Continuous Compound Interest Calculator
Using FV = P·e^(rt), compute the future value, interest and equivalent effective annual rate under continuous compounding.
輸入資料
計算結果
重點速覽：Continuous compounding is the limit as frequency → infinity; FV = P x e^(r·t), e ≈ 2.71828; interest = FV - P; equivalent EAR = e^r - 1. It is the theoretical upper bound of compounding, used in mathematical finance (Black-Scholes, rate models). Example: 1,000 at 12%, 1y → ≈1,127.50. WARNING: No real deposit compounds truly every instant; daily is the practical max and is nearly identical. Theoretical estimate, not advice.
計算公式
未來值 FV = P × e^(r × t)，其中 P 為本金、r 為年利率、t 為年期、e ≈ 2.71828。
總利息 = FV − P。
等效有效年利率 = e^r − 1。
$$未來值：$FV = P\,e^{rt}$$$$$總利息：$Interest = FV - P$$$$$等效有效年利率：$EAR = e^{r}-1$$$$$示例 (12%、1 年)：$1000\times e^{0.12}\approx1127.50$$$使用說明
- Enter the principal.
- Enter the annual rate and years.
- View the future value, interest and equivalent EAR.
本金 HK$100,000、年利率 5% 在不同複利頻率下的一年後金額
| 複利頻率 | 一年後金額 (HK$) | 有效年利率 |
|---|---|---|
| 每年 (1 次) | 105,000.00 | 5.0000% |
| 每月 (12 次) | 105,116.19 | 5.1162% |
| 每日 (365 次) | 105,126.75 | 5.1267% |
| 連續複利 | 105,127.11 | 5.1271% |
理財情境案例
案例一：連續複利的未來值與利息
本金 HK$50,000，年利率 8%,以連續複利計 10 年。
未來值 = 50,000 × e^(0.08×10) = 50,000 × e^0.8 ≈ HK$111,277.05；總利息 = 111,277.05 − 50,000 ≈ HK$61,277.05；等效有效年利率 = e^0.08 − 1 ≈ 8.3287%。
解讀：10 年後本利和約為本金的 2.2 倍，賺得利息約 HK$61,277。注意等效有效年利率 (8.3287%) 高於名目的 8%,這多出來的部分正是『連續不斷把利息滾入本金再生息』的複利威力。
案例二：連續複利 vs 每日複利，差多少？
以年利率 5% 比較：連續複利的等效有效年利率 = e^0.05 − 1 ≈ 5.1271%；每日複利 (365 次) 的有效年利率 = (1 + 0.05/365)^365 − 1 ≈ 5.1267%。
兩者只差約 0.0004 個百分點——以 HK$100,000 本金計，一年下來連續複利只比每日複利多賺約 HK$0.36。
解讀：連續複利雖然是理論上限，但在常見利率水平下，它與每日複利的差距微乎其微。因此現實中的存款產品用『每日／每月複利』就已極接近連續複利的效果，連續複利更多是用於金融模型 (如 Black-Scholes 期權定價) 的數學工具。本計算器讓你直觀看到連續複利的結果與等效年利率。
常見問題
What is continuous compounding?
It assumes interest is reinvested at every instant — the limit as frequency → infinity. At the same nominal rate, it produces the highest possible future value.
How much does it differ from daily compounding?
Very little. At 12%, HK$1,000 daily for a year ≈ HK$1,127.47; continuous ≈ HK$1,127.50 — cents apart. Continuous is mainly for theoretical analysis.
Does continuous compounding really exist, and where is it used?
Strictly, no bank compounds truly every instant; the most frequent is daily, already near the continuous limit. Continuous compounding is used in mathematical finance (Black-Scholes option pricing, interest-rate and bond models), academia, and as a benchmark to convert rates (force of interest) for fair comparison.
What is the constant e, and why does it appear here?
e ≈ 2.71828 is the natural constant, the limit of (1 + 1/n)^n as n → infinity — exactly the continuous-compounding limit. Generalised, FV = P x e^(r·t). It also appears in natural growth/decay, probability and calculus. The tool has e built in; just enter P, r, t.
What is the equivalent EAR for?
It converts continuous compounding to an 'equivalent annual rate compounded once' (e^r - 1) so different compounding bases can be compared fairly. Example: '8% continuous' → EAR ≈ 8.33%, which beats '8.2% annual'. Like APY for deposits or the true annual rate for loans, EAR strips out frequency to reveal the real yearly rate. This tool outputs the equivalent EAR for easy comparison.
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。