1. Notes – Theory of Choice
In economics, utility is a measure of the relative satisfaction from or desirability of consumption of goods. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility. For illustrative purposes, changes in utility are sometimes expressed in units called utils.

2. Theory of Choice
While preferences are the conventional foundation of microeconomics, it is convenient to represent preferences with a utility function and reason indirectly about preferences with utility functions. Let X be the consumption set, the set of all mutually-exclusive packages the consumer could conceivably consume . The consumer's utility function ranks each package in the consumption set.
If u(x) >u(y), then the consumer strictly prefers x to y, while if u(x) =u(y), then the consumer is indifferent between them.
Example 1:, suppose a consumer's consumption set is
X = {nothing, 1 apple, 1 orange, 1 apple and 1 orange, 2 apples, 2 oranges},
and its utility function is u(nothing) = 0,
u (1 apple) = 1, u (1 orange) = 2, u (1 apple and 1 orange) = 4, u (2 apples) = 2 and u (2 oranges) = 3.
Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges or 2 apples.

3. Theory of Choice
Example 2: suppose that a person is twice as much satisfied with apple than with bananas. We can define the individual utility function as where stands form the number of apples and the number of bananas. Using such an individual utility function allows us to rank different combinations of apples and bananas for that individual. It also allows us to draw indifference curves, which represent different combinations of apples and bananas that provide the same utility for that person.
In Finance, we are concerned with one type of good, money (or wealth). Further, our main concern is about making decision under uncertainty (when we do not really know what the exact outcome will be). To this end, one would need tools that allow analyzing how people make choices under uncertainty.

4. Expected Utility
Theory states that based on a few axioms of rational behaviour (comparability, transitivity, strong independence, measurability, and ranking), one can always find a utility function that allows the individual to make choices when faced with uncertainty.
Expected utility can be used to rank combinations of risky alternatives. Mathematically, this means that,
, where G () is the individual’s utility function for the gamble, x and y the two possible outcomes, and α and (1- α ) the probability that x and y will occur, respectively (note the sum of probabilities for the different outcomes is 1).
When there are more than two states, we can write the expected utility of wealth as follows: where is the
probability of ending at state i and is the terminal wealth in state i.

5. Expected Utility
Mathematically stated, we are only interested in utility functions where wealth has a positive marginal utility (first derivative of the utility function is positive),
A person who prefers the gamble to the expected value, we call a risk lover and for
her,
2) A person who is indifferent between the expected value and gamble, we call a risk neutral and for her,
3) A person who prefers the expected value to the gamble we call a risk averse and for her,
Example

6. Expected Utility
Mathematically, a risk-averse individual has a utility function whose second derivative is negative
Mr. Smith has a log utility and is faced with the gamble that pays off $8 with a 50% probability and $2 with a 50% probability? How much would Mr. Smith be willing to pay in order to avoid the gamble and receive $5 (the expected value of the gamble) for sure.

7. Risk Premium, Certainty equivalent
Quantifying Risk Premium
The Risk premium is the difference between the individual’s expected wealth, given the gamble, and the level of wealth the individual would accept with certainty if the gamble were removed, i.e., his or her certainty equivalent wealth.
Example: A risk-averse individual has logarithmic utility and current wealth of $10, but the gamble is a 10% chance of winning $10 and a 90% change of winning $100. What is the risk premium of the gamble?

8. Risk Premium, Certainty equivalent
Note that in the previous example, the risk premium (call it π) of a gamble was a function of the level of wealth, and the gamble. Mathematically, the risk premium, can be defined as the value that satisfies the following equality:
The expected utility of wealth with a gamble equals the utility of (wealth + the expectation of the gamble – the risk premium of the gamble).

9. Risk Premium
The risk premium is a function of the variance of the gamble and the utility function of the individual

10. Risk Aversion We define the measure of absolute risk aversion as ARA.
We define the measure of relative risk aversion as RRA.

11. Risk Aversion
Example: Quantify the ARA and RRA of the utility function.

12. Mean-Variance (MV) Preferences
We assume that individuals are concerned only with the mean and the standard deviation of their investments.
We shall focus on a special case of a quadratic utility function.
Where A is a constant (“risk aversion parameter”) which depends on the individual and E and σ are in decimal form.
Example of indifference curves

13. Optimal Portfolio Choice Problem
Feasible set: portfolios created by one risky security and the riskless rate
Suppose that an individual has a risk aversion parameter A and utility function form Suppose we are given two securities: a risky security (or portfolio) X (with mean and variance: ) and a riskless asset R. We will assume that
Suppose that the individual's choice is restricted to all portfolios of positive investments in these two securities (i.e., no short selling). Then we can describe the individual’s problem as:

14. Optimal Portfolio Choice Problem
II. Feasible set: portfolios created by two risky securities
Now suppose, instead, that the choice is among all portfolios created by two risky securities, denoted #1 and #2 with mean and standard deviations, respectively. To calculate the variance of the portfolio we also need the covariance (or correlation) between these securities, defined to be
Then we can describe the individual’s problem as:

15. 11.5 Risk-Free Saving and Borrowing
Risk can be reduced by investing a portion of a portfolio in a risk-free investment, like T-Bills. However, doing so will likely reduce the expected return.
On the other hand, an aggressive investor who is seeking high expected returns might decide to borrow money to invest even more in the stock market.

16. Investing in Risk-Free Securities
Consider an arbitrary risky portfolio and the effect on risk and return of putting a fraction of the money in the portfolio, while leaving the remaining fraction in risk-free Treasury bills.
The expected return would be:

17. Figure 11.9 The Risk–Return Combinations from Combining a Risk-Free Investment and a Risky Portfolio

18. Chapter 11, problem 32
Suppose you have $100,000 in cash, and you decide to borrow another $15,000 at a 4% interest rate to invest in the stock market. You invest the entire $115,000 in a portfolio J with a 15% expected return and a 25% volatility.
What is the expected return and volatility (standard deviation) of your investment?
What is your realized return if J goes up 25% over the year?
What return do you realize if J falls by 20% over the year?

19. Identifying the Tangent Portfolio
To earn the highest possible expected return for any level of volatility we must find the portfolio that generates the steepest possible line when combined with the risk-free investment.

20. Figure 11.10 The Tangent or Efficient Portfolio

21. Identifying the Tangent Portfolio (cont'd)
Combinations of the risk-free asset and the tangent portfolio provide the best risk and return tradeoff available to an investor.
This means that the tangent portfolio is efficient and that all efficient portfolios are combinations of the risk-free investment and the tangent portfolio. Every investor should invest in the tangent portfolio independent of his or her taste for risk.

22. Identifying the Tangent Portfolio (cont'd)
An investor’s preferences will determine only how much to invest in the tangent portfolio versus the risk-free investment.
Conservative investors will invest a small amount in the tangent portfolio.
Aggressive investors will invest more in the tangent portfolio.
Both types of investors will choose to hold the same portfolio of risky assets, the tangent portfolio, which is the efficient portfolio.

23. 11.6 The Efficient Portfolio and Required Return
How to Improve a Portfolio: Beta and the Required Return
Assume there is a portfolio of risky securities, P. To determine whether P has the highest possible Sharpe ratio, consider whether its Sharpe ratio could be raised by adding more of some investment i to the portfolio.
The contribution of investment i to the volatility of the portfolio depends on the risk that i has in common with the portfolio, which is measured by i’s volatility multiplied by its correlation with P.

24. 11.6 The Efficient Portfolio and Required Return (cont'd)
How to Improve a Portfolio: Beta and the Required Return
If you were to purchase more of investment i by borrowing, you would earn the expected return of i minus the risk-free return. Thus adding i to the portfolio P will improve our Sharpe ratio if:

25. 11.6 The Efficient Portfolio and Required Return (cont'd)
How to Improve a Portfolio: Beta and the Required Return
Increasing the amount invested in i will increase the Sharpe ratio of portfolio P if its expected return E[Ri] exceeds the required return ri , which is given by:

26. Problem
You are currently invested in the Farrallon Fund, a broad-based fund of stocks and other securities with an expected return of 12% and a volatility of 25%. Currently, the risk-free rate of interest is 4%. Your broker suggests that you add a venture capital fund to your current portfolio. The venture capital fund has an expected return of 20%, a volatility of 80%, and a correlation of 0.2 with the Farrallon Fund. Calculate the required return and use it to decide whether you should add the venture capital fund to your portfolio.

27. Expected Returns and the Efficient Portfolio
Expected Return of a Security
A portfolio is efficient if and only if the expected return of every available security equals its required return.