## Morris: How to Win Your Bonus

Sometimes rules set up to achieve one result can have exactly the opposite result.

Your company has a bonus scheme, just to make sure your personal interests and the greater good are aligned.  As if that were necessary!

Let’s see just how that bonus scheme works.

Ten percent of the company’s profits are paid in bonuses to its traders.  Of course if there is no profit then no one gets a bonus.

Each trader’s bonus is in proportion to their profit.  Only traders that make a profit are included.

There are 100 traders in the company, including yourself.  Every trader has the same \$100 to invest.

Let’s look at how this could work out:

80 traders trade in company A and double their money.   Total profit = 80 x \$100 = \$8,000

20 traders trade in company B and lose their stake.  Total loss = 20 x \$100 = \$2,000

Overall profit = \$6,000

10% of profit available for bonuses = \$600

80 traders qualify for bonuses.  As they have achieved the same profit they each get the same share.

It all seems to be working fine so far, the traders who made the good trade made money for the company and got a bonus.

How could this possibly go wrong?

Suppose that the other 99 traders all trade in Company C.  If it does well they will double their money.  If it does badly they will lose their stake.  Each is equally likely, it is as though they were all betting on the same coin-flip coming up heads.

You are a bit better than the other guys.  But can you turn it to your advantage?

You have spotted Company D and have worked out that it has a 75% chance of success.  Again if it does well you double the money, if it does badly you lose the original stake.

What should you do?  Should you follow the crowd and trade in Company C or take the better trade in Company D?

What difference does your choice make to your bonus and to the company’s expected profit?  Are they actually aligned as intended?

Thanks to Paul Wilmott on More or Less for this great insight into the current financial meltdown excitement.

1. ### jyoak said,

March 20, 2009 at 11:45 pm

[spoiler] That’s cool.  So if you invest in C, 50% of the time the company makes \$10,000 for \$1000 in commissions or \$10 each.  Your expectation is \$5. If you invest in D, the outcomes are: 1: 12.5 % D loses, C loses. 2: 12.5 % D loses, C wins. 3: 37.5 % D wins, C loses. 4: 37.5 % D wins, C wins. The commission in each case is: 1: 0 (the company loses money) 2: 0 (the company makes money, but you aren’t eligible for a bonus) 3: 0 (the company loses money) 4: 10 Since this occurs only 37.5% of the time, the expectation is \$3.75 . So what we really need to do to be smart is to discover not a different company that doubles more often, but how to borrow on margin.  Suppose you can double the amount you can invest at a cost of 10%.  You thus invest \$200 in company C. It now looks like this: 1: 50% – you lose \$210 for the company. (and everyone else loses \$100) 2: 50% – you make \$190 for the company (and everyone else makes \$100.) Clearly this is worse for the company.  Instead of having zero expectation before bonus, you have -\$10 for the company.  On the other hand, the company makes \$10,090 when you profit.  This makes improves your commission to \$10.09 when you win, or an expectation of \$5.045 — an improvement! This is so smart to do, in fact, that you should tell the other 99 traders.  :-) [/spoiler]

2. ### Stephen Morris said,

March 30, 2009 at 6:36 pm

jyoak, that’s exactly the solution I had!
[spoiler] I didn’t spot that they can play the system by borrowing, no wonder we are all broke now! It seems that unless the risk/reward calculation is exactly the same for the brokers as for the company then they will always have a different angle.
The main problem is that they can only win a bonus, never lose one, whereas the company can win and lose.  So the expected result will always be distorted.
Actually I don’t think that is the main problem.  I think the only way for the trader’s interest to match the companies interest is if they are entirely independent, so the trader can make money when the company doesn’t.
It seems perverse but seems to be true.
It means we should be paying bonuses to succesful traders even when their companies have been bailed out by poor tax-payers like us! [/spoiler]

Can anyone see another way to make this work?