Effective yield

Deposit-taking institutions quote two numbers when they advertise a product. The first is the nominal or stated rate — the actual annualised interest rate on the label. The second is the effective yield.

They differ whenever interest is paid more often than once a year. A rate of 4.20% "per annum, paid monthly" does not leave you with 4.20% at the end of the year, because each monthly payment starts earning on its own. The effective yield is the single annual figure that produces the same end-of-year amount.

It exists so that products with different payment frequencies can be compared on one scale. Without it, a monthly-paying deposit and an annually-paying one cannot honestly be set side by side.

The workbook's worked figure is unambiguous: a bond paying a 4.20% annual coupon would be worth 4.28% if the coupon is paid every month.

That 8 basis point gap is produced by compounding within the year:

(1 + 0.0420/12)^12 - 1 = (1.0035)^12 - 1 = 0.042818 = 4.28%

Each of the twelve payments of 0.35% is received early enough to earn for the rest of the year, and the accumulated effect is 4.28% rather than 4.20%.

A caution on the workbook's wording. Section 9.5.3 describes the effective yield as the rate producing the same final amount after one year "if simple interest is applied" — yet its own 4.20% to 4.28% figure can only be obtained by compounding the monthly payments, and under simple interest the answer would be 4.20% exactly. The prose and the number point in different directions. Learn the number, which is checkable, and do not build an answer on the phrase.

The same section is also visibly damaged in transcription around this sentence. The figures are clear; the sentence is not.

Effective yield = (1 + nominal rate / m)^m - 1

m = number of compounding or payment periods in the year

Worked:  (1 + 0.0420/12)^12 - 1 = 4.28%

A retired client is shown three deposits, all quoting a nominal 4.20% but paying at different frequencies. He has Rs 20,00,000 to place for one year and intends to reinvest every payment as it arrives.

The monthly option is worth Rs 1,636 more than the annual one on the same headline rate — not a large sum, but obtained for nothing, and the ranking is what the question tests.

Where it matters more. The gap widens with the rate. On a Rs 20,00,000 deposit at a nominal 9%:

Annual   : 9.0000%  ->  Rs 1,80,000 interest
Monthly  : (1 + 0.09/12)^12 - 1 = 9.3807%  ->  Rs 1,87,614 interest

Difference: Rs 7,614 on the same quoted 9%

The adviser's use of it. A client comparing a bank deposit quoting 7.50% paid quarterly against a corporate deposit quoting 7.65% paid annually should compare 7.7136% against 7.65% — and the bank, which looked 15 basis points worse on the label, is 6 basis points better in fact.

Chapter 9 (Investing in Fixed Income Securities), section 9.5.3, sitting between the day-count and yield conventions of 9.5.2 and yield to call at 9.5.4. It also connects directly to the compounding-frequency arithmetic of Chapter 2 (Time Value of Money). Questions give a nominal rate and a payment frequency and ask for the effective rate, or ask which of several quoted products is genuinely the best.

• The effective yield is always at least the nominal rate, and equals it only when interest is paid once a year. An answer below the nominal rate is wrong.
• More frequent payment means a higher effective yield, never lower — the direction is fixed.
• The workbook's definition and its own example disagree. Section 9.5.3 says "if simple interest is applied", but 4.20% to 4.28% is a compounding result; simple interest gives 4.20%. State the arithmetic, not the phrase.
• Divide the nominal rate by m and raise to the power m. Dividing without raising, or raising without dividing, are the two standard slips.
• Nominal and stated rate are the same thing in this section. The "second rate" the institution quotes is the effective one.
• Effective yield is not yield to maturity. It adjusts for payment frequency within a year; YTM discounts every cash flow to the market price over the whole life of the bond.
• Conventions differ between markets in the same country — the workbook warns that India's money market convention differs from its bond market convention, so a question's stated basis governs.