Effective and Nominal Interest Rates for Exam FM

Effective Interest Rate

The effective interest rate i is the actual interest earned per period as a fraction of the balance at the start of the period. If compounding occurs annually, i is the annual effective rate. The key equation: 1 + i equals the ratio of the balance at the end of one year to the balance at the start.

The effective interest rate is the "true" rate that allows direct comparison between different compounding frequencies. Exam FM problems often require converting between nominal and effective rates.

Nominal Interest Rate

A nominal interest rate i^(m) compounded m times per year means the effective rate per compounding period is i^(m)/m. The relationship between the nominal rate and the annual effective rate is: 1 + i = (1 + i^(m)/m)^m. Solving for the nominal rate: i^(m) = m * ((1+i)^(1/m) - 1).

Example: a nominal rate of 12% compounded monthly means i^(12) = 0.12, the monthly effective rate is 0.01, and the annual effective rate is (1.01)^12 - 1 = 0.1268 or 12.68%.

Nominal Discount Rate

The nominal discount rate d^(m) compounded m times per year satisfies: 1 - d = (1 - d^(m)/m)^m. The conversion between nominal interest and nominal discount rates uses the identity (1 + i^(m)/m) * (1 - d^(m)/m) = 1, which holds for any m. As m approaches infinity, both i^(m) and d^(m) approach delta, the force of interest. These conversions appear frequently on Exam FM and must be performed quickly and accurately.