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### Last Day

### International Diversification

Utility Analysis

Expected Utility Theory:

- The nice thing about this theory is that it is consistent with choice that would be made by examining the investment directly
- That is, if people obey certain postulates of behavior (act in a certain way) they expected utility theory will yield identical results as if we had engaged in direct analysis

Expected Utility: Real World

- A number of investment firms employ expected utility models as part of their security selection process for clients
- Through the use of questionnaires and computer programs, they are able to discern what investor preferences (utility functions) are
- Problems:
- Investors do not always behave rationally

Expected Utility: Real World

- Even if firms do not derive, formally, utility functions, the good news is that utility analysis still has a lot of value-added for analysts
- Understanding the properties of the analysis can lead analysts to insightful conclusions that would otherwise not be obvious
- For instance, insight into the process of rational choice can eliminate certain portfolios from consideration at the outset, thereby reducing the number of opportunities needed to be considered and the probability that a really bad decision will be made

Properties of Utility Functions

- Property 1: Nonsatiation -- More is preferred to less (we saw this last day)
- Property 2: Risk Preferences
- Risk Averse
- Risk Loving
- Risk Neutral
- Property 3: Preference changes with a change in wealth
- As wealth, say, increases, will the investor invest more or less in risky assets?

Property 2: Risk Preferences

- What are the investors’ taste for risk?
- To show the differences in individual taste for risk, it is often explained in terms of a fair gamble

Property 2 cont’d:

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

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 not invest payoff)

Property 2 cont’d:

The point of all of this?

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

Property 2 cont’d:

- Where does the notion of a fair gamble come in?
- It is the fact that the EXPECTATION of the investor is the same, regardless of whether or not they make the bet
- Make note: the investor’s position may be improved or hurt if the investment is made
- But, the expectation (again) is that his/her position won’t change

So what does this all mean?

- It means that we can now illustrate the difference between individual taste towards risk
- Risk Averse:
- If the investor is risk averse, she/he would not make the bet
- Risk Loving:
- Would choose to make the bet
- Risk Neutral:
- Individual is indifferent between making the bet and not

Risk Averse:

- It the context of our previous example, this individual prefers $1 with certainty, over an equal chance to increase that $1 to $2.
- In terms of modeling this type of behavior, it means that the second derivative of utility with respect to wealth is negative.

Risk Averse cont’d:

- Another way to put risk aversion is:
- The disutility that comes with losing is greater than the utility that is associated with winning.
- If we examine risk neutral and risk loving individuals, we would find that their second derivatives are also able to explain peoples’ behavior

Risk Loving and Risk Neutral:

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

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, is the same as a change from 1 to 2.

Risk Neutral and Risk Loving cont’d

- Given the preceding analysis, we can say that an individual who is risk neutral would have a second derivative of zero.
- Using the above analysis, but applying the notion of risk loving would yield a second derivative that is positive.

Property 3: Preference and Wealth

- Will the investor change her/his behavior as wealth changes?
- Will more wealth imply greater investment in risky assets? Will the individual invest less in risky assets?
- To deal with these changes, we introduce the concepts of absolute and relative risk aversion.

Absolute/Relative Risk Aversion:

- If, as the investor’s wealth increases, the investor increases the amount of money she/he places in risky assets than this person is said to exhibit
- Decreasing Absolute Risk Aversion
- If there is no change in investment patterns, the investor exhibits
- Constant Absolute Risk Aversion
- If he/she invests less is risk assets as wealth increases then the investor exhibits
- Increasing Absolute Risk Aversion

Absolute Risk Aversion: Mathematically

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

Relative Risk Aversion

- Like is cousin absolute risk aversion, relative risk aversion measures the investors attitude toward risky assets as wealth changes
- The difference being that in the case of relative risk aversion, we are examining the percentage invested in risky assets, as opposed to a $ amount.
- That is, we are looking at the change in the percentage invested in risk assets as wealth changes

In general:

- Generally agreed that investors, for the most part, exhibit decreasing absolute risk aversion
- While there is a moderate consensus surrounding absolute risk aversion, the same cannot be said of relative risk aversion
- Often researches will use constant relative risk aversion for ease of exposition
- While it may describe certain individuals’ behavior, this assumption is certainly not born out of the need for descriptive accuracy.

Example 1:

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

Example 2

- Given the following utility function

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

Portfolio Diversification in a World Context

Risk and Returns of Foreign Securities

International Investing:

- When we talk about international diversification it is often in the context of North American investors
- From this point of view, international diversification is a relatively new concept
- BUT, investors in other countries have historically invested a large portion of their money in international securities (that is, securities issued in countries other than their home country)

International Investing:

- Is investing international a sound option for investors?
- What types of risk to individuals incur international, that they would otherwise not?
- Do international investing strategies have to be actively managed or can they be passively managed?
- These are some of the questions we will attempt to address as we move through this section of the course

World Equity Markets in 2000

- If we took a total of all the equity markets all over the world and then computed proportions, North America would have accounted for roughly 52% of the world equity market (Canada only accounts for roughly 2%)
- Aside: As far as bond markets go, the U.S. bond market is the largest; it accounts for roughly 47% of the world’s bond market

World Markets cont’d:

- Given the previous #’s it’s easy to see why international portfolio managers have included foreign securities in their portfolios for a much longer time, than U.S. portfolio managers
- The reason being, an extremely large portion of world’s wealth lies outside foreign investors’ home countries
- This represents huge opportunities for these investors, in terms of diversifying their portfolio, since many of these options will be new.

World Markets Cont’d:

- Of course, not all of these opportunities will be new.
- Sometimes international assets are simply duplicates of what could be purchased in the home country
- If the latter case holds, then international diversification may not be all that valuable.
- If a majority of the opportunities are new (that is, not available in the home country) then diversifying internationally could have (potentially) large payoffs.

Returns on Foreign Investments:

- We will now analyze the situation analysts face when incorporating foreign securities into a portfolio
- Namely, the return calculation needs to be slightly modified to take into consideration the “international context” of the investment
- Return on asset in home country
- Exchange rates

An example to illustrate:

- Consider the case where a German mark is worth $0.50 U.S. in period 0, and declines by 10% in period 1 and that the following table holds.

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

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