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In spring 2020, the Houston Texans traded star wide receiver, DeAndre Hopkins, to the Arizona Cardinals for David Johnson and a second-round draft pick. Johnson came with a $13 million salary and a worrisome injury history. Many think the Cardinals picked the Texans’ pocket.
Why did the Texans make such a one-sided trade?
Three years earlier, Hopkins and the Texans signed a contract that guaranteed him $49 million for three seasons. The deal gave the Texans the right, but not the obligation, to retain Hopkins at fixed salaries for the following three seasons. A contract that confers the right, but not the obligation, to purchase is a call option. If Houston exercised its options, Hopkins would receive $40 million over the next three seasons, while comparable players were negotiating new contracts in the range of $63 million for three seasons. The Texans would pay Hopkins at least $23 million less than his peers. The call options created friction between the two parties. The Texans decided they needed to either renegotiate the contract or trade Hopkins. They chose the trade.
What’s It Worth, and What’s the Risk?
Although sports contracts often include options, no one discusses what they are worth or how they affect risk. The principles developed in valuing options on common stocks provide insight into both questions.
NFL contracts vary widely in their values, the number of years, and the division between pay-for-performance and guaranteed payments. Aldon Smith recently signed a one-year deal with the Dallas Cowboys that is a pure pay-for-performance contract. It may pay up to $4 million, but every payment is contingent on performance. For example, Smith receives $300,000 for attending training camp, $600,000 if he is on the active roster and up to $2 million for 14 sacks.
At the other extreme, Kenyan Drake, a running back for the Cardinals, has a fully guaranteed one-year contract for $8.5 million. Other agreements offer combinations of guarantees and incentives. The NFL’s average career is short—about three or four seasons—for most players. Due to those abbreviated careers, call options appear in Pro Bowl-level player contracts, who average over 11 seasons.
What Determines the Value of a Call Option?
Suppose a team expects a player’s services to be worth $5.0 million for the coming season, and his contract includes an option to pay the player $4.5 million. In that case, that call option is worth $0.5 million. Suppose the option is for the following season. In that case, the value of the player’s services is variable, and the value of the option changes.
Suppose the value of the player’s services for the following season is equally likely to be $6.0 million or $4.0 million. If the services are worth $6.0 million, the option will be worth $1.50 million. If they are worth $4.0 million, the team will cut the player, and the option is worth $0.0.
On average, the option is worth $0.75 million—the more variable the future value of the player’s services, the more valuable the call option.
After three years, Hopkins remained an elite player producing services worth $21 million per year. It is easy to estimate that the value of the Texans’ call options was approximately $23 million, the $63 million minus the $40 specified by the options. It is much more difficult to calculate what the Texans paid for the options when the player and team negotiated the original contract. There was greater uncertainty about Hopkins’ level of play in three years. That is where financial models help.
Consider a six-year contract with terms similar to Hopkins’ contract. The contract’s total value is $90 million, with $51 million guaranteed over the first three seasons. The team has options for seasons four, five, and six at $13 million per season. Calculating the value of the options requires a model of how a player’s services vary over time. As a base case, assume the player’s services’ value goes randomly up or down on a game-by-game basis with an average change of zero. Also, consider the value of the player’s services has a ceiling of $25 million per season, a value higher than any non-quarterback player’s current salary. Monte Carlo simulation is the best way to value options in this model.1
The more variable the future value of the player’s services, the more valuable the call option.
Four Simulations of Player Value Over Six Seasons
Here, we display four examples of how a player’s services’ value might vary over time in a Monte Carlo simulation. At the start of the contract, date 0, the player’s services’ value is $17 million per season. Each game, the value of the player’s services is the value produced in the previous game, plus or minus a random amount. At the beginning of each of seasons four, five, and six, the team retains or cuts the player.
- The green line is an example of a six-year player who plays under the contract for six years. The team retains him to start seasons four, five, and six. His starting service value is above his $13 million annual salary.
- The purple line follows an example of a five-year player retained for seasons four and five. He is cut at the start of season six because his service value declined to $10 million.
- The red line represents an example of a four-year player. The team retains him at the start of season four but cuts him at the beginning of season five when his services’ value is less than $10 million.
- The blue line gives us an example of a three-year player. The team cuts him at the start of season four because his services’ value is less than $8 million.
The team decides to retain or cut a player for season six based on a simple comparison of the expected value of services and the contractual salary. At the start of season five, the decision is more complicated. The team must calculate whether the expected value of the player’s services in season five, plus the option to retain the player at a salary of $13 million for season six, are worth more than the player’s $13 million in season five.
At the beginning of season four, it is still more complicated. To retain the player for seasons five and six, the team must compare:
- The expected value of the player’s services in season four,
- Plus the value of the options,
to the player’s $13 million salary during season four.
The Monte Carlo simulation includes 100,000 examples, like the four shown in the first chart. For each simulation example, the team decides to retain or cut a player at the start of seasons four, five, and six.2 Given those decisions, each example produces a total value of a player’s team’s services and a total salary paid to the player. The call option’s value is the average of the difference between the value of services received and the total compensation paid for the 100,000 examples. In the base case, the value of the team’s options is $6.4 million.
Modeling ‘What if?’
The base case model does not include the effects on the value of a player’s injury or aging, both of which will increase the value of the team’s call options. The injury model adds a 5% injury probability at the beginning of each season to the basic model. Injuries reduce the value of a player’s services by 50%. The ability to release an injured player without the obligations for salaries in later seasons increases the value of the team’s call options from $6.4 to $7.4 million. The aging model adds a decrease in the value of a player’s services of 5% per season to the basic model. This aging effect also increases the value of the team’s options from $6.4 to $7.4 million. A model with both injury and aging increases the value of the team’s options to $8.5 million.
Should the player enter into a contract with a guaranteed minimum and team options?
It depends on the player’s risk tolerance. Suppose both sides bargain on an equal footing. In that case, the guarantee and call options’ primary effect is to reduce the player’s risk. Our second graph compares the probability distribution of salaries under the six-year contract to distribute a pure pay-for-performance deal.
Two comparisons illustrate the downside protection of the six-year contract. With pay-for-performance, the player has a 10% probability of earning less than $51 million over six seasons, the guaranteed minimum under the six-year contract. Pay-for-performance has a 50% probability of paying less than $80 million. In comparison, the six-year contract upside has only a 40% probability of paying less than $80 million.
Conversely, the six-year contract limits the player’s upside. There is a 27% probability that the pay-for-service contract will exceed $100 million. In comparison, there is only a 2% probability that the six-year contract will exceed $100 million.
In the injury and aging model, the player pays $8.5 million for risk reduction that creates a lower and a higher limit on his earnings. If the player has remained an elite performer at the end of three seasons and could sign a three-season contract for $63 million, the options would be worth $23 million to the team. The player would have lost $14.5 million by selling those options for $8.5 million when he signed his six-year contract. This assumes the team can realize the value of the options, which did not appear to happen in the Hopkins case.
The Functionality of Monte Carlo Simulations
Corporate finance and accounting professionals, attorneys, and auditors may find distinct advantages in Monte Carlo simulations’ functionality. This algorithmic procedure, which outputs a wide range of values, is a specialized approach often used by valuation professionals when developing an analysis of companies holding complex securities or other hard-to-value assets.
We welcome you to contact article author, Dwight Grant, for a deeper discussion and to learn how he can assist with your valuation analysis requirements.
1 Analysts use Monte Carlo simulation to value complex financial options. They also use it to model the outcomes of complex sporting events such as March Madness, where, for example, the probability of a team winning in round five depends on which of its prospective opponents won in games four, three, two, and one.
2 Each decision has a minimum acceptable player service value that maximizes the value of the options for the team. To understand the process of calculating these minimum values, please refer to this research article, courtesy of Dwight Grant Consulting: Monte Carlo Simulation and the Early Exercise Problem.
Author’s note: “Thank you to Colin Grant for your many helpful suggestions!”