Necessary conditions – the necessary conditions for FOSD, SOSD and TOSD to hold are that the lower bound of the cumulative distribution function for the dominant prospect not be less than for the dominated prospect, and the mean of the dominant prospect not be less than that of the dominated prospect.

88. Fine Point Revisited Wealth Preference Risk Aversion Ruin Aversion
The stochastic dominance
orders are nested in
reverse order.
Third-Order
Second-Order
First Order

89. Empirical Implementation
The above illustration depends on continuous probability distributions. We need to come up with ways of using SD when the distributions are discrete. We follow the following 4-step procedure:
Step 1: Take the outcomes for all probability distributions and arrange them in order.
Step 2: Write the frequencies of observations against each of the x levels for each distribution. Some of these frequencies will usually be zero if for example an x level is observed in one distribution and not in the other.
Step 3: Form the CDF starting at the first value of x.
Step 4: Do the comparisons.

90. Example 1
Suppose you are given two investment alternatives with returns (outcomes) and probabilities as given in the following table. Which one do you prefer?
We shall use FOSD to answer this question.

91. Two Investment Alternatives: Example 1
Investment B
Outcome
Probability 12
1/3 11 10 9 8 7

92. Calculations for Example 1
Investment A
Investment B
Return
CDF A
CDF B 7
1/3 8 9
2/3 10 11 1 12

93. INSURANCE AND CERTAINTY EQUIVALENCE
Individuals desire insurance because they are risk averse. Risk aversion implies that an individual is willing to pay a premium that exceed his or her expected loss.
To show this, suppose W is an individual’s current level of wealth, and she is faced with a random loss of amount If she is uninsured, her expected utility is
where U is her utility function. A risk averse individual has a utility function that is increasing and concave in wealth. Jensen’s inequality implies
This means that she would be happier paying a sure premium equal to the expected loss, , than be uninsured.

94. INSURANCE AND CERTAINTY EQUIVALENCE
Furthermore, due to risk-aversion, this individual may be happier being insured even if the insurance premium is somewhat higher than the fair premium,
In contrast to individuals, firms are not thought to have utility functions (or to be risk-averse). However, there may be value-maximizing reasons why firms seek insurance.
Firms desire for insurance is related to their desire for risk-management, a topic that we will cover shortly.

95. Life Insurance Companies
Many products offered by life insurers provide financial remuneration at the time of a policyholder’s death. Mortality risk is pooled and diversified away by writing policies for large numbers of individuals.
Life insurers can accurately predict claims for death benefits by using actuarial tables that forecast life expectancies for certain types of individuals (e.g., 25 year-old, female non-smoker).
By writing policies for a large numbers of individuals, the Law of Large Numbers states that with high probability, the insurance company’s average policyholder claim will be close to the true mortality risk of the entire population. Moreover, the Central Limit Theorem states that this average claim will be normally distributed, making the risk of deviations quantifiable.

96. Normal Distribution of Insurer’s Average Policyholder Claim
Standard Deviation,
Declines as Number
of Policyholders, n,
Increases
Claim amount
Population Average, 
However, a factor that could invalidate this statistical model is adverse selection. Insurers must always be conscious that applicants for policies may not be representative of the entire population of individuals.
Individuals that know they have poor health (and lower life expectancy) have a greater incentive to apply for life insurance.

97. Hence, insurers will put conditions in place to prevent adverse selection.
Individuals applying for life insurance will be required to submit to a medical exam.
Group life insurance paid for by an employer and covering all employees avoids adverse selection because of its mandatory coverage. Medical exams are unnecessary if the employees are representative of an overall population.
As individuals’ financial and tax planning has become more complex, and as insurance companies have faced greater competition from banks and other financial institutions, the variety of life insurers’ products has expanded.

98. Insurance policies
Term life: In return for periodic premium payments over a fixed term of coverage, the policyholder receives a (tax-exempt) payment contingent on death. Most term contracts are renewable (do not require additional medical exam after the first coverage period) up to a maximum age. Competition, especially from internet, has reduced premiums.
Whole life: Combines mortality insurance and a savings plan. Policy holder makes level periodic premium payments whose value is above (below) the fair term premium in early (late) years. The accumulated overpayments accrue interest at a fixed rate and equal the policy’s “cash value.” Policyholder receives a payment at death or at the maturity of the policy. Policy can be “surrendered” for cash value. Cash value in excess of premiums paid is taxable.

99. Universal life: Like whole, but policyholder can vary premium
Universal life: Like whole, but policyholder can vary premium. Accumulated overpayments (cash value) accrue interest at a rate that varies with market interest rates. Policyholder can withdraw some of cash value or change the amount of premium allocated to mortality insurance.
Variable life: Accumulated overpayments (cash value) are invested in mutual funds chosen by the policyholder. Hence, the cash value’s rate of return is risky.
Disability: Provides monthly income benefit in the event of loss of income from an accident or illness.
Health: Insurers often write health insurance policies, competing with Blue Cross/Blue Shield Associations (nonprofits sponsored by hospitals). To prevent moral hazard, insurers administer and cover only catastrophic costs for employer-sponsored plans. For HMO plans, insurers pay HMOs a fixed payment per person, giving HMOs the incentive to reduce health care costs.

100. Annuities
In return for the annuity holder (annuitant) making an investment, the insurance company promises a series of payments starting immediately (immediate annuity) or at some future date (deferred annuity).
The return on this investment is tax deferred: taxes are due when payments are received. These payments are specified over a fixed period of time or for the life of the annuitant.
Fixed annuity: Fixed interest is paid on the annuity’s principal.
Variable annuity: Return on principal varies with a mutual fund return, thereby affecting the value of the ultimate payments made by the insurance company to the annuitant.
To avoid adverse selection, some pension plans require all employees to purchase lifetime annuities.

101. Pension Fund Management:
Because of their expertise in managing long-maturity liabilities (e.g., life insurance policies), life insurance companies manage some employer’s pension plans.
A popular product that is often offered to pension plans is a Guaranteed Investment Contract (GIC). Essentially, it is a long-maturity, zero-coupon bond issued by the insurance company to the pension plan. It guarantees a specific rate of return over the life of the contract (maturity of the bond).