Yakk

Me, Maple and a friend of mine got it.

Time = ln((payment_rate+goal*interest_rate)/(payment_rate+start*interest_rate))/interest_rate

where
"goal" is your desired goal (saving goal or pay-off-debt goal).
"interest_rate" is your interest rate per period of time (compounded instantly, which is wierd[1]).
"payment_rate" is your payment rate per period of time.[2]
"start" is the start state (how much money you have saved up, or how deep your debt is. Debt is negative.)
"Time" is the number of periods of time it will take.

By plugging numbers into this, you will get ballpark values for real-life loan repayment times and the like. It won't be accurate, because in the real world you only make payments every month, and interest doesn't work quite the same.

Footnotes:
[1] if you take 12% yearly interest, and you compound it every month, you get 1% per month. This works out to 12.68% interest over a year. If you compound every instant, you end up with something like 13% interest.

[2] Ie, if your period of time is years, your interest rate and the units the answer is in will also be in years.

A prettier version:
Define X to be:
X =
payment_rate+goal*interest_rate
-------------------------------------------
payment_rate+start*interest_rate

Then the amount of time it takes to pay off your loan is:
ln(X)/interest_rate

Example:
you have 10,000 in debt at 9% interest/year. You are paying 4800/year to pay off the debt.

Your goal is 0$ of debt. You start at -10000.

X = (4800 + 0*0.09)/(4800 + -10000*0.09)
X = 1.23

So, it will take about:
ln(1.23)/0.09
or 2.3 years to pay off your debt.

Testing against the morgage equation from a few posts ago. . .
28 monthly payments, about .8% monthly interest, initial debt of 10000.

Payment per month =
0.008*10,000*(1.008)^28)/((1.008)^28-1)
= $400.06

sweetness, it worked!

Edit:
If you care, the rate of change of the repayment time, with respect to the payment rate, is:
(start*interest_rate - goal*interest_rate)/interest_rate
*
1
--------------------------------------------------------
(payment_rate+goal*interest_rate)(payment_rate+start*interest_rate)

but I doubt you care much, and it may be wrong.
Edit2:

Best version of the formula yet!

First concept: surplus. This is
payment_rate - debt*interest_rate
basically, how much you are reducing the debt by.

So, if your interest payments are 100$, and you are putting 400$ down on the debt, your surplus is 300$.

If your interest payments are 0$ and you are putting 400$ down on the debt, your surplus is 400$.

then, the amount of time it takes to pay off the debt is:
ln(desired_surplus)-ln(current_surplus)
--------------------------------------------------
interest_rate

Note you have to calculate the interest rate per payment period, not the annual interest rate.

To pay off the debt, you want the desired surplus to be the entire payment.

Example:
Your interest payments are 100$. You are paying 200$ a month to pay off the debt. Interest rates are 1%/month.

Time to pay off it:
ln(200)-ln(100)
------------------
0.01

which is 69.3 months, or about 5 years 10 months.

__________________
Last edited by JHVH : 10-29-4004 BC at 09:00 PM. Reason: Time for a rest.