2.11 Effective Annual Rate When Compounded More Than Once a Year
Free study material · concepts, shortcuts & solved questions
When interest is compounded n times a year at nominal annual rate R%, the effective annual rate (the equivalent once-a-year compounding rate that gives the same yearly growth) is:
R_(eff) = [(1+(R)/(100n))^(n) - 1]× 100%
The effective rate is always higher than the nominal rate whenever n > 1, because more frequent compounding means interest starts earning interest sooner.
Solved Example 2.11.1: Find the effective annual rate for a nominal rate of 10% p.a. compounded half-yearly.
Solution: R_eff = [(1.05)² − 1] × 100 = (1.1025 − 1) × 100 = 10.25%.
Solved Example 2.11.2: Find the effective annual rate for 20% p.a. compounded half-yearly.
Solution: R_eff = [(1.10)² − 1] × 100 = (1.21 − 1) × 100 = 21%.
Solved Example 2.11.3: Find the effective annual rate for 8% p.a. compounded quarterly.
Solution: R_eff = [(1.02)⁴ − 1] × 100 ≈ (1.08243 − 1) × 100 ≈ 8.24%.
Solved Example 2.11.4: Find the effective annual rate for 6% p.a. compounded monthly (edge case: n = 12, the most compounding periods tested at this level).
Solution: R_eff = [(1 + 6/1200)¹² − 1] × 100 = [(1.005)¹² − 1] × 100. (1.005)¹² ≈ 1.061678, so R_eff ≈ (1.061678 − 1) × 100 ≈ 6.17% (approx). The more frequent the compounding, the closer the effective rate creeps toward the continuous-compounding limit — but SSC/RRB never requires you to compute beyond a direct power like this.
Solved Example 2.11.5: Which is more beneficial to an investor for a 1-year deposit: 10% p.a. compounded annually, or 9.9% p.a. compounded half-yearly? (edge case: comparing a lower nominal rate against a higher one via effective rates)
Solution: The 10% annual option simply gives an effective rate of 10%. For 9.9% compounded half-yearly: R_eff = [(1 + 9.9/200)² − 1] × 100 = [(1.0495)² − 1] × 100 = (1.10145025 − 1) × 100 ≈ 10.145%. Since 10.145% > 10%, the 9.9% half-yearly option is actually better, despite having a lower quoted nominal rate — a good reminder that the nominal rate alone can be misleading; always compare effective rates when compounding frequencies differ.