🚀 go-pugleaf

Rod & Shannon Smith (rodsmith@nf.sympatico.ca) wrote:
about the math for the Ford Lease:
: 
: On the form, they take the net price multiply it with an acquisition
: factor to give A.  They then take the Actual lease end value and multiply
: it by a lease end factor to give B.  To get the Monthly payments: A-B.
: What I would like to know, is where or how does a person figure out
: those two factors!

Let N = Net price (What you owe when the lease begins.)
Let L = Lease end value (What is still owing when the lease ends.)
Let m = Months in the lease.
Let r = Monthly Interest Rate (Annual Interest rate divided by 12.)
Let p = Monthly Payments.

Each month, the outstanding balance for that month is multiplied
by the interest rate to get that month's interest.  The interest
is added to the outstanding balance and the monthly payment is
deducted:
  (New Balance) = (Old Balance) + (Interest) - (Payment)
                = (Old Balance) + (Old Balance)*r - p
                = (Old Balance)*(1+r) - p
So if you initially owe N$, after one month you owe:  N*(1+r) - p
After two months you owe:  (N*(1+r) - p) * (1+r)  -  p
After three months:  [(N*(1+r) - p) * (1+r)  -  p] * (1+r)  - p
...
At the end of the Lease (m months):

  L  =  {...{[(N*(1+r) - p) * (1+r)  -  p] * (1+r)  - p]...}*(1+r) - p
     Multiply it all out:
     =  N*(1+r)^m - p*(1+r)^(m-1) - p*(1+r)^(m-2) - ... - p*(1+r) - p
     Factor out the -p:
     =  N*(1+r)^m - p*[(1+r)^(m-1) + (1+r)^(m-2) + ... + (1+r) + 1]
     The stuff in square brackets is a "geometric series".
     Use the standard formula to get the sum:
     = N*(1+r)^m - p*[(1+r)^m - 1]/[(1+r) - 1]
     = N*(1+r)^m - p*[1+r)^m - 1]/r

Now solve for p:

  p  =  N * r*(1+r)^m/[(1+r)^m - 1]  -  L * r/[(1+r)^m - 1] = A - B

So Ford's "Acquisition Factor" is:  r*(1+r)^m/[(1+r)^m - 1]
and their "Lease End Factor" is:  r/[(1+r)^m - 1]

For a more detailed treatment of compound interest look for the
FAQ (Frequently Asked Questions) from the Sci.Math newsgroup.

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