Uncertainty and Information

1
Chapter 5

• Uncertainty and Information

2
Probability

• The probability of an event happening is, roughly
  speaking, the relative frequency with which an
  event occurs.
• For example, the probability of a head coming up
  on a flip of a fair coin is ½.
• That is, when a coin is flipped a large number of
  times, we can expect a head to come up in
  approximately one-half of the flips.

3
Expected Value

• The expected value of a game with a number of
  uncertain outcomes is the size of the prize the
  player will win on average.
• If, for example, on a single flip of a coin,
  Jones pays Smith 1
• (X1 1) if a tail comes up and Smith
  will pay Jones 1
• (X2 -1) if a head comes up, the expected
  value of the game is

4
Expected Value

• If the game was changes so that, from Smiths
  point of view, X1 10, and X2 -1, the
  expected value for Smith would be
• Because Smith would stand to win 4.50 on
  average, she might be willing to pay Jones up to
  this amount to play.
• Fair games are games that cost their expected
  value.

5
APPLICATION 15.1 Blackjack Systems

• Each player in Blackjack is dealt two cards (with
  the dealer playing last).
• The dealer asks each player if he or she wishes
  another card.
• The player getting a hand that totals closest to
  21, without going over 21, is the winner.
• If the receipt of a card puts a player over 21,
  that player automatically loses.

6
APPLICATION 15.1 Blackjack Systems

• Played this way, the game offers a number of
  advantages to the dealer.
• The dealer plays last, so other players can go
  over 21 (and lose) before the dealer plays.
• Usually, the dealer also wins ties.
• The dealer has a winning margin of 6 percent on
  average, with the player winning 47 percent of
  the hands and the dealer 53 percent.

7
APPLICATION 15.1 Betting Systems

• To entice more people to play, casinos have eased
  the rules.
• At many Las Vegas casinos, ties result in bets
  being returned to the players.
• With these rules, the dealers advantage falls to
  as little as 0.1 percent.
• If players use systems, such as card counting,
  they may win over 50 percent of the time.

8
APPLICATION 15.1 Casinos Response

• To deal with card counting, Las Vegas casinos
  have made several rule changes.
• They use multiple decks to make counting more
  difficult.
• They refuse admission to known system players.
• This illustrates that small changes in expected
  values can have important implications.

9
Risk Aversion

• When people are faced with a risky but fair
  situation, they will usually choose not to
  participate.
• Risk aversion is the tendency for people to
  refuse to accept fair games.
• A Swiss mathematician, Daniel Bernoulli,
  theorized that it is not strictly the monetary
  payoff of a game that matters to people, but the
  expected utility from the games prizes.

10
Diminishing Marginal Utility

• Specifically, Bernoulli assumed that the utility
  associated with the payoffs in a risky situation
  increases less rapidly than the dollar value of
  these payoffs.
• The extra (or marginal) utility that winning an
  extra dollar in prize money provides is assumed
  to decline as more dollars are won.

11
Diminishing Marginal Utility

• Diminishing marginal utility is reflected in
  Figure 5.1, which shows the utility associated
  with possible prizes (or incomes) from 0 to
  15,000.
• The concave shape of the curve reflects assumed
  diminishing marginal utility.
• The gain in utility due to an increase in income
  from 1000 to 2000 exceeds the gain from
  14,000 to 15,000.

12
FIGURE 5.1 Risk Aversion
1. retain current level of income (10,000)
without taking any risk 2. take a fair bet with
a 50-50 chance of winning or losing 2,000 3.
take a fair bet with a 50-50 chance of winning
or losing 5,000
Utility
U
0
10
12
15
8
5
Income (thousands of dollars)
13
A graphical Analysis of Risk Aversion

• The current 10,000 provides utility of U3.
• The utility of the 2,000 bet is the average of
  the utility of 12,000 (if he or she wins) and
  the utility of 8,000 (if he or she loses).
• This average utility is U2 lt U3.
• The utility (U1 lt U2) of the 5000 bet is the
  average of the utility of winning (15,000) and
  losing (5,000).

14
FIGURE 15.1 Risk Aversion
Option 1 The current 10,000 provides utility of
U3.
Utility
U
U3
U2
U1
0
10
12
15
5
8
Income (thousands of dollars)
15
FIGURE 15.1 Risk Aversion
Option 2 The utility of the 2,000 bet is the
average of the utility of 12,000 (if he or she
wins) and the utility of 8,000 (if he or she
loses). This average utility is U2 lt U3.
Utility
U
U3
U2
U1
0
10
12
15
5
8
Income (thousands of dollars)
16
FIGURE 15.1 Risk Aversion
Option 3 The utility (U1 lt U2) of the 5000 bet
is the average of the utility of winning
(15,000) and losing (5,000).
Utility
U
U3
U2
U1
0
10
12
15
5
8
Income (thousands of dollars)
17
Willingness to Pay to Avoid Risk

• With risk aversion and equal expected values
  people will prefer risk-free incomes to risky
  incomes which offer less utility.

18
FIGURE 5.2 Risk Aversion
A risk-free income of 9,500 provides the same
utility as the 2,000 gamble.
Utility
U
The person would pay up to 500 to avoid the risk
of the 2000 gamble.
U3
U2
U1
0
10
12
15
8
5
Income (thousands of dollars)
9.5
19
Methods of Reducing Risk Insurance

• Definition An insurance premium is a (regular)
  payment by an insurance policy customer who gets
  compensated for the expenses he/she has to bear
  in case of an accident.
• Types of risk insured against
• Hands of surgeons / basketball players /
  musicians
• Injuries to falling parachutists for the swimming
  pool owners
• Earthquake insurance
• Definition Insurance for which the premium is
  equal to the expected value of the loss is called
  fair insurance

20
Methods of Reducing Risk Insurance

• Assume that a person with a 10,000 current
  income, faces a 50 percent chance of having
  4,000 in unexpected medical bills.
• Without insurance, this persons utility would be
  U1, the utility of the average of 6000 and
  10,000.

21
Insurance Reduces Risk
Utility
Expected Utility
U
U1
Current Income
Income in case of an accident
Income (thousands of dollars)
0
6
10
22
Fair Insurance

• What is the fair insurance premium in our
  example?
• The insurance company will pay 4000 with
  probability of 50 each year
• Expected value of payment to the insured person
  is 2000 every year
• Thus, fair insurance premium must also be equal
  to 2000

23
Fair Insurance

• Only two outcomes are possible
• No accident the insured pays 2000 to the
  company, and his income is 800010000-2000
• Accident happens
• the insured pays 4000 in medical expenses
• The insurance company pays 4000 back to the
  insured
• Annual premium is still 2000
• The income of the insured is the same
  800010000-2000
• Main point under fair insurance, the insured
  enjoys a certain income (8000) whether an
  accident happens or not

24
Insurance Reduces Risk
Utility
U
U2
U1
Utility in case of fair insurance
Certain income in case of fair insurance
Utility in case of no insurance
Income (thousands of dollars)
0
8
7.5
6
10
25
Unfair Insurance

• Insurance companies have costs
• Records maintenance
• Collection of premiums
• Investigate against fraud
• Computing probabilities of accidents
  (complicated!)
• These costs have to be added to the fair
  insurance premium
• As a result, the actual insurance premiums are
  unfair
• How much would an insured person be willing to
  pay on top of the fair premium to still prefer
  being insured (compared to no insurance)?

26
Insurance Reduces Risk
Utility
Utility if insurance premium is 2500
U
U2
U1
Utility if no insurance
U0
Utility if insurance premium is 3500
2500
Income (thousands of dollars)
0
8
7.5
6.5
6
10
27
Insurance and Risk Aversion

• Risk-averse individuals will always buy insurance
  unless the insurance premium exceeds the expected
  value of a loss by too much
• Risk lovers, however, will prefer not to get
  insured

28
Uninsurable Risks

• Three types of events are typically uninsurable
• Rare events
• Mars invasion
• Collision with another planet
• Since probability of an accident is not
  computable, no insurance is provided
• Adverse selection
• Insurance buyers have more information about the
  risks involved
• Insurance companies charge a lower premium than
  is justified by the risks
• If controlling for adverse selection is
  impossible, insurance will not be sold
• Moral hazard
• Ex-post behavior taking less care of ones
  health, drunk driving etc
• Insurance premiums may be so high that there will
  be no demand for insurance

29
Deductibles in Insurance

• Definition. A requirement in an insurance policy
  that the insured pay the first X dollars in case
  of an accident, after which the rest of the
  expenses is covered by the insurance company, is
  called a deductible provision.
• Example Your expenses in case of a car accident
  are 2000. If your deductible provision is 500,
  the insurance company only pays you 1500, while
  you have to pay the deductible of X500
• Rationale for deductibles
• Administrative costs may exceed the size of the
  claim by a lot (e.g. 100 in paper work for a
  claim of 12)
• Such imbalance is likely to inflate the insurance
  premium in case the policy also covers small
  claims
• Deductible provisions make applying for small
  claims less attractive

30
Methods of Reducing Risk Diversification

• Diversification is the economic principle
  underlying the adage, Dont put all your eggs in
  one basket.
• Definition Diversification is the spreading of
  risk among several options rather than choosing
  only one.
• Suitably spreading risk around may increase
  utility above that obtain by a single action.

31
Investing in One Stock
Buy 4000 stocks of one company at 1 One year
later the price is 2 or 0 with a 50 probability
Utility
U
U1
Expected utility of investing in one stock
Income if stock is worthless
Income if stock price is 2
Income (thousands of dollars)
0
10
14
6
32
Possible Outcomes from Investing in Two Companies

• Company A is identical to company B in terms of
• Initial stock price of 1
• End-of-year stock price of
• 2 with probability 50
• 0 with probability 50

33
Diversification

• Diversification means investing in two stocks
  simultaneously by e.g. splitting the 4000 into
  two equal parts for stock A and stock B
• Diversification makes sense even if the two
  companies are performing in an identical fashion
  in terms of the payoffs and probabilities

34
Diversification Reduces Risk
Utility
Expected utility in all cases the average of
utility at C and at D
D
U
U2
C
U1
Expected utility if B stock is 2
Expected utility if B stock is 0
Expected utility of investing in a single stock
Income (thousands of dollars)
0
10
14
6
35
Mutual Funds Diversification

• Definition A financial institution that pools
  money from many investors to buy shares in
  several different companies is called a mutual
  fund.
• Investors pay an annual management fee to the
  fund managers of 0.51.5 of invested money
• Rationales to invest with a mutual fund
• Expertise (VERY doubtful)
• Diversification brokerage commissions would eat
  up most of investment funds if one investor were
  to buy shares in 100 companies
• With many investors, brokerage commissions get
  spread
• Brokerage commissions are also lower because of
  the large volumes involved
• Diversification reduces risk!

36
Mutual Funds Portfolio Management

• Reducing risk
• Diversification (mentioned already)
• Choosing stocks moving in opposite directions
• Stocks of mining companies tend to rise when
  stock market in general falls
• Choosing stocks from different countries
• Financial derivatives
• Put and call options
• Stock index options
• Interest rate futures
• Computer trading bots

37
Index Funds

• 1970s index funds were introduced to mimic the
  performance of market average
• Standard and Poors 500
• Dow Jones Industrial Average
• Wiltshire 5000 Stock Average
• Very low management cost
• Index funds incur lt0.25 of their assets in
  management fees versus 1.3 in the actively
  managed funds

38
Options

• Definition. A contract offering the right, but
  not the obligation, to complete an economic
  transaction over a specific period.
• Options are another way to reduce risk.
• Imagine buying a second-hand car as-is
• Buy a second-hand car with a money-back guarantee
  (i.e. an option to sell it back in case of
  problems) you would like to pay more for the
  reduction in uncertainty
• Options increase investors flexibility since
  purchasing options does not involve any
  obligations.

39
Options Attributes

• Transaction Specification
• What is being bought or sold
• At what price
• Any other details
• Example. Used cars
• the resale of a car is the specified transaction
• original purchase price is the price
• Additional details may specify the place of
  re-sale, depreciation coverage, gas tank full or
  not etc

40
Options Attributes

• Period during which an option may be exercised
• One-month money-back guarantee for a car
• A stock may be bought (at a specific price) on
  the 1st of June, 2015
• Price of the option
• Explicit
• Implicit when the price of an option is part of
  a larger transaction

41
Valuing an Option Expected Value

• Value of an option has two general dimensions
• Expected value of the transaction
• Variability (risk) of the value of the
  transaction
• Example expected value of the transaction. The
  chance of your car needing a 500 repair is 50
  during the next month.
• Expected value of a repair is 25050x50050x0
  
• If the value of repair were 1000, its expected
  value would be 500
• You are willing to pay more for a transaction to
  resell in the second case

42
Valuing an Option Variability

• Example variability of the transactions value.
  The chance of your car needing a 500 repair is
  50 during the next month.
• Expected value of a reselling transaction is
  25050x50050x0
• Suppose now that 50 chance of a 1000 repair,
  and a 50 chance that your car is worth more than
  you paid for it by 500.
• The car needs a repair re-selling transaction is
  worth 1000 to you with a 50 chance
• The car is great re-selling transaction has a
  negative value to you of -500 with a 50 chance
• Total expected value of a reselling transaction
  is
• 
  50x100050x(-500)250
• Both situations yield the same expected value of
  the transaction. However, the option to resell is
  more valuable in the second case!

43
Valuing an Option Variability

• Remember 50 chance of a 1000 repair, and a 50
  chance that your car is worth more than you paid
  for it by 500.
• Because the transaction of re-selling your car is
  an OPTION, you dont have to resell your car if
  you discover it is worth 500 more than you paid
  for it!
• The expected value of an option after the
  uncertainty is resolved is 50x100050x0500.
  
• 50x0 means you do not re-sell your car if you
  discover it is great
• The expected value of the transaction after
  uncertainty resolution is 500
• This is a greater value compared to 250 in case
  there is a 50 chance that the car needs a 500
  repair
• The more variable the value of the transaction
  specified in the option, the more valuable will
  be the ability to choose to exercise the option
  or not.

44
Duration of an Option

• Longer duration increases an options value
• More time gives you more flexibility
• Compare
• Option to buy 1 gallon of gas tomorrow at todays
  price
• Option to buy 1 gallon of gas one year later at
  todays price
• Interest rates may also affect an options value
• Opportunity costs of investing your spare funds
  elsewhere are higher when the interest rates are
  higher

45
Options and Risk Reduction

• Financial options some definitions
• You may buy a specific stock at a specific price
  in the future call options
• You may sell a specific stock at a specific price
  in the future put options
• European options are the ones with a specific
  exercise date
• American options are the ones that can be
  exercised any time during a specific time period
  e.g. one month
• The price at which an option is exercised is
  called the strike price
• Other options
• Money-back guarantees
• Allowances for upgrades
• After-sale service
• Getting education you pay for an option to get a
  better job in the future

46
Black-Scholes Options Model

• One Microsoft stock is trading now at 25.
• In one month, there are two possibilities
• Microsoft stock is trading at 30 with a 50
  probability
• Microsoft stock is trading at 20 with a 50
  probability
• Consider the following call option
• You have the right (not obligation) to buy one
  Microsoft stock at 27 in one month. Its value to
  you is 0 if stock price is 20, and its value is
  30-273 to you if the Microsoft price is
  trading at 30.

47
Black-Scholes an Equivalent Portfolio

• Definition. A set of financial assets is called a
  portfolio of assets.
• General idea look for a portfolio whose
  performance is the same with your option. This
  kind of portfolio is called an equivalent
  portfolio.
• The value of this equivalent portfolio is often
  considered to be the price of an option.
• .

48
Black-Scholes an Equivalent Portfolio
Example Consider buying k of one Microsoft
stock and taking out a loan of L USD at the same
time. In case the Microsoft stock trades at
20 In case the stock price is 30 Solving
this system results in k30, L6 The cost of
this equivalent portfolio (i.e. 30 of one
Microsoft stock and a 6 loan) is
49
Black-Scholes Theorem

• Black-Scholes theorem allows one to derive the
  value of an option based on
• A variety of stock prices after one month
• American option possibility
• Interest rate of a loan
• Theorem introduced in 1973

50
Pricing of Risk in Financial Assets

• Because people are willing to pay something to
  avoid risks, it seems that one should be able to
  study the process directly.
• We could treat risk like any other commodity
  and study the factors that influence its demand
  and supply.
• With financial assets, the risks people face are
  purely monetary and relatively easy to measure.

51
Risk as Volatility
Standard deviation of annual returns is a range
of those returns around the mean in which roughly
2/3 of all annual returns will be
found. Example. For common stock, the average
annual return is 12.2 for 1926-1994 with the
standard deviation of 20.2 (percentage points,
not percent). Two-thirds of all annual returns
on common stocks during 1926-1994 fell between
12.2-20.2-8, and 12.220.232.4.
52
Market Options for Investors
Market Line
Annual return
C

• Each point is a financial asset
• Risk is standard deviation of an asset
• Market line is the maximum of return for each
  level of risk

B
A
Risk

• Point A is the risk-free asset (bank deposit)
• Mixing bank deposits with risky assets, investors
  move up the market line

53
Equity Premium Puzzle
Total Annual Returns, 1926-1994

• 7 equity premium difference in returns between
  stocks and long-term bonds
• Theory predicts a 1 to 2 premium
• Explanations
• Business cycles low stock returns mean low
  investment returns
• Higher transaction costs on stocks

Financial Asset Average Annual Rate of Return Standard Deviation of Rate of Return
Common stocks 12.2 20.2
Long-term government bonds 5.2 8.8
Short-term government bonds 3.7 3.3
54
Choices by Individual Investors
UIII
UII
Market Line
Annual return

• Indifference curves are positively sloped risk
  is a bad, not a good
• Convex curvature since risk aversion grows
  increasingly fast
• Variety of risk attitudes
• Investor 1 low aversion
• Investor 2 modest toleration
• Investor 3 real speculator

N
UI
M
L
A
Risk
55
The Economics of Information

• Uncertainty originates in lack of information
• Lottery
• Casino
• Weather
• Stock returns
• Foreign currency exchange
• Additional information to reduce uncertainty has
  value
• Car mechanic to evaluate a second-hand car
• Consumer reports magazine
• Consultancy firms for marketing or investment
  advice

56
Uncertain States of the World

• Two future outcomes, or states of the world, are
  possible state 1 and state 2
• Uncertainty as to which outcome will occur
• Consumption is if state 1 occurs
• Consumption is if state 2 occurs
• A particular risk is represented by some
  combination of and

57
Uncertain States of the World and Utility

• At point E there is no uncertainty

Certainty line
C2

• State A is dangerous since in state 2 consumption
  is too low
• Giving up some consumption in state 1 to increase
  consumption in state 2 is one way to decrease the
  uncertainty

E
E
C2
U2
A
A
C2
U1
C1
CE1
CA1
58
Utility Maximization under Uncertain States of
the World
Certainty line
C2
Insurance premium Insurance payment
E
E
C2
Effectively, this person moves from A to E where
there is complete certainty.
U2
Utility grows from to
A
A
C2
U1
C1
E
A
C1
C1
59
Balancing Gains and Costs of Information
Certainty line
C2

• Gathering information costs time and money
• Point B represents investment in information that
  increases utility
• Point D corresponds to overly expensive
  information

E
E
C2
U2
B
D
A
A
C2
U1
C1
E
A
C1
C1
60
Information Cost Differences

• Costs of information acquisition differ among
  individuals
• Professionals versus laymen
• Sellers versus buyers
• Repeat buyers versus first-time buyers
• Previous investment in information services
• Consumer reports
• Consultancy services

61
Preferences and Information Levels

• Some people care about the best buy
• Some people have strong aversion to seeking
  bargains
• Normally economists assume all market
  participants possess the same level of
  information
• However, in many cases this assumption is
  untenable
• Asymmetric information