Master the concept of Effective Rate of Interest for CBSE Class 11 Applied Mathematics. Learn to calculate annual yields with compounding frequencies.
The effective rate of interest (ERI) is the actual annual rate of interest earned or paid after accounting for the effects of compounding within a year. While the nominal rate is the stated annual percentage, the ERI reflects the true growth of an investment when interest is compounded periodically. It allows for a direct comparison between different investment schemes with varying compounding frequencies.
Step 1: Identify nominal rate r = 0.12 and frequency n = 4.
Step 2: Apply formula rₑ = (1 + 0.12/4)⁴ − 1.
Step 3: Calculate (1.03)⁴ − 1 = 1.12550881 − 1.
Answer: 0.1255 or 12.55%
The nominal rate is the base rate, but the effective rate increases as the frequency of compounding (n) increases. For a fixed nominal rate, the effective rate is always greater than or equal to the nominal rate. As n approaches infinity, the effective rate approaches the limit of continuous compounding.
Step 1: Given r = 0.08 for continuous compounding.
Step 2: Apply rₑ = e⁰.⁰⁸ − 1.
Step 3: Using e⁰.⁰⁸ ≈ 1.083287.
Answer: 0.0833 or 8.33%
Investors often compare different banks by calculating the ERI. A bank offering 10% compounded semi-annually yields a different return than 10% compounded monthly. By converting all rates to their effective annual equivalents, one can objectively choose the most profitable investment.
Step 1: Scheme A: 8% compounded semi-annually (n=2). rₑ = (1 + 0.08/2)² − 1 = 0.0816.
Step 2: Scheme B: 7.9% compounded monthly (n=12). rₑ = (1 + 0.079/12)¹² − 1 = 0.0819.
Step 3: Compare 8.16% and 8.19%.
Answer: Scheme B is better.