Torre and the evil empire

It’s disappointing that I feel obligated to comment on the Joe Torre contract insult situation. You might ask, don’t the Yanks get enough attention as it is? Certainly, but Yankees spending can seem so ridiculous at times it’s an easy target. Sorry friend, already off topic. Anyways, let’s take all the politics and personal feelings out of this. An incentive rich, one year, $5 million contract for Torre with possible $1 million bonuses for reaching each the Division, League Championship, and World Series. Incentive clauses that were similar to other contracts Torre has agreed upon. It is a $2.5 million pay cut from last season, yet Torre would still be the highest paid manager in the game by $1.5 million with this new contract. For the 2007 season Torre was the highest paid manager by $4 million over Lou Pinella and is four standard deviations above the mean salary. Take a look at the distribution of managerial salaries for the 2007 season. Torre is all alone in the tail of the distribution. The new offer is still generous for what is considered heavily incentive based. Without the bonuses the contract offer is nearly 2.5 standard deviations above the 2007 average. This is not an insult contract from the Yankees. Torre would still be the highest paid manager by a significant amount and the contract is probably over valued given his true marginal value product. If he does decide to manage somewhere else, it is unlikely he’ll find a similar offer.

Rule of 72

The Rule of 72 is a helpful approximation used to determine the time frame required for an investment to double. It’s a nice trick to remember when a quick calculation is necessary. Simply divide 72 by the prevailing interest rate (expressed as 100*r) and the solution is the resulting time necessary for the investment to double. The derivation is simple for compound and continuous compound problems. The future value of an asset with an interest
rate compounded per annum for t years is given by,

To determine the time period at which the future value of the asset is doubled, we simply solve the previous equation for t while setting the future value to twice that of the present value.

A first order Taylor series approximation for ln(1+r) is simply r, and
easily simplifies our result,

Given that the ln(2)=0.693, the rule can be applied as follows%


72 or 70 are commonly used for the convenience of division, and the approximation is relatively accurate. For cases involving continuous compounding the derivation is even more straight forward.




Because interest is compounded continuously a Taylor series approximation isn’t necessary, and the rule applies similarly,


As an example, assume you want to put $1,000 in an account accruing an annual percentage rate of 8%. According to our rule, it would take around 9 years to double the investment to $2,000. The actual calculation is 8.66 years.