Successive Approximations Example:
Automobile-Lease Interest Rate (real solutions only)

If you lease a car and make monthly payments to the dealer, then the car's purchase price will decrease from start-of-lease to end-of-lease. The sum of your payments will exceed the change in purchase price, and the extra money that you pay is effectively interest. However, the dealer might call the excess payment "fees" instead of interest, and avoid stating an interest rate.

Instead of leasing, you could borrow enough money to pay the initial price of the car, and then make monthly payments to the lender at his interest rate. In order to choose between leasing and borrowing, you would need to compare the lender's interest rates and the car-dealer's effective interest rate.

We will compute the effective interest rate of a 3-year car lease, given the follwing lease parameters:

Co = Original cost of car (including sales tax) = $64,307.47
Cf = Final cost of car at end-of-lease (including sales tax) = $34,957.22
M = Monthly payment (including taxes) = $988
N = Number of months in 3-year lease-term = 36

The car cost at the beginning of the lease (Co) equals the present-value of the final cost (Cf) plus the present-value of monthly payments (M), discounted at annual-interest rate R:

Co = Cf*(1 + R/12)-N + (12M/R)*(1 - (1 + R/12)-N)

We can subtract Co from each side to get an equation that equals zero:

0 = Cf*(1 + R/12)-N + (12M/R)*(1 - (1 + R/12)-N) - Co

This equation lacks exponents greater than zero, and thus it is non-optimal for successive approximations. In particular, the right-hand-side approaches a finite limit as R approaches plus and minus infinity:

Limit R-->+infinity = [Cf*(1 + R/12)-infinity + (12M/R)*(1 - (1 + R/12)-N) - Co] = -Co
Limit R-->-infinity = [Cf*(1 + R/12)-infinity + (12M/R)*(1 - (1 + R/12)-N) - Co] = -Co

This property sometimes causes successive approximations to diverge to infinity, rather that converge on finite solutions. Fortunately we can easily fix the equation by multiplying both sides by R, thus creating a new equation whose limits are infinitely negative as R becomes infinite. This new and optimal equation is:

0 = Cf*R*(1 + R/12)-N + 12M*(1 - (1 + R/12)-N) - Co*R

We can approximate interest rate R from estimate z using

R ~ z - F(z)/S(z)
F(z) = Cf*z*(1 + z/12)-N + 12M*(1 - (1 + z/12)-N) - Co*z
S(z) = Cf*(1 + z/12)-N - (N*Cf*z/12)*(1 + z/12)-N - 1 + N*M*(1 + z/12)-N - 1 - Co

When entering these equations in a spreadsheet, I recommend entering the values of Co, Cf, M and N near the top of the sheet, and then naming these cells Co, Cf, M and N. Then you can use the cellnames in the formulas, and you can also change the values in the named cells to compare different leases. If the initial balance, final balance, monthly payment, and number of months are named Cf, Co, M, and N, and if estimate z is located in cell A7, then the Excel formulas would look like this:

F = Cf*A7*(1 + A7/12)^-N + 12*M*(1-(1 + A7/12)^-N) - Co*A7
S = Cf*(1 + A7/12)^-N - N*Cf*A7/12*(1 + A7/12)^(-N-1) + N*M*(1 + A7/12)^(-N-1) - Co

Interest rates are typically around 10% (more or less), so I tried z = 0.1 as a first estimate. The approximations converged on R = 0.041170679 by the sixth approximation. This 4.117% lease-rate is less than many banks' lending rates, so leasing might be the best option.



Unpublished Work. © Copyright 2001 Pat Russell. Updated April 17, 2009.