A Risk Based Analysis of the Dustin McGowan Extension

The Jays have signed Dustin McGowan to a two-year extension, for a total of $3.5-million guaranteed including a 2015 option buyout of $500K. Essentially, this is the cost of a decent reliever in the free agent market on a one year deal, and consequently it's setting a fairly low performance bar for the Jays to get value - if he produces 1 WAR over those two years, they'll have done fine. On the other hand, we're talking about a 30 year old reliever who has suffered many serious injuries and has barely pitched in the last 3 years, which is what some detractors have focussed on. What I want to do here is try to measure and analyze the risks and benefits a little more, in order to get a better understanding of the potential costs and benefits to both sides.

What the Blue Jays Gain

As I indicated above, the production needed to justify the deal is quite small. But that doesn't tell us anything about the distribution of those outcomes, or the likelihood of achieving each outcome.. Thus I tried to do some crowdsourcing in the comments section of the announcement post, by defining 6 buckets of WAR production over the next three years, and asking people to estimate the chance of McGowan producing varying amounts of WAR. The results were as follows:


In order to estimate an expected value of 3.85 WAR, I took the midpoint of each bucket. For the 10 WAR+ bucket, I just used 11 WAR, which is slightly above the minimum. The median projection would be somewhere in the 3-5 bucket, near the low end, so the mean is greater than the median. That makes sense considering the skew in the distribution of value - a lot of it is realized from low probability favourable outcomes. At this point, there are two adjustments to make. The extension was for the 2 years following this year, so we should only look at the value attributable to those two years. To avoid confusion, I didn't pose the question that way, since it's a fairly easy adjustment to make. Due to aging factor, it shouldn't be equally weighted. I'm using 4/3/2 weighting factor, which ends up closely approximating the standard -0.5/year aging factor. So around 56% of the 2012-14 production allocated to the latter two years. The second adjustment is to convert this to dollars. I'm just going to use $5-million per WAR, no inflation adjustment. This essentially means the alternative to spending on this deal is free agent spending, which is reasonable given the new caps in spending on amateurs. We then get the following:


So what we see is that there's something like a 25-35% chance the Jays end up behind here, whereas there's about a 25% chance that they end up with a surplus value of more than $10-million. Those are really good odds for the club, given the vagaries of spending on free agents. But maybe those projections are too optimistic. But even if I discount them by 50% (chopping them in half!), that means there's still roughly a 50/50 chance of the Jays getting positive surplus value, and the distribution again favours the Jays since it skews towards the upside. Additionally, there is a fairly cheap option that adds some value we have not accounted for. Given these odds, it's hard to see this as anything but a good deal for the Jays.

Looking from McGowan's Point of View

The important thing to remember here is that McGowan, especially with his history, is exposed to a lot of risk. Unlike the team, which holds a portfolio of players across which to distribute injury risk and other risks (variation in performance, variation in player aging, etc), the player only gets one shot at his career, and serious injuries greatly impair career earnings. Thus players, especially one like McGowan, want to transfer that risk where possible to a team. The question is of course, at what price?

But looking at it in monetary terms does't tell us the whole story. Money is only useful insofar as it buys goods and services that we desire - we call this utility. How does money relate to utility? It depends on the individual, and broadly we can group people into three categories - risk averse, risk neutral and risk loving. Contrary to how the term is usually used, risk averse means that individuals will will only take on more risk if there is more reward, in terms of expected utility. Most people fall into this category in their everyday dealings. Risk neutral refers to being individuals being indifferent between two options with the same expected utility (think being indifferent to getting $1 or a 50/50 shot at $2 or nothing), and risk loving means an individual would accept more risk for a lower expected utility - the thrill of the uncertainty is worth the loss of utility. For a risk averse individual, monetary wealth has diminishing returns in terms of utility. Your first $X has the most utility, the next $X less utility, but still a considerable amount, and so on. We can denote this mathematically using the function Utility equals Wealth to the exponent x (where x is a variable), or U = W^x. In the case of a risk averse individual, x has to be a positive number less than 1 (0 < x < 1), for a risk neutral person x = 1 and for a risk loving individual, x is greater than 1.

So typically, for a risk averse individual, we use x = 0.5 (the square root of monetary wealth). A quick example shows how this works. For wealth of $10K, the utility is 100. If we add another $10K of wealth, we get the square of root of $20K, which is utility of 141.4 . We doubled monetary wealth, but didn't double the utility.

How does this apply to Dustin McGowan? I estimate his career earnings (including 2012, which was already negotiated), including signing bonus but not negligible minor league salaries, at around $3-million. If we figure 50% in taxes and agent fees, that leaves $1.5 million. Let's figure $500K in spending along the way along with any investment returns, leaving a nest egg of $1-million. Using our model above, that means a utility of 1,000. This was the status quo before signing the extension. Adding in the guaranteed $3.5-million, net of 50% for the same reasons, we get wealth of $2.75-million and utility of 1,658.

Now we can integrate the scenarios from above and the expected production in what would have been McGowan's free agent years. In translating the WAR production to dollars, I'm going to discount the values by 25% to account for the continuing effect of possible injuries. The expected earnings value over 2012-13 is $8-million, which net of the same 50% would leave a total wealth of $5-million, which has an associated utility value of 2,236. However, the expected value is just that, an expected value. But the player faces a number of possible outcomes and their associated utilities, not an expected value and utility based off that. We calculate this as the expected utility of the wealth at every outcome, weighted by the probability::


We then get, with this model, an expected utility of 2,151 by going the free agent route, which is about 30% higher than taking the deal, despite the fact that the expected earnings being almost double. Of course, we have made assumption about McGowan's risk aversion, and we do not know exactly how risk averse McGowan is. The point is to show that for a risk averse individual, changes in wealth will have much smaller changes in utility, so the individual can be better guaranteeing themselves a smaller monetary value and eliminating risk.

Would it Have Been to Wait?

One final thing to think about is whether the Jays would have been best off waiting, seeing what McGowan was capable of, and making the decision then, with more information. However, that's a double edged sword. If you wait, and McGowan succeeds, a fair bit of uncertainty is removed, and the past injuries are less of a concern. This is called the expected value of perfect information (EVPI). Without getting into a lot of gory details, I think it's enough to say that the Jays have had a lot of their people watching him for a long time, and so they're in a pretty good place to be making this talent evaluation call. Given the past track record of this front office, I see no reason to doubt their ability to decide if it makes more sense to wait, or lock in the deal now. They simply have far more information than anyone else.The only yellow flag might be that no one really has better information than McGowan himself, particularly about how he feels. I believe his willingness to sign can be attributed to risk aversion, but there's always some risk of asymmetric information, namely adverse selection.


This deal was the rare example of a truly win/win deal. In this case, we have a player who faces even higher amounts of risk than the typical player selling that risk to the team, and buying a large amount of financial security. We have a team who is in a much better place to bear that risk buying it fairly cheaply, and putting itself in position to earn a better expected return on that guaranteed money than if likely invested elsewhere. In hindsight of course, there will likely be a clear winner and a loser, but that's not this extension should be evaluated. Each side got something very valuable in this deal, due to the fairly unique circumstances. .

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