## MATH318: Utility

The purpose of this exercise is to practice determining indifference values, an important step in determining the utility of money. Please work with one or two partners on this activity.

Suppose your grandmother gives you \$1,000 as a graduation gift, with the condition that you must invest it for at least one year before spending it. You have the following investment options:

A. Invest in a friend's startup company. This has a 40% chance of succeeding and returning \$5,000 at the end of the year and a 60% chance of returning nothing.
B. Invest in the stock market. This has a 70% chance of returning \$1200 and a 30% chance of returning \$900.
C. Deposit the money in your checking account which earns no interest.

1. Which investment do you think would be best for you? Why?

2. What are the payoffs of each of the options above?

3. What is the expected value of each investment above?

4. If you were to decide based solely on expected value, which investment would you choose?

The following exercise should establish the indifference values of the payoffs in the problem above. These should enable you to calculate the expected utility of each investment option, which should indicate which is the best decision for your current financial circumstances and personal risk tolerance. Please try to answer the questions below as accurately as possible.

 Payoff Indifference Value Utility Value \$5,000 n/a 10 \$1,200 \$1,000 \$900 \$0 n/a 0

The table above lists the payoffs possible for your investment. A utility value of 10 has been assigned to the best possible payoff, and a utility value of 0 has been assigned to the worst one. (This is typical; it ensures that the calculations in steps 3(c) and 4 on page 159 of your text are trivial.)

Help your partner determine the indifference value corresponding to the payoff of \$1,200 as follows:

1. Ask your partner if he or she would pay \$1,200 for a 25% chance (one in four) of winning \$5,000.
• If your partner replies "yes", decrease the chance of winning -- would he or she pay \$1,200 for a 20% chance (1 in 5) of winning \$5,000?
• If your partner replies "no", increase the chance of winning -- would he or she pay \$1,200 for a 33% chance (1 in 3) of winning \$5,000?
2. Continue to adjust the probability of winning until you have identified thepercentage at which their answer changes from "yes" to "no". (For example, they might pay \$1,200 for a 90% chance at \$5,000 but not for an 85% chance for an indifference value of roughly 87% = .87.)
3. Enter this indifference value in the table above.
Repeat this exercise to find indifference values for the other payoffs shown in the table. Then multiply each entry by 10 to determine the utility value of each payoff. (This is the calculation in step 3(c) on page 159.)

The expected utility of investing in a friend's startup is the weighted average of the utility values of the payoffs: .4 (10) + .6 (0) = 4.

Compute the expected utility (to you) of the other two investment options. Which option has the highest expected utility for you personally? Is this the option you chose at the start of this exercise?