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Utility AnalysisUtility Analysis

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Utility Analysis (cont’d)Utility Analysis (cont’d)

International DiversificationInternational Diversification

Expected Utility Theory:Expected Utility Theory:

The nice thing about this theory is that it is The nice thing about this theory is that it is consistent with choice that would be made by consistent with choice that would be made by examining the investment directlyexamining the investment directly That is, if people obey certain postulates of That is, if people obey certain postulates of

behavior (act in a certain way) they expected behavior (act in a certain way) they expected utility theory will yield identical results as if we utility theory will yield identical results as if we had engaged in direct analysishad engaged in direct analysis

Expected Utility: Real Expected Utility: Real WorldWorld A number of investment firms employ A number of investment firms employ

expected utility models as part of their expected utility models as part of their security selection process for clientssecurity selection process for clients Through the use of questionnaires and computer Through the use of questionnaires and computer

programs, they are able to discern what investor programs, they are able to discern what investor preferences (utility functions) arepreferences (utility functions) are

Problems:Problems: Investors do not always behave rationallyInvestors do not always behave rationally

Expected Utility: Real Expected Utility: Real WorldWorld Even if firms do not derive, formally, utility Even if firms do not derive, formally, utility

functions, the good news is that utility functions, the good news is that utility analysis still has a lot of value-added for analysis still has a lot of value-added for analystsanalysts Understanding the properties of the analysis can Understanding the properties of the analysis can

lead analysts to insightful conclusions that would lead analysts to insightful conclusions that would otherwise not be obviousotherwise not be obvious

For instance, insight into the process of rational For instance, insight into the process of rational choice can eliminate certain portfolios from choice can eliminate certain portfolios from consideration at the outset, thereby reducing the consideration at the outset, thereby reducing the number of opportunities needed to be considered number of opportunities needed to be considered and the probability that a really bad decision will and the probability that a really bad decision will be madebe made

Properties of Utility Properties of Utility FunctionsFunctions Property 1: Nonsatiation -- More is preferred Property 1: Nonsatiation -- More is preferred

to less (we saw this last day)to less (we saw this last day) Property 2: Risk PreferencesProperty 2: Risk Preferences

Risk AverseRisk Averse Risk LovingRisk Loving Risk NeutralRisk Neutral

Property 3: Preference changes with a Property 3: Preference changes with a change in wealthchange in wealth As wealth, say, increases, will the investor invest As wealth, say, increases, will the investor invest

more or less in risky assets?more or less in risky assets?

Property 2: Risk Property 2: Risk PreferencesPreferences What are the investors’ taste for risk?What are the investors’ taste for risk? To show the differences in individual taste for risk, it To show the differences in individual taste for risk, it

is often explained in terms of a fair gambleis often explained in terms of a fair gamble

InvestInvest Do not investDo not invest

OutcomeOutcome ProbProb OutcomeOutcome ProbProb

22 0.500.50 11 11

00 0.500.50

Property 2 cont’d:Property 2 cont’d:

InvestInvest Do not investDo not invest

OutcomeOutcome ProbProb OutcomeOutcome ProbProb

22 0.500.50 11 11

00 0.500.50

The above example has an expected value of 1 if the individual The above example has an expected value of 1 if the individual decides to invest.decides to invest.

Now assume that to invest, the individual would have to pay $1. If Now assume that to invest, the individual would have to pay $1. If she/he chooses not to invest, then the dollar is kept (hence, the do she/he chooses not to invest, then the dollar is kept (hence, the do not invest payoff)not invest payoff)

Property 2 cont’d:Property 2 cont’d:

InvestInvest Do not investDo not invest

OutcomeOutcome ProbProb OutcomeOutcome ProbProb

22 0.500.50 11 11

00 0.500.50

The point of all of this?The point of all of this?

The expected value of making the gamble is exactly The expected value of making the gamble is exactly offset by the costoffset by the cost

Property 2 cont’d:Property 2 cont’d: Where does the notion of a fair gamble come Where does the notion of a fair gamble come

in?in? It is the fact that the It is the fact that the EXPECTATIONEXPECTATION of the of the

investor is the same, regardless of whether or not investor is the same, regardless of whether or not they make the betthey make the bet

Make noteMake note: the investor’s position may be : the investor’s position may be improved or hurt if the investment is madeimproved or hurt if the investment is made But, the expectation (again) is that his/her position won’t But, the expectation (again) is that his/her position won’t

changechange

So what does this all So what does this all mean?mean? It means that we can now illustrate the It means that we can now illustrate the

difference between individual taste towards difference between individual taste towards riskrisk Risk Averse:Risk Averse:

If the investor is risk averse, she/he would not make the If the investor is risk averse, she/he would not make the betbet

Risk Loving:Risk Loving: Would choose to make the betWould choose to make the bet

Risk Neutral:Risk Neutral: Individual is indifferent between making the bet and notIndividual is indifferent between making the bet and not

Risk Averse:Risk Averse: It the context of our previous example, this individual It the context of our previous example, this individual

prefers $1 with certainty, over an equal chance to prefers $1 with certainty, over an equal chance to increase that $1 to $2.increase that $1 to $2.

In terms of modeling this type of behavior, it means In terms of modeling this type of behavior, it means that the second derivative of utility with respect to that the second derivative of utility with respect to wealth is negative.wealth is negative.

0)( WU

Risk Averse cont’d:Risk Averse cont’d: Another way to put risk aversion is:Another way to put risk aversion is:

The disutility that comes with losing is greater The disutility that comes with losing is greater than the utility that is associated with winning.than the utility that is associated with winning.

If we examine risk neutral and risk loving If we examine risk neutral and risk loving individuals, we would find that their second individuals, we would find that their second derivatives are also able to explain peoples’ derivatives are also able to explain peoples’ behaviorbehavior

Risk Loving and Risk Risk Loving and Risk Neutral:Neutral: RecallRecall

A risk neutral investor would be indifferent between taking A risk neutral investor would be indifferent between taking the fair gamble and notthe fair gamble and not In other words,In other words,

)1()2()0()1(

rearrange and 2

)0()2/1()2()2/1()1(

UUUU

UUU

So, a change in utility from a one unit change in wealth is So, a change in utility from a one unit change in wealth is independent of the relative magnitude; that is a change from 0 to 1, independent of the relative magnitude; that is a change from 0 to 1, is the same as a change from 1 to 2. is the same as a change from 1 to 2.

Risk Neutral and Risk Loving Risk Neutral and Risk Loving cont’dcont’d Given the preceding analysis, we can say Given the preceding analysis, we can say

that an individual who is risk neutral would that an individual who is risk neutral would have a second derivative of zero.have a second derivative of zero.

Using the above analysis, but applying the Using the above analysis, but applying the notion of risk loving would yield a second notion of risk loving would yield a second derivative that is positive.derivative that is positive.

Summary Table:Summary Table:

Risk AttitudeRisk Attitude Fair GambleFair Gamble UtilityUtility

Risk AversionRisk Aversion Reject Fair Reject Fair GambleGamble

U”(0)<0U”(0)<0

Risk LovingRisk Loving Accept Fair Accept Fair GambleGamble

U”(0)>0U”(0)>0

Risk NeutralRisk Neutral Indifferent to Fair Indifferent to Fair GambleGamble

U”(0)=0U”(0)=0

Property 3: Preference Property 3: Preference and Wealthand Wealth Will the investor change her/his behavior as Will the investor change her/his behavior as

wealth changes?wealth changes? Will more wealth imply greater investment in Will more wealth imply greater investment in

risky assets? Will the individual invest less in risky assets? Will the individual invest less in risky assets?risky assets?

To deal with these changes, we introduce the To deal with these changes, we introduce the concepts of concepts of absolute absolute andand relative risk relative risk aversion.aversion.

Absolute/Relative Risk Absolute/Relative Risk Aversion:Aversion: If, as the investor’s wealth increases, the If, as the investor’s wealth increases, the

investor increases the amount of money investor increases the amount of money she/he places in risky assets than this person she/he places in risky assets than this person is said to exhibitis said to exhibit Decreasing Absolute Risk AversionDecreasing Absolute Risk Aversion

If there is no change in investment patterns, If there is no change in investment patterns, the investor exhibitsthe investor exhibits Constant Absolute Risk AversionConstant Absolute Risk Aversion

If he/she invests less is risk assets as wealth If he/she invests less is risk assets as wealth increases then the investor exhibitsincreases then the investor exhibits Increasing Absolute Risk AversionIncreasing Absolute Risk Aversion

Absolute Risk Aversion: Absolute Risk Aversion: MathematicallyMathematically

)(

)()(

WU

WUWA

Taking the derivative of this equation would show how an Taking the derivative of this equation would show how an investor’s absolute risk aversion varies as we change investor’s absolute risk aversion varies as we change wealth.wealth.

Relative Risk AversionRelative Risk Aversion Like is cousin absolute risk aversion, relative risk Like is cousin absolute risk aversion, relative risk

aversion measures the investors attitude toward aversion measures the investors attitude toward risky assets as wealth changesrisky assets as wealth changes The difference being that in the case of relative risk The difference being that in the case of relative risk

aversion, we are examining the percentage invested in aversion, we are examining the percentage invested in risky assets, as opposed to a $ amount.risky assets, as opposed to a $ amount. That is, we are looking at the change in the percentage That is, we are looking at the change in the percentage

invested in risk assets as wealth changesinvested in risk assets as wealth changes

)()( WAWWR

Relative Risk Aversion:Relative Risk Aversion:

RRARRA Def’nDef’n PropertyProperty

IncreasingIncreasing % invested in % invested in risky assets risky assets declines as declines as wealth increaseswealth increases

R’(W) > 0R’(W) > 0

ConstantConstant % invested % invested remains remains unchangedunchanged

R’(W) = 0R’(W) = 0

DecreasingDecreasing % invested % invested increasesincreases

R’(W) < 0R’(W) < 0

In general:In general: Generally agreed that investors, for the most Generally agreed that investors, for the most

part, exhibit part, exhibit decreasingdecreasing absolute risk absolute risk aversionaversion

While there is a moderate consensus While there is a moderate consensus surrounding absolute risk aversion, the same surrounding absolute risk aversion, the same cannot be said of relative risk aversioncannot be said of relative risk aversion Often researches will use constant relative risk Often researches will use constant relative risk

aversion for ease of expositionaversion for ease of exposition While it may describe certain individuals’ behavior, this While it may describe certain individuals’ behavior, this

assumption is certainly not born out of the need for assumption is certainly not born out of the need for descriptive accuracy.descriptive accuracy.

Example 1:Example 1:

What are the characteristics of the utility function What are the characteristics of the utility function given below with respect to absolute and relative given below with respect to absolute and relative risk aversion?risk aversion?

2/1)( WWU

Example 2Example 2 Given the following utility functionGiven the following utility function

bWaeWU )(

Where Where aa and and bb are constants, what are the signs of are constants, what are the signs of aa and and bb, assuming the investor prefers more to , assuming the investor prefers more to less and is risk averse.less and is risk averse.

International International DiversificationDiversification

Portfolio Diversification in a World Portfolio Diversification in a World ContextContext

Risk and Returns of Foreign SecuritiesRisk and Returns of Foreign Securities

International Investing:International Investing: When we talk about international When we talk about international

diversification it is often in the context of diversification it is often in the context of North American investorsNorth American investors From this point of view, international From this point of view, international

diversification is a relatively new conceptdiversification is a relatively new concept BUT, investors in other countries have BUT, investors in other countries have

historically invested a large portion of their historically invested a large portion of their money in international securities (that is, money in international securities (that is, securities issued in countries other than their securities issued in countries other than their home country)home country)

International Investing:International Investing: Is investing international a sound option for Is investing international a sound option for

investors?investors? What types of risk to individuals incur What types of risk to individuals incur

international, that they would otherwise not?international, that they would otherwise not? Do international investing strategies have to Do international investing strategies have to

be actively managed or can they be passively be actively managed or can they be passively managed?managed?

These are some of the questions we will attempt These are some of the questions we will attempt to address as we move through this section of the to address as we move through this section of the coursecourse

World Equity Markets in World Equity Markets in 20002000 If we took a total of all the equity markets all If we took a total of all the equity markets all

over the world and then computed over the world and then computed proportions, North America would have proportions, North America would have accounted for roughly 52% of the world accounted for roughly 52% of the world equity market (Canada only accounts for equity market (Canada only accounts for roughly 2%)roughly 2%)

Aside: As far as bond markets go, the U.S. bond Aside: As far as bond markets go, the U.S. bond market is the largest; it accounts for roughly 47% market is the largest; it accounts for roughly 47% of the world’s bond marketof the world’s bond market

World Markets cont’d:World Markets cont’d: Given the previous #’s it’s easy to see why Given the previous #’s it’s easy to see why

international portfolio managers have international portfolio managers have included foreign securities in their portfolios included foreign securities in their portfolios for a much longer time, than U.S. portfolio for a much longer time, than U.S. portfolio managersmanagers The reason being, an extremely large portion of The reason being, an extremely large portion of

world’s wealth lies outside foreign investors’ home world’s wealth lies outside foreign investors’ home countriescountries This represents huge opportunities for these investors, This represents huge opportunities for these investors,

in terms of diversifying their portfolio, since many of in terms of diversifying their portfolio, since many of these options will be new.these options will be new.

World Markets Cont’d:World Markets Cont’d: Of course, not all of these opportunities will Of course, not all of these opportunities will

be new.be new. Sometimes international assets are simply Sometimes international assets are simply

duplicates of what could be purchased in the duplicates of what could be purchased in the home countryhome country If the latter case holds, then international diversification If the latter case holds, then international diversification

may not be all that valuable. may not be all that valuable. If a majority of the opportunities are new (that is, not If a majority of the opportunities are new (that is, not

available in the home country) then diversifying available in the home country) then diversifying internationally could have (potentially) large payoffs.internationally could have (potentially) large payoffs.

Returns on Foreign Returns on Foreign Investments:Investments: We will now analyze the situation analysts We will now analyze the situation analysts

face when incorporating foreign securities face when incorporating foreign securities into a portfoliointo a portfolio Namely, the return calculation needs to be slightly Namely, the return calculation needs to be slightly

modified to take into consideration the modified to take into consideration the “international context” of the investment“international context” of the investment Return on asset in home countryReturn on asset in home country Exchange ratesExchange rates

An example to illustrate:An example to illustrate: Consider the case where a German mark is worth Consider the case where a German mark is worth

$0.50 U.S. in period 0, and declines by 10% in $0.50 U.S. in period 0, and declines by 10% in period 1 and that the following table holds.period 1 and that the following table holds.

TimeTime Cost of one Cost of one markmark

Value of Value of German SharesGerman Shares

Value in Value in DollarsDollars

00 0.500.50 40 DM40 DM 2020

11 0.400.40 45 DM45 DM 1818

What is the return to a German Investor? What about the U.S. What is the return to a German Investor? What about the U.S. investor?investor?