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There is a discussion on this page regarding the back-calculation of the interest rate, given the other parameters of a typical finance calculation (present value, repayments and term length).

Three methods are given - one is a brute force search (Newton's method) and the other two methods (provided by one David Cantrell) provide estimates of the rate using formulas.

Unfortunately the formulas given do not take into account future value (aka residual). Does anyone know, or can anyone help me work out, how the future value could be factored into these formulas?

cbp
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  • What do you mean by future value? The future value of the principle? Because if that is what you mean the future value of the principle is just the $Pe^{rt}$ – meh Jun 03 '14 at 15:19
  • By future value I mean the remaining principle at the end of the term. In some types of finance the repayments are made at a lower rate through the term. At the end of the term there is an outstanding amount left, which is paid out in one lump sum. Take a look at the explanation in the Excel docs for example: http://office.microsoft.com/en-au/excel-help/pmt-HP005209215.aspx – cbp Jun 04 '14 at 02:45
  • Do you mean a balloon payment? If so then look up coupon bonds. Basically if you are making 100 payments every month and a big 10000 payment at the end then you are going to have the annuity(formula they give you) + 10000 at the end. Because you are paying the 10000 last its future value is 10000. – meh Jun 04 '14 at 16:05
  • Hi yes balloon payments is another name for future value. I don't understand your point about coupon bonds. I assume David Cantrell's formula must be different if the future value is taken into account, but I'm not sure what the new formula would look like. – cbp Jun 06 '14 at 02:59
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    The link you provided is no longer valid. Here is a snapshot for the Internet Archive: https://web.archive.org/web/20060318052036/http://oakroadsystems.com/math/loan.htm – Kevin Feb 27 '17 at 12:56

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F = Face value of loan, C = regular payment, i = interest rate, N = number of payments, B = balloon payment

So $ F = \frac{C}{i}[1 - (1+i)^{-N}] + B(1 + i)^{-N}$

Now if you want to find what the interest rate would be if you know everything else

You get ${C}[1 - (1+i)^{-N}] + Bi(1 + i)^{-N} - Fi = 0 = f(i)$

Then you have to have to take the the derivative of the above which is

$NC(1+i)^{-N-1} - F + B(1+i)^{-N-1}(-NB + B + 1) = f'(i) $

where $ i_{new} = 1 - \frac{f(i)}{f'(i)} $

and iterate over this until you get convergence.

The reason I relate this to coupon bonds is because F is also the value of a coupon bond paying coupon C and having final payment B.

meh
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  • Hi Matthew, thanks. The iterative method works but can fail if the initial guess is too far from the actual value. The article I linked to discusses this problem. The article provides alternative solutions that use a formula to estimate the rate, without relying on iteration. These formulas are useful as they allow you to more accurately select the initial guess, meaning there is a higher likelihood that the iterative process succeeds. – cbp Jun 06 '14 at 08:51
  • Use this formula to get a good starting point $ \frac {C + \frac{B - F}{N}}{\frac{B + P}{2}}$ – meh Jun 06 '14 at 15:41