Having trouble with APR calculation

Question

I am using the formula listed on the following site to try to calculate APR:

http://www.fdic.gov/regulations/laws/rules/6500-1950.html#fdic6500appendixjtopart226 _{go to appendix J, if it doesn't auto scroll}

In step 9, they are trying to find the value of A = (33.61 * )/ (i+1)

and that is calculated as 1 + 1/(i+1) + 1/(i+1)^2 +......1/(i+1)^n

Taking n = 36, I find = 28.83,

and I am getting the value of A = 33.61 * 28.83 / 1.0104 = 941.3

But, the link somehow arrives at the number of 1004.674391

Can anyone explain how they arrive at this number and what A' and A'' mean?

I am trying to code for this in C++, but before I do, I need to get the math straight on pencil.

Answer 1

I have answered your question in detail here http://stackoverflow.com/questions/12396422/apr-calculation-formula

The annuity formula in FDIC document is at first finding PVIFAD present value annuity due factor and multiplying it with annuity payment and then dividing it by an interest factor of (1+i) to reduce the annuity to an ordinary annuity with end of period payments

They could have simply used PVIFA and multiplying it with annuity payment to find the present value of an ordinary annuity

In any case, you should not follow the directions in FDIC document to find interest rate at which the present value of annuity equals the loan amount. The method they are employing is commonly used by Finance Professors to teach their students how to find internal rate of return. The method is prone to lengthy trial and error attempts without having any way of knowing what rate to use as an initial guess to kick off the interest rate calculations

So this is what I would suggest if you are not short on time and would like to get yourself familiar with numerical methods or iterative techniques to find internal rate of return

There are way too many methods at disposal when it comes to finding interest rates some of which include

1. Newton Raphson method
2. Modified Newton Raphson method
3. Muller's method
4. Bairstow's method
5. Bisection method

All of the above methods use a seed value as a guess rate to start the iterative calculations and if results from successive calculations tend to converge within a certain absolute Error bound, we assume that one of the rates have been found as there may be as many rates as the order of the polynomial in this case 36

There are however some other methods that help find all rates by making use of Eigenvalues, but for this you would need a lengthy discourse of Linear Algebra

One of the methods that I have come across which was published in the US in 1969 (the year I was born :) ) is called the Jenkins Traub method named after the two individuals who worked jointly on finding a solution to all roots of a polynomial discarding any previous work on the same subject

I been trying to go over the Jenkins Traub algorithm but am having difficulty understanding the complex nature of the calculations required to find all roots of the polynomial

In summary you would be better of reading up on this site about the Newton Raphson method to find IRR

Answer 2

Check your calculation of A**. I was able to duplicate their calculations using excel. Make you sure have accounted for all the terms, it can be easy to be one off.

They are making a guess at the interest rate which will be wrong, then they are adjusting it to see how wrong it is, then making another adjustment. They will repeat until they see no movement in the guesses.