nominal rates and conversion

Banks love to say things like 12 percent per year, compounded monthly. That headline 12 percent is not the rate you actually earn in a year — it is a nominal rate, a convenient label that has to be divided up before it means anything. The trick is that the 12 percent is split into twelve equal monthly slices of 1 percent each, and those slices compound. The label and the truth differ, and knowing how to convert between them is a core skill.

A nominal rate, written i with a superscript (m) meaning convertible m times a year, is by convention the rate per year before compounding. To use it, divide by m to get the rate per compounding period: i^(12) of 12 percent means 1 percent per month. The effective annual rate is then (1 + i^(m) divided by m) raised to the power m, minus 1. Going the other way, if you know the effective annual rate i, the equivalent nominal rate convertible m times is m times the quantity (1 + i) to the power one-over-m, minus 1. The more frequently a nominal rate compounds, the higher the effective rate it hides.

Actuaries convert constantly because contracts, bonds, and loans quote rates in many conventions — annually, semi-annually, quarterly, monthly — and nothing can be compared or summed until all rates sit on a common footing. Mishandling the conversion is one of the most frequent and costly errors in practice. The rule of thumb worth memorising: a nominal rate convertible m times is always larger than its effective annual equivalent (except when m equals one, where they coincide).