Continuous compound calculator
1,000 at 5% compounded continuously for 10 years grows to 1,648.72, or 648.72 of interest, a little more than the same sum compounded once a year. Enter principal, annual rate and years to apply the exponential formula directly. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Principal amount
- 1,000
- Interest rate
- 5
- Time
- 10
- Growth factor
- 1.6487212707
- Final amount
- 1,648.72
- Interest amount
- 648.72
Continuous compounding is the growth model that appears when interest is posted infinitely often, so the balance follows the exponential A equals P times e to the r t instead of a discrete power with a finite n. Students, treasurers, and product teams meet the same closed form in force-of-interest problems, continuously compounded returns, and quotes that advertise continuous posting. This continuous compound calculator accepts a starting principal, a nominal annual rate as a percent, and a time horizon in years, then returns the final amount together with the interest earned as the difference from principal. It does not schedule extra deposits, invert a growth rate into CAGR, or evaluate discrete compounding with monthly or daily periods.
What continuous compounding means and how the exponential works
Ordinary compound interest posts a finite number of times per year: A equals P times one plus r over n, raised to n times t. Continuous compounding is the limit of that identity as n goes to infinity, which calculus turns into the exponential A equals P times e to the r t.
P is the starting principal in currency units. r is the nominal annual rate as a decimal, so a five percent quote is r equals 0.05. t is the horizon in years and may be a fraction such as two point five. The growth factor e to the r t is always positive for a finite exponent, and the interest earned is simply A minus P.
This capability does not add regular contributions, subtract fees, or switch back to a discrete frequency n; those are different identities and different search intents. A zero-year horizon returns the original principal with zero interest because e to the zero is one. A zero rate likewise leaves the principal unchanged.
Negative rates are allowed when they stay finite, which models a continuous decay or a negative force of interest rather than a typing error.
What you can do with it
Continuously compounded savings checks
Project the future amount of a lump-sum deposit quoted with continuous compounding from principal, the annual percent, and years held.
Classroom force-of-interest oracles
Generate reproducible A = P e^(rt) answer keys that reject a negative time period and never invent a discrete n.
FAQ
How is continuous compounding calculated?
The final amount is A = P × e^(r×t), where P is principal, r is the annual rate as a decimal (percent divided by 100), and t is years. Interest earned is A minus P.
Is the rate a percent or a decimal?
A percent. Send 5 for five percent per year. Sending 0.05 would be treated as five hundredths of a percent, not five percent.
What happens if the time period is negative?
A negative time period is not a valid horizon for this growth formula and is not billed.
How is this different from ordinary compound interest?
Ordinary compound interest uses a finite frequency n: A = P(1+r/n)^(nt). Continuous compounding is the n→∞ limit, A = P e^(rt). There is no compounding-frequency field here.
Where is continuous compounding actually used?
In force-of-interest problems, in continuously compounded returns used by analysts and treasurers, and in deposit quotes that advertise continuous posting. It is also the standard exercise in finance and calculus courses.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/fin/continuous-compound \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"principal":1000,"rate":5,"time":10}'const res = await fetch("https://api.kit.forhosting.com/fin/continuous-compound", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"principal": 1000,
"rate": 5,
"time": 10
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/fin/continuous-compound",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"principal": 1000,
"rate": 5,
"time": 10
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/fin/continuous-compound", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"principal":1000,"rate":5,"time":10}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"principal":1000,"rate":5,"time":10}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/fin/continuous-compound", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"principal": 1000,
"rate": 5,
"time": 10
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "fin.continuous_compound",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
It evaluates the exponential identity in pure arithmetic so a worksheet, a coaching session, and an API client never disagree. A negative time period is rejected as invalid_input rather than a silent countdown into the past.
What continuous compounding means and how the exponential works
A negative time period is not a horizon: it would ask the exponential to run backwards without stating a present-value convention, so the task fails with invalid_input instead of inventing a discount.
Input contract, response shape, and error behaviour
Missing, empty, or non-finite values fail naming the field. Example: one thousand principal, five percent, and ten years yields a growth factor of e to the one half, a final amount of one thousand six hundred forty-eight point seventy-two, and interest of six hundred forty-eight point seventy-two—slightly above the same lump sum compounded once per year, which is the point of the continuous limit. Failed validation does not consume balance. POST a JSON body with three required finite numbers. principal is the starting amount in one currency; aliases amount, p, and starting_balance are accepted so spreadsheets can keep familiar column names, and the value must sit between zero and one trillion inclusive. rate is the nominal annual percent—five means five percent per year—not a decimal 0.05 unless you truly intend five hundredths of a percent; aliases annual_rate, annual_percent, and interest_rate map to the same field. time is t in years, a finite number from zero through one hundred twenty; fractional years are allowed, and aliases t, years, and periods_years are accepted. If time is negative the task fails with invalid_input naming the time period; that is the contracted error and it is not billed. Each field may arrive as a JSON number or a numeric string. On success the body echoes principal, rate, and time, reports growth_factor as e to the r t rounded to twelve decimal places, and returns final_amount and interest rounded half-up to two decimal places for stable currency-style goldens, plus a formula string restating A equals P times e to the r t.
Where a continuous compound API earns its keep
Households and students use a continuous compound calculator when a homework problem, an actuarial force-of-interest exercise, or a product quote asks how much a lump sum grows when interest is compounded continuously. Personal-finance coaches need a deterministic oracle that rejects a negative time period instead of silently showing a discounted figure that a buggy form might treat as success. When you later need regular deposits or a finite posting frequency, switch to those identities rather than stretching this three-field continuous contract. Product teams wire the same contract into onboarding demos so a user who types a principal, a percent, and a horizon sees the identical final_amount the API will later return in production tests. Keep requests small, pin rounded final_amount and interest in tests, and you get reproducible future values for worksheets, LMS plugins, and edge workers without standing up a spreadsheet service of your own.
Integration use cases
Product demos and CI fixtures
Seed finance apps and automated tests with deterministic final_amount and interest values that match the free browser widget byte-for-byte.
Technical questions
What does the continuous compound calculator cost?
The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.