The True Value of a Home Run

I have been meaning to do a post of the true value of a home run for awhile, but unfortunately I put it on the back burner for awhile until I was asked this question: which is worth more, 30 home runs or 30 saves? In this post, I am going to examine the true value of a home run, and in the next post I will examine exactly how much a save is worth, so I can compare the two.

The data I am going to use is for all teams in the 2010 regular season. The first thing to do is to find the number of home runs hit in each base-out state, which can be found from baseball-reference:

RUNNERS	HR_OUTS_0	HR_OUTS_1	HR_OUTS_2
None	1220	811	617
1st	248	318	312
2nd	74	133	147
3rd	9	39	52
1st and 2nd	54	119	142
1st and 3rd	27	49	47
2nd and 3rd	9	30	30
Bases Loaded	23	43	60

The total number of home runs hit last year was 4613, and over half of those were solo home runs. It was very rare for players to hit home runs with no outs and runners on third, as it would usually require a triple, or a double and steal.

The next step is to find the expected runs matrix for 2010 (from Baseball Prospectus):

RUNNERS	EXP_R_OUTS_0	EXP_R_OUTS_1	EXP_R_OUTS_2
None	0.49154	0.26151	0.10374
1st	0.85877	0.50512	0.2282
2nd	1.10113	0.67765	0.3215
3rd	1.35798	0.93308	0.34192
1st and 2nd	1.42099	0.88181	0.45503
1st and 3rd	1.80042	1.0982	0.46571
2nd and 3rd	1.96584	1.38849	0.58205
Bases Loaded	2.36061	1.51185	0.77712

We can use these two matrices together to determine the true value of a home run. The equation we will use is: value of a home run = Expected runs at the end of the play - Expected runs at the beginning of the play + the number of runs scored during the play. What this means is that we are taking the expected runs after - before to determine the value of the play (e.g. a leadoff out would be calculated as 0.26151 - 0.49154 = -0.23003, meaning the expected runs for the team in that inning would decrease by 0.23 runs), and then adding the number of runs that were scored.

This matrix shows the true value of a home run for each base-out state. Obviously, when there are no runners on base, the value of a home run will be 1, as the beginning and end states will be the same.

RUNNERS	Value_OUTS_0	Value_OUTS_1	Value_OUTS_2
None	1	1	1
1st	1.63277	1.75639	1.87554
2nd	1.39041	1.58386	1.78224
3rd	1.13356	1.32843	1.76182
1st and 2nd	2.07055	2.3797	2.64871
1st and 3rd	1.69112	2.16331	2.63803
2nd and 3rd	1.5257	1.87302	2.52169
Bases Loaded	2.13093	2.74966	3.32662

The most valuable home runs, obviously, are grand slams, as they score 4 runs, while home runs hit with two outs are more valuable than those hit with 0 or 1 out as there will be fewer chances remaining in the inning to drive in the runners or base, thus making the home run more valuable.

Finally, we need to multiply the matrix containing the number of home runs hit by the matrix showing the true value of a home run for each base-out state to find the run values for each base-out state.

RUNNERS	Value_OUTS_0	Value_OUTS_1	Value_OUTS_2
None	1220	811	617
1st	404.92696	558.53202	585.16848
2nd	102.89034	210.65338	261.98928
3rd	10.20204	51.80877	91.61464
1st and 2nd	111.8097	283.1843	376.11682
1st and 3rd	45.66024	106.00219	123.98741
2nd and 3rd	13.7313	56.1906	75.6507
Bases Loaded	49.01139	118.23538	199.5972

To find the true value of a home run, we simply add up all of the runs (6485) and divide by the total number of home runs hit (4613) to find the average value of a home run: 1.406 runs. What this means is that the average home run hit in 2010 was worth 1.406 runs for the player's team. We will use this number later to figure out exactly how much each home run is worth in a dollar amount, and whether or not it is worth more than a save.