{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-present-value-discounting",
  "semantic_family_id": "sf-present-value-discounting",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "現值與折現"
  },
  "summary": {
    "zh-Hant": "固定正累積因子與期數時，未來金額除以累積因子冪得到現值。"
  },
  "statement": {
    "zh-Hant": "相對固定每期利率i、正累積因子1+i與期數n，未來金額F的現值定義為F/(1+i)^n。此式與固定期率複利互逆；現金流、變動利率與費用需另建模型。"
  },
  "formula": {
    "tex": "P(F)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>P</mi><mo>(</mo><mi>F</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "P",
        "glyph": "P"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "F",
          "glyph": "F"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "87eb7397007e456976bc5d2c45cac36d8a9bf96c47a5fb98cf43b31ef30c7696"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "期別、利率慣例與n一致，1+i>0，期間內無額外現金流、費用、稅或四捨五入。"
    },
    {
      "id": "a2",
      "zh-Hant": "題庫使用非負精確有理金額；模型計算不是報價、投資建議或市場預測。"
    }
  ],
  "symbols": [
    {
      "token": "P",
      "role": "present_value_operator",
      "role_zh": "現值算子",
      "scope": "representative_formula"
    },
    {
      "token": "F",
      "role": "future_amount",
      "role_zh": "未來金額",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-compound-interest",
      "title_zh": "複利模型",
      "reason_zh": "折現是固定累積因子複利的反向運算。"
    },
    {
      "id": "mko-fraction",
      "title_zh": "分數",
      "reason_zh": "現值使用精確除法。"
    },
    {
      "id": "mko-equation",
      "title_zh": "方程式",
      "reason_zh": "可由未來值方程反解現值。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "一期折現把F除以1+i；n期則除以其n次冪。",
      "先複利再以同一因子與期數折現應回到原本金，形成精確roundtrip核對。",
      "若利率期別不同、因子非正或存在多筆現金流，本條目公式不直接適用。"
    ],
    "common_misconception_zh": "把折現乘上累積因子，或混用年率、月率及不同期數。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "只核固定期率無現金流模型；不是投資建議或市場預測。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "加入現金流時點、年金、名目／有效利率與敏感度。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-present-value-discounting.md 整理；2026-10-13 脈衝 R02（使用者手動接續 2026-10-22 工作序列；實際日期仍為 2026-10-13），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-13"
  }
}