Historical Interest Rate Calculator
Methodology
Interest is the amount paid over and above the principal in return for the use of money. The interest rate is this payment expressed as a fraction (generally given as a percentage) of the principal over a given period of time (usually on an annual basis).
A simple interest rate equation is shown in equation (1) below.
Equation (1)
where L is the sum loaned (the principal); F is the final amount repaid (i.e. the principal plus the interest); r is the interest rate; and T is the period of time over which the loan was outstanding.
Most interest rate calculators available online today are designed to help modern borrowers make comparisons between different financial products. They assume that the interest rate, the principal and the length of loan are all known, and they then calculate the total repayment (or monthly/annual repayments) due. The current project aproaches this problem from a different angle, where the interest rate is the unknown factor. In order to calculate the interest rate, therefore, it is necessary to know the values of the other three variables:
- the value of the principal;
- the value of total sum repaid (i.e. the principal plus interest);
- the length of time during which the loan was outstanding (i.e. the period between the advancing of the loan and the full repayment of the principal, including the timing of any intermediate interest payments).
Where these data are known, this historical interest rate calculator can be used to calculate an annualised interest rate (incorporating compounding on an annual basis). More technical details are given below.
The approach used to determine the interest rates on the loans works by calculating the rate that would set the present value of the sum of the interest payments and of the repayments to the value of the loan. This accounts for the time value of money by measuring all the cashflows in terms of what they would have been worth at the time of the loan, so that all payments in the formula are included on a like-for-like basis. It also supports multiple repayments, which means that the timing of any intermediate payments and the effect of compounding (the interest received in subsequent years on the interest paid in previous years) will be included in the interest rate calculated. The formula works with years (and fractions of years) as the unit of time and therefore the rate that emerges is automatically an annualised one. The general formula for a loan of L made now with a final repayment of F in T years (T need not be a whole number) and N interest payments of P1, P2, PN at times T1, T2, …, TN is shown in equation (2) below.
Equation (2)
This formula is sufficiently flexible that it can handle interest or other intermediate payments that do not occur at regular intervals. Of course, a loan may also be advanced in installments, and a slightly generalised version of this formula could be used for such calculations.
It is important to note that the general formula above calculates compounded interest on an annual basis, so that from the second year interest is paid on the accrued interest from the first year as well as on the principal. However, this means that an adjustment to the calculated interest rate is needed to attain annually compounded rates for debts with a maturity of less than one year. This is shown below in equation (3).
Equation (3)
where T is the term of the loan expressed as a fraction of a year, r is the interest rate (expressed as a proportion rather than a percentage) obtained from equation (2), and r' is the annually compounded rate. This second formula is automatically applied during the calculation process in those cases where the maturity of the loan is less than one year.