Game Theory suggests that you can get pretty far simply looking at expected values. But the expected values need to be done right, looking at both upside and downside.

Looking at the consequences of the compensation paid by financial institutions suggests that boards who approved the compensation did not do their Game Theory homework. Management is given a huge share of the upside and their incentives are thought to be aligned with shareholders because of a stock component to their compensation.

Maybe that works, maybe not. The math is simple enough. They should check. A Game Theory, risk adjusted compensation analysis (simplified for this post) would look like this:

Proposed Transaction:

Upside: $100,000,000 – likelihood 60%

Downside: ($100,000,000) – likelihood 40%

Expected Value: $20,000,000

So far so good. Now the firm already has a risk adjusted compensation system. Under that system, there is a cost of capital charge assessed against profits before calculating bonus. In this case, the capital is based upon the $100,000,000 downside. The cost of capital is 5%, so the “risk adjustment” is 5%. The bonus formula will pay out 40% of the risk adjusted gain, half in cash and half in stock. In the past, the compensation committee has seen this process and stopped there. It seems that they took care of every angle.

But this year, one comp committee member hears a lecture on Game Theory and asks for additional analysis:

Risk Adjusted Expected Value Analysis:

Management:

Upside: 40% of $100,000,000 less $5,000,000 equals $38,000,000. Pay $19,000,000 cash and $19,000,000 stock. Stock is purchased at time of award. Likelihood 60%

Downside: Zero Current award. Loss in value of stock holdings from past awards. Back to that in a minute.

Shareholders:

Upside: $62,000,000 of gains plus risk charge times 10 equals $620 million. Likelihood of 60%

Downside $$100,000,000 of losses times 10 equals ($1 Billion). Likelihood of 40%

Expected value: ($2.8 million)

Now back to the employees…

The downside from their 0.1% of stock is ($1 million) so their expected value is $22 million positive.

So a real Game Theory based risk adjusted analysis would show that there is huge upside to management for risky deals and much smaller risk adjusted expectation for the shareholders. (In this example an expected loss).

Perhaps every deal should be presented on this risk adjusted basis. It might take a few of these presentations, but sooner or later the lopsided deal will sink in.

But then the game will shift. Already, the game is to present these deals optimistically, so that the likelihood of upside is overstated and the downside is underestimated. If compensation is skewed as drastically as the above example, highly risky deals look just fine on a risk adjusted expected value basis to employees. If the board insists that the shareholders really have a positive expected value, then the deals will need to be much less risky – at least on paper.

The stress testing that is being promoted as a major risk management tools (in part because of this very problem of over optimistic risk models) needs to then also be done to the risk adjusted compensation analysis. The stress tests for this purpose do not need to be as drastic as the stress tests for solvency management. What you should be looking for is the inflection point where the deal starts to fall into the situation where the management and shareholders are no longer on the same side, where their expected values are of opposite signs. If that inflection point is found with a stress test that is somewhat close to the base model assumptions, then that is a flashing red light for the risk manager and the board.

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