What a lease payment is actually built from, and what a down payment (called a "cap cost reduction" in lease paperwork) mathematically does to it.
A lease payment is built from a depreciation charge plus a rent charge. The depreciation charge is (net capitalized cost − residual value) ÷ term in months. The rent charge is (net cap cost + residual value) × the money factor — a small decimal like 0.00125 that functions like an interest rate spread across the whole balance rather than the declining balance. Multiply the money factor by 2400 to get its rough APR equivalent.
A down payment on a lease is called a capitalized cost reduction. It lowers the net cap cost you're financing, which lowers both parts of the payment formula above — smaller depreciation charge and smaller rent charge, since the rent charge is calculated on the reduced balance too. That's the mechanism behind why skipping the down payment raises the monthly payment by more than just "the down payment divided by the term."
Federal Reserve Regulation M requires it be disclosed in the lease contract itself, even though dealers aren't always forthcoming with it verbally before you sign. Ask for it directly, or divide the rent charge line on the disclosure by (net cap cost + residual value).
Set by the leasing company using industry guides like the Automotive Lease Guide (ALG) or Black Book, not negotiated. A higher residual means lower monthly payments, since you're financing less depreciation.
The Reg M disclosure box states this explicitly — it's the one number that actually lets you compare a zero-down offer against a with-down offer on equal footing.