# Risk-Averse Utility Function

```Fall 2008 Version
Professor Dan C. Jones
FINA 4355
Risk Management and Insurance: Perspectives in a Global Economy
2. Risk Perceptions and Reactions
Professor Dan C. Jones
FINA 4355
Study Points
Decision-making under uncertainty:
Individuals
Society
Economic Theories
3
Individual Decision-making under Uncertainty
The Case
John
Received an inheritance of \$10,000 from an uncle
Unsure of what to do with the money for investment purpose
Decision-making theories
Expected value (function)
Expected utility (function)
Risk-averse expected utility (function)
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Risk Decisions and the Expected Value
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Risk Decisions and the Expected Value
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Risk Decisions and the Expected Value
The expected value (EV) of a set of n possible outcomes
equals the sum of these outcomes, each outcome (xi)
weighted by the probability of its occurrence (pi):
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Risky Decisions and the Expected Utility Rule
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Risk-Averse Utility Function
A utility function for the risk-averse person exhibits two
characteristics:
Increasing wealth leads to increasing levels of satisfaction.
The marginal utility of wealth decreases as wealth increases.
 The law of diminishing marginal utility as shown Figure 2.1
Risk-averse individuals will select the option with the highest
expected utility.
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Risk-Averse Utility Function (Figure 2.1)
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Risk-Averse Utility Function
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Case
Maria
Owns a home worth \$150,000 in the Philippines
Has \$50,000 in a savings account
The earthquake
Probability of 10% in any given year
An earthquake will totally destroy her house
XYZ Insurance Company
Fully indemnifies insureds for earthquake-related losses
Charges a premium equaling the expected value of the loss
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Expected Utility with Insurance – Maria
Premium equaling the expected value of the loss
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Expected Utility with Insurance – Maria
Should Maria buy the insurance policy?
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Maximization
Insurance Demand and Lifetime Utility Maximization
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Theories of Consumption
Life cycle hypothesis  Figure 2.4
The typical individual’s income:
Low in the beginning and end stages of life
High during the middle stage of life
The individual maintains a constant or increasing level of
consumption.
Consumption theories and insurance
“Risk-averse” individuals increase their expected lifetime utility by the
purchase of:
Life insurance to provide payments on death and
Annuities to provide payments during retirement
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Life Cycle of Consumption and Saving (Figure 2.4)
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Economic Theories of Consumption
Prospect Theory: Loss Aversion and the Value Function
Kahneman and Tversky
Each of us has a personal value function that reflects our degree of
satisfaction derived from gains and losses from some reference point.
The reference point is each individual’s point of comparison or
standard against which risky decisions are contrasted.
This point is not static and can vary for the same individual
depending on how the choices are framed or presented to the
individual.
The Kahneman–Tversky value function
Concave in gains, as with a utility function, but convex in losses
Figure 2.5
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Illustrative Prospect Theory Value Function
(Figure 2.5)
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Probability Weighting Function (Figure 2.6)
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Other Anomalies
Decision regret
our failures to have made smart choices.
Mental accounting
We tend to separate a whole into components.
Endowment effect
The tendency to set a higher price to sell that which we already own
than what we would be willing to pay to purchase the identical item if
we did not own it.
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Sole proprietorship
Partnership
General partners
Limited partners
Corporation
Limited liability of its owners
Easy transfer of ownership
Continuity of existence
Closely held corporations
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Sole Proprietorship and Partnership Reactions to Risk
Sole Proprietorship
Being indistinguishable from the individual who owns it, the sole
proprietorship’s risk perception and behavior logically is that of its
owner.
Partnership
Its risk perception and behavior reflect some combination of the
owners’ degrees of risk aversion. An exception can occur
If the partnership interest is held by many partners, has a ready
market or is a small portion of the partners’ overall investment
portfolio, or
If the partners have a sound understanding of how the value of
their partnership interest changes with changes in the value of
other investments
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Corporate Reactions to Risk
The goal of the firm and owner risk profiles
By assembling their own portfolio of financial assets, owners can
determine their own consumption and risk profiles.
They want managers simply to maximize net present values, acting
as risk-neutral agents for the owners and undertaking every project
whose net present value is positive, irrespective of its risk.
Risk-neutral utility function
Increasing wealth leads to increasing levels of satisfaction.
The marginal utility of wealth is constant as wealth increases.
Figure 2.7
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Risk-Neutral Utility Function (Figure 2.7)
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Why Do Corporations Manage Risks?
Managerial self-interest
Principal-agent problems
Corporate taxation
Refer also to Chapter 19
(the Economic
Foundations of Insurance)
Cost of financial distress
Bankruptcy risk and cost vs. the cost of insurance
Capital market imperfections
High transaction costs associated with external finance
Imperfect information about firm riskiness by those who might provide
the external finance
The high cost of financial distress
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Societal Decision-making under Uncertainty
Economic Efficiency as a Social Goal
Just as individuals seek to maximize their welfare (i.e., their
individual utilities), so do societies want to maximize their
welfare.
Because of differences in individual preferences, there is no such
thing as a societal utility function. Economics offers an alternative.
Efficient allocation of resources
Individuals and businesses should undertake risk management (and
all other) actions so long as the marginal benefit is greater than the
marginal cost.
Economists measure benefits based on the concept of “willingness to
pay.”
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Economic Efficiency as a Social Goal
Two problems
Even if it’s efficient, is it fair?
Society may not necessarily prefer a Pareto efficient allocation of
its resources
Even if it’s fair, is it efficient?
Market prices do not always equal opportunity costs.
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Imperfections in Markets – Market Failures
Some goods or services may be unavailable or available
only in some suboptimal way.
Four general classes of problems
Market power
Externalities
Free rider problems
Information problems
These classes of
problems are discussed
throughout the book.
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Market Power
The ability of one or a few sellers or buyers to influence the
price of a product or service
Causes
Governmentally created barriers to entry
Economies of scale
Product differentiation/price discrimination
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Market Power – Governmentally Created Barrier
Market power arises when a market has entry or exit barriers
and few sellers.
Monopoly
Oligopoly
National tax regimes can lead to the creation of market
power.
In financial services, national licensing requirements
technically are entry barriers, although they may be justified
on consumer protection grounds.
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Market Power – Economies of Scale
Economies of scale exist when a firm’s output increases at a
rate faster than attendant increases in its production costs.
Minimum efficient scale (MES) at which long-run average
costs are at a minimum; further growth yields no additional
efficiencies
If efficiency increases over an industry’s entire relevant output range,
the MES is so large relative to market size that only one firm can
operate efficiently—a so-called natural monopoly case.
A monopolist or oligopolist unable to exercise market power
if the market is contestable
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Market Power – Product/Price Differentiation
Product differentiation
When a large number of firms produce similar but not identical
products, they are engaged in monopolistic competition which gives
the firms an element of monopoly power (i.e., the ability to influence
price).
Price discrimination
Firms offering identical products at different prices to different groups
of customers
Predatory pricing  also known as dumping
Lowering prices to unprofitable levels to weaken or eliminate
competition with the idea of raising prices after competitors are driven
from the market
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Externalities
Benefits or costs that occur when a firm’s production or an
individual’s consumption has direct and uncompensated
effects on others
Positive externalities
Negative externalities
Societal risk management is particularly concerned about
negative externalities.
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Externalities – Nature
The purely competitive economic model does not
accommodate externalities easily, because the market prices
of goods and services that carry externalities fail to reflect
their true (opportunity) costs.
Here lies the problem with allowing competitive markets to
deal freely with goods and services that carry externalities.
With negative externalities  too much of the good or service
produced or consumed  the price becoming too low  too little
effort and resources devoted to correct/reduce the externality.
With positive externalities  too little of the good or service produced
 the price becoming too high  too little effort devoted to
enhancing the externality.
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Externalities – The Importance of Property Rights
Negative externalities (e.g., pollution) can persist in
competitive markets because the persons adversely affected
by the negative spillovers have poorly defined, dispersed or
no property rights.
Too often, property rights are not well established or they are
widely dispersed, thus precluding meaningful actions by
private citizens against the polluter.
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Free Rider Problems
Some collectively consumed goods and services that are
desired by the public—called public goods—carry extensive
positive externalities. Examples are:
Public education
Lighthouse
Police and fire protection services
When such goods and services are available to others at low
or zero cost, they can cause a free rider problem.
Left to itself, a competitive market is unlikely to provide as
much of public goods as society really wants.
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Information Problems
Information problems occur when buyers (or sellers) lack
sufficient information to make an informed purchase (or
sales) decision.
Markets that suffer such information asymmetries often are
regulated if the goods or services involved are important
elements of our lives or the economy.
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Information Problems – Asymmetric Information
The problems arise when one party to a transaction has
relevant information that the other does not have.
A Lemons problem when the buyer knows less than the seller about
the seller’s products
A principal-agent (or agency) problem when the buyer of services
knows less about its agent’s actions than does the agent
An adverse selection problem when the seller knows less than the
Moral hazard is the propensity of individuals to alter their behavior
when risk is transferred to a third party.
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Information Problems – Asymmetric Information
The additional expenses incurred to become better informed and
The additional costs inherent in making decisions with less
information
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Information Problems – Non-existent Information
Neither the buyer nor the seller has complete information
because desired information simply does not exist.
This uncertainty leads them to take ameliorating actions intended to
reduce their risk exposure.
These offsetting actions require the expenditure of additional
resources, thus decreasing overall benefits to society.
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Information Problems – Non-existent Information
These situations can be addressed through actions such as
diversification and creation by governments of various
“safety nets” for its citizens
One of the premises for social insurance programs is that
individuals will not or cannot fully arrange for their own
financial security
 So government must force them to do so.
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Discussion Questions
Discussion Questions 1 &amp; 2
“If individuals were not risk averse, insurance would not
exist.” Do you agree with this statement?
“If individuals were not risk averse, risk management would
not exist.” Do you agree with this statement?
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Discussion Question 3
Why would we ordinarily expect corporations whose shares
were widely held to be risk neutral?
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Discussion Question 4
Even corporations whose shares are widely held often seem
to be risk averse.
Offer some sound economic reasons for such corporate behavior.
Offer some practical reasons for such corporate behavior.
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Discussion Question 5
Justify the following statement: “If externalities did not exist,
society would have no worry about pollution.”
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Discussion Question 6
“If a market is operating with reasonable efficiency,
government should leave it alone.” Under what
circumstances would you (a) agree and (b) disagree with this
contention?
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