Math Considerations
Purpose
This section is for those who want more information about the mathematical side of finding the best method for paying off multiple loans. It is intended as informational and to generate discussion among those interested in the problem.
The Problem
To clearly state the problem it is necessary to define a variable called Excess. The variable Excess is the total amount you can pay each month minus the sum of the minimum payments of your loans. Once the minimum payments are set, the problem is to find the Best Method of Distributing the Excess. The Best Method of Distributing the Excess is defined as the distribution of the Excess, at all monthly payment cycles, that minimizes the total interest one will have paid when all loans are paid off.
Use of the Term Best
Currently, methods that are advocated on the internet, such as the Avalanche or Snowball methods as well as the other methods tested in our workbook, use a single rule for distributing the Excess among the loans each month. From a mathematical point of view, because the complexity of the problem is ignored, these single-rule approaches should not be considered definitive in defining the payoff method which is better than all others in finding the minimum total interest. In the context of these current single-rule approaches, the term best can only be used when comparing one single-rule approach against others. The term best can not be used for all possible approaches to distribution of Excess at each payment cycle. Because of the large number of ways that one can distribute Excess among multiple loans, just assigning one method of allocation of Excess to each payment cycle may be, as I will illustrate, a gross oversimplification.
How many payoff methods are there?
If one takes the Excess (E) to be an integer and distributes the Excess in single dollar amounts among loan payments (L = the number of loans), the number of possible payment options for the first month (N) is: N = (L + E - 1)!/[E!*(L-1)!].
This is a standard formula from combinatorial mathematics. This problem is known as the Stars and Bars Problem. (ref example 18).
Consider only 3 loans and an excess of just $100.00.
The formula above shows that, at the first monthly payment cycle, there would be 5152 ways to distribute the Excess in one dollar amounts among the three loans.
If no loans were paid off when second set of monthly payments came due, each of the 5152 different payment options from the first month would have 5152 ways to distribute the Excess. Rounding to 5000, at the second monthly payments there would be 25 million ways that the Excess could be distributed in one dollar amounts among all the payments made to that time. So, at the time of the third monthly loan payments, there would be more than 625 trillion ways that Excess could have been distributed among all of the payments made to that time. So, for payoff of 5, 10, or 20 loans with several hundred dollars of Excess there would be an uncountable number of payoff approaches. Finding the Best Method of Distributing the Excess by a Monte Carlo approach would require evaluation of each of the payoff options.
Clearly, when increments of one dollar are used, a brute force search for the best payoff method of multiple loans is not feasible.
An attempt was made to simplify this computational issue with brute force by making distributions of Excess in larger lump-sum amounts and, at each payment cycle, only considering the distribution of Excess that resulted in the lowest total interest payment up to that point. This approache did not yield satisfactory results. There were long computation times and it seemed probable that the way down the surface of total interest payment to the true minimum total interest payment could have been missed.
Independent Validation
It is always reasonable to ask: “How do you know your results are valid?” A separate calculator was set up using a different programing approach. Both calculators agreed in multiple runs. My Debt Killer was also validated against calculations done on a spread sheet using standard financial formulas.
An Observation
One observation deserves comment. For all of the data sets that have been run with our nine-method approach, assigning the Excess, at each payment cycle, to the loan with the highest interest rate has yielded the lowest total interest payment. Others have had the same experience, albeit, generally, with comparison against fewer single-rule approaches. To the author’s knowledge, no one has reported a single-rule approach that is superior to the assignment of Excess to the highest rate loan at each payment cycle. Considering the universe of possible distributions of Excess for all possible loans and for all possible Excess, this observation, in the absence of a rigorous mathematical proof, does not allow one to conclude that assignment of Excess from highest to lowest according to interest rate is The Best Method of Distributing Excess. Displaying the results, for the various commonsense single-rule approaches, allows users to see the results of choices based on emotion or various advice and to experiment to find the optimal approach for them with in the choices offered. And, of course, no one knows for sure if the highest-interest-rate assignment will yield the lowest total interest payment for all users.
Conclusion
The conclusion is that, for the present, the best we can do is make comparisons within a reasonable set of single-rule approaches for distribution of Excess in each payment cycle. In this webpage the user is allowed to find the best way of distributing Excess from a set of nine single-rule methods for finding total interest after all loans are paid off. These are approaches that have been noted on the internet or seem reasonable based on common sense. My Debt Killer is made available without charge to help people escape the awful burden of debt related to multiple loans. For comments and suggestions please email me at dr.sullivan@mydebtkiller.com.