{
  "id": "discount-rate",
  "version": "a9ef4fcc5fa8",
  "status": "published",
  "name": "Discount Rate Calculator",
  "question": "What is the discount rate?",
  "summary": "Computes the yearly discount rate that turns a future value into a present value over a number of years, with the effective rate and the discount factor.",
  "category": "finance",
  "subcategory": "investing",
  "url": "https://www.acalculator.org/finance/discount-rate-calculator",
  "markdown": "https://www.acalculator.org/finance/discount-rate-calculator.md",
  "kind": "function",
  "method": "r = m × ((FV ÷ PV)^(1 ÷ (m × t)) − 1) for m compounding periods a year over t years; continuous: r = ln(FV ÷ PV) ÷ t. Effective rate = (FV ÷ PV)^(1 ÷ t) − 1. Discount factor = PV ÷ FV.",
  "assumptions": [
    "One amount today and one amount at the end, with no payments in between.",
    "The rate is the same every year. A future value below the present value gives a negative rate."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "pv": {
        "title": "Present value",
        "description": "What the amount is worth today.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0.01,
        "maximum": 1000000000000
      },
      "fv": {
        "title": "Future value",
        "description": "The amount at the end of the period.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0.01,
        "maximum": 1000000000000
      },
      "years": {
        "title": "Years",
        "description": "The time between the present and the future value, in years. Decimals are fine.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 1000
      },
      "compounding": {
        "title": "Compounding",
        "description": "How often the rate compounds.",
        "type": "string",
        "enum": [
          "annually",
          "semiannually",
          "quarterly",
          "monthly",
          "daily",
          "continuous"
        ]
      }
    }
  },
  "outputs": {
    "rate": {
      "label": "Discount rate (yearly)",
      "description": "The nominal yearly rate, compounded as chosen.",
      "format": "percent"
    },
    "effective": {
      "label": "Effective yearly rate",
      "description": "The rate compounded once a year that gives the same growth.",
      "format": "percent"
    },
    "factor": {
      "label": "Discount factor",
      "description": "Present value ÷ future value: what $1 at the end is worth today.",
      "format": "number"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "pv": 15000,
      "fv": 25000,
      "years": 8.5,
      "compounding": "annually"
    },
    "outputs": {
      "rate": 6.193969008260831,
      "effective": 6.193969008260831,
      "factor": 0.6
    },
    "text": "Discounting $25,000.00 back to $15,000.00 over 8.5 years takes a discount rate of 6.194% a year."
  },
  "examples": [
    {
      "given": {
        "pv": 1000,
        "fv": 1125.51,
        "years": 4,
        "compounding": "annually"
      },
      "expect": {
        "rate": 3.000027225453571,
        "factor": 0.8884861085196933
      },
      "source": "OpenStax, Principles of Finance, 7.3 Methods for Solving Time Value of Money Problems (solving for the rate: $1,000 to $1,125.51 in 4 years is 3%; $15,000 to $25,000 in 8.5 years is 6.194%). https://openstax.org/books/principles-finance/pages/7-3-methods-for-solving-time-value-of-money-problems: 3% (Python 3: (1,125.51 ÷ 1,000)^(1 ÷ 4) − 1 = 3.00003%)"
    },
    {
      "given": {
        "pv": 15000,
        "fv": 25000,
        "years": 8.5,
        "compounding": "annually"
      },
      "expect": {
        "rate": 6.193969008260831
      },
      "source": "OpenStax, Principles of Finance, 7.3 Methods for Solving Time Value of Money Problems (solving for the rate: $1,000 to $1,125.51 in 4 years is 3%; $15,000 to $25,000 in 8.5 years is 6.194%). https://openstax.org/books/principles-finance/pages/7-3-methods-for-solving-time-value-of-money-problems: 6.1940% (Python 3: (25,000 ÷ 15,000)^(1 ÷ 8.5) − 1)"
    },
    {
      "given": {
        "pv": 100,
        "fv": 200,
        "years": 10,
        "compounding": "continuous"
      },
      "expect": {
        "rate": 6.931471805599452,
        "effective": 7.177346253629316
      },
      "source": "hand calculation in content.mdx: ln(2) ÷ 10 = 6.9315%; e^0.069315 − 1 = 7.1773%"
    },
    {
      "given": {
        "pv": 10000,
        "fv": 15000,
        "years": 5,
        "compounding": "monthly"
      },
      "expect": {
        "rate": 8.136764313761281,
        "effective": 8.447177119769862,
        "factor": 0.6666666666666666
      },
      "source": "hand calculation in content.mdx: 12 × (1.5^(1 ÷ 60) − 1) = 8.1368%"
    }
  ],
  "sources": [
    "OpenStax, Principles of Finance, 7.3 Methods for Solving Time Value of Money Problems: solving for the rate ($1,000 to $1,125.51 in 4 years is 3%; $15,000 to $25,000 in 8.5 years is 6.194%). https://openstax.org/books/principles-finance/pages/7-3-methods-for-solving-time-value-of-money-problems (retrieved 2026-10-05)",
    "OpenStax, Principles of Finance, 7.2 Time Value of Money (TVM) Basics: FV = PV × (1 + i)^n and present value as discounting. https://openstax.org/books/principles-finance/pages/7-2-time-value-of-money-tvm-basics (retrieved 2026-10-05)"
  ],
  "related": [
    "present-value",
    "future-value",
    "npv",
    "wacc",
    "cagr"
  ],
  "changelog": []
}