Here is the problem:

L: original loan amount

B: current balance after P periods

P: number of periods that have been paid

A: period payment

**you do not know the number of periods remaining, so you do not know N (the total number of periods for the loan), and you do not know the interest rate (that is what is to be found). Each payment includes both principal and interest.

Example: 56,000 loan, 460.54 period payment, $24,052 current balance, 120 periods have been paid, calculate the interest rate note: the correct answer will result in 5.6% and a total N = 180 periods

Because I do not know the real maturity date, I am getting stuck in the maths on how to calculate the interest rate. All that is known is the original amount, the current amount, period payment, and how many payments have been made. Any suggestions would be great!

  • $\begingroup$ Have you been given any formulas relating all these quantities? $\endgroup$ – Gerry Myerson Feb 17 '14 at 9:45
  • $\begingroup$ Just the common expressions known for finding the Balance and for finding the period payments such as: A = L[i(1 + i)^n]/[(1 + i)^n - 1], and B = L[(1 + i)^n - (1 + i)^p]/[(1 + i)^n - 1], where n= total #of periods. But in the problem, I don't know the total number of periods, so I got stuck in the substitution. $\endgroup$ – user129306 Feb 21 '14 at 3:12

There are two ways to solve a loan balance question: The Prospective (looking forward) and the Retrospective (looking backward) approach.

In this case, we do not know the number of periods remaining, so we cannot use the prospective method to solve this problem, as we are unable to determine the present value of the remaining payments. Instead, we should use the retrospective method.In this case, we calculate the loan balance as an accumulated value of the loan at the time of evaluation minus the accumulated value of all the payments made at the time of evaluation.

We can easily see that the accumulated value of the loan at time 120 is

$ B_{120} = 56000(1+i)^{120} $

In addition, the accumulated value of the annuity payments should be equal to

$ As_{120|i} = 460.54 * ((1+i)^{120} - 1)/i $

The first value minus the second value should result in 24,052, the current balance.

Solving using a financial calculator or linear approximation, it seems that $i^{(12)} = 5.6$% as required.

In an exam situation, you would never be asked to solve such a high exponential polynomial. They will more than likely use smaller cash flow periods, or solve using a financial calculator.

  • $\begingroup$ So it resolves to an algorithmic solution to find the best approximation right? Your explanation is considerably better than the 3 engineering text books I have been reviewing. This had me completely stumped and I knew there had to be a better understanding than just saying "no formula exists, use interpolation". $\endgroup$ – user129306 Feb 21 '14 at 3:08
  • $\begingroup$ Yes, unfortunately, unknown interest rate problems are extremely difficult to solve. A better way the problem could have been stated was: "given that the interest rate i is between 5 and 6%, use linear approximation to determine the best approximation" $\endgroup$ – David L Feb 21 '14 at 14:13

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