Question

Tom sold his old house in upstate New York and now has a sum of $175,000 for investment. Tom is considering select only one of the three possible investments to invest his $175,000: a mutual fund (MF), a technology stock (TS), or a certificate of deposit (CD). The investment industry estimates the probability of a good, average, and poor market to be 42%, 27%, and 31%, respectively. The CD is guaranteed to pay a 3.5% return regardless of the market condition. The financial consultant estimates the return on the mutual fund (MF) as 7%, 2%, or -3%, depending on whether market condition is good, average, or poor, respectively. Under the same condition, the financial consultant estimates the technology stock (TS) would yield 9%, 5%, or -11%.

Create a regret table for Tom. What decision should be made according to the minimax approach? What decision should be made according to the expected value approach?

Answer 1

Please have a look at below tables.

We are given three strategies to choose from: Mutual Fund, Technology Stock and CD

We are also give the three possible states of nature as Good, Average and Poor with their respective probabilities.

We are also given the return we would generate from each strategy for every State of nature.

Given the above information we first create a Payoff Table (First Table in the screenshot). This table lists the payoffs that each strategy (columns) would generate given the respective States of Natures (Rows).

Regret Table

Regret Table displays the opportunistic loss for a strategy given a state of nature occured. In simple terms it is what could have been earned given a state of nature occured but is not earned because of our choice of strategy. For example if state of nature happened to be Good, then from the payoff table we see that we could have earned a maximum return of 9% on choosing Technology stock. However, if we had chosen Mutual Fund and the state of nature was good so we earned 7% which is 2% less than what we could have earned if we had chosen technology stock. So, in regret table we write 2% under Mutual Fund in a Good scenario. Similarly we fill the other cells in the regret table.

So to make the regret table we first look at the payoff table to find the maximum return for each state of nature (highlighted in blue)

Then for each state of nature we subtract the return from each strategy from the maximum return for that state of nature.

We now apply the minimax approach to the regret table to make our decision

We find out in the regret table the maximum value under each strategy. For example for Mutual fund the maximum value in 6.5%. This is the maximum regret. We wish to minimise the maximum regret and so we chose the strategy with minimum value of maximum regret. So maximum regret/loss for Mutual fund is 6.5%, for Technology stock the maximum regret is 14.5% and for  CD the maximum regret is 3.5%. The minimum of these three values is for CD. Hence, CD is our answer.

Expected Value Approach

The expected values from the three strategies have been shown in the payoff table bottom line.

The Expected Value approach utilises the probabilities of the states of nature. For each strategy we simply mutiply the return under each state of nature with the probability of occurence of that state of nature. This gives the expected value of return of that strategy.

For example the expected value of retiurn for Mutual Fund is:

7% * 42% + 2% * 27% - 3% * 31% = 2.5%

The strategy with maximum value of expected value will be chosen under this approach. We find that CD will generate the maximum expected value (3.5%) hence CD should be the choice.