Although he’s dragging a bit in comparison to previous years (currently hitting .049 home runs per plate appearance, as opposed to last year’s .057 and 2010′s .06), there’s an argument to be made that he’s the victim of bad luck. For example, the league’s batting average on balls in play (BABIP) is .292, and Pujols’ is a full .016 below that at .276. In his 401 at-bats, that’s about 6 hits that average defense wouldn’t have fielded. Mike Trout, on the other hand, is up at a BABIP of .400. That’s about 36 hits on his 333 at-bats that are above his expectation if he had the league’s average BABIP. This is emphatically not to say that Trout’s season is a fluke, or that Pujols’ is, but sometimes the human element of the game has odd results.

So, a friend of mine made the following comment on Facebook the other day:

Someone has to look this up for me. Has any player, ever, made the last out for his team more often than A-Rod? He’s like the bizarro Mo.

Of these five, Kelly was the most versatile, playing at every position except second base and shortstop this year. Maier and McDonald were the least: each played three outfield positions in addition to pitching and hitting, while Cuddyer played first base, second base, and right field. McCoy, a typical utilityman, played second, third, short, center, and right. Kelly’s tenure on the mound was the shortest (one batter, one out), with everyone else pitching a full inning. McDonald gave up two runs on a hit and two walks in six batters faced; Maier faced four and gave up one hit, but no runs; Cuddyer allowed one hit and walked one for six batters faced and no runs; and McCoy pitched a perfect inning.

There’s no telling who will join these fellows next year – Skip Schumaker and Wilson Valdez each pitched an inning this year, but  while Valdez is a journeyman, he’s unlikely to DH, and Schumaker is locked in with the Cardinals for the next two years.

*not a guarantee

I made some odd choices in modeling run production in that post. The first big questionable choice was to detrend according to raw time. That might make sense starting with a brand-new league, where we’d expect players to be of low quality and asymptotically approach a true level of production – a quadratic trend would be an acceptable model of dynamics in that case. That’s not a sensible way to model the major leagues, though; even though there’s a case to be made that players being in better physical condition will lead to better production, there’s no theoretical reason to believe that home run production will grow year over year.

So, let’s cut to the chase: I’m trying to capture a few different effects, and so I want to start by running a linear regression of home runs on a couple of controlling factors. Things I want to capture in the model:

• The DH. This should have a positive effect on home runs per game.
• Talent pool dilution. There are competing effects – more batters should mean that the best batters are getting fewer plate appearances, as a percentage of the total, but at the same time, more pitchers should mean that the best pitchers are facing fewer batters as a percentage of the total. I’m including three variables: one for the number of batters and one for the number of pitchers, to capture those effects individually, and one for the number of teams in the league. (All those variables are in natural logarithm form, so the interpretation will be that a 1% change in the number of batters, pitchers, or teams will have an effect on home runs.) The batting effect should be negative (more batters lead to fewer home runs); the pitching effect should be positive (more pitchers mean worse pitchers, leading to more home runs); the team effect could go either way, depending on the relative strengths of the effects.
• Trends in strategy and technology. I can’t theoretically justify a pure time trend, but I also can’t leave out trends entirely. Training has improved. Different training regimens become popular or fade away, and some strategies are much different than in previous years. I’ll use an autoregressive process to model these.

My dependent variable is going to be home runs per plate appearance. I chose HR/PA for two reasons:

1. I’m using Baseball Reference’s AL and NL Batting Encyclopedias, which give per-game averages; HR per game/PA per game will wash out the per-game adjustments.
2. League HR/PA should show talent pool dilution as noted above – the best hitters get the same plate appearances but their plate appearances will make up a smaller proportion of the total. I’m using the period from 1955 to 2010.

After dividing home runs per game by plate appearances per game, I used R to estimate an autoregressive model of home runs per plate appearance. That measures whether a year with lots of home runs is followed by a year with lots of home runs, whether it’s the reverse, or whether there’s no real connection between two consecutive years. My model took the last three years into account:

Since the model doesn’t fit perfectly, there will be an “error” term, , that’s usually thought of as representing a shock or an innovation. My hypothesis is that the shocks will be a function of the DH and talent pool dilution, as mentioned above. To test that, I’ll run a regression:

The results:

The DH and batter effects aren’t statistically different from zero, surprisingly; the pitching effect and the team effect are both significant at the 95% level. Interestingly, the team effect and the pitching effect  have opposite signs, meaning that there’s some factor in increasing the number of teams that doesn’t relate purely to pitching or batting talent pool dilution.

For the record, fitted values of innovations correlate fairly highly with HR/PA: the correlation is about .70, despite a pretty pathetic R-squared of .08.

By any measure, his 7 innings, 3 runs (2 earned), no walks and 7 strikeouts were a quality start. (They gave him a game score of 58, and matched the 6-inning, 3-run criterion MLB uses for a quality start.) Three innings was enough, though, to give the Mets the loss. The Mets have given up an average of 4.57 runs per game this season, putting them .39 above the NL average and 13th in the league. That’s not too bad – except that they only score 4.44, and that extra 13% of a run adds up over time. (Note that when I crunched numbers for home field advantage, the Mets’ home advantage was quite high, at 1.4 more runs scored at home, so last night’s performance was quite a letdown.) The Mets weren’t running a September callup lineup, either – Mike Nickeas was at catcher, but the rest of the lineup was pretty consistent.

Dickey’s had a rough year. A loss in a quality start is called a Tough Loss, and he’s had six of them. That doesn’t lead the league – Hiroki Kuroda and Jeremy Hellickson split that honor with eight each – but it’s tough to pin all of the blame on Dickey when he’s pitched to six tough losses. Worse, he has seven Quality No-Decisions, which are, predictably, no-decisions in quality starts. Those are more common, but it means that of Dickey’s 30 starts, with 19 of them quality starts, a whopping 13 of them haven’t gotten him a win. By contrast, of his 8 wins, only 2 came in non-quality starts. (We call those Cheap Wins.) That kind of breakdown shows a lack of support from the team.

It’s not like the Mets are this unsupportive all the time, though – Dickey’s six Tough Losses were over one-third of the 17 Tough Losses earned by the team this year, and his seven Quality No-Decisions are around one-third of the Mets’ 22 quality starts with no-decisions for the pitcher. His two Cheap Wins? The Mets have sixteen.

Dickey just can’t get lucky this year.

On August 23, Skip Schumaker took the mound for his St. Louis Cardinals in a blowout loss to the Los Angeles Dodgers. Though Schumaker has started the majority of his games this season, he was on the bench that night and Tony La Russa used him in relief. Kyle Lohse, who ironically has played a bit of left field himself, only managed three innings as the starter and allowed eight runs, all of them earned. He was relieved by Mitchell Boggs, who allowed two earned and one unearned in two innings, followed by the competent Marc Rzepczynski for two scoreless innings on two hits, and a one-hit, one-K inning from Octavio Dotel. Skip came in to an 11-0 deficit, then promptly struck out Trent Oeltjen, walked Andre Ethier, and gave up a home run to the light-hitting infielder Aaron Miles. Rod Barajas flied out to deep center and relief pitcher Blake Hawksworth, batting for himself, struck out looking. The Cardinals scored two runs in the bottom of the ninth, but that wasn’t enough to save Skip. He hadn’t pitched since college.

Three days later, the Oakland As were visiting the Boston Red Sox and the game wasn’t going well for the home team. Journeyman outfielder Darnell McDonald had started the game at right field, as is his custom. In the top of the ninth, the Athletics were leading 13-4. Terry Francona had only gotten four innings out of starter Tim Wakefield, followed by three competent innings from Scott Atchison and a painful four-earned-run inning from Matt Albers. Since Albers wasn’t really a good option to stay in the game, McDonald moved from the field to the pitcher’s mound. Of course, this being the American League, that meant the team had to give up its designated hitter, so David Ortiz had a seat and Josh Reddick came in to play right. McDonald finished the game, giving up two runs, both earned, on one hit and two walks. Unsurprisingly, a game started by a knuckleballer had two wild pitches; surprisingly, one was Wakefield’s and one was Atchison’s. The position player and the guy who gave up four earned runs? No wild pitches at all.

Finally, Wesley Wright did things a little backwards. On the 23rd, manager Brad Mills called on the left-handed Wright to pitch to the Rockies’ lefty outfielder, Carlos Gonzalez. Gonzalez popped out to catcher Humberto Quintero and was followed in the linup by right-hander Troy Tulowitzki. This season, Troy is hitting .288/.361/.518 against right-handers and .345/.415/.634 against left-handers, so Mills was faced with a strategic decision: after Tulowitzki came first baseman Todd Helton, whose splits are in the opposite direction (.314/.402/.491 against right-handers, .292/.356/.438 against left-handers), so it was nonoptimal to lift Wright for a right-hander and then have the righty face Helton. The only other lefty in the bullpen was starter Wandy Rodriguez. Mills took Brian Bogusevic out of right field, moved Wright to right, and put in right-hander David Carpenter to face the right-handed Tulowitzki. Troy grounded out to the middle infield, and then Mills brough Wright back in to face Helton (putting J.B. Shuck in right field to complete the switch). That’s mainly notable because of the density of words pronounced like ‘right’ in that description.

But I digress.

A lot of people have been finding The World’s Worst Sports Blog by searching for “who’s next to 600 home runs”, which usually brings up this old post of mine. Of course, since it determines that Thome is next, it’s not terribly useful. The leaderboard for home runs (active players only) gives us Manny Ramirez, who retired early in the season, as following Thome up with 555. Chipper Jones follows with 448, followed by Vladimir Guerrero with 446, Albert Pujols with 439, Jason Giambi with 427, and Andruw Jones rounding out the top 5 with 416. Suffice to say it’ll be a while before anyone hits #500, much less #600. (No one else is above 400.)

The ages of the players involved – Chipper is 39, Giambi is 40, Guerrero is 36, Andruw is 34, and Pujols is a sprightly 31 – make things a little more interesting. Let’s take a look at them one by one:

1. Chipper Jones, age 39, 448 home runs. His production was about .03 home runs per plate appearance in 2009 and 2011, but dropped to about .026 in 2010. If he has 52 home runs to hit, and he only hits a home run about 3% of the time, it’ll take about 1733 plate appearances to hit #500, and about triple that to hit #600. 1733 plate appearances is about three years of full-time play (600 plate appearances over three years would be 1800), and I can’t imagine Chipper maintaining his hitting ability with his history of injuries. He may stick around until he’s 42 to hit #500, but he won’t hit #600.
2. Vladimir Guerrero, age 36, 446 home runs. Vlady’s a high-variance hitter. His home run production over the past few years has been .041, .045, .037, .045, and then .022 this year. Let’s downgrade him to about .04 to account for age. That gives him 135 plate appearances to #500 and about 385 to #600. If he normalizes down to about .03, his numbers will be similar to Jones’. For Vlad, consistency is going to be the biggest hurdle to making a milestone.
3. Albert Pujols, age 31, 439 home runs. Albert is hitting at a career low .027 clip this year. His previous years were .047, .057, .067, .06, and this year he’s way down at .027. That’s probably due to his fractured wrist, so let’s credit him with .05 home runs per plate appearance over the next few years. That means he’ll take about 1220 plate appearances to #500, or about 3220 to #600. He’ll need to play to the ripe old age of 36 to hit his 600th home run, so I think he’s a pretty safe bet.
4. Jason Giambi, age 40, 427 home runs. Giambi has hit at a .09 rate this season, but as he’s a full-time pinch hitter, he’s only made 122 plate appearances. Even assuming he hits at a .05 home run per plate appearance rate, and assuming he played a full season of 600 plate appearances as a DH, he’d need almost two and a half seasons (1460 plate appearances) to make his 500th home run. That would make him over 42. He won’t make it that far.
5. Andruw Jones, age 34, 416 home runs. Jones’ production has consistent – in 2009 he hit .051 home runs per plate appearance, and this year and last have both been around .058. Let’s call his average production going forward .055. He’ll need about 1527 plate appearances before he hits his 500th, or about 3345 before #600. Those normalize to about two and a half years and about five and a half years, respectively. Of course, Jones hasn’t made 600 plate appearances in a while – he’s made about 300 for the past few years. Still, doubling the time to hit 500 and 600 still put #500 within striking distance for Jones, who would be about 39 five years from now.

Realistically, Albert Pujols is the only one of this group who’s likely to make 600 home runs at all, much less within a few years.

Bonilla’s arrangement mimics a financial instrument called an annuity, where a constant payment is made at specific time periods after a specific present sum is invested. The annuity formula is:

where r is the annualized interest rate and t is the number of years of payment. Keep in mind, though, that the present value of the annuity isn’t $5.9 million – it’s $5.9 million compounded annually at some rate of interest agreed to by Bonilla and the team for the ten years between the deal and the first payout. In general, that means

Since we know Bonilla’s payout, we can substitute in:

and that solves out neatly to the 8% that the team and Bonilla agreed to. The math checks out so far.

At the time the deal was made, the 8% was 50 basis points (0.5%) below the Prime Rate, the reference rate used by banks in making loans. The average prime rate over the previous year was about 8.16%, and rates had hovered within 75 basis points since September of 1994*, so while interest rates are expected to move, it was very likely that rates would stay similar, at least in the short term. For the record, a 30-year fixed rate mortgage would have cost between 8.15% and 8.25%, so taking into account the long maturity of the loan, it wasn’t a bad deal.

Let’s look at how good a prediction it was. Annualizing prime rates, the Mets could have earned a (full prime) rate of return as follows:

So, the actual value of the $5.9 million on January 1, 2011, was $10,891,903.26, but the agreement pegged the value at

for a difference of about $1.85 million. Bobby’s already better off because historical interest rates didn’t keep up with 8%.

My biggest question is why the Mets agreed to an 8% interest rate then and there to be in effect for the next 35 years. Since I’m not a finance professional, I don’t know whether that’s an industry standard agreement or not, but it seems like the risk of setting an interest rate that far in the future would be far too high. What if the Mets had agreed to the 8% interest rate for ten years and then offered Bonilla a menu of financially equivalent options? All of them would rely on the payment formula:

where t is the number of periods and r is the newly figured interest rate.

One option would be to take the $12,737,657.48 as a lump sum, although that wouldn’t necessarily be a good idea for the Mets. (We know they’re cash strapped.)

The current prime rate is 3.25%, so if we took the lump sum $12,737,657.48 from the original agreement and reamortized it today at 2.75%, Bobby could receive a payment of $711,270.46 over the next 25 years. Similarly, at 2.75%, $1,047,789.14 per year for 15 years or $2,761,502.75 for five years would be equivalent options. Each has a different total cash outlay, but the discount rate means that each of them is worth the same $12,737,657.48 in 2011 dollars.

Bringing it all back, that’s why it’s a little silly to talk about the Mets paying $30 million to defer $6 million in compensation. It’s true that they’ll end up putting more dollars into Bonilla’s hands, but that simply represents Bonilla’s forebearing on the ability to invest that money at current interest rates. It doesn’t matter when you pay him – the money is worth the same amount, and that’s all that matters.

* Historical prime rates here, thanks to the St. Louis Fed and Federal Reserve Economic Data

Raphy at Baseball Reference has already done the honors of digging out no-hit non-shutout games, including Jered Weaver‘s heartbreaking 1-0, 0 ER 6-hitless-inning loss. I would, however, like to recognize the efforts of the other seven guys who started a game and left with no hits. Miguel Batista, Kyle Davies, and John Danks did it with style, leaving due to injury; Jon Lester, Aaron Harang, and Chris Tillman, meanwhile, came out early for various other reasons. As for Santana, he pitched four no-hit innings in May, but was replaced by Rich Thompson in the fifth.