Lease payments vs. loan payments

A lease contract is signed for a $10,000 machine. The contract stipulates a down payment of 15 % and a residual value of 10 %.
The debt is to be repaid with 48 monthly payments and interest is at 6.9 % compounded monthly.

How much is the monthly payment for the lease?

As the 15 % or $1,500 is paid immediately, the lease amount or the present value of the debt is $8,500. The residual value to purchase the car at lease-end is 10 % or $1,000.

Answer: The monthly payment for the lease is $183.95 with a total of $9,829.67 (including the residual payment).

How does this compare to an ordinary loan for $8,500 at the same interest rate?

Just switch to the amortization calculation...

Answer: Monthly payments for the loan will be $203.15 and the total paid will be $9,751.15.

Which is cheaper: the lease or the loan?

As a matter of fact they are equivalent: remember that money has a time value, and that to compare both options correctly, you'll have to replace all payments by equivalent values dated on the same day.

For the loan the present value of the periodic payments is exactly $8,500 as this was entered into the Amortization calculation.

For the lease the present value also equals $8,500: it's the total of the present values of both the periodic payments and of the residual payment.

This means that from a mathematical point of view both series of payments are equivalent if interest rates are equal. Whichever option should be preferred will be dictated mainly by other factors, such as tax returns, extra costs or higher insurance premiums.

If the term is different both series of payments will still be equivalent, as the periodic payments will change accordingly.

When interest rates are low it could be advantageous to opt for the lease: if interest rates rise during the term of the lease, you would be able to make a profit as the present value of the residual payment drops.