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Refinance is a Ruby gem that provides a collection of finance algorithms. Currently, it contains algorithms for calculating the properties of ordinary annuities: principal, interest rate, number of payment periods, and payment amount.


Refinance does calculations related to annuities in finance theory. In general, an annuity is a finite series of regular payments; an installment loan for a car is an example that might be familiar to you.

Refinance allows you to answer questions like these:

  1. Say you have a car loan and are paying a fixed amount each month to pay it off. If you got a reduced interest rate, exactly how much lower would your monthly payments be?

  2. If you get a reduced interest rate but continue to pay the same monthly payments, exactly how much shorter will the loan duration be?

  3. Say you want to borrow money to buy a car. If your bank will lend you money at an X% annual interest rate, and you're willing to make monthly payments as high as $Y, how expensive of a car can you buy?

  4. Say you're making monthly payments of $X to pay off a $Y loan. What interest rate are you being charged?

  5. Say your loan has an interest rate of X% per month. What is its effective annual interest rate?

Annuities have four defining properties (at least for our purposes), and the first four questions above provide examples of each: The first question is asking for the annuity's periodic payment amount. The second is asking for the number of remaining payments. The third is asking for the principal. And the fourth is asking for the interest rate.

In general, if you know three of those properties, Refinance can calculate the fourth.

The fifth question is about converting an interest rate. Perhaps you think this is trivial: if you're paying 1% per month, then it's equivalent to paying 12% per year, right? Wrong. It's more like 12.68%. You need to account for the effects of compound interest.

Usage example

I'll use Example 2 from Stan Brown's paper:

You are buying a $250,000 house, with 10% down, on a 30-year mortgage at a fixed rate of 7.8%. What is the monthly payment?

We want to calculate the monthly payment, so we'll use the method Refinance::Annuities.payment. That method requires three arguments: the interest rate (per month, since we're calculating the monthly payment), the total number of monthly payments, and the principal. These are easy to determine from the example:

  • The interest rate is 7.8%, but there are two things to note here. First, 7.8% is a nominal annual rate, and we want the monthly interest rate. So we divide 7.8% by 12 (because there are 12 months in a year) and get 0.65%. Second, the method wants the argument in decimal form, not percent form, so 0.65% becomes 0.0065.

  • The loan lasts 30 years, and we want the number of payment periods -- that is, the number of months. 30 years is 360 months.

  • The house's purchase price is $250,000, and we're making a 10% down payment, so the loan's principal will be 90% of $250,000, or $225,000.

Now we can call the method:

Refinance::Annuities.payment 0.0065, 360.0, 225000.0

(You can name the arguments "interest rate", "periods", and "principal". For this and related methods, the arguments go in alphabetical order.)

The return value is 1619.7086268909618. So the monthly payment is about $1,619.71.


Here's a summary of the public-facing methods provided by this library. They are all stateless "functions".

Arguments to these methods are not directly converted to a particular numeric type; they can be any objects that support the necessary mathematical methods. Instances of Float and BigDecimal will work for real-valued arguments.

Interest rates must be in decimal form, not percent form.


This method calculates an annuity's interest rate over its payment period. The result will be given as a decimal number, not as a percent. There is no closed-form solution for this, so the answer is iteratively approximated with the Newton-Raphson method.


  1. The periodic payment amount.
  2. The total number of payment periods.
  3. The principal.
  4. (Optional) The initial guess at the interest rate.
  5. (Optional) The precision; the algorithm will stop if the magnitude of the last improvement was less than this.
  6. (Optional) The maximum number of decimal places. After each iteration, the guess will be rounded to this many decimal places.
  7. (Optional) The maximum number of iterations to allow.


This method calculates an annuity's periodic payment amount.


  1. The interest rate over one payment period.
  2. The total number of payment periods.
  3. The principal.


This method calculates the total number of payment periods for an annuity.


  1. The interest rate over one payment period.
  2. The periodic payment amount.
  3. The principal.


This method calculates the principal of an annuity.


  1. The interest rate over one payment period.
  2. The periodic payment amount.
  3. The total number of payment periods.


This method calculates the effective annual interest rate.


  1. The nominal annual interest rate.
  2. The number of compounding periods per year.


The algorithms are simple rather than fast and numerically stable. At present, they deal only with annuities immediate (in which the interest is accumulated before the payment), not annuities due (in which the interest is accumulated after the payment).


This library is tested with the following versions of MRI (Matz's Ruby Interpreter): 1.9.3, 2.0.0, 2.1.0, 2.1.1, and 2.1.2.

If you wish you to use BigDecimal, we recommend using the bigdecimal gem instead of the version of BigDecimal that ships in the standard library. Refinance's test suite does not pass with the version of BigDecimal that ships with MRI 2.1.0.


Add this line to your application's Gemfile:

gem 'refinance'

And then execute:

$ bundle

Or install it yourself as:

$ gem install refinance


  1. Fork it
  2. Create your feature branch (git checkout -b my-new-feature)
  3. Commit your changes (git commit -am 'Add some feature')
  4. Push to the branch (git push origin my-new-feature)
  5. Create new Pull Request

Authorship and copyright information

This software was written by Reinteractive, a software consulting company in Sydney, Australia. It is distributed under the MIT License; see LICENSE.txt for details.


Thanks to Stan Brown for his paper Loan or Investment Formulas. It is an excellent introduction to the mathematics of annuities, and we used some of its examples as test cases.