One more thought on pay...
The columns of the FT have been reverberating with comments on banker's pay recently (see here, here, here and here) with a predictably strident reaction from the blogosphere (see, if you must, here, here and here). One issue here that hasn't received attention, though, is the wonderful effect of diversification for managers.
Suppose you are a junior trader with a budget of $5M and you get paid 10% of your excess P/L over $3M. Your bonus is therefore 10% x max(0, P/L - $3M). Suppose the average trader makes budget, just, and the SD of returns is $2M. Then while the average trader makes $200K, a bad or unlucky guy 2 SD from the mean makes 10% x max(0, 5M - 2 x 2M - 3M) = 0. Fair enough.
Note though that the trader owns a call, and you maximise the value of the call by increasing volatility. So the trader is incentivised to make their P/L volatility as large as possible.
Now consider the desk heads. Suppose each employs ten traders, so their budget is $50M. They too get paid on the same basis, so their bonus is 10% x max(0, P/L - $30M). The mean is $50M but the SD goes as the square root of the number of traders employed, so it is root ten times $2M or about $6M assuming zero correlation. Therefore while the average desk head takes home $2M every Christmas, a bad one 2 SD from the mean has a bonus of 10% x max(0, 50M - 2 x 6M - 30M) = $800K. So even a pretty bad (or unlucky) desk head makes money. Of course the zero correlation assumption is a stretch but note that the desk head owns a call on a basket, so they should maximise both volatility of the basket components and their correlation.
Finally consider the head of the business employing a thousand traders. By the same logic, their budget is $500M, and they get paid 10% x max(0, P/L - $300M). Their volatility though is only root thousand times $2M or $20M. So a really terrible manager four SDs from the mean still earns 10% x max(0, 500 - 4 x 20 - 300) = $12M. Very nice. Diversification works beautifully in executive pay, at least if you are one of the executives. Shareholders may have a different perspective.
Suppose you are a junior trader with a budget of $5M and you get paid 10% of your excess P/L over $3M. Your bonus is therefore 10% x max(0, P/L - $3M). Suppose the average trader makes budget, just, and the SD of returns is $2M. Then while the average trader makes $200K, a bad or unlucky guy 2 SD from the mean makes 10% x max(0, 5M - 2 x 2M - 3M) = 0. Fair enough.
Note though that the trader owns a call, and you maximise the value of the call by increasing volatility. So the trader is incentivised to make their P/L volatility as large as possible.
Now consider the desk heads. Suppose each employs ten traders, so their budget is $50M. They too get paid on the same basis, so their bonus is 10% x max(0, P/L - $30M). The mean is $50M but the SD goes as the square root of the number of traders employed, so it is root ten times $2M or about $6M assuming zero correlation. Therefore while the average desk head takes home $2M every Christmas, a bad one 2 SD from the mean has a bonus of 10% x max(0, 50M - 2 x 6M - 30M) = $800K. So even a pretty bad (or unlucky) desk head makes money. Of course the zero correlation assumption is a stretch but note that the desk head owns a call on a basket, so they should maximise both volatility of the basket components and their correlation.
Finally consider the head of the business employing a thousand traders. By the same logic, their budget is $500M, and they get paid 10% x max(0, P/L - $300M). Their volatility though is only root thousand times $2M or $20M. So a really terrible manager four SDs from the mean still earns 10% x max(0, 500 - 4 x 20 - 300) = $12M. Very nice. Diversification works beautifully in executive pay, at least if you are one of the executives. Shareholders may have a different perspective.
Labels: Organisation Structure, Organisational Culture, Rules
posted by David Murphy at 08:44
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