Nominal rate against effective yield, worked on a savings balance at five compounding frequencies, with the Excel functions and a three-account comparison.
APY = (1 + APR ÷ m)m − 1, where m is the number of times a year interest is added. An illustrative 4.5 per cent APR paid monthly is an APY of 4.59 per cent: 10,000 earns 459.40 in a year, not 450. Compounding adds 9.40 in year one and 140.23 over ten years, and it decides which of two similar-looking accounts actually pays more.
Worked in full in Personal Finance for Beginners by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The APR (annual percentage rate) is the nominal rate: the periodic rate times the number of periods. The APY (annual percentage yield, called AER, the annual equivalent rate, in the UK) is what a full year of compounding produces. Banks advertise whichever looks better, so the only safe comparison is APY against APY.
| Input | Value |
|---|---|
| Balance | 10,000 |
| Nominal rate (APR) | 4.50% |
| Interest added | monthly, m = 12 |
| Withdrawals | none; interest left in the account |
The bank pays one twelfth of the APR each month: 4.5 per cent ÷ 12 = 0.375 per cent, or 0.00375. In the first month 10,000 earns 37.50. In the second, interest is paid on 10,037.50, and so on.
APY = (1 + APR ÷ m)m − 1
= (1 + 0.045 ÷ 12)12 − 1 = 1.0037512 − 1 = 1.045940 − 1 = 4.59% (4.5940% unrounded)
Interest on 10,000 in a year: 10,000 × 0.045940 = 459.40
Excel: =EFFECT(4.5%, 12) returns 4.5940%; =(1+4.5%/12)^12-1 does the same.
The extra 9.40 over simple interest is interest earned on interest during the year. It is small at this rate, but it is free, and it grows with both the rate and the time.
| Frequency | Periodic rate | APY | Interest in one year |
|---|---|---|---|
| Annually | 4.5000% | 4.5000% | 450.00 |
| Quarterly | 1.1250% | 4.5765% | 457.65 |
| Monthly | 0.3750% | 4.5940% | 459.40 |
| Daily | 0.0123% | 4.6025% | 460.25 |
| Continuously | 4.6028% | 460.28 |
Most of the benefit arrives with the move from annual to monthly. Daily compounding adds only 0.85 more in the year, and continuous compounding, the mathematical limit eAPR − 1, adds almost nothing beyond daily.
| Years | Simple interest | Compounded annually | Compounded monthly | Monthly over annual |
|---|---|---|---|---|
| 1 | 10,450.00 | 10,450.00 | 10,459.40 | 9.40 |
| 5 | 12,250.00 | 12,461.82 | 12,517.96 | 56.14 |
| 10 | 14,500.00 | 15,529.69 | 15,669.93 | 140.23 |
The bigger gap is between the first column and the others: leaving interest in the account rather than withdrawing it. Taking 37.50 out each month leaves you on the simple-interest column, 1,169.93 behind after ten years.
The gap between APR and APY grows faster than the rate itself.
| APR | APY, monthly | Gap, points | APY, daily |
|---|---|---|---|
| 2.0% | 2.02% | 0.02 | 2.02% |
| 4.5% | 4.59% | 0.09 | 4.60% |
| 6.0% | 6.17% | 0.17 | 6.18% |
| 22.9% | 25.46% | 2.56 | 25.73% |
The last row is why the same arithmetic matters for debt. In the United States a card APR is a nominal rate, so a balance left unpaid on a 22.9 per cent card costs 25.46 per cent a year with monthly compounding, and 25.73 with the daily compounding most US cards use. In the UK the APR quoted on borrowing is already an annual equivalent rate, so no further conversion is needed there. For how a card actually charges interest day by day, see how credit card interest is calculated.
Account A pays 4.5 per cent APR monthly. Account B pays 4.55 per cent once a year. Account C advertises an APY of 4.60 per cent. B has the highest headline nominal rate of the first two, but on 10,000 for a year A pays 459.40, B 455.00 and C 460.00. Convert everything to APY first: 4.59, 4.55 and 4.60 per cent. To go the other way, C's 4.60 per cent APY is a monthly nominal rate of 12 × (1.0461/12 − 1) = 4.51 per cent (=NOMINAL(4.6%, 12)).
APY describes one year at one rate. Many accounts pay a bonus for twelve months and then drop. An account paying 5.2 per cent APR monthly for a year (APY 5.33) and 2.5 per cent after that (APY 2.53) turns 10,000 into 10,798.93 over two years. The steady 4.5 per cent account reaches 10,939.90. Over the two years the bonus account has yielded 3.92 per cent a year: compute the APY for each period you expect to hold the money, then compound them, rather than reading the headline. At a steady 4.59 per cent APY a balance doubles in 15.43 years, against 15.75 with interest added once a year.
For an emergency fund held in savings, the APY is the rate to plug into any projection. See how much an emergency fund should be and how long it takes to build.
APY = (1 + APR ÷ m)m − 1: 4.59 per cent for a 4.5 per cent APR paid monthly. Compare accounts only on APY or AER, leave the interest in, and remember the gap widens with the rate, to 2.56 points at 22.9 per cent. The budgeting and saving plan the book builds, and the debt workbook that prices interest month by month, are on the free companion page for this book.
Yes, in substance. Both state the interest a balance would earn in a year with interest compounded and left in the account. US banks call it APY, UK banks AER. A 4.5 per cent nominal rate paid monthly has an APY or AER of 4.59 per cent, so 10,000 earns 459.40 over the year.
Use =EFFECT(APR, periods per year). =EFFECT(4.5%, 12) returns 4.5940 per cent for monthly compounding. To go back, =NOMINAL(APY, periods): an APY of 4.60 per cent is a nominal 4.51 per cent compounded monthly. The manual formula is (1 + APR / m)^m minus 1.
Very little at savings rates. On 10,000 at 4.5 per cent, monthly compounding pays 459.40 in a year and daily compounding 460.25, a difference of 0.85. The move from annual to monthly matters more: 450.00 to 459.40.
This article is one calculation from Personal Finance for Beginners. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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