{
  "version": "2026.1",
  "generated_at": "2026-09-07",
  "generated_by": "scripts/gen_annuity_study.mjs",
  "study": "annuity-vs-4-percent-rule-2026",
  "url": "https://quantcalc.app/research/annuity-vs-4-percent-rule-2026/",
  "license": "CC-BY-4.0",
  "method": {
    "annuity": "Single-premium immediate annuity, life-only, level nominal payout, bought at 65: income = payout rate x premium, paid at the start of each year while the annuitant is alive; real value in year t = nominal / (1 + inflation)^(t - 1); break-even = smallest number of payments whose cumulative real value reaches the premium; hybrid = premium sized to a nominal essential floor, the rest of the target funded from the residual.",
    "longevity": "Survival from 65 read from the SSA 2022 period life table (lx by exact age, male and female) embedded in the repository behind the published 2032 claiming study; unisex = 50/50 blend of the two conditional curves; expected payments truncated at age 100. Cross-checked against the published age-at-death percentiles of the Social Security claiming Monte Carlo (SSA 2021 table) within one year.",
    "four_percent_rule": "Quoted from the published QuantCalc Monte Carlo studies listed under quoted_sources; not re-simulated here.",
    "age_at_purchase": 65,
    "horizon_years": 30,
    "capital": 1000000,
    "target_income": 40000,
    "essential_floor": 30000,
    "payout_cases": [
      0.06,
      0.065,
      0.07
    ],
    "inflation_base": 0.025,
    "inflation_sensitivity": 0.035,
    "note": "Payout rates are illustrative assumptions, not quotes; actual rates vary by insurer, age, sex, interest rates and contract features."
  },
  "annuity": [
    {
      "payout_rate": 0.06,
      "nominal_income_per_1m": 60000,
      "real_income_at_2_5pct": {
        "year_1": 60000,
        "year_10": 48043.701706132226,
        "year_20": 37531.6629858181,
        "year_30": 29319.67513862973
      },
      "real_income_at_3_5pct": {
        "year_1": 60000,
        "year_10": 44023.85833137685,
        "year_20": 31209.34142320674,
        "year_30": 22124.889298402974
      },
      "first_year_real_income_below_target_at_2_5pct": 18,
      "premium_to_cover_target": 666666.6666666667,
      "residual_for_portfolio": 333333.33333333326,
      "residual_share": 0.33333333333333326,
      "hybrid": {
        "premium_to_cover_floor": 500000,
        "residual": 500000,
        "residual_share": 0.5,
        "withdrawal_rate_for_remaining_slice": 0.02,
        "portfolio_burden_real_at_2_5pct": {
          "year_1": {
            "amount": 10000,
            "share_of_original_residual": 0.02
          },
          "year_10": {
            "amount": 15978.149146933887,
            "share_of_original_residual": 0.03195629829386777
          },
          "year_20": {
            "amount": 21234.16850709095,
            "share_of_original_residual": 0.0424683370141819
          },
          "year_30": {
            "amount": 25340.162430685137,
            "share_of_original_residual": 0.05068032486137027
          }
        }
      },
      "break_even": {
        "nominal": {
          "payments": 17,
          "age": 81,
          "p_reach": {
            "male": 0.579054804785406,
            "female": 0.6884994955410467,
            "unisex": 0.6337771501632263
          }
        },
        "real_2_5pct": {
          "payments": 22,
          "age": 86,
          "p_reach": {
            "male": 0.3758171623472262,
            "female": 0.4997622664702949,
            "unisex": 0.4377897144087606
          }
        },
        "real_3_5pct": {
          "payments": 25,
          "age": 89,
          "p_reach": {
            "male": 0.25180227901087826,
            "female": 0.36748964989388966,
            "unisex": 0.30964596445238396
          }
        }
      },
      "expected_real_income_per_dollar_premium_at_2_5pct": {
        "male": 0.8455661396927421,
        "female": 0.9449508414563413,
        "unisex": 0.8952584905745417
      }
    },
    {
      "payout_rate": 0.065,
      "nominal_income_per_1m": 65000,
      "real_income_at_2_5pct": {
        "year_1": 65000,
        "year_10": 52047.34351497658,
        "year_20": 40659.301567969604,
        "year_30": 31762.98140018221
      },
      "real_income_at_3_5pct": {
        "year_1": 65000,
        "year_10": 47692.513192324914,
        "year_20": 33810.119875140634,
        "year_30": 23968.630073269887
      },
      "first_year_real_income_below_target_at_2_5pct": 21,
      "premium_to_cover_target": 615384.6153846154,
      "residual_for_portfolio": 384615.3846153846,
      "residual_share": 0.38461538461538464,
      "hybrid": {
        "premium_to_cover_floor": 461538.4615384615,
        "residual": 538461.5384615385,
        "residual_share": 0.5384615384615385,
        "withdrawal_rate_for_remaining_slice": 0.01857142857142857,
        "portfolio_burden_real_at_2_5pct": {
          "year_1": {
            "amount": 10000,
            "share_of_original_residual": 0.01857142857142857
          },
          "year_10": {
            "amount": 15978.149146933887,
            "share_of_original_residual": 0.0296737055585915
          },
          "year_20": {
            "amount": 21234.16850709095,
            "share_of_original_residual": 0.039434884370311764
          },
          "year_30": {
            "amount": 25340.162430685137,
            "share_of_original_residual": 0.04706030165698668
          }
        }
      },
      "break_even": {
        "nominal": {
          "payments": 16,
          "age": 80,
          "p_reach": {
            "male": 0.6164569390971809,
            "female": 0.7202978047337965,
            "unisex": 0.6683773719154886
          }
        },
        "real_2_5pct": {
          "payments": 20,
          "age": 84,
          "p_reach": {
            "male": 0.4593808945505284,
            "female": 0.5811367141747168,
            "unisex": 0.5202588043626226
          }
        },
        "real_3_5pct": {
          "payments": 22,
          "age": 86,
          "p_reach": {
            "male": 0.3758171623472262,
            "female": 0.4997622664702949,
            "unisex": 0.4377897144087606
          }
        }
      },
      "expected_real_income_per_dollar_premium_at_2_5pct": {
        "male": 0.9160299846671374,
        "female": 1.0236967449110366,
        "unisex": 0.969863364789087
      }
    },
    {
      "payout_rate": 0.07,
      "nominal_income_per_1m": 70000,
      "real_income_at_2_5pct": {
        "year_1": 70000,
        "year_10": 56050.985323820925,
        "year_20": 43786.94015012111,
        "year_30": 34206.287661734685
      },
      "real_income_at_3_5pct": {
        "year_1": 70000,
        "year_10": 51361.16805327298,
        "year_20": 36410.89832707453,
        "year_30": 25812.370848136805
      },
      "first_year_real_income_below_target_at_2_5pct": 24,
      "premium_to_cover_target": 571428.5714285714,
      "residual_for_portfolio": 428571.42857142864,
      "residual_share": 0.42857142857142866,
      "hybrid": {
        "premium_to_cover_floor": 428571.4285714285,
        "residual": 571428.5714285715,
        "residual_share": 0.5714285714285715,
        "withdrawal_rate_for_remaining_slice": 0.017499999999999998,
        "portfolio_burden_real_at_2_5pct": {
          "year_1": {
            "amount": 10000,
            "share_of_original_residual": 0.017499999999999998
          },
          "year_10": {
            "amount": 15978.149146933887,
            "share_of_original_residual": 0.0279617610071343
          },
          "year_20": {
            "amount": 21234.16850709095,
            "share_of_original_residual": 0.03715979488740916
          },
          "year_30": {
            "amount": 25340.162430685137,
            "share_of_original_residual": 0.044345284253698984
          }
        }
      },
      "break_even": {
        "nominal": {
          "payments": 15,
          "age": 79,
          "p_reach": {
            "male": 0.6521666106818945,
            "female": 0.7500550845983464,
            "unisex": 0.7011108476401204
          }
        },
        "real_2_5pct": {
          "payments": 18,
          "age": 82,
          "p_reach": {
            "male": 0.5402444381282138,
            "female": 0.6546949472927369,
            "unisex": 0.5974696927104753
          }
        },
        "real_3_5pct": {
          "payments": 20,
          "age": 84,
          "p_reach": {
            "male": 0.4593808945505284,
            "female": 0.5811367141747168,
            "unisex": 0.5202588043626226
          }
        }
      },
      "expected_real_income_per_dollar_premium_at_2_5pct": {
        "male": 0.9864938296415325,
        "female": 1.1024426483657317,
        "unisex": 1.0444682390036322
      }
    }
  ],
  "longevity": {
    "source": {
      "table": "SSA 2022 period life table",
      "ssa_url": "https://www.ssa.gov/oact/STATS/table4c6.html",
      "repo_file": "research/ss_claiming/compute.py",
      "study_url": "https://quantcalc.app/research/social-security-claiming-age-2032/"
    },
    "from_age": 65,
    "p_reach": {
      "age_80": {
        "male": 0.6164569390971809,
        "female": 0.7202978047337965,
        "unisex": 0.6683773719154886
      },
      "age_85": {
        "male": 0.4178186610165112,
        "female": 0.5413714325474598,
        "unisex": 0.4795950467819855
      },
      "age_90": {
        "male": 0.21322446448412186,
        "female": 0.32270297224895916,
        "unisex": 0.26796371836654054
      },
      "age_95": {
        "male": 0.06530838996408361,
        "female": 0.12718164001345225,
        "unisex": 0.09624501498876793
      }
    },
    "expected_annual_payments_from_65_truncated_at_100": {
      "male": 17.968695899330772,
      "female": 20.574723707251458,
      "unisex": 19.271709803291113
    },
    "share_alive_at_100": {
      "male": 0.009172889589416294,
      "female": 0.025570850390230892,
      "unisex": 0.017371869989823593
    },
    "median_age_at_death_from_65": {
      "male": 83,
      "female": 85,
      "unisex": 84
    },
    "cross_check_from_62": {
      "table": {
        "p25": {
          "male": 74,
          "female": 78
        },
        "median": {
          "male": 82,
          "female": 85
        },
        "p75": {
          "male": 88,
          "female": 91
        },
        "reach80": {
          "male": 0.5880721733343193,
          "female": 0.6996012705278097,
          "unisex": 0.6438367219310646
        }
      },
      "published_claiming_study": {
        "url": "https://quantcalc.app/research/ss-claiming-mc-2026/",
        "dataset": "https://quantcalc.app/research/ss-claiming-mc-2026/data/summary.json",
        "mortality": "SSA 2021 Period Life Table",
        "regime": "spend_it",
        "percentiles": {
          "male": {
            "p25": 75,
            "median": 82,
            "p75": 88
          },
          "female": {
            "p25": 78,
            "median": 85,
            "p75": 90
          }
        }
      }
    }
  },
  "quoted_sources": {
    "cme_stress_test": {
      "url": "https://quantcalc.app/research/4-percent-rule-cme-stress-test/",
      "dataset": "https://quantcalc.app/research/4-percent-rule-cme-stress-test/data.json",
      "title": "Does the 4% rule survive forward-looking forecasts? Six CMEs stress-tested"
    },
    "sequence_of_returns": {
      "url": "https://quantcalc.app/research/sequence-of-returns-risk-2026/",
      "dataset": "https://quantcalc.app/research/sequence-of-returns-risk-2026/data/summary.json",
      "title": "Retiring into a bear market: same plan, 46% failure vs 0%"
    },
    "historical_cohorts": {
      "url": "https://quantcalc.app/research/safe-withdrawal-by-retirement-cohort-2026/",
      "dataset": "https://quantcalc.app/research/safe-withdrawal-by-retirement-cohort-2026/data/summary.json",
      "title": "The 4% rule across 123 historical retirement cohorts"
    },
    "social_security_claiming_mc": {
      "url": "https://quantcalc.app/research/ss-claiming-mc-2026/",
      "dataset": "https://quantcalc.app/research/ss-claiming-mc-2026/data/summary.json",
      "title": "Social Security at 70 beats 62 only 42% of the time, if you invest"
    }
  },
  "four_percent_rule_quoted": {
    "plan": {
      "initial_savings": 1000000,
      "annual_withdrawal": 40000,
      "horizon_years": 30,
      "allocation": {
        "us_equity": 0.45,
        "intl_equity": 0.15,
        "bonds": 0.4
      },
      "num_simulations": 10000,
      "sampling": "pseudo-random (Mersenne Twister)"
    },
    "by_forecast_source": [
      {
        "source": "Charles Schwab",
        "key": "schwab",
        "success_rate_30yr": 0.9085,
        "failure_share_30yr": 0.09150000000000003,
        "paths_ran_out": 915,
        "of_paths": 10000,
        "median_terminal": 691352.411254595,
        "p10_terminal": 17750.90702516851,
        "p90_terminal": 1928559.934035635,
        "outlier": false
      },
      {
        "source": "J.P. Morgan",
        "key": "jpmorgan",
        "success_rate_30yr": 0.8556,
        "failure_share_30yr": 0.14439999999999997,
        "paths_ran_out": 1444,
        "of_paths": 10000,
        "median_terminal": 766828.284102446,
        "p10_terminal": 0,
        "p90_terminal": 2708856.0657152077,
        "outlier": false
      },
      {
        "source": "Invesco",
        "key": "invesco",
        "success_rate_30yr": 0.8535,
        "failure_share_30yr": 0.14649999999999996,
        "paths_ran_out": 1465,
        "of_paths": 10000,
        "median_terminal": 498085.82550159364,
        "p10_terminal": 0,
        "p90_terminal": 1564213.9260337828,
        "outlier": false
      },
      {
        "source": "BlackRock",
        "key": "blackrock",
        "success_rate_30yr": 0.8437,
        "failure_share_30yr": 0.1563,
        "paths_ran_out": 1563,
        "of_paths": 10000,
        "median_terminal": 471499.305962846,
        "p10_terminal": 0,
        "p90_terminal": 1513260.4207602516,
        "outlier": false
      },
      {
        "source": "Vanguard",
        "key": "vanguard",
        "success_rate_30yr": 0.7846,
        "failure_share_30yr": 0.21540000000000004,
        "paths_ran_out": 2154,
        "of_paths": 10000,
        "median_terminal": 339503.49489676976,
        "p10_terminal": 0,
        "p90_terminal": 1261060.1294072142,
        "outlier": false
      },
      {
        "source": "GMO",
        "key": "gmo",
        "success_rate_30yr": 0.0246,
        "failure_share_30yr": 0.9754,
        "paths_ran_out": 9754,
        "of_paths": 10000,
        "median_terminal": 0,
        "p10_terminal": 0,
        "p90_terminal": 0,
        "outlier": true
      }
    ],
    "mainstream_range": {
      "failure_share_min": 0.09150000000000003,
      "failure_share_max": 0.21540000000000004,
      "median_terminal_min": 339503.49489676976,
      "median_terminal_max": 766828.284102446
    },
    "sequence_of_returns": {
      "pooled_failure_rate_pct": 6.02,
      "pooled_median_terminal": 1823819,
      "bottom_decile_failure_rate_pct": 45.58,
      "bottom_decile_median_terminal": 49522,
      "top_decile_failure_rate_pct": 0,
      "top_decile_median_terminal": 5355467,
      "n_paths": 50000
    },
    "historical_cohorts": {
      "cohort_count": 123,
      "allocation": "60/40",
      "lasted_30yr_at_4pct_share_pct": 95.9,
      "worst_cohort_year": 1966,
      "worst_cohort_max_swr_pct": 3.69,
      "median_max_swr_pct": 6.15
    }
  }
}