1. Risk & Uncertainty in Decision Making1

2. Topics to be discussed
Describing & Differentiating Risk from Uncertainty
Role of Risk in Crime Deterrence
Preferences Toward Risk
Mechanisms for Reducing Risk
Demand for Risky Assets (optional) 2

3. Describing & Interpreting Risk
To measure risk one must know:
1) All of the possible outcomes.
2) The likelihood that each outcome will occur (its probability).
In case of uncertainty, either (1) or even both (1) and (2) are unknown.
Objective Interpretation
Based on the observed frequency of past events
Subjective Interpretation
Based on perception or experience with or without an observed frequency
Different information or different abilities to process the same information can influence the subjective probability 4

4. Describing Risk: Expected Value
The weighted average of the payoffs or values resulting from all possible outcomes.
The probabilities of each outcome are used as weights
Expected value measures the central tendency; the payoff or value expected on average
E(X) = p1x1 + p2x2 + …..+ pnxn 7

5. Standard deviation measuring risk
While the expected values are the same, the variability is not.
Greater variability from expected values signals greater risk.
Deviations or differences between expected payoff and actual payoff are important
The standard deviation measures the square root of the average of the squares of the deviations of the payoffs associated with each outcome from their expected value. 16

6. Why SD important? f(X), g(Y) = prob. densities X, Y
Which distribution would you
prefer, if you are risk-averse?
X, Y
E(X)=E(Y), but sd(X) >sd(Y)

7. Example of a risky situation
Job 1 is a job in which the income ranges from $1000 to $2000 in increments of $100 that are all equally likely.
Job 2 is a job in which the income ranges from $1300 to $1700 in increments of $100 that, also, are all equally likely.
Job 1: greater spread & standard deviation
Job 2: peaked distribution - extreme payoffs are less likely
Decision Making
A risk avoider would choose Job 2: same expected income as Job 1, but with less risk. 24

8. Role of Risk in Crime Deterrence
Suppose, Extent of crime=r
Gain per unit of crime=g
Probability of detection & conviction =P(r)
Cost to guilty if detected & convicted=C
Hence, optimizing exercise of the criminal is to
Maximize E(net gain) = g.r – P(r).C w.r.t. r
FOC=> g – P’(r).C = 0 => g = P’(r).C => r = r (g, C, P)
(+) (-) (-)
SOC=> -P’’(r)<0 => P’’(r)>0 => you need some curb to keep crime rate r at finite level (check C(r) with C’’(r)>0 and/or g(r) with g’’(r)<0 can serve similar purpose)
Observations: (i) A rise in P and drop in C or a drop in P with rise in C can serve the same purpose; (ii) With very high C, E(net gain) < 0 even with low P(r) close to zero; (iii) Societal cost to make P(r) = 1 or even close to 1 is very high; (iv) If P(r) = 0 then E(net gain) >0 for all values of r & r tends to be infinite.
Why then capital punishment sometimes doesn’t work as deterrent to even heinous crimes? (look into text of example 6.2 for answer)

9. Role of Risk in Crime Deterrence: Figure
g.r, P.C
P(r).C with P’(r)>0 & P’’(r)>0
g.r r
r*=equilibrium level of crime

10. Preferences Toward Risk
Assumptions for choosing among risky alternatives
Consumption of a single commodity
The consumer knows all probabilities
Payoffs measured in terms of utility
Utility function given
Defining risk averseness – i.e., preferences toward risk
A person who prefers a certain given income to a risky income with the same expected value.
A person is considered risk averse if he has a diminishing marginal utility of income
The use of insurance demonstrates risk aversive behavior
If an individual gets more utility from his present lower-paid job than a higher-paid risky job, he is said to be risk averse 34

11. Risk Averse Preferences Toward Risk
Utility
The consumer is risk
averse because she
would prefer a certain
income of $20,000 to a
gamble with a 0.5 probability
of $10,000 and a 0.5
probability of $30,000. E 10 15 20 13 14 16 18 30 A B C D
Risk averseness means
gain valued less than
loss at the margin
Income ($1,000) 46

12. Risk Neutral Preferences Toward Risk6 A E C 12 18
The consumer is risk
neutral as he is indifferent
between certain events
and uncertain events
with the same
expected income.
Utility
A person is said
to be risk neutral
if he shows no
preference
between a certain
income, and an
uncertain one
with the same
expected value.
Risk neutrality means
gain valued same as
loss at the margin
Income ($1,000) 10 20 30 49

13. Risk Loving Preferences
A person is said to be risk loving if he shows a
preference toward an uncertain income over a certain income with the same expected value. Examples: gambling, criminal activity
Utility 3 10 20 30 A E C 8 18
The consumer is risk
loving because she
would prefer the gamble
to a certain income.
Read example 6.3
carefully to understand why the US is becoming a nation of gamblers & poor nations too
are prone to the
same virus!
Risk loving behavior means
gain valued more than
loss at the margin
Income ($1,000) 52

14. Notion of Risk Premium
The risk premium is the amount of money that a risk-averse person would pay to avoid taking a risk.
R = U[E(Y)] – E[U(Y)] in utility terms
It can also be measured along the horizontal axis in terms of money
The greater the variability – i.e., the spread of values of the random variable around mean, the greater the risk premium.
The greater the concavity of the utility curve, the greater the risk premium – it means a much higher weight is assigned to a decline in income as compared to an equal amount of rise in income 53

15. Estimating Risk Premium10 16
Here , the risk premium
is $4,000 because a
certain income of $16,000
gives the person the same
expected utility as the
uncertain income that
has an expected value
of $20,000. 18 30 40 20 14 A C E G F
Risk Premium
in money terms
Utility
Risk premium in
utility terms
Income ($1,000) 57

16. Meaning of High Degree of Risk Aversion
Here increase in
Standard deviation
requires a large
increase in income
to maintain
satisfaction. U1 U2 U3
Expected
Income
Standard Deviation of Income

17. Meaning of Low Degree of Risk Aversion
Expected
Income
Slightly Risk Averse =>
A large increase in standard
deviation requires only a
small increase in income
to maintain satisfaction. U1 U2 U3
Standard Deviation of Income

18. Indifference curves of risk-neutral & risk-loving people
Risk-neutrality means horizontal indifference curve showing zero premium the consumer is willing to pay to buy hedge against risk (i.e., he doesn’t mind bearing more risk)
Risk-lover willing to pay (i.e., make sacrifice in terms expected income) to enjoy the thrill of greater risk-bearing – thus making indifference curves usual downward sloping

19. Utility function of a person who buys insurance and gambles too!
Total utility
TU curve
Near point of inflection, A (MU
at its minimum), a rise averter
will spend small amount to buy
insurance against small chance of
large loss, while as a risk seeker
spend a small amount (on lottery)
to seize small chance of a large
gain.
Convex to horizontal axis A
Concave to horizontal axis
Income

20. Risk Reduction Strategies
Three ways consumers attempt to reduce risk are:
1) Diversification
2) Insurance
3) Obtaining more information
Firms can reduce risk by diversifying among a variety of activities that are not closely related. 67

21. Reducing Risk thro’ Diversification
Reducing risk by allocating resources to a variety of activities whose outcomes are not closely related
Example:
Suppose a firm has a choice of selling air conditioners, heaters, or both
The probability of it being hot or cold is 0.5
How does a firm decide what to sell? 68

22. Income from Sales of Appliances
Hot Weather
Cold Weather
Air conditioner sales
$30,000
$12,000
Heater sales
12,000
30,000 69

23. Diversification – Example
If the firm sells only heaters or air conditioners their income will be either $12,000 or $30,000
Their expected income would be:
1/2($12,000) + 1/2($30,000) = $21,000
If the firm divides their time evenly between appliances, their air conditioning and heating sales would be half their original values
If it were hot, their expected income would be $15,000 from air conditioners and $6,000 from heaters, or $21,000
If it were cold, their expected income would be $6,000 from air conditioners and $15,000 from heaters, or $21,000
With diversification, expected income is $21,000 with no risk
Better off diversifying to minimize risk
Firms can reduce risk by diversifying among a variety of activities that are not closely related 70

24. Reducing Risk – The Stock Market
If invest all money in one stock, then take on a lot of risk
If that stock loses value, you lose all your investment value
Can spread risk out by investing in many different stocks or investments
Ex: Mutual funds 73

25. Choosing an Investment Portfolio
Rate of return(%)
Highest possible indifference curve
Risk-return frontier
F=high return, high risk investment
C=optimum portfolio choice
E= less return, less risk investment
Risk

26. Reducing Risk – Insurance
Risk averse are willing to pay to avoid risk
If the cost of insurance equals the expected loss, risk averse people will buy enough insurance to recover fully from a potential financial loss
For the risk averse consumer, guarantee of same income regardless of outcome has higher utility than facing the probability of risk
Expected utility with insurance is higher than without
Purchases of insurance transfers wealth and increases expected utility
Actual Fairness: When the insurance premium = expected payout 74

27. Value of Title Insurance When Buying a House : An Example
In the absence of title insurance, a risk averse buyer would pay much less for the house
By reducing risk, title insurance thus increases the value of the house by an amount far greater than the premium. 77

28. The Decision to Insure – An example75

29. How does insurance work? The Law of Large Numbers
Insurance companies know that although single events are random and largely unpredictable, the average outcome of many similar events can be predicted
When insurance companies sell many policies, they face relatively little risk
Insurance companies can be sure total premiums paid will equal total money paid out
Companies set the premiums so money received will be enough to pay expected losses 77

30. Why no insurance against certain events?
Some events with very little probability of occurrence such as floods and earthquakes are no longer insured privately
Cannot calculate true expected values and expected losses
Governments have had to create insurance for these types of events
Ex: National Flood Insurance Program 77

31. The Value of Information
Risk often exists because we don’t know all the information surrounding a decision
Because of this, information is valuable and people are willing to pay for it
The value of complete information
The difference between the expected value of a choice with complete information and the expected value when information is incomplete 79

32. The Value of Information – Example
Per capita milk consumption has fallen over the years
The milk producers engaged in market research to develop new sales strategies to encourage the consumption of milk
Milk advertising increases sales most in the spring
Allocating advertising based on this information in New York increased profits by 9% or $14 million
The cost of the information was relatively low, while the value was substantial (increased profits)
Findings
Milk demand is seasonal with the greatest demand in the spring
Price elasticity of demand is negative and small
Income elasticity is positive and large 85

33. Demand for Risky Assets
Assets: Something that provides a flow of money or services to its owner. However, the flow of money or services can be explicit (dividends) or implicit (capital gain).
Capital Gain: An increase in the value of an asset, while a decrease is a capital loss.
Risky Asset: Provides an uncertain flow of money or services to its owner. Examples: apartment rent, capital gains, corporate bonds, stock prices
Risk-free Asset: Provides a flow of money or services that is known with certainty. Examples: short-term government bonds, short-term certificates of deposit
Return on an Asset: The total monetary flow of an asset as a fraction of its price.
Real Return of an Asset: The simple (or nominal) return less the rate of inflation.
Asset Return: Return = Money flow / bond price 86

34. Trade-Off Between Risk and Return
Higher returns are associated with greater risk.
The risk-averse investor must balance risk relative to return
An investor is choosing between T-Bills and stocks:
T-bills (risk-free) versus Stocks (risky)
Rf = the return on risk free T-bills
Expected return equals actual return when there is no risk
Rm = the expected return on stocks
rm = the actual returns on stock
The risky asset will have a higher expected return than the risk free asset (Rm > Rf).
Otherwise, risk-averse investors would buy only T-bills. 94

35. The Investment Portfolio
How to allocate savings:
b = fraction of savings in stock market
1 - b = fraction in T-bills
Expected Return:
Rp: weighted average of the expected return on the two assets
Rp = bRm + (1-b)Rf
If Rm = 12%, Rf = 4%, and b = ½, then
Rp = 1/2(.12) + 1/2(.04) = 8% 97

36. The Investment Portfolio
Risk (standard deviation) of the portfolio is the fraction of the portfolio invested in the risky asset times the standard deviation of that asset: σp = bσm
Rp = bRm + (1-b)Rf => Rp = Rf + b(Rm-Rf)
=> Rp = Rf + (σp/σm)(Rm-Rf), which is a budget line describing the trade-off between risk (σp) and expected return (Rp).
Observations:
This is a straight line with Rp, Rf and σm as constants;
Its slope= (Rm-Rf)/σm;
Expected return, RP, increases as σp - i.e., risk increases. 99

37. Choosing between Risk and Return
U2 is the optimal
choice of those
obtainable, since it
gives the highest
return for a
given risk and is
tangent to the
budget line.
Expected
Return,Rp Rf
Budget Line Rm R* U2 U1 U3
109

38. The Choices of Two Different Investors
Can equilibrium occur at a point to the right
of point M? Under what circumstances?
Expected
Return,Rp
Given the same budget
line, investor A chooses
low return-low risk,
while investor B
chooses high return-
high risk. UA RA UB RB Rm
Budget line
How will this figure
change for a
risk-lover? Indifference curves will have the usual shape for two goods, rather than one good and one bad. M Rf
113

39. Hedging against Foreign Exchange Risk
Read the example in pp for details about
Forward contract &
Futures contract (standardized forward contract for predetermined quantities & selected calendar dates)
What happens if US$ & Yen continues to depreciate vis-à-vis Indian rupee?

40. Investing in the Stock Market: Example
Observations
Percent of American families who had directly or indirectly invested in the stock market
1989 = 32%
1995 = 41%
Share of wealth in the stock market
1989 = 26%
1995 = 40%
Participation in the stock market by age
Less than 35
1989 = 23%
1995 = 29%
More than 35
Small increase

41. Behavioral Economics
Sometimes individuals’ behavior contradicts basic assumptions of consumer choice
More information about human behavior might lead to better understanding
This is the objective of behavioral economics
Behavioral economics is a developing field to help explain and elaborate on situations not well explained by the basic consumer model
Improving understanding of consumer choice by incorporating more realistic and detailed assumptions regarding human behavior

42. Behavioral Economics
There are a number of examples of consumer choice contradictions
You take at trip and stop at a restaurant that you will most likely never stop at again. You still think it fair to leave a 15% tip rewarding the good service.
You choose to buy a lottery ticket even though the expected value is less than the price of the ticket