Saving 1.25% Mathmatics

On a recent post on the Lending Club blog, in which I compared an unsecured loan to a Home Equity Line of Credit (HELOC), an interesting point came up. Below, a mathematical description of why the savings in interest, on a reduced interest rate, is dependent on the starting interest rate.

Variables:
Rate - APR as a whole number (i.e. Rate=5 means 5%)
Red - Reduction in interest rate (amount saved)
J - Monthly interest
N - Number of payment months
a - principal amount (a(1) is starting principal, a(2) principal after 1st payment, etc)
mp - monthly payment
int - interest per month (int(1)= interest in month 1, etc)

The monthly payment can be calculated as:




Principal, after each months interest and monthly payment is:

Interest is:

Expansion of the first three interest terms yields:



As the expansion shows, the first term of the interest varies with the rate. The second term varies with the rate and the square of the rate. The third term varies with the rate, the square of the rate, and the cube of the rate. That means that the 36th term of the expansion would contain J terms, J^2 terms, etc all the way up to J^36 terms.

Clearly, the exponentiation of the J term will yield varying results when subtracting the interest paid when J=Rate/(12*100) from the interest paid when J=(Rate-Red)/(12*100). Keeping Red fixed, to say a 1% reduction in interest, will not reduce the interest paid by the same amount regardless of Rate.

Here's the raw data that generated the graph on the main post: