Chicago Bulls
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Max starting salaries run $41.3M to $57.8M depending on experience (projected), and the Bulls' cap position is -$99.1M. The one path: renouncing all their free-agent holds would open $48.2M, enough for the $41.3M tier, at the cost of their Bird rights.
Without room, a max-level player can still arrive by trade (matching salary) or by the Bulls re-signing their own free agent with Bird rights, which works over the cap.
The rule behind this
A player's maximum starting salary is a percentage of the salary cap, set by his years of service: 25% with 0–6 years, 30% with 7–9, and 35% with 10 or more.
More Bulls questions
Numbers come straight from the live Bulls cap sheet, recomputed whenever the data updates.