How do I calculate the effective rate of interest when interest is compounded quarterly or monthly?

Question

In Business Maths, Time Value of Money, I get confused between nominal rate and effective rate. If a bank says 12% per annum compounded quarterly, is the real rate still 12%? How do I calculate it quickly in the exam?

GradeHunt_Team · Answer

No, it is more than 12%. The 12% is the nominal (stated) rate. When interest is added more than once a year, each round of interest also earns interest, so the rate you actually earn in a year is higher. That actual rate is the effective rate.

Formula:
E = (1 + i)^n - 1
where i = rate per conversion period and n = number of conversion periods in a year.

Worked example: 12% p.a. compounded quarterly.
i = 12% / 4 = 3% = 0.03, n = 4
E = (1.03)^4 - 1 = 1.12550881 - 1 = 0.1255, so about 12.55%.

Check with money: Rs 10,000 at 3% for 4 quarters = 10,000 x 1.12550881 = Rs 11,255.09. Interest = Rs 1,255.09, which is 12.55% of 10,000.

Same 12% nominal rate, different compounding:
- Yearly: 12.00%
- Half-yearly: (1.06)^2 - 1 = 12.36%
- Quarterly: (1.03)^4 - 1 = 12.55%
- Monthly: (1.01)^12 - 1 = 12.68%

So the more often interest is compounded, the higher the effective rate, though the increase gets smaller each time.

Exam tips:
1. Always convert the annual rate to the per-period rate first (divide by 2, 4 or 12).
2. Keep at least 4 decimals in (1 + i)^n before subtracting 1. Rounding 1.1255 to 1.13 too early is where answers drift.
3. If the question gives you the effective rate and asks for the nominal rate, reverse it: i = (1 + E)^(1/n) - 1, then multiply by n.

How do I calculate the effective rate of interest when interest is compounded quarterly or monthly?

1 answer