APY Calculator
Convert nominal Annual Percentage Rate (APR) to effective Annual Percentage Yield (APY), analyze daily, monthly, and continuous compounding schedules, and solve reverse interest rates.
Interest & Compounding Settings
Nominal advertised annual percentage rate
How often interest is calculated and credited to balance
Starting deposit amount to calculate exact monetary return
Step-by-Step Mathematical Derivation Expand
Annual Percentage Yield (APY)
Annual Percentage Yield (APY) represents the true annualized rate of return earned on a deposit balance or investment fund, accounting for the compounding of interest across all intra-year crediting cycles. As compounding frequency accelerates from annually to monthly or daily, APY expands progressively above the nominal stated APR.
Mathematical Formulations: Discrete & Continuous APY
The Annual Percentage Yield (also designated as the Effective Annual Rate or EAR) normalizes different compounding cadences into a standardized annual metric. The governing mathematical equations depend on whether interest is credited across finite discrete intervals or compounded continuously:
| Compounding Model | Mathematical Formula | Variables & Constraints |
|---|---|---|
| Discrete Periodic Compounding | APY = (1 + r / n)^n − 1 | r = nominal APR (decimal), n = cycles per year |
| Continuous Compounding | APY = e^r − 1 | e = Euler's constant ≈ 2.7182818, r = nominal rate |
| Reverse APR from APY (Discrete) | APR = n × [(1 + APY)^(1/n) − 1] | Solves unadvertised nominal rate from bank APY |
| Reverse APR (Continuous) | APR = ln(1 + APY) | Natural logarithm transformation |
| Future Balance Growth | A = P × (1 + APY)^t = P × (1 + r / n)^(n × t) | P = principal deposit, t = time in years |
Statutory Basis: Truth in Savings Act (TISA) Regulation DD
In the United States, 12 CFR Part 1030 (Regulation DD) mandates that depository institutions quote APY using an exact 365-day annual basis, computing interest on the full balance daily: $\text{APY} = 100 \left[ \left( 1 + \frac{\text{Interest Earned}}{\text{Principal}} \right)^{365 / \text{Days}} - 1 \right]$. This ensures retail consumers can directly compare products across different institutions regardless of whether interest is credited daily, monthly, or quarterly.
APR vs APY: Stated Rate vs Effective Annual Compound Yield
While both acronyms describe annual interest percentages, their economic implications represent fundamentally opposite sides of the compounding equation:
Annual Percentage Rate (APR) • Simple Cost Metric
APR is the annualized simple interest rate. It represents what you pay or earn without considering intra-year compounding. When evaluating loans, mortgages, and auto financing, APR provides a standardized baseline of borrowing cost before compounding interest snowballs the debt.
- Ignores interest accrued on previously credited interest
- Always lower than or equal to APY ($APR \le APY$)
- Governed by Truth in Lending Act (TILA / Regulation Z)
Annual Percentage Yield (APY) • Compound Earning Metric
APY reflects the true net return generated over 365 days because it factors in the frequency of interest crediting. Every time interest is added to your principal, subsequent cycles earn interest on that larger sum, elevating your effective yield above the simple rate.
- Fully integrates intra-year compounding accumulation
- Always greater than APR for any $n > 1$ schedule
- Governed by Truth in Savings Act (TISA / Regulation DD)
Compounding Frequency Yield Sensitivity & Euler Limits
How much does compounding frequency actually matter? As the interval $n$ increases from annual to daily, returns climb rapidly at first, then experience asymptotic diminishing returns toward Euler's limit $e^r - 1$:
| Compounding Schedule | Periods ($n$) | Effective APY on 5.00% APR | 1-Yr Earnings on $10,000 | Compounding Bonus |
|---|---|---|---|---|
| Annual | 1 | 5.0000% | $500.00 | $0.00 (Baseline) |
| Semi-Annual | 2 | 5.0625% | $506.25 | +$6.25 |
| Quarterly | 4 | 5.0945% | $509.45 | +$9.45 |
| Monthly | 12 | 5.1162% | $511.62 | +$11.62 |
| Daily (365) | 365 | 5.1267% | $512.67 | +$12.67 |
| Continuous Compounding | ∞ | 5.1271% | $512.71 | +$12.71 |
Real-World Banking: HYSAs, CD Ladders & Debt Costs
High-Yield Savings Accounts (HYSA)
Leading online financial institutions compound interest daily and credit monthly. Because deposit balances fluctuate with withdrawals, daily compounding ensures every dollar deposited earns interest from its exact transaction date.
Certificates of Deposit (CDs)
CDs lock in fixed interest rates for terms ranging from 3 months to 5 years. Building a CD ladder across maturities allows investors to capture peak APYs while maintaining regular liquidity tranches.
Revolving Credit Cards & Debt Costs
Card issuers advertise nominal APRs (e.g. 24.99%), but compound interest daily using a Daily Periodic Rate (DPR = APR / 365 = 0.06847%). Carried balances experience an effective APY exceeding 28.3%, compounding debt exponentially.
Graded Worked Numerical Solutions
Convert an 18.00% nominal APR credit card compounding monthly ($n = 12$) into its true effective APY.
1. Convert nominal APR to decimal: $r = 18 / 100 = 0.18$.
2. Calculate monthly periodic rate: $r / 12 = 0.18 / 12 = 0.015$ (1.50% per month).
3. Apply APY formula: $\text{APY} = (1 + 0.015)^{12} - 1$.
4. Evaluate exponential factor: $(1.015)^{12} \approx 1.195618$.
Effective APY = 1.195618 − 1 = 19.56% (+1.56% compounding surcharge).
A bank offers 4.50% APR compounding daily ($n = 365$) on a $25,000 balance. Determine the 1-year yield and final balance.
1. Convert rate: $r = 0.045$, cycles $n = 365$.
2. Daily rate: $r / 365 = 0.045 / 365 \approx 0.00012328767$.
3. $\text{APY} = (1 + 0.00012328767)^{365} - 1 \approx 1.046025 - 1 = 0.046025$ (4.603%).
4. Total interest earned: $I = \$25,000 \times 0.046025 = \$1,150.63$.
Final Maturity Balance = $26,150.63 ($25.63 more than simple interest).
A certificate of deposit advertises a 5.25% APY compounding monthly ($n = 12$). What nominal APR is the bank paying?
1. Target APY in decimal: $\text{APY} = 0.0525$, compounding $n = 12$.
2. Inversion formula: $\text{APR} = n \times \left[ (1 + \text{APY})^{1/n} - 1 \right]$.
3. Calculate $(1 + 0.0525)^{1/12} = (1.0525)^{0.083333} \approx 1.0042735$.
4. Multiply periodic rate by 12: $\text{APR} = 12 \times 0.0042735 = 0.051282$.
Nominal Stated APR = 5.128%.
Common Calculation Pitfalls & Consumer Marketing Traps
Comparing APR Directly Against APY Across Different Institutions
A common consumer error is comparing an APR quote from one bank directly against an APY quote from another. Because APY accounts for compounding, a 4.95% APY may actually generate less revenue than a 4.90% APR that compounds daily. Always standardize both quotes into APY before making a deposit decision.
Overestimating the Value of Continuous vs Daily Compounding
While shifting from annual ($n = 1$) to daily ($n = 365$) compounding provides a noticeable yield bump, shifting from daily to continuous compounding yields an incremental difference of mere pennies (less than $0.04 per $10,000 deposited per year) due to mathematical Euler limit convergence.
The 360-Day Commercial Banking Convention ("Banker's Year")
Certain commercial money market accounts and wholesale commercial loans operate on a 360-day year (12 months of 30 days each). When interest is credited on a 360-day divisor but collected over 365 actual calendar days, the true effective borrowing rate is approximately 1.39% higher than stated.
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