As I was casually looking through and viewing the panels and videos of the Sloan Sports Analytics Conference the other weekend, I realized a common theme for the 2018 Conference to the previous ones. For the NBA, there were plenty of applications of analytics¹ to player evaluations, injuries, in-game strategies, and ticket pricing. All of these topics were present in 2018 as they have been throughout the conference’s lifetime. However, there was only limited discussion of the NBA’s Salary Cap and any analytical applications applied to the Salary Cap was non-existent.

¹ I use the term analytics liberally here as there were panelists invoking “the eye test” in their discussions. Why has this become a phrase? It’s just someone talking out of their ass.

Panelists certainly know about the Salary Cap as this was brought up in at least 2 panels. Steve Ballmer made comments on his front office’s analytics at around the 31 minute mark. He had an interesting tidbit on the Salary Cap where he effectively thinks of it in a game theoretic context:

One actually that has become most interesting to me is salary cap planning. It’s most fundamental to the product. It’s not as statistical and involved but it’s like a big chess game: Where do you want to have powder? How do you use the powder? Will you be successful at taking the risks that you want to take? I find it a very interesting numeric chess game, if you will, and there are teams that absolutely do it better and do it worse. I see Darrell sitting in the front row, we made a trade: Chris Paul for a number of guys back for Houston. I thought they did something very clever in having a whole set of non-guaranteed contracts that really gave him more flexibility in putting the deal together with us than they would have had otherwise. But that is kind of numeric, strategic chess game like planning that I think is an important part of let me call it the numeric exercise as opposed to what people think about is classic analytics.

And in the penultimate presentation, Sam Hinkie also touched on the Salary Cap at around the 20 minute mark through about the 23 minute mark with the same analogy of treating cap space like powder² but offered little insight beyond how to utilize the cap space. Neither of these panels discussed the possibility of using analytics to better predict the Salary Cap. It is hard to determine if this is because no one is engaged in this form of analytics or if they simply do not want to tip their hand and give others an edge on this topic.

² When I hear people use the same analogy to describe a topic, I assume this is because the people learned the topic in the same manner – ie had the same teacher or read the same book.

Whatever the reason is, it dawned on me that I have yet to see anyone publish key components to the Salary Cap from the 2016-17 NBA Season. Larry Coon used to keep track of certain financial values in his CBAFAQ for the 2011 CBA, but has yet to do so for the 2017 CBA. Albert Nahmad over at HeatHoops.com has also tracked financial components for the NBA and even discussed the NBA’s 2016-17 Salary Cap projections last year. But Albert’s last update for NBA financials was in June of 2017 a few weeks before the Salary Cap was officially set.

Let me provide some of the important figures for the NBA’s finances in the 2016-17 Season. However, I will not just give the values but I will provide a description for how I came to these values as a way to “show my work” and as an educational exercise for others interested in the Salary Cap.

Revenues Side

Let’s start with what we know: the NBA set the Salary Cap for 2017-18 Season at $99,093,000 and the Luxury Tax at $119,266,000. That allows us to know pretty close to what the BRI for 2016-17 is and lets us know exactly what Projected BRI is. To get to these values, we need to define how the Salary Cap is set:

\[ {SC}_t = \frac{0.4474*\text{Projected BRI}_t - \text{Projected Benefits}_t}{\text{\# of teams (30)}} + \frac{\text{Adjustment}_t}{\text{\# of teams (30)}} \]

Where I will eventually get to defining what goes into Projected BRI, Projected Benefits, and Adjustment throughout this article. As of right now, all three of those components are unknown to us but can and will be determined. The Luxury Tax is set in the same fashion, except the factor multiplied by Projected BRI is 53.51% instead of 44.74%:

\[ {LC}_t = \frac{0.5351*\text{Projected BRI}_t - \text{Projected Benefits}_t}{\text{\# of teams (30)}} + \frac{\text{Adjustment}_t}{\text{\# of teams (30)}} \]

This leaves us with 2 equations and 3 unknowns: Projected BRI, Projected Benefits, and the Adjustment. We can subtract the Salary Cap from the Luxury Tax and do some algebra to redefine Projected BRI as:

\[ \text{Projected BRI}_t = \frac{\text{\# of teams (30)}}{(0.5351 - 0.4474)}*(LT_t - SC_t) = \frac{\text{\# of teams (30)}}{(0.0877)}*(LT_t - SC_t) \]

We have eliminated one equation and two of the unknowns. Now we are left with one equation and one unknown which becomes a plug-n-chug to figure out that the Projected BRI used in the 2017-18 Salary Cap calculation is $6,900,684,151. We can use Projected BRI as well as the NBA’s TV contract to back out what actual BRI was for 2016-17. This comes from the definition of Projected BRI in the CBA VII 7.2.c where the known value of the National TV payout for the year of interest is added to 1.045 times the non-TV portion of BRI for the previous year. Or in other words, BRI is split into the TV and non-TV portion for every year. The TV portion is known in advance due to the nature of the contract having a specific yearly payout. However, the non-TV portion of BRI covers revenue sources that are variable and not well known for the future. Because of the variable nature, the NBA will simply project forward a 4.5% increase in this amount for the Projected BRI:

\[ \\ \text{Projected BRI}_t = TV_t + 1.045*\text{non-TV BRI}_{t-1} \\ \text{Projected BRI}_t = TV_t + 1.045*({BRI}_{t-1} - {TV}_{t-1}) \]

And doing a bit more arithmetic gymnastics, we can write BRI as a function of Projected BRI and TV revenues:

\[ {BRI}_{t-1} = \frac{\text{Projected BRI}_t - TV_t}{1.045} + {TV}_{t-1} \]

Or in other words, the 2016-17 BRI is a function of the Projected BRI in 2017-18 and the TV values of 2017-18 and 2016-17. We know projected BRI here, thus to get BRI in 2016-17 we need to inject what the National TV deal values are. Larry Coon reported a value of $2.1 billion for the 2016-17 season in an insiders article that was confirmed by Zach Lowe. I took a educated guess of the 2017-18 value as $2.225 billion based on Zach Lowe’s description of the payouts for the national TV contract. Plugging these values in gives us an estimate of BRI for 2016-17 of $6,574,338,900.

That’s what we’re going with for 2016-17 BRI: $6,574,338,900 with $2.1 billion representing the TV contract and $4,474,338,900 in non-TV BRI. But keep in mind, this is an educated guess because we don’t know exactly what the national TV deal values are. If we did, then we’d have complete certainty on this value.

Costs Side

At this point, we have a good guess at how the revenues in 2016-17 went for the NBA. We need to go into the cost side of things to get a full picture for the trends in the NBA Salary Cap. Since we already know Projected BRI for 2017-18, we can actually back out what the Projected Benefits less the Adjustment (also known as the shortfall) were. Doing some plug-n-chug from the Salary Cap equation gives us a value of $114,576,089 for Projected Benefits less the Adjustment for 2017-18. We’ll get to how the adjustment is calculated in a second.

Projected Benefits are defined within the CBA IV-9 and are effectively an increase of 4.5% from the previous year’s Benefits plus an additional amount equal to 1% of Projected BRI. The previous year’s Benefits for this calculation exclude the 1% of BRI which gets added on and I term this Residual Benefits – which are Benefits less 1% of the BRI for that year. This can be summarized as follows:

\[ \text{Projected Benefits}_t = 0.01*\text{Projected BRI}_t + 1.045*\text{Residual Benefits}_{t-1} \]

Although another way to define Projected Benefits would be:

\[ \text{Projected Benefits}_t = 0.01*\text{Projected BRI}_t + 1.045*(\text{Benefits}_{t-1} - 0.01*{BRI}_{t-1}) \]

In trying to back out the Projected Benefits used for the 2017-18 Salary Cap, we need to know Projected BRI of 2017-18 (got it), Benefits in 2016-17 (missing), and BRI in 2016-17 (got it). Almost there! We have two unknowns in this equation. Even though we know that Projected Benefits less Adjustment is $114,576,089 we don’t know the Adjustment. We’re going to have to go about this a different way.

In the 2015-16 season, the value of Benefits less the Adjustment ended up being $105,593,330. Per Larry Coon, the players received a total amount of Salaries and Benefits of $2,558,000,000 and of that amount, $2,346,000,000 was Salaries and $212,000,000 was Benefits. The players were short-changed $130.5 million in their Guaranteed BRI amount – ie the shortfall – as the players were guaranteed to receive $2,688,495,000 (50.83% of BRI) and yet their total Salaries and Benefits only made up $2,558,000,000. The $130.5 million shortfall was added onto the Salary Cap for the 2016-17 Season which increased the Salary Cap by $4.35 million.

For the 2016-17 Season, we know that the BRI was $6,574,338,900 and the Forecasted BRI was $5,089,000,000. The players are guaranteed between 49% and 51% of BRI where the difference is a result of the actual BRI and Forecasted BRI. For each dollar over (or under) the Forecasted BRI, 60.5% is added (subtracted) to the guaranteed share. Per CBAFAQ:

Collectively, the players are guaranteed to receive 50% of forecasted revenues, plus (or minus) 60.5% of the amount by which revenues exceed (or fall short of) the forecasts, with a lower limit of 49% of BRI and an upper limit of 51% of BRI.

Forecasted BRI in 2016-17 was $5.089 billion – the 50% limit was $2.5445 billion. We already determined that the BRI in 2016-17 was $6.574 billion. Actual BRI exceeded Forecasted BRI by $1,485,338,900. The 60.5% amount of the excess would be $898,630,035 to be added onto the guaranteed percentage to give a total of $3,443,130,035 in Guaranteed BRI. Except this amount exceeds the 51% limit. The players were guaranteed a total of $3,352,912,839 in Salaries and Benefits for the 2016-17 Season.

Now we have another equation we can introduce in order to get to Projected Benefits and the Adjustment. When the NBA fails to pay out collectively in Salaries and Benefits the guaranteed amount, a check is cut to the NBPA for the Adjustment amount and the Adjustment (divided by the number of teams) is then added onto the Salary Cap and Luxury Tax. Using this knowledge, we know that this relationship must hold:

\[ \text{Guaranteed BRI}_t = \text{Adjustment}_t + \text{Salaries}_t + \text{Benefits}_t \]

We just figured out the Guaranteed BRI amount but still don’t know what the Actual Benefits and Adjustment are. But we can estimate the Salaries paid out for the 2016-17 Season were to put this together.

Spotrac’s value³ of money spent for the 2016-17 Season was $2,962,716,058. In comparison, his 2015-16 value was $2,374,265,169 and Larry Coon reported $2.346 billion in his CBAFAQ. That’s good enough for government work.

³ While I do not condone listing Spotrac as a primary source, they actually do a good job of compiling salary data. Spotrac gathers their data from some combination of Mark Deeks, Bobby Marks, and Eric Pincus.

We know Guaranteed BRI less Salaries (per Spotrac) in 2016-17 is $390,196,781. That’s another way of saying that Benefits plus Adjustment is $390,196,781. Now we can go way back to a relationship of:

\[ \text{Projected Benefits}_t = 0.01*\text{Projected BRI}_t + 1.045*(\text{Benefits}_{t-1} - 0.01*BRI_{t-1}) \]

Now we subtract the Adjustment from each side and realize we already figured out that the Projected Benefits less Adjustment is equal to $114,576,089:

\[ \$114,576,089 = 0.01*\text{Projected BRI}_{2017} + 1.045*(\text{Benefits}_{2016} - 0.01*{BRI}_{2016}) - {Adjustment}_{2017} \]

Going to start adding in some more values, but also keep in mind that $6,900,684,151 is our Projected BRI amount:

\[ \\ \$114,576,089 = \$69,006,842 + 1.045*({Benefits}_{2016} - \$52,890,000) - {Adjustment}_{2017} \\ \$114,576,089 = \$69,006,842 - \$55,270,050 + 1.045*{Benefits}_{2016} - {Adjustment}_{2017} \\ \$100,839,297 = 1.045*{Benefits}_{2016} - {Adjustment}_{2017} \]

Per the Larry Coon values, the Benefits in the 2015-16 Season were $212,000,000, which will give us:

\[ {Adjustment}_{2017} = 1.045*{Benefits}_{2016} - \$100,839,297 = \$221,540,000 - \$100,839,297 \]

Giving us an adjustment value of $120,700,703. Which implies that the NBA Salary Cap was increased by approximately $4 million due to not paying the players enough in the form of Salaries and Benefits.

Our last bit of arithmetic is then to back out the Benefits for the 2016-17 Season based on our derived value of the Adjustment. This gives us $269,496,078 in Benefits paid out by the NBA.