Q.Consider two investment offers: Investment A pays 10% interest, compounded monthly. Investment B pays 10.1% interest, compounded semi-annually. Using the effective annual interest rate, which is the better offer?
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🔒 Start your 14-day free trial to unlock the full solution →Concept understanding — Effective Rate of Interest
Effective Rate of Interest – The Real Cost of Borrowing
When you borrow money, the lender quotes a nominal interest rate — say, 12% per annum. But if interest is compounded more than once a year (monthly, quarterly, half-yearly), the actual interest you pay over one year ends up being more than 12%. That actual, real rate is the Effective Rate of Interest.
Think of it this way: if the bank charges you interest every month, then after the first month you owe a little more, and the next month's interest is calculated on that bigger amount. So you're paying "interest on interest" even within a single year. The effective rate captures this compounding effect.
The nominal rate is the "headline" rate. The effective rate is what you actually end up paying (or earning) after compounding is taken into account.
The Precise Definition
The Effective Annual Rate (EAR) — also called the Effective Rate of Interest — is the annual interest rate that would give the same final amount if compounding were done only once per year, as the nominal rate gives when compounded m times per year.
Effective Rate=(1+mr)m−1
where:
- r = nominal annual interest rate (as a decimal)
- m = number of compounding periods per year
Why It Matters
Suppose you invest ₹1,00,000 at a nominal rate of 12% p.a., compounded monthly.
- Nominal rate = 12% → you might think you'll earn ₹12,000 in one year.
- Actual calculation: monthly rate = 12%/12=1%=0.01. After 12 months:
Final amount=1,00,000×(1.01)12≈1,12,682.50
So you actually earned ₹12,682.50 — that's an effective rate of 12.6825%.
The difference of ₹682.50 is the "interest on interest" you wouldn't get if compounding were annual.
A common mistake: students think the effective rate is simply r×m divided by something. It is not. The formula (1+r/m)m−1 is the only correct one.
Key Exam Points
- If compounding is annual (m=1), effective rate = nominal rate.
- As m increases (more frequent compounding), the effective rate increases — but it approaches a limit (continuous compounding gives er−1).
- For loans, the effective rate is always greater than or equal to the nominal rate (equal only when m=1). …
Compare Effective Annual Rates (EAR) of two offers to find the better one.
EARA=(1+0.10/12)12−1≈10.47%. EARB=(1+0.101/2)2−1=10.355%. Since 10.47%>10.355%, A is better. …
Convert both nominal rates to their Effective Annual Rate (EAR) and compare, since the two offers use different compounding frequencies.
EAR=(1+mr)m−1, where r = nominal annual rate, m = number of compounding periods per year.
Given: Investment A: r=10%, monthly compounding (m=12). Investment B: r=10.1%, semi-annual compounding (m=2).
- Investment A: mr=120.10=0.0083333.
EARA=(1.0083333)12−1
- Compute (1.0083333)12 by squaring: 1.00833332=1.0167361, 4=1.0337520, 8=1.0686513, 12=8×4=1.0686513×1.0337520=1.104720.
EARA=1.104720−1=0.104720=10.472%
- Investment B: mr=20.101=0.0505. EARB=(1.0505)2−1=1.10355025−1=0.10355=10.355% …
- CA Foundation 2026Set jan-20261 markMCQQ.Mr. Ravi allocates a corpus of ₹ 50,000 into a term deposit account which accrues interest at a nominal annual rate of 10%, compounded on a quarterly basis. What will be the effective annual rate of interest? (A) 10 % (B) 10.25 % (C) 10.38 % (D) 10.50 %
›Reveal solutionSolution
EAR=(1+0.10/4)4−1=(1.025)4−1≈0.1038=10.38%.
Step 1 — Formula
Effective annual rate for a nominal rate compounded m times a year:
EAR=(1+mi)m−1.
Step 2 — Substitute
With i=10%=0.10, m=4 (quarterly):
EAR=(1+0.025)4−1=(1.025)4−1.
Step 3 — Evaluate
(1.025)4=1.103813,
so
EAR=1.103813−1=0.103813≈10.38%.
The ₹50,000 principal is not needed — the effective rate is independent of the amount. …
- CA Foundation 2026Set may-20261 markMCQQ.Find the effective interest rate, if nominal rate is 12% per annum, quarterly compounding. (A) 12% (B) 12.36% (C) 12.55% (D) 13%
›Reveal solutionSolution
Effective rate =(1+r/m)m−1=(1.03)4−1≈12.55%.
Step 1 — Write the effective-rate formula
For nominal annual rate r compounded m times per year:
E=(1+mr)m−1
Step 2 — Substitute
r=0.12, m=4 (quarterly), so the periodic rate is 0.03:
E=(1.03)4−1
Step 3 — Compute
(1.03)4=1.125509 ⇒ E=0.125509≈12.55% …
- CA Foundation 2025Set jan-20251 markMCQQ.The effective rate of interest corresponding to a nominal rate of 8% per annum payable quarterly is (Given that (1.02)4=1.08243216) (A) 6.24% (B) 5.38% (C) 8.24% (D) 82.4%
›Reveal solutionSolution
Effective rate =(1.02)4−1=0.0824=8.24%.
Step 1 — Find the periodic rate
Nominal 8% payable quarterly → rate per quarter =48%=2%=0.02, with 4 quarters a year.
Step 2 — Apply the effective-rate formula
E=(1+kr)k−1=(1.02)4−1
Step 3 — Substitute the given value
E=1.08243216−1=0.08243216≈8.24%
Why the other options are wrong: (A) 6.24% and (B) 5.38% understate the growth; (D) 82.4% misplaces the decimal by a factor of 10. The effective rate must be just above the nominal 8%. …
- CA Foundation 2025Set may-20251 markMCQQ.The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly is (A) 6.06% (B) 6.07% (C) 6.08% (D) 6.09%
›Reveal solutionSolution
E=(1+0.03)2−1=6.09%.
Step 1 — Find the periodic rate
Nominal 6% payable half-yearly → rate per half-year =26%=3%, with n=2 compounding periods per year.
Step 2 — Apply the effective-rate formula
E=(1+ni)n−1
E=(1+0.03)2−1=1.0609−1=0.0609
Step 3 — Express as a percentage
E=6.09%
Why the other options are wrong: 6.06%, 6.07%, 6.08% are near-misses from rounding (1.03)2 incorrectly; the exact value 1.0609 gives 6.09%. …
- CA Foundation 2025Set may-20251 markMCQQ.Relationship between annual nominal rate of interest and annual effective rate of interest, if frequency of compounding is greater than one (A) Effective rate < Nominal rate (B) Effective rate > Nominal rate (C) Effective rate = Nominal rate (D) Effective rate = 0.9 times Nominal rate
›Reveal solutionSolution
Compounding more than once a year makes the effective rate exceed the nominal rate.
Step 1 — Recall the effective-rate relation
E=(1+ni)n−1
where i is the nominal annual rate and n>1 is the compounding frequency.
Step 2 — Reason about the inequality
Expanding (1+ni)n for n>1 gives 1+i+(positive higher-order terms), so
E=i+(positive terms)>i
The 'positive terms' are interest-on-interest — they cannot be negative, so E strictly exceeds i.
Step 3 — Confirm with a number
At i=6%, n=2: E=(1.03)2−1=6.09%>6% — exactly the pattern. …
- CA Foundation 2025Set sep-20251 markMCQQ.If Mr. XYZ is investing ₹ 86,000 in a bank fixed deposit scheme where interest will be payable at 12% per annum, compounded half-yearly, what will be the effective rate of interest in a year ? (A) 12.36% (B) 12.24% (C) 12.12% (D) 12.48%
›Reveal solutionSolution
Effective rate =(1+20.12)2−1=(1.06)2−1=12.36%.
Step 1 — Find the periodic rate
Half-yearly compounding ⇒ rate per half-year =212%=6%, with 2 periods a year.
Step 2 — Apply the effective-rate formula
EAR=(1+mi)m−1=(1.06)2−1
Step 3 — Compute
(1.06)2=1.1236, so EAR =0.1236=12.36%. (The principal ₹86,000 is irrelevant to the rate.) …
- CA Foundation 2024Set sep-20241 markMCQQ.What is the effective rate of interest when principal amount of ₹ 50,000 deposited in a nationalized bank for one year, corresponding to a nominal rate of interest 6% per annum payable half yearly ? (A) 6.06% (B) 6.07% (C) 6.08% (D) 6.09%
›Reveal solutionSolution
Effective rate =(1+i/m)m−1; with i=6%, m=2: (1.03)2−1=6.09%.
Step 1 — Recall the effective-rate formula
Effective rate=(1+mi)m−1
where i = nominal annual rate, m = number of compounding periods per year.
Step 2 — Substitute half-yearly compounding
i=0.06, m=2⇒ half-yearly rate =3%:
(1.03)2−1=1.0609−1=0.0609=6.09% …
- CA Foundation 2023Set jun-20231 markMCQQ.The Nominal rate of interest is 10% per annum. The interest is compounded quarterly. The effective rate of interest per annum will be: (A) 10 % (B) 10.10 % (C) 10.25 % (D) 10.38 %
›Reveal solutionSolution
Effective rate =(1+0.10/4)4−1=(1.025)4−1=10.38%.
Step 1 — Effective-rate formula
ieff=(1+mr)m−1.
Step 2 — Substitute quarterly compounding
=(1+40.10)4−1=(1.025)4−1.
Step 3 — Evaluate
(1.025)4=1.103813 ⇒ ieff=0.103813≈10.38%.
Watch out10.25% (C) is the semi-annual effective rate (1.05)2−1. Use m=4 for quarterly, giving 10.38%. The effective rate is always higher than the 10% nominal because of intra-year compounding. …
- CA Foundation 2022Set dec-20221 markMCQQ.The effective annual rate of interest corresponding to a normal rate of 6% per annum payable half yearly is: (A) 6.06 % (B) 6.07 % (C) 6.08 % (D) 6.09 %
›Reveal solutionSolution
EAR = (1.03)² − 1 = 6.09%.
Step 1 — Find the periodic rate
Half-yearly ⇒ rate per period =6%/2=3%, and number of periods per year m=2.
Step 2 — Apply the effective rate formula
EAR=(1+mi)m−1=(1+0.03)2−1=1.0609−1=0.0609=6.09%
Watch outThe nominal 6% is not the effective rate — half-yearly compounding raises it to 6.09%. Options 6.06/6.07/6.08 are near-miss distractors from rounding errors. …
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