An equation writing challenge...

I would like to come up with an equation that compares how effectively one person pays off debt in comparison to another with a different income and with respect to different frames of time or payoff periods. Variables include the following:

- Paid-off Debt Amount with respect to Payoff Period = d
- Income Handicap with respect to Payoff Period = i
- Payoff Period = p
- Unforeseen Handicaps (Inheritance [+] or Unexpected Loss of Income [-]) = h

The Inheritance handicap would give the debt payer an unfair advantage, so it should affect the payoff factor negatively. Conversely, the Unexpected Loss of Income handicap (such as an unseen large expense) would make it harder to pay down the debt in a shorter time period, therefore, it should affect the payoff factor positively.

I would like to set this up so that it can be approached from any time frame for the payoff period, be it monthly, quarterly, semi-annually, or annually. I would also like the individual's income for said time frame to be factored in where a larger income becomes a negative handicap and a smaller income becomes a positive handicap.

I was thinking it may be easiest to come up with a factor that the debt amount can simply be multiplied or divided by. It would be a two-step equation:

Result = d*FACTOR or d/FACTOR

FACTOR = something with i, p, and h

Any help much appreciated!

Re: An equation writing challenge...

Hi there. It sounds like you are looking for the loan balance equation. It has two parts.

First we must figure out what our accrued interest is.

That is done like so:

$\displaystyle loan(1+i)^n$

Interest rate is dependent on how often you make payments and how often accrual periods occur. For sake of simplicity, lets assume that payments and accrual periods are the same length. If you are given an interest rate of 12% and it is compounded monthly, then n is 12 and interest rate(i) is 12/1 = 1% or 0.01

We then take into consideration what you are paying and how much interest it has negated. $\displaystyle \frac{payment}{i}[(1+i)^n -1]$Subtract this from the first number and you get your current balance after n payments. If you want to include a loss of income or lump sum payments, simply calculate your balance to the point of the event, and if it is a lump sum subtract it, or if is a nonpayment, accrue however many periods of interest you want and then continue with the first formula. Hope this helps,

Chris