Yakk

How is it Useful?

So, what does all this mean, and how is it useful?

TimeToPayOffDebt =
ln(desired_surplus)-ln(current_surplus)
--------------------------------------------------
interest_rate

where
surplus = payment_rate - debt*interest_rate


payment =
(r*D*(1+r)^m)
-------------
((1+r)^m)-1

payment is "payment per period of time"
r is "interest rate per period of time"
m is "number of payments"
D is "starting debt"
^ means "to the power of". So, (1+r)^m means "(1+r) to the power m".

In general, this is how you should use the above.

The first rule is, interest isn't a percentage: If you have 12% interest, your interest is 0.12. If you have 20% interest, your interest is 0.20.

Next, figure out your monthly interest rate. There are two ways you can do this.

Accurate way:
MonthlyInterest = (1+Annual_Interest)^(1/12)-1

Easy way:
MonthlyInterest = Annual_Interest / 12

The above is easy to calculate if you turn the windows calculator into "scientific" mode. =)

Example: 9% annual interest.
Monthly_Interest = (1+0.09)^(1/12) - 1
= 1.09^(1/12) - 1
= 1.007207 - 1
= 0.007207

Work out your monthly interest payments.
Monthly_Interest_Payments = Monthly_Interest * Debt

Suppose your debt was 9000$. Then
Monthly_Interest_Payments = 0.007207 * 9000
Monthly_Interest_Payments = 64.87

The next thing you do is decide how much you can afford to pay off your debt. Lets say 200$ / month.

Now, what is your monthly surplus? Take the amount you are paying off and subtract your Monthly_Interest_Payments.

Initial_Monthly_Surplus = 200 - 64.87
Initial_Monthly_Surplus = 135.13$

Eventually, you want to be paying no interest. At this point, your Interest_Payments will be 0.

End_Monthly_Surplus = 200 - 0
End_Monthly_Surplus = 200

Now, pull out the 'how long to pay off debt' equation:
TimeToPayOffDebt =
ln(desired_surplus)-ln(current_surplus)
--------------------------------------------------
interest_rate

"ln" is a function on most calculators. It is also known as "the natural logarithm".

Plug in the numbers
TimeToPayOffDebt =
ln ( End_Monthly_Surplus ) - ln (Initial_Monthly_Surplus)
---------------------------------------------------------
Monthly_Interest_Rate

(notice everything is monthly, if they are inconsistent things break)

TimeToPayOffDebt =
ln( 200 ) - ln( 135.13 )
------------------------
0.007207
=
0.39208
------------------------
0.007207
=
54.4

So, this says it will take about 54.4 months to pay off the debt at this rate.

After you have an answer, use the 2nd equation to verify it. This is important: not only did you do alot of math, the first equation isn't prerfectly accurate for the real world either. Round the number of months up.

payment =
r*D*(1+r)^m
-------------
((1+r)^m)-1
=
Monthly_Interest* Debt * (1+Monthly_Interest)^NumberOfMonths
------------------------------------------------------------
((1+Monthly_Interest)^NumberOfMonths) - 1
=
0.007207 * 9000 * (1+0.007207)^55
---------------------------------
((1+0.007207)^55) - 1
=
64.863 * 1.4843
---------------
1.4843 - 1
=
96.276
---------------
0.4843
=
198.80

which is really close to 200$, close enough that we can be confident we didn't make a serious mistake. =)

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