Effective interest rate

A bank advertises 6%. It pays half of it after six months. You reinvest that half, and it earns interest for the remaining six months. At the end of the year you are holding more than 6%.

The nominal rate is what was quoted: 6%. The effective rate is what you ended up with: 6.09%. The gap is compounding, and it is not optional — it happens automatically to anyone who reinvests the interim payments.

This is why two deposits quoting the same rate are not the same deposit, and why any comparison between instruments paying at different frequencies has to be put on an effective basis first.

Work it through in rupees rather than symbols, which is how the workbook introduces it.

Put Rs 1,000 into a 6% bond compounding semi-annually. After six months you receive Rs 1,000 × 3% = Rs 30. Reinvest it at the same rate and the second half-year earns 3% on Rs 1,030 = Rs 30.90. The year's total is Rs 60.90 on Rs 1,000 — an effective rate of 6.09%.

The more often interest is paid, the more of the year it spends earning interest of its own, so the effective rate climbs with frequency and the increments shrink:

Pushing the frequency to its limit gives continuous compounding, which the workbook notes is only marginally above daily compounding and rarely worth the trouble in practice.

The same relationship runs backwards. A semi-annual yield cannot be annualised by doubling it — doubling understates the effective annual yield, because it ignores the interest the first payment earns.

Effective rate = (1 + r ÷ n)ⁿ − 1

   r = nominal annual rate
   n = compounding periods per year


Effective annual yield = (1 + semi-annual rate)² − 1

Semi-annual yield      = 2 × [ (1 + annual rate)^½ − 1 ]

The workbook's base case. A 6% bond compounding semi-annually:

(1 + 0.06/2)² − 1 = 1.03² − 1 = 0.0609 = 6.09%

Rs 1,000 becomes Rs 1,060.90, not Rs 1,060. Thirty extra rupees on a lakh; Rs 9,000 on a Rs 1 crore deposit.

Monthly compounding. A bond paying a 4.20% annual coupon, paid monthly:

(1 + 0.042/12)¹² − 1 = 1.0035¹² − 1 = 0.042818 = 4.28%

Eight basis points of free yield, for no change in the coupon.

Where it decides a real choice. A treasury has Rs 25 crore to place for a year and two quotes on the desk:

The lower-coupon instrument wins by Rs 2,42,250, and the nominal rates say the opposite. That inversion is the whole reason the effective rate exists.

And the annualisation trap. A bond quoting a 3.50% semi-annual yield is not a 7.00% bond:

(1.035)² − 1 = 7.1225%

Twelve basis points that doubling would have thrown away.

Chapter 1 raises it twice. Section 1.1.1 (Effective Rate) gives the Rs 1,000 illustration and the 6% to 6.09% conversion; section 1.10.2.4 (Effective Yield) returns to it as a yield measure, supplies the annualisation formulas and warns that doubling a semi-annual yield underestimates the effective annual yield. The frequency table in section 1.7.3 is the same idea applied to a 12% investment.

Questions are computational — convert a nominal rate at a stated frequency into an effective rate, or annualise a semi-annual yield properly — and comparative: which of two differently compounded instruments actually pays more.

• Never annualise by multiplying. Doubling a semi-annual yield always understates the effective annual yield; the exponent, not the multiplier, is the right operation.
• The effective rate is never below the nominal rate, and the two are equal only when interest is paid once a year.
• Divide the nominal rate by n before raising to the power n. (1 + 0.06)² is a different and wrong number from (1 + 0.06/2)².
• Higher frequency raises the effective rate with diminishing returns. Annual to semi-annual buys far more than quarterly to monthly, and continuous compounding adds almost nothing over daily.
• Compare like with like. Two instruments quoting different nominal rates at different frequencies can only be ranked on their effective rates, and the ranking sometimes reverses.
• This is a compounding adjustment, not an inflation adjustment. Stripping out inflation gives the real interest rate, which is a separate calculation entirely.