30 HRs or 30 saves?

I have done two posts, the true value of a home run and the true value of a save. These posts sprung out of the question: which is worth more, 30 home runs or 30 saves?

We found that the true value of a home run was worth 1.406 runs, and that the true value of a save was 0.11415 WPA. So how do we compare these two variables in different units? Eventually, we want to set a dollar value to each event, but we must first translate each into a win value.

It has been estimated that a win is worth somewhere between 9.5 and 10 runs. There are many different explanations how that was calculated and why it is so, but for simplicity I am just going to accept the argument that 10 runs = 1 win. We can now change the run value of a home run into a win. One home run is worth 0.1406 wins, so 30 home runs would be worth 4.218 wins.

Although it is usually not helpful to sum up WPA, in this case, it is the best we can do to approximate the value of a save. We found that the average save is worth 0.114 wins, so 30 saves would be worth 3.425 wins, using WPA. If we use WPA/LI, the average save was worth 0.0614 wins, so 30 saves would now only be worth 1.841 wins.

We have found out that, mathematically, 30 home runs are clearly worth more than 30 saves. We can now figure out how much each are worth in dollars. It has been estimated each win is worth about $4.5 million on the open market (so each win above replacement will cost approximately $4.5 million to replace, obviously a player with 8 wins above replacement is not going to be paid $36 million per year). So the value of 30 home runs, on the open market, is $18.98 million. This seems to be an unrealistic number, but there are players such as Jayson Werth who hit 27 home runs last year and received a 7-year, $126 million contract (average of $18 million/year) this offseason from the Nationals.

30 saves measured by WPA are worth 3.425 wins, or $15.41 million, and 30 saves measured by WPA/LI are worth $8.28 million. This dollar amount for WPA/LI is much more realistic than the amount for home runs. One example is Bobby Jenks, who compiled 27 saves last year and got a 2-year, $12 million contract this offseason.

So, the answer to the question of 30 home runs or 30 saves has clearly been answered. Home runs are either only slightly more valuable, or much more valuable than saves, depending on your view of relief pitchers. I believe that the math agrees with intuition here, as it seems as though it would be much easier (and cheaper) to acquire a player that will get 30 saves as opposed to a player that will hit 30 home runs. The marginal difference between an average closer (like Frank Fransisco for the Jays) and another pitcher in the bullpen (say, Jason Frasor) is much smaller than the marginal difference between a player like Aaron Hill and a bench player, such as John MacDonald.

In conclusion, I want to show one more example. This is a list of the 18 players who hit at least 30 home runs last year. The average Wins Above Replacement for the players was 4.39. If we look only at Batting Wins (WAR with the defense and running statistics removed), the players still have an average of 3.58 wins. This is a list of the 14 pitchers who saved at least 30 games last year. They have an average WAR of 2.09 wins. I believe that this shows that the pitchers who save 30 games are less valuable to their teams than the players who hit 30 home runs, which we have seen over the past three posts.