Interest Rate Calculator
Back-solve the deposit rate from principal, final balance, term and compounding frequency.
Input Data
Results
At a glance:The Interest Rate Calculator back-solves the 'nominal annual rate, periodic rate and EAR' needed to reach a given balance from principal, term and compounding frequency. Based on F = A × (1 + r/m)^(m·t): periodic rate i = (F/A)^(1/(m·t)) − 1; nominal r = i × m; EAR = (1 + i)^m − 1. Useful for assessing real rates of time deposits/savings plans or comparing deposit products.
Formula
Deposit model: F = A × (1 + r/m)^(m·t).
Periodic rate: i = (F/A)^(1/(m·t)) − 1; nominal r = i × m.
EAR = (1 + i)^m − 1, the real annual return after compounding.
$$F = A \times \left(1 + \dfrac{r}{m}\right)^{m t}$$$$i = \left(\dfrac{F}{A}\right)^{\frac{1}{m t}} - 1,\; r = i \times m,\; EAR = (1+i)^m - 1$$How to Use
- Enter the initial principal and final balance.
- Enter the term and choose the compounding frequency.
- View nominal, periodic and effective annual (EAR) rates instantly.
Back-solving the rate to reach a balance (deposit compound model)
| Initial | Final | Term | Compound | Nominal | EAR |
|---|---|---|---|---|---|
| HK$1,000 | HK$2,000 | 10 yr | Yearly (1) | 7.1773% | 7.1773% |
| HK$30,000 | HK$36,000 | 5 yr | Monthly (12) | 3.6520% | 3.7137% |
| HK$100,000 | HK$150,000 | 8 yr | Quarterly (4) | 5.1006% | 5.1990% |
Doubling in 10 years ≈ 7.18%, matching the Rule of 72 (72 ÷ 10 ≈ 7.2). Yearly compounding: nominal = EAR.
Case Studies
Case 1: Rate to double principal
How much annual rate (yearly) to double HK$1,000 to HK$2,000 in 10 years?
Periodic i = (2,000/1,000)^(1/(1×10)) − 1 = 2^0.1 − 1 ≈ 7.1773%. Yearly compounding → nominal = EAR ≈ 7.18%.
This matches the Rule of 72: 72 ÷ 10 = 7.2%, nearly identical to the exact 7.18%. The Rule of 72 is the quick mental version of this compound formula.
Case 2: Compare two deposits by EAR
May compares two deposits: A HK$100,000 for 8 years → HK$150,000, quarterly; B labelled 'nominal 5%, monthly'.
Back-solve A: i = (150,000/100,000)^(1/(4×8)) − 1 ≈ 1.27514%, nominal ≈ 5.1006%, EAR = (1.0127514)^4 − 1 ≈ 5.1990%. B's EAR = (1 + 0.05/12)^12 − 1 ≈ 5.1162%.
EAR: A 5.20% > B 5.12%, so A is slightly better. Compare by EAR including fees; for annuities/loans use other methods. Pair with the compound-interest and EAR calculators.
FAQ
Nominal vs effective annual rate (EAR)?
Nominal is the periodic rate times compounding frequency, ignoring compounding effect; EAR reflects the real annual return after compounding. More frequent compounding makes EAR exceed nominal more visibly.
Why must principal and balance both be > 0?
Back-solving needs F/A and a root; if principal or balance is 0 or negative the compound rate is mathematically invalid. Both must be positive.
Does this rate apply to loans?
This tool targets a lump-sum principal rolling to maturity (deposit). Loans with periodic repayment (annuity) need a different solving method (personal-loan/mortgage calculator).
Why require positive principal and balance; why not negative or zero?
The solver i = (F/A)^(1/(m·t)) − 1 has division (F/A) and a root, both requiring valid values. A=0 is division by zero (undefined) and financially meaningless (no money deposited). Negative A or F makes F/A negative, and an even root of a negative is not real — the calculator cannot return a valid real rate. Edge: F=A gives 0% (no growth); F<A gives a negative rate (meaningful in some fee/negative-rate contexts). Most savings assume F ≥ A. For recurring contributions use an annuity/savings calculator instead.
Is the back-solved rate fixed; are real deposit rates floating?
This tool returns an equivalent annual rate assuming the rate is fixed over the whole term — precise but an average/equivalent concept. Real rates float (time deposits reset on renewal, savings rates adjust, HK rates follow the Fed). Feeding actual past amounts gives the real equivalent annual rate (accurate); using it to predict the future gives only a target threshold, not a guaranteed fixed rate. Advice: use for review (actual annualised return) it is accurate; for projection treat it as a reference hurdle and adjust if far above market; lock fixed rates via time deposits (rate resets on renewal); the model assumes no mid-term deposits/withdrawals. Pair with compound-interest and fixed-deposit calculators.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.