Archive / Investment Adviser (Level 1) / Evaluating the Financial Position of Clients

nism-series-xa-effective-annual-rate-ear-quarterly-compounding[ INVESTMENT ADVISER (LEVEL 1) ]

A bank deposit advertises a nominal interest rate of 12.0% p.a. compounded QUARTERLY. What is the true Effective Annual Rate (EAR) earned by the depositor?

[ A ]12.00% p.a. (Nominal rate without compounding)

[ B ]12.55% p.a. [ (1 + r/m)^m - 1 = (1 + 0.03)^4 - 1 ][ CORRECT ]

[ C ]13.50% p.a.

[ D ]15.00% p.a.

Takeaway

Option B is correct: 12.55% p.a. [ (1 + r/m)^m - 1 = (1 + 0.03)^4 - 1 ]

Derivation

Nominal annual rate r = 12.0% = 0.12, compounded m = 4 times a year (quarterly). Periodic quarterly rate = 0.12 / 4 = 0.03, or 3% per quarter. EAR = (1 + 0.03)^4 − 1 = (1.03)^4 − 1. Raising 1.03 to the 4th power gives approximately 1.125508, so EAR = 12.55% p.a. The gap versus the quoted 12.0% nominal rate, though smaller than what monthly or daily compounding would produce, is still a real 0.55-percentage-point uplift purely from interest earning interest four times a year rather than once. This is the same underlying formula tested with monthly compounding elsewhere in the syllabus, just with m = 4 instead of m = 12.

Why the others fail

A — 12.00% p.a. (Nominal rate without compounding)

States the nominal rate itself without applying any compounding adjustment; quarterly compounding lifts the realized effective yield above 12.0% to 12.55%.

C — 13.50% p.a.

Overstates the effect of quarterly compounding on a 12.0% nominal rate; the correctly compounded figure works out to 12.55%, not this higher number.

D — 15.00% p.a.

Is an arbitrary figure with no basis in the quarterly compounding formula applied to a 12.0% nominal annual rate over four periods.