{
  "id": "present-value",
  "version": "5cd076338c98",
  "status": "published",
  "name": "Present Value Calculator",
  "question": "What is the present value?",
  "summary": "Computes the present value of a future amount, of level payments (an annuity), or of both, at a yearly discount rate, with payments at the end or the start of each period.",
  "category": "finance",
  "subcategory": "investing",
  "url": "https://www.acalculator.org/finance/present-value-calculator",
  "markdown": "https://www.acalculator.org/finance/present-value-calculator.md",
  "kind": "function",
  "method": "Present value = (future value + payment × (1 + i × t) × ((1 + i)^N − 1) ÷ i) ÷ (1 + i)^N, with i the rate per period and t = 1 for payments at the start of each period, 0 at the end.",
  "assumptions": [
    "The interest rate is the same in every period, and every payment is the same size.",
    "The yearly rate is nominal, compounded as many times a year as set; payments are made as many times a year as set.",
    "The payments and the future amount are money received, and the present value is what they are worth today.",
    "No fees or taxes are included. This is an estimate for planning, not financial advice."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "solve": {
        "title": "Find the",
        "description": "Which value to work out; type the others.",
        "type": "string",
        "enum": [
          "n",
          "rate",
          "pv",
          "pmt",
          "fv"
        ]
      },
      "n": {
        "title": "Number of periods",
        "description": "How many periods until the future amount, and how many payments. Years when P/Y is 1.",
        "type": "number",
        "exclusiveMinimum": 0,
        "maximum": 12000
      },
      "rate": {
        "title": "Discount rate (yearly)",
        "description": "The yearly interest rate used to discount, in percent, compounded C/Y times a year.",
        "type": "number",
        "x-unit": "percent",
        "exclusiveMinimum": -100,
        "maximum": 1000
      },
      "pv": {
        "title": "Present value",
        "description": "What the future amount and the payments are worth today.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "pmt": {
        "title": "Payment each period",
        "description": "A level amount received every period (an annuity). 0 for a single future amount.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "fv": {
        "title": "Future value",
        "description": "A single amount received after the last period. 0 for payments only.",
        "type": "number",
        "x-unit": "USD",
        "minimum": 0,
        "maximum": 1000000000000
      },
      "py": {
        "title": "Periods a year",
        "description": "How many periods (and payments) there are in a year: 1 for yearly, 12 for monthly.",
        "type": "integer",
        "minimum": 1,
        "maximum": 365
      },
      "cy": {
        "title": "Compounding a year",
        "description": "How many times a year the rate compounds. Usually the same as the periods a year.",
        "type": "integer",
        "minimum": 1,
        "maximum": 365
      },
      "due": {
        "title": "Payments at the",
        "description": "The end of each period (an ordinary annuity) or the start of each period (an annuity due).",
        "type": "string",
        "enum": [
          "end",
          "begin"
        ]
      }
    }
  },
  "outputs": {
    "n": {
      "label": "Number of periods",
      "description": "How many periods until the future amount, and how many payments. Years when P/Y is 1.",
      "format": "number"
    },
    "rate": {
      "label": "Discount rate (yearly)",
      "description": "The yearly interest rate used to discount, in percent, compounded C/Y times a year.",
      "format": "percent"
    },
    "pv": {
      "label": "Present value",
      "description": "What the future amount and the payments are worth today.",
      "format": "money"
    },
    "pmt": {
      "label": "Payment each period",
      "description": "A level amount received every period (an annuity). 0 for a single future amount.",
      "format": "money"
    },
    "fv": {
      "label": "Future value",
      "description": "A single amount received after the last period. 0 for payments only.",
      "format": "money"
    },
    "periodRate": {
      "label": "Rate per period",
      "description": "The interest rate for one payment period, i = (1 + I/Y ÷ C/Y)^(C/Y ÷ P/Y) − 1.",
      "format": "percent"
    },
    "ear": {
      "label": "Effective yearly rate",
      "description": "The rate over a whole year once interest compounds: (1 + i)^(P/Y) − 1.",
      "format": "percent"
    },
    "totalPmt": {
      "label": "Total of payments",
      "description": "N × the payment each period.",
      "format": "money"
    },
    "interest": {
      "label": "Interest (the discount)",
      "description": "Everything received (N × payment + future value) minus the present value.",
      "format": "money"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "solve": "pv",
      "py": 1,
      "cy": 1,
      "due": "end",
      "pv": 6139.13,
      "n": 10,
      "rate": 5,
      "pmt": 0,
      "fv": 10000
    },
    "outputs": {
      "pv": 6139.132535407594,
      "periodRate": 5,
      "ear": 5,
      "totalPmt": 0,
      "interest": 3860.867464592406
    },
    "text": "At 5% a year, $10,000.00 after 10 periods plus $0.00 each period is worth $6,139.13 today."
  },
  "examples": [
    {
      "given": {
        "solve": "pv",
        "n": 10,
        "rate": 5,
        "pmt": 0,
        "fv": 10000,
        "py": 1,
        "cy": 1,
        "due": "end"
      },
      "expect": {
        "pv": 6139.132535407592,
        "interest": 3860.867464592408
      },
      "source": "the compound interest formula A = P(1 + r)^n solved for P (SEC Investor.gov, https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator); hand calculation in content.mdx; Python 3 cross-check"
    },
    {
      "given": {
        "solve": "pv",
        "n": 240,
        "rate": 8,
        "pmt": 500,
        "fv": 0,
        "py": 12,
        "cy": 12,
        "due": "end"
      },
      "expect": {
        "pv": 59777.14585118838,
        "totalPmt": 120000
      },
      "source": "Microsoft Excel PV function example: ($59,777.15) (https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd); Python 3 cross-check"
    },
    {
      "given": {
        "solve": "pv",
        "n": 5,
        "rate": 6,
        "pmt": 1000,
        "fv": 0,
        "py": 1,
        "cy": 1,
        "due": "begin"
      },
      "expect": {
        "pv": 4465.1056126996555
      },
      "source": "present value of an annuity due, Broverman (2017), Mathematics of Investment and Credit, section 2.1; hand calculation in content.mdx; Python 3 cross-check"
    },
    {
      "given": {
        "solve": "pv",
        "n": 20,
        "rate": 4,
        "pmt": 50,
        "fv": 1000,
        "py": 2,
        "cy": 2,
        "due": "end"
      },
      "expect": {
        "pv": 1490.543000337913
      },
      "source": "present value of an annuity plus a lump sum, Broverman (2017), Mathematics of Investment and Credit, chapter 2; hand calculation in content.mdx; Python 3 cross-check"
    }
  ],
  "sources": [
    "U.S. Securities and Exchange Commission, Investor.gov compound interest calculator and formula. https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator",
    "Microsoft, Excel PV function, with a worked example. https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd",
    "S. A. Broverman (2017), Mathematics of Investment and Credit, 7th edition, chapters 1 and 2: present value, annuities immediate and annuities due."
  ],
  "related": [
    "future-value",
    "npv",
    "tvm",
    "financial",
    "annuity"
  ],
  "changelog": []
}