Minimax Regret

You are sometimes given a table with stocks and there possible payoffs (i.e. 0.6 payoff will indicate a 0.6 cents profit on every dollar invested) at consecutive times.

 Stock Year1 Year2 Year3 Year 4 Intel -0.3 0.9 0.4 0.5 Microsoft 0 0.5 0.1 0.4 Disney 0.6 0.4 0 0.6 Shell 0.3 -0.1 -0.7 0.7 Barclay -0.2 -0.6 0.4 0

In the minimax regret problem, you are required to minimize your highest regret when you choose one stock over the other.

Step 1: We look at the first payoff time in this case Year 1, and find the stock with the highest payoff. In this case it is the Disney stock which has a payoff of 0.6 (shaded in blue).

 Stock Year1 Year2 Year3 Year 4 Intel -0.3 0.9 0.4 0.5 Microsoft 0 0.5 0.1 0.4 Disney 0.6 0.4 0 0.6 Shell 0.3 -0.1 -0.7 0.7 Barclay -0.2 -0.6 0.4 0

Now we ask ourselves, suppose we bought Intel Stock instead of Disney Stock, how much I would have lost (or regretted) making this decision.

Since Disney Stock is 0.6 and Intel Stock is -0.3, then the amount we regret will be:

0.6 – (-0.3) = 0.9

We have thus potentially lost a 0.9 payoff.

Similarly, we would ask how much I would regret buying Microsoft Stock with 0 payoff versus Disney Stock with a 0.6 payoff in the first year.

0.6 – 0 = 0.6

My regret will thus be 0.6 if I bought Microsoft stock.

We continue similarly for the rest of the stocks in that year. You will note there is no regret in buying Disney stock.

I.E. Regret for Disney Stock in Year 1

0.6 – 0.6 = 0

We now construct a table call a “Regret Table” to put these values in:

 Stock Year1 Year2 Year3 Year 4 Intel 0.9 Microsoft 0.6 Disney 0 Shell 0.3 Barclay 0.8

Step 2. Similarly to Step 1, we look under the second column, in this case Year 2, and look for the stock with the highest payoff. In this case, it is Intel stock with a 0.9 payoff.

We now, ask ourselves, how much we will regret investing in Microsoft instead of Intel which has a payoff of 0.5. The regret will then be

0.9 - 0.5 = 0.4

We continue similarly for Disney, Shell and Barclay Stock. Obviously, there will be no regret in investing in Intel Stock. We can thus fill in the regrets for Year 2 (in mauve).

 Stock Year1 Year2 Year3 Year 4 Intel 0.9 0 Microsoft 0.6 0.4 Disney 0 0.5 Shell 0.3 1.0 Barclay 0.8 1.5

Step 3. We now look at the Year 3 column. You will notice that Intel and Barclay has the same payoff of 0.4. Therefore, we will have no regret in investing Barclay vs Intel or vice versa. We can choose between Intel and Barclay arbitrarily to be our highest payoff stock, in this case I will choose Barclay.

The regret of Intel vs Barclay is:

0.4 (Intel) – 0.4 (Barclay) = 0

We can now fill in our regret for Year 3 in a similar fashion (in mauve).

 Stock Year1 Year2 Year3 Year 4 Intel 0.9 0 0 Microsoft 0.6 0.4 0.3 Disney 0 0.5 0.4 Shell 0.3 1.0 1.1 Barclay 0.8 1.5 0

Step 4. We now look at our last column Year 4, the highest payoff in this column is for the Shell Stock with a 0.7 payoff. We therefore calculate the regret of buying Intel, Microsoft, Disney and Barclay versus that of Shell.

These values are filled in the Year 4 column (in mauve)

 Stock Year1 Year2 Year3 Year 4 Intel 0.9 0 0 0.2 Microsoft 0.6 0.4 0.3 0.3 Disney 0 0.5 0.4 0.1 Shell 0.3 1.0 1.1 0 Barclay 0.8 1.5 0 0.7

Step 5. We now insert a new column in our table and call it the Maximum Regret column. Let us take the Intel Stock firstly, and look at the regret of not investing in this stock over the Year 1 to Year 4 (shaded in pink). We see that we will most regret not investing in Intel in Year 1 since it gives a potential payoff of 0.9 as compared to the other stocks. Thus, our maximum regret for Intel is 0.9, we will insert this number in our Maximum Regret column (shaded in blue).

 Stock Year1 Year2 Year3 Year 4 Maximum Regret Intel 0.9 0 0 0.2 0.9 Microsoft 0.6 0.4 0.3 0.3 Disney 0 0.5 0.4 0.1 Shell 0.3 1.0 1.1 0 Barclay 0.8 1.5 0 0.7

Looking at the Microsoft Stock (in green) row for the four years, our maximum regret is 0.6 in Year 1. We thus enter this value in the Maximum Regret column (in peach).

 Stock Year1 Year2 Year3 Year 4 Maximum Regret Intel 0.9 0 0 0.2 0.9 Microsoft 0.6 0.4 0.3 0.3 0.6 Disney 0 0.5 0.4 0.1 Shell 0.3 1.0 1.1 0 Barclay 0.8 1.5 0 0.7

We continue in this same manner for the Disney, Shell and Barclay Stock, to fill in the Maximum Regret Column (in grey).

 Stock Year1 Year2 Year3 Year 4 Maximum Regret Intel 0.9 0 0 0.2 0.9 Microsoft 0.6 0.4 0.3 0.3 0.6 Disney 0 0.5 0.4 0.1 0.5 Shell 0.3 1.0 1.1 0 1.1 Barclay 0.8 1.5 0 0.7 1.5

Step 6. We now want to choose the stock that will minimize regret i.e. the stock that we invest in when compared to the others won’t make us feel too bad about its payoff being lower than the others in Year 1 to Year 4. We now look for under the Maximum Regret column, for the stock with the lowest Maximum Regret, in this case it is the Disney Stock (in pink), with a Maximum Regret of 0.5.

 Stock Year1 Year2 Year3 Year 4 Maximum Regret Intel 0.9 0 0 0.2 0.9 Microsoft 0.6 0.4 0.3 0.3 0.6 Disney 0 0.5 0.4 0.1 0.5 Shell 0.3 1.0 1.1 0 1.1 Barclay 0.8 1.5 0 0.7 1.5

Thus, we will invest in the Disney Stock based on the Minimax Regret Method. If you go back to our payoff table, you will observe that the Disney Stock offered a somewhat consistent and decent payoff.