What Is a Run Worth?

I recently began thinking about how teams can know that they are efficiently spending their money, or where teams actually get the runs that they spend all their money on. With players signing massive contracts in the 2018-19 offseason, I began to wonder if any players were really worth that much money. The process begins with one big question: What is a run worth? I quickly realized that each team theoretically needs to manufacture the same number of runs as all the other teams do if they want a better chance to make the postseason. What is different from team to team is budget. This means that a run is worth a different monetary value to each team, and that each team would be willing to pay a different amount of money for the same number of runs. The problem is that to each player, a run costs the same amount, causing Billy Beane, played by Brad Pitt in the movie Moneyball, to claim that “It’s an unfair game”.

Figuring out what each team values their runs at would enable me to evaluate how efficient the signing of certain contracts was for each team and furthermore would allow me to figure out where the most value comes from in the payroll of a team. First, I had to figure out how to convert the basic statistics of a player into the number of runs that player actually contributed to the team. I eventually came across the Estimated Runs Produced statistic from the 1985 Bill James abstract. Below is the calculation.

ERP = (2 (TB + BB + HBP) + H + SB – (.605 (AB + CS + GDP – H))) .16

This is a stat created by Paul Johnson in order to obtain more accuracy than Runs Created, which he succeeded in doing. I then fired up R and ran some tests on team statistics to see how well it lined up with the actual number of runs that each team scored. I graphed ERP against Runs Scored first for every team dating back to the beginning of the 30-team era in MLB:

Residual Standard Error: 22.42

R^2: .9309

As you can see, the statistic is still quite accurate, and because I wanted this to be used to evaluate pitchers’ contracts as well, I graphed ERP calculated from Batting Against stats to Runs Against for every team, once again dating back to the beginning of the 30-team era in MLB:

Residual Standard Error: 23.79

R^2: .9285

Estimated Runs Produced seemed to work very well for both hitting and pitching, so I proceeded to develop the monetary side of my investigation. In order to figure out how much a run is worth to each team, I had to find the budget of each team, for which data is not available. I was able to use past payroll info and market size in order to group teams together and estimate the team budget for payroll. Below are my estimates:

Estimated Team Budgets
Team Budget Estimate
Boston Red Sox $275,000,000
New York Yankees $275,000,000
Chicago Cubs $275,000,000
San Francisco Giants $235,000,000
Washington Nationals $235,000,000
Houston Astros $235,000,000
Los Angeles Angels $235,000,000
Philadelphia Phillies $225,000,000
Los Angeles Dodgers $215,000,000
St. Louis Cardinals $215,000,000
New York Mets $215,000,000
Colorado Rockies $200,000,000
Seattle Mariners $170,000,000
Atlanta Braves $170,000,000
Milwaukee Brewers $170,000,000
Minnesota Twins $160,000,000
Cleveland Indians $160,000,000
Texas Rangers $150,000,000
Cincinnati Reds $150,000,000
Arizona Diamondbacks $150,000,000
San Diego Padres $150,000,000
Detroit Tigers $130,000,000
Toronto Blue Jays $130,000,000
Kansas City Royals $130,000,000
Chicago White Sox $130,000,000
Baltimore Orioles $130,000,000
Oakland Athletics $115,000,000
Miami Marlins $115,000,000
Pittsburgh Pirates $115,000,000
Tampa Bay Rays $95,000,000

To figure out the number of runs a team needs to score and to allow, I took the records of the top 30 teams in the last five years and averaged them out to find that a team would need to win approximately 97 games and score around 814 runs in order to win their division. From those numbers, I calculated the Runs Allowed with Bill James’ win expectancy and came up with 654. But before I assign monetary value to the runs, I needed to know if there was a significant difference in the money teams spend on hitters versus the money that teams spend on hitters. I took all 544 player contracts from Spotrac and found that on average, hitter contracts are nearly 20% larger than pitcher contracts, so that is how I split up the budget of each team for pitching and hitting. I ended up with these numbers:

Estimated Team Budgets With Hitting/Pitching Split
Team Budget Hitting Budget Pitching Budget
Boston Red Sox $275,000,000 $149,792,924.90 $125,207,075.10
New York Yankees $275,000,000 $149,792,924.90 $125,207,075.10
Chicago Cubs $275,000,000 $149,792,924.90 $125,207,075.10
San Francisco Giants $235,000,000 $128,004,863.10 $106,995,136.90
Washington Nationals $235,000,000 $128,004,863.10 $106,995,136.90
Houston Astros $235,000,000 $128,004,863.10 $106,995,136.90
Los Angeles Angels $235,000,000 $128,004,863.10 $106,995,136.90
Philadelphia Phillies $225,000,000 $122,557,847.70 $102,442,152.30
Los Angeles Dodgers $215,000,000 $117,110,832.21 $97,889,167.79
St. Louis Cardinals $215,000,000 $117,110,832.21 $97,889,167.79
New York Mets $215,000,000 $117,110,832.21 $97,889,167.79
Colorado Rockies $200,000,000 $108,940,309.03 $91,059,690.97
Seattle Mariners $170,000,000 $92,599,262.68 $77,400,737.32
Atlanta Braves $170,000,000 $92,599,262.68 $77,400,737.32
Milwaukee Brewers $170,000,000 $92,599,262.68 $77,400,737.32
Minnesota Twins $160,000,000 $87,152,247.23 $72,847,752.77
Cleveland Indians $160,000,000 $87,152,247.23 $72,847,752.77
Texas Rangers $150,000,000 $81,705,231.77 $68,294,768.23
Cincinnati Reds $150,000,000 $81,705,231.77 $68,294,768.23
Arizona Diamondbacks $150,000,000 $81,705,231.77 $68,294,768.23
San Diego Padres $150,000,000 $81,705,231.77 $68,294,768.23
Detroit Tigers $130,000,000 $70,811,200.87 $59,188,799.13
Toronto Blue Jays $130,000,000 $70,811,200.87 $59,188,799.13
Kansas City Royals $130,000,000 $70,811,200.87 $59,188,799.13
Chicago White Sox $130,000,000 $70,811,200.87 $59,188,799.13
Baltimore Orioles $130,000,000 $70,811,200.87 $59,188,799.13
Oakland Athletics $115,000,000 $62,640,677.69 $52,359,322.31
Miami Marlins $115,000,000 $62,640,677.69 $52,359,322.31
Pittsburgh Pirates $115,000,000 $62,640,677.69 $52,359,322.31
Tampa Bay Rays $95,000,000 $51,746,646.79 $43,253,353.21

Then to determine how much money a run is worth to each team, I divided the hitting budget by 814 and the pitching budget by 654 to get the numbers below:

WoRS: Worth of a run scored

WoRA: Worth of a run against

Estimated Team Budgets: How Much Is A Run Worth?
Team Budget Hitting Budget Pitching Budget WoRS WoRA
BOS $275,000,000.00 $149,792,924.92 $125,207,075.08 $183,861.65 $191,508.37
NYY $275,000,000.00 $149,792,924.92 $125,207,075.08 $183,861.65 $191,508.37
CHC $275,000,000.00 $149,792,924.92 $125,207,075.08 $183,861.65 $191,508.37
SFG $235,000,000.00 $128,004,863.11 $106,995,136.89 $157,118.14 $163,652.61
WAS $235,000,000.00 $128,004,863.11 $106,995,136.89 $157,118.14 $163,652.61
HOU $235,000,000.00 $128,004,863.11 $106,995,136.89 $157,118.14 $163,652.61
LAA $235,000,000.00 $128,004,863.11 $106,995,136.89 $157,118.14 $163,652.61
PHI $225,000,000.00 $122,557,847.66 $102,442,152.34 $150,432.26 $156,688.67
LAD $215,000,000.00 $117,110,832.21 $97,889,167.79 $143,746.38 $149,724.73
STL $215,000,000.00 $117,110,832.21 $97,889,167.79 $143,746.38 $149,724.73
NYM $215,000,000.00 $117,110,832.21 $97,889,167.79 $143,746.38 $149,724.73
COL $200,000,000.00 $108,940,309.03 $91,059,690.97 $133,717.57 $139,278.82
SEA $170,000,000.00 $92,599,262.68 $77,400,737.32 $113,659.93 $118,387.00
ATL $170,000,000.00 $92,599,262.68 $77,400,737.32 $113,659.93 $118,387.00
MIL $170,000,000.00 $92,599,262.68 $77,400,737.32 $113,659.93 $118,387.00
MIN $160,000,000.00 $87,152,247.23 $72,847,752.77 $106,974.05 $111,423.05
CLE $160,000,000.00 $87,152,247.23 $72,847,752.77 $106,974.05 $111,423.05
TEX $150,000,000.00 $81,705,231.77 $68,294,768.23 $100,288.17 $104,459.11
CIN $150,000,000.00 $81,705,231.77 $68,294,768.23 $100,288.17 $104,459.11
ARI $150,000,000.00 $81,705,231.77 $68,294,768.23 $100,288.17 $104,459.11
SDP $150,000,000.00 $81,705,231.77 $68,294,768.23 $100,288.17 $104,459.11
DET $130,000,000.00 $70,811,200.87 $59,188,799.13 $86,916.42 $90,531.23
TOR $130,000,000.00 $70,811,200.87 $59,188,799.13 $86,916.42 $90,531.23
KCR $130,000,000.00 $70,811,200.87 $59,188,799.13 $86,916.42 $90,531.23
CWS $130,000,000.00 $70,811,200.87 $59,188,799.13 $86,916.42 $90,531.23
BAL $130,000,000.00 $70,811,200.87 $59,188,799.13 $86,916.42 $90,531.23
OAK $115,000,000.00 $62,640,677.69 $52,359,322.31 $76,887.60 $80,085.32
MIA $115,000,000.00 $62,640,677.69 $52,359,322.31 $76,887.60 $80,085.32
PIT $115,000,000.00 $62,640,677.69 $52,359,322.31 $76,887.60 $80,085.32
TBR $95,000,000.00 $51,746,646.79 $43,253,353.21 $63,515.84 $66,157.44

I then used this info to evaluate player contracts. For hitters, I calculated the expected number of runs that a player would be worth based on their contract by dividing their average annual value by the cost of a run to their team, as seen below.

xR: Expected Runs

xR = AAV / WoRS

Then to determine whether the team spent their money effectively, I calculated what I called their Run Plus Minus (rPM). This tells how many runs the player under- or overperformed their contract by based on the money available to the team. The formula is Estimated Runs Produced (ERP) minus Expected Runs (xR).

rPMhit = ERP – xR

This was relatively simple for hitters because it was just about scoring runs, while with pitchers it’s about stopping runs from being scored or saving them. The formula is below.

rPMpit = (((654 – ERP) / 654) * PitchingBudget * (IP / 1458) – AAV) / WoRA

It takes the number of runs that the pitcher “saved” out of the 654 that the team is theoretically supposed to allow and multiplies it by the amount of money available to the team’s pitching and then reduces this number according to the portion of the team’s innings that he pitched. This first part of the equation gives the salary that the team should be giving the pitcher based on his performance. I then subtract his actual salary to get the difference between the two and divide by the worth of a run against for the team. This gives the difference between how a pitcher performs and how they should be performing based on their salary. An rPM of 0 for either pitching or hitting means that the team paid the exact amount that the player is worth to the team in runs in order to reach the goal of 814 runs scored and 654 runs allowed. A positive rPM means the player performed better than what the team paid him for and put the team on track to surpass 814 runs or allow fewer than 654. A negative rPM would indicate the opposite.

If we believe that a budget can be split into money for the purpose of pitching or hitting, a team can overflow into the money required for the other category. For example, if a team uses more for pitching than is allotted, that team would need a positive rPM for offense in order to reach the goal of 814 runs.

What I was most curious about is the low-budget teams who really can’t afford to offer big contracts, because if they were to sign a star player out of free agency for north of $25 million, that would theoretically be nearly half of their available money for either hitting or pitching. I took a look at the entire Rays roster for 2019. Below are their salaries, ERP, Expected Salary based on their production, and rPM.

Tampa Bay Rays Hitters
Name Salary ERP xSalary rPM
Mike Zunino $4,412,500.00 19.0832 $1,212,085.48 -50.38765955
Ji-Man Choi $850,000.00 69.7016 $4,427,155.67 56.31911147
Brandon Lowe $1,000,000.00 50.8344 $3,228,789.62 35.09029585
Willy Adames $562,400.00 69.5624 $4,418,314.27 60.70791583
Yandy Diaz $558,400.00 48.8352 $3,101,808.75 40.04369224
Tommy Pham $4,100,000.00 95.5216 $6,067,134.66 30.97077299
Kevin Kiermaier $8,166,666.00 51.5856 $3,276,502.72 -76.99124004
Avisail Garcia $3,500,000.00 72.664 $4,615,315.00 17.55963548
Austin Meadows $557,400.00 103.4352 $6,569,773.61 94.65943635
Travis d’Arnaud $555,000.00 48.284 $3,066,798.82 39.5460222
Joey Wendle $570,400.00 24.648 $1,565,538.42 15.66756299
Daniel Robertson $570,300.00 19.54 $1,241,099.51 10.5611374
Guillermo Heredia $782,000.00 23.1648 $1,471,331.73 10.85291056
Matt Duffy $2,675,000.00 16.868 $1,071,385.19 -25.2474786
Nate Lowe $555,000.00 22.0512 $1,400,600.49 13.3132222
Mike Brosseau $555,000.00 18.7368 $1,190,083.59 9.998822198
Eric Sogard $555,000.00 14.3024 $908,428.95 5.564422198
Jesus Aguilar $637,500.00 13.4808 $856,244.34 3.443933606
Christian Arroyo $555,000.00 5.9848 $380,129.60 -2.753177802
Michael Perez $556,700.00 5.7952 $368,087.00 -2.969542779
TOTAL $32,274,266.00 794.0792 $50,436,607.41 285.949795
Tampa Bay Rays Pitchers
Name Salary ERP xSalary RPM
Jose Alvarado $570,600.00 18.0416 $865,427.30 4.456449703
Nick Anderson $555,000.00 4.4616 $621,685.67 1.007984398
Anthony Banda $556,800.00 2.9984 $118,120.68 -6.630838854
Jalen Beeks $556,500.00 61.996 $2,795,410.33 33.84215484
Mike Brosseau $555,000.00 3.1584 $118,091.64 -6.604069987
Diego Castillo $561,500.00 30.716 $1,928,182.45 20.65803102
Yonny Chirinos $561,800.00 55.5832 $3,612,880.90 46.11848501
Jose De Leon $558,000.00 1.7816 $118,341.53 -6.645639132
Oliver Drake $566,100.00 18.7776 $1,613,594.10 15.83335306
Peter Fairbanks $555,000.00 8.608 $354,235.13 -3.034652894
Jake Faria $555,000.00 6.9192 $293,522.61 -3.952350493
Wilmer Font $555,000.00 8.4144 $409,981.82 -2.192016234
Ian Gibaut $555,000.00 0.956 $59,245.69 -7.49355344
Tyler Glasnow $566,700.00 13.5824 $1,748,804.89 17.8680567
Andrew Kittredge $555,000.00 23.8504 $1,406,332.79 12.86828492
Adam Kolarek $555,000.00 20.2792 $1,238,954.52 10.33828582
Brendan McKay $555,000.00 29.6104 $1,387,809.21 12.58829263
Hoby Milner $555,000.00 2.4552 $94,575.41 -6.959528444
Charlie Morton $15,000,000.00 69.2328 $5,151,106.83 -148.8705303
Emilio Pagan $555,000.00 21.5288 $2,008,254.04 21.9666003
Ricardo Pinto $555,000.00 3.1624 $61,997.73 -7.451955097
Colin Poche $555,000.00 20.7728 $1,470,650.74 13.84048025
Austin Pruitt $555,000.00 24.052 $1,343,018.05 11.91125362
Trevor Richards $570,000.00 10.2472 $674,548.91 1.580304599
Chaz Roe $1,275,000.00 26.6928 $1,451,206.25 2.663438157
Casey Sadler $555,000.00 6.672 $560,842.43 0.088310932
Aaron Slegers $555,000.00 1.8488 $88,747.00 -7.047627626
Blake Snell $1,600,000.00 50.2112 $2,930,501.66 20.11114189
Ryne Stanek $555,000.00 22.8448 $1,580,355.54 15.49871849
Cole Sulser $555,000.00 1.6472 $210,099.51 -5.213328825
Hunter Wood $555,000.00 12.2816 $844,159.25 4.370774411
Ryan Yarbrough $563,400.00 53.1632 $3,848,253.17 49.65205981
TOTAL $34,051,400.00 636.5472 $41,008,937.77 105.1663693

The Rays were incredibly efficient with their spending this past season, which is why they were a playoff team and gave the Astros a run for their money. They ended up with a hitting rPM of about 286, meaning that they got a little over $18 million worth of runs for free. They also excelled in pitching with an rPM of about 105, meaning they spent nearly $7 million less than what their players are worth based on their production. The key to their success was loading up on pre-arbitration players who could consistently be better than what they are paid.

When teams sign a lot of players to massive deals, they are taking a big risk. With the star players eating up much of the budget, the team has to hope that they can bring in good low-cost players to fill out the rest of the roster. They also have to hope that their stars continue to produce big numbers or else they’ll find themselves searching for low-cost players to make up for the lost production, which there are very few of. Good players are going to cost a lot of money. If you have the money, it’s okay to lay down $30 million for a star. You just need to have some pre-arbitration or cheaper talent that can bring the rPM back to 0. I tested out rPM on several well-known players to see who is really worth what they are being paid. Some of the results for the hitters are below. For each of them I have included ERP, contract AAV, expected AAV, rPM, and rPM162, which calculates what the hitter’s rPM would be had they played the full season.

rPM For Top Hitters
Name ERP Salary xSalary rPM rPM162
Mike Trout 128.048 $35,541,667.00 $20,118,664.00 -98.16182 -71.40553
Nolan Arenado 120.4008 $32,525,000.00 $16,099,702.00 -122.8358 -117.3983
Ronald Acuna Jr. 119.4976 $12,500,000.00 $13,582,089.00 9.520409 14.11647
Eloy Jimenez 73.4128 $7,166,667.00 $6,380,778.00 -9.041898 15.02787
Christian Yelich 130.9616 $7,081,429.00 $14,885,086.00 68.65795 100.8946
Anthony Rendon 125.132 $18,800,000.00 $19,660,507.00 5.476816 19.18991
Juan Soto  118.1432 $578,300.00 $18,562,440.00 114.4625 123.914
Rafael Devers 120.2384 $614,500.00 $22,107,231.00 116.8962 121.5208
Aaron Judge 75.8016 $684,300.00 $13,937,008.00 72.07978 116.669
DJ Lemahieu 106.7672 $12,000,000.00 $19,630,394.00 41.50074 54.01827
Nelson Cruz 103.7696 $14,300,000.00 $11,100,655.00 -29.90768 6.411684

As you can see, not even Mike Trout or Nolan Arenado are worth their giant contracts. However, good players cost a lot of money, so the Angels have the ability to sign Rendon and Trout to big deals, but they need to have some players that they get for less than they are really worth to balance everything out. The Brewers are clearly getting a bargain on Christian Yelich until 2023 as long as the team exercises their option in 2022. In addition, it is clear that it pays to have pre-arbitration stars like Soto, Devers, and Judge who outperform their contracts by over 100 runs each. Another interesting thing I spotted is that Nelson Cruz underperformed his contract by 29 runs, but if he had played the whole season he would have over performed it by 9. I did this for starting pitchers and relievers as well, as seen below. The starting pitchers have had their Expected Salaries and rPMs adjusted as if they had each pitched 200 innings, approximately a starter’s full season of pitching.

rPM For Top Starting Pitchers
Name ERP Salary xSalary rPM
Jacob DeGrom 63.356 $21,780,000.00 $12,152,155.00 -64.30364
Jose Berrios 94.8208 $620,000.00 $8,545,944.00 71.13379
Max Scherzer 64.4848 $28,689,376.00 $12,996,925.00 -95.88879
Gerrit Cole 63.1424 $13,500,000.00 $13,341,808.00 -0.9666334
Stephen Strasburg 74.8936 $26,000,000.00 $13,068,093.00 -79.02048
David Price 57.4704 $31,000,000.00 $14,361,899.00 -86.87923
Hyun-Jin Ryu 60.8776 $17,900,000.00 $12,058,923.00 -39.0121
Walker Buehler 68.1112 $570,000.00 $11,893,406.00 75.62816
Robbie Ray 95.4656 $6,050,000.00 $7,798,917.00 16.7426
Kyle Hendricks 76.3208 $13,875,000.00 $14,909,704.00 5.402916
Justin Verlander 65.8008 $33,000,000.00 $13,352,169.00 -120.0582
rPM For Top Relievers
Name ERP Salary xSalary rPM
Brandon Workman 12.7248 $1,150,000.00 $6,034,937.00 25.50769
Ryan Pressly 14.0152 $2,900,000.00 $3,901,532.00 6.119864
Liam Hendriks 23.2568 $2,150,000.00 $2,943,915.00 9.91336
Aroldis Chapman 15.7672 $17,200,000.00 $4,776,879.00 -64.86986
Josh Hader 23.4664 $687,600.00 $3,872,904.00 26.90586
Joe Kelly 25.6992 $8,333,333.00 $3,310,798.00 -33.54513

Similar to what was seen with the hitters, even the very best pitchers are not worth their contracts. The difference-makers are the guys like Berrios and Buehler who make up for the money lost in those large contracts.

So how do you keep a good pre-arbitration player for the long run but at a low cost? The answer is in offering deals to players before they reach arbitration. This was done with Blake Snell and Brandon Lowe, and now the Rays have an elite pitcher for just $10 million AAV until 2024 and a very valuable hitter for his AAV of $4 million through 2026 if they choose. This is an excellent move, especially if Lowe continues to improve after his first half-year in the majors. This is how low-budget teams now need to survive, and thanks to Chaim Bloom’s leadership before he left town, the Rays are set up for the future very well. Now the Red Sox have hired Bloom to work the same magic he did with the Rays. Bloom starts his journey in the Sox organization with a team that is a result of lots of big long-term deals that haven’t panned out the way the Sox would have hoped as well as very few up-and-coming players that have been able to produce in the majors. A big problem for the club has been their inability to develop starting pitchers.

We have seen more than just the Rays switching to the early extension mentality. More recently, the Mariners have extended Evan White on a six-year, $24 million deal, giving him what they think he could exceed with his production when he reaches the majors in the next couple years. Additionally, the White Sox have begun to implement this strategy by signing top prospect Luis Robert to a six-year, $50 million deal. Robert could be starting 2020 in the lineup for the White Sox and has the potential to put up much bigger numbers than what they are paying him for.

The moral of the story is not to spend money just because you have it. Even if you have one of the largest media markets it does not mean that you are financially invincible. From this investigation, I have come up with a way to effectively evaluate player contracts while more objectively finding reasons to stress the development of players more than before. Either that or trade for young players who can exceed their contract value in run production. Players are now going to be paid for their futures with low-value deals more than before, many of which end up being low-risk and high-reward. Once those players gain value, they can be traded for prospects and then wash, rinse, repeat.

Sources

“ESTIMATED RUNS PRODUCED.” Paul Johnson’s Estimated Runs Produced, www.baseballthinkfactory.org/btf/pages/essays/jameserp.htm.

“MLB Stats, Scores, History, & Records.” www.Baseball-Reference.com

“Capology.” www.Capology.com

“Spotrac.com.” Sports Contracts, Salaries, Caps, Bonuses, & Transactions, 4 Jan. 1970, www.spotrac.com/.

“FanGraphs Baseball: Baseball Statistics and Analysis.” www.FanGraphs.com

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Freshman Computer Science and Statistics double major. I am a Red Sox Diehard. I have used R to analyze baseball stats and am also proficient in Java and familiar with Python.

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John Edwards
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channelclemente
Member

I wonder if it’s possible to calculate the NPV (Net Present Value) of a run toward a win for each team?

Clueless Joe Jackson
Member
Clueless Joe Jackson

Fascinating!! I had always suspected the conclusions you came to about big money contracts, but it took your math skills to prove it. I would speculate: Players on long-term, big money deals are either about to enter, or are already in, their decline phase. Therefore, in general at least, the runs they create or prevent shrink as they age, while their salaries, in general, either stay the same or go up, making whatever value they might provide at the beginning of their deal shrink considerably toward the end of the deal. I’m thinking Pujols as the poster boy example of… Read more »

sportsfreak2744
Member
sportsfreak2744

One. A run is worth one.