This is simpler than you think. Capital cost reduction is anything used to reduce the amount your financing, above and beyond your sales price. So if you buy a $50k car and you agree on a purchase price of $45k before incentives and any down payment, etc., then your gross cap cost is $45k. But if there is also a $5k seasonable factory rebate from Volvo and you choose to put $5k as a down payment, then your net capitalized cost was $35k — this is the amount that your lease payments will ultimately be based on. Your residual value is based on the MSRP no matter what is negotiated.
So when they say cap cost reduction, they’re usually referring to “customer cash,” commonly known as a down payment. Now, you shouldn’t make a down payment on a lease - ever. Better to have higher payments and use that money from your savings account to help you make a portion of the payments each month.
Now, I haven’t factored in taxes and DMV fees and the like - those items will also raise your cap cost, but I didn’t want to make the example too complicated.
What you want to do is negotiate the best sales price - before rebates/incentive - but quote the car out based only on “drive offs only” aka no customer cash aka no cap cost reduction (except for any applicable factory rebates - those are wonderful cap cost reductions).
In other words, if they’re quoting this much in “cap cost reduction,” they’re assuming you’re paying a bunch of money as a down payment to reduce your payments, which is probably also why your advertised payment looks reasonably attractive. And that’s how they get you
So my advice again - best way to get a good deal on the car is to get a good deal on the car, before any of the fuzzy math. Know the market (find out the base money factor, confirm the residual value, and know how much off MSRP people are paying for the same car before you solve the rest of the math problem).
Happy to elaborate if any of this didn’t make sense!
Edit: in case you already knew all of this, and this wasn’t your question, hopefully the info I wrote will be helpful to someone else reading!
Second edit: trying to figure out the inconsistency in their offer that you cite - if $4450 is DAS but cap cost reduction is listed at $10k plus first month payment, I would have to assume that the delta is Volvo incentives straight from the factory. Problem with these offers, as I mention above, is that they’re juicing the leasing number to make the payment look good while probably not discounting the car that much, so make sure to know what the gross cap cost is before you factor in incentives, customer cash/customer paid cap reduction, and the like.