{
  "id": "continuous-compound-interest",
  "version": "dc06b6424c82",
  "status": "published",
  "name": "Continuous Compound Interest Calculator",
  "question": "Continuous compound interest",
  "summary": "Calculates what a deposit grows to with interest compounded continuously, A = Pe^(rt), with the interest earned, the effective yearly rate, and monthly and yearly compounding for comparison.",
  "category": "finance",
  "subcategory": "investing",
  "url": "https://www.acalculator.org/finance/continuous-compound-interest-calculator",
  "markdown": "https://www.acalculator.org/finance/continuous-compound-interest-calculator.md",
  "kind": "function",
  "method": "A = P × e^(r × t), with r the yearly rate as a decimal and t in years. Interest = A − P. Effective yearly rate = e^r − 1.",
  "assumptions": [
    "The rate is a nominal yearly rate compounded continuously, and it stays the same for the whole time.",
    "No deposits or withdrawals after the start, and no taxes or fees."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "p": {
        "title": "Starting amount",
        "description": "The deposit or principal at the start.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0,
        "maximum": 1000000000
      },
      "r": {
        "title": "Yearly interest rate",
        "description": "The nominal yearly rate, compounded continuously.",
        "type": "number",
        "x-unit": "percent",
        "minimum": 0,
        "maximum": 100
      },
      "t": {
        "title": "Years",
        "description": "How long the money grows, in years; decimals allowed (1.5 is 18 months).",
        "type": "number",
        "minimum": 0,
        "maximum": 100
      }
    }
  },
  "outputs": {
    "amount": {
      "label": "Final amount",
      "description": "P × e^(r × t).",
      "format": "money"
    },
    "interest": {
      "label": "Interest earned",
      "description": "The final amount minus the starting amount.",
      "format": "money"
    },
    "effective": {
      "label": "Effective yearly rate",
      "description": "e^r − 1: the yearly rate that gives the same growth compounded once a year.",
      "format": "percent"
    },
    "monthly": {
      "label": "With monthly compounding",
      "description": "P × (1 + r/12)^(12t), for comparison.",
      "format": "money"
    },
    "yearly": {
      "label": "With yearly compounding",
      "description": "P × (1 + r)^t, for comparison.",
      "format": "money"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "p": 10000,
      "r": 5,
      "t": 10
    },
    "outputs": {
      "amount": 16487.212707001283,
      "interest": 6487.212707001282,
      "effective": 5.127109637602404,
      "monthly": 16470.09497690283,
      "yearly": 16288.946267774414
    },
    "text": "$10,000.00 at 5% compounded continuously for 10 years grows to $16,487.21."
  },
  "examples": [
    {
      "given": {
        "p": 1000,
        "r": 10,
        "t": 1
      },
      "expect": {
        "amount": 1105.1709180756477,
        "interest": 105.17091807564763,
        "effective": 10.517091807564762
      },
      "source": "OpenStax, College Algebra 2e, section 6.1 Exponential Functions: continuous growth A(t) = P·e^(rt); Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18); Python 3: 1000 * math.exp(0.1)"
    },
    {
      "given": {
        "p": 10000,
        "r": 5,
        "t": 10
      },
      "expect": {
        "amount": 16487.212707001283,
        "interest": 6487.212707001282,
        "monthly": 16470.0949769028,
        "yearly": 16288.946267774414
      },
      "source": "OpenStax, College Algebra 2e, section 6.1 Exponential Functions: continuous growth A(t) = P·e^(rt); Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18); Python 3: 10000 * math.exp(0.5); 10000 * (1 + 0.05/12)**120; 10000 * 1.05**10"
    },
    {
      "given": {
        "p": 5000,
        "r": 0,
        "t": 20
      },
      "expect": {
        "amount": 5000,
        "interest": 0,
        "effective": 0
      },
      "source": "OpenStax, College Algebra 2e, section 6.1 Exponential Functions: continuous growth A(t) = P·e^(rt); Example 11: $1,000 at 10% compounded continuously for 1 year is about $1,105.17. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions (retrieved 2026-10-18); hand calculation in content.mdx: e^0 = 1"
    }
  ],
  "sources": [
    "OpenStax, College Algebra 2e, section 6.1 Exponential Functions: the continuous growth formula A(t) = P·e^(rt) for principal P, rate r and time t; Example 11, $1,000 at a nominal 10% compounded continuously for 1 year is about $1,105.17. Retrieved 2026-10-18. https://openstax.org/books/college-algebra-2e/pages/6-1-exponential-functions"
  ],
  "related": [
    "compound-interest",
    "savings",
    "apy",
    "simple-interest"
  ],
  "changelog": []
}