How is APR calculated?
Annual Percentage Rate (APR) is converted to Monthly Percentage Rate (MPR) for interest charges to occur during the monthly payment schedule. APR ÷ 12 = MPR. Example: 6% APR ÷ 12 = 0.5% MPR. Notate this as 0.005 MPR to remove the percentage and avoid errors in the car payment formula. Annual Term is converted to Monthly Term (T) so all factors relate accurately. Instead of using 6 years, compute with 72 months. How is a car payment calculated? The car payment formula is M=LX. The monthly payment (M) equals the loan amount (L) times the APR and term factor (X) in a car payment. 

What is the X Factor in the formula? The X Factor is the equation used to charge interest to the unpaid portion of the loan every month as the buyer pays it down over the term. X = (MPR x Z) ÷ (Z  1). What is the Z factor in the formula? The Z Factor is complex because it must account for the reduction of principal each month and the finance charge. It is calculated by adding 1 to the Monthly Percentage Rate (MPR) and then raising it by the power of the Term (T). Z = (1 + MPR)^{T} 

Can I calculate by hand?
Yes! Once the X Factor is calculated, the rest of the car payment equation is easy. 
Sample APR Calculation by Hand
$30000 Vehicle. 60 Months Term. 6% APR.
APR ÷ 12 = MPR 0.06 ÷ 12 = 0.005 Z = (1 + MPR)^{T} Z = 1.005 raised to the power of 60 = 1.34885 X = (MPR x Z) ÷ (Z  1) X = (0.005 x 1.34885) ÷ (0.34885) X = 0.00674425 ÷ 0.34885 X = 0.01933281 MLX Calc M=LX Monthly Payment (M) = 30000 x 0.01933281 = $579.98 Since the X factor was calculated for 6% APR at 60 months, any loan amount can be entered to find the payment. For example, here is the calculation for a $40000 loan amount. Monthly Payment (M) = 40000 x 0.01933281 = $773.31 