Compound Interest Rate Calculator
From principal, final balance and years, solve the compound annual rate (with continuous compounding supported).
輸入資料
計算結果
重點速覽：Given PV, FV, years and frequency, solve the rate. Discrete: r = m x ((FV/PV)^(1/(m·t)) - 1); continuous: r = ln(FV/PV)/t; EAR = (1+r/m)^m - 1 (or e^r - 1). Example: 1,000 → 2,000 in 10y annual → ≈7.18% nominal and EAR. KEY: same FV/years always gives the same EAR regardless of frequency; nominal falls as frequency rises. WARNING: PV, FV > 0; compare on EAR. Education, not advice.
計算公式
離散複利模型：FV = PV × (1 + r/m)^(m·t)，反解得 r = m × ((FV/PV)^(1/(m·t)) − 1)。
連續複利模型：FV = PV × e^(r·t)，反解得 r = ln(FV/PV) ÷ t。
有效年利率 EAR = (1 + r/m)^m − 1 (離散) 或 e^r − 1 (連續)。
$$離散複利反推：$r=m\left(\left(\dfrac{FV}{PV}\right)^{\frac{1}{mt}}-1\right)$$$$$連續複利反推：$r=\dfrac{\ln(FV/PV)}{t}$$$$$有效年利率：$EAR=\left(1+\dfrac{r}{m}\right)^{m}-1$ (離散) 或 $EAR=e^{r}-1$ (連續)$$使用說明
- Enter the initial principal and final balance.
- Enter the years and pick the compounding frequency.
- View the nominal rate and EAR.
PV 1,000 → FV 2,000、10 年，不同複利頻率下反推的名目年利率與 EAR
| 複利頻率 | 名目年利率 | 有效年利率 (EAR) |
|---|---|---|
| 每年 (m=1) | 約 7.1773% | 約 7.1773% |
| 每月 (m=12) | 約 6.9515% | 約 7.1773% |
| 連續複利 | 約 6.9315% | 約 7.1773% |
| PV 10,000→15,000 / 5 年 / 每季 | 約 8.1921% | 約 8.4472% |
理財情境案例
案例一：反推十年翻倍的年化複利利率
某投資十年前投入 HK$1,000，如今結餘 HK$2,000 (剛好翻倍)，以每年複利一次計算，想知道這相當於多少年化複利利率。
名目年利率 = 1 × ((2,000 ÷ 1,000)^(1 ÷ (1 × 10)) − 1) = 2^0.1 − 1 ≈ 7.1773%；因每年複利一次，EAR 與名目利率相同，約 7.18%。
解讀：這符合著名的『72 法則』直覺——本金翻倍所需年數 × 年利率 ≈ 72，即 10 年翻倍對應約 7.2% 的年利率，與精算出的 7.18% 非常接近。反推利率讓你把一筆『投入多少、變成多少、花了幾年』的投資，換算成可比較的年化複利回報。
案例二：複利頻率如何影響名目利率與 EAR
同樣 10 年由 HK$1,000 增至 HK$2,000，但分別以『每年』與『每月』複利計算：每年複利名目利率約 7.1773%,每月複利名目利率約 6.9515%。
然而兩者的有效年利率 (EAR) 都約 7.1773%——因為最終同樣是翻倍，實際年回報必然相同。差別只在名目利率的『報價方式』:複利頻率越高，達成同一 EAR 所需的名目利率越低。
解讀：這帶出一個實用提醒——比較不同存款或投資產品時，不能只看名目利率的數字大小，因為它們的複利頻率可能不同。應該統一換算成 EAR 再比較，才能公平反映真實的年回報。這正是本計算器同時輸出名目利率與 EAR 的原因。
常見問題
What is the difference between nominal rate and EAR?
Nominal rate ignores compounding effect; EAR reflects the true annual return after compounding. Higher frequency makes EAR exceed the nominal rate more noticeably.
What is continuous compounding?
It assumes interest is reinvested at every instant — the limit as frequency → infinity, using e. It is the theoretical upper bound of compounding and the basis of many finance models (e.g. derivatives pricing).
Why must principal and balance be positive?
Solving the rate requires taking the root or log of FV/PV; zero or negative values are mathematically invalid for a meaningful compound rate.
How does this differ from normal compound calculation?
Normal compound calc is forward (known rate → FV). This is reverse (known PV, FV, years → rate). Very useful when you have 'invested X, ended with Y, over N years' (a fund, policy or business) and want the annualised return. Mathematically it needs the FV/PV ratio rooted (discrete) or logged (continuous), hence positivity required.
Why is the continuous nominal rate lowest — is it the best?
Not better — just a different quote. Higher frequency means stronger compounding, so a lower nominal rate achieves the same FV. Continuous is the limit, so its nominal is lowest. But the EAR is identical across frequencies for the same FV/years, because EAR reflects the real return set by the outcome. Always compare on EAR, not nominal. Continuous is a theoretical cap; real products use discrete compounding.
相關工具
參考資料
內容審核：香港計算器財經團隊。計算邏輯與公式參考香港金融管理局（HKMA）及投資者及理財教育委員會（IFEC）之個人理財計算指引，結果僅供參考，實際以相關機構公佈為準。