Derek plans to buy a $$31,558\mathrm{.}00$ car. The dealership offers zero percent financing for $48\mathrm{.}00$ months with the first payment due at signing $($today$).$ Derek would be willing to pay for the car in full today if the dealership offers him $ cash back. He can borrow money from his bank at an interest rate of $4\mathrm{.}20\%.$

Answer format: Currency: Round to: $2$ decimal places.