{
 "model": "r = mu + sigma * (T(z) - m) / s, T(z) = (exp(g z) - 1) / g * exp(h z^2 / 2)",
 "references": [
  "Tukey, J. W. (1977). Modern techniques in data analysis. NSF-sponsored regional research conference, Southeastern Massachusetts University.",
  "Hoaglin, D. C. (1985). Summarizing shape numerically: the g-and-h distributions. In Exploring Data Tables, Trends, and Shapes, Wiley.",
  "Martinez, J. and Iglewicz, B. (1984). Some properties of the Tukey g and h family of distributions. Communications in Statistics - Theory and Methods 13(3).",
  "Dutta, K. and Perry, J. (2006). A tale of tails: an empirical analysis of loss distribution models for estimating operational risk capital. Federal Reserve Bank of Boston Working Paper 06-13.",
  "Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance 1(2)."
 ],
 "min_years_to_fit": 90,
 "max_h": 0.24,
 "engine_equity_parameters": {
  "mu_log": 0.088,
  "sigma_total": 0.175
 },
 "fits": {
  "equity": {
   "method": "maximum likelihood",
   "chosen": {
    "g": -0.268,
    "h": 0.0
   },
   "maximum_likelihood": {
    "g": -0.268,
    "h": 0.0
   },
   "quantile_estimate": {
    "g": -0.224,
    "h": 0.007
   },
   "years": [
    1871,
    2024
   ],
   "n": 154,
   "log_return_mean": 0.0881,
   "log_return_volatility": 0.1748,
   "skewness": -0.815,
   "excess_kurtosis": 1.05,
   "moment_interval_95": {
    "skewness": [
     -1.245,
     -0.314
    ],
    "excess_kurtosis": [
     -0.404,
     2.518
    ]
   },
   "candidate_implied_moments": {
    "skewness": -0.84,
    "excess_kurtosis": 1.279
   },
   "severe_candidate": {
    "g": -0.3,
    "h": 0.08
   },
   "log_likelihood": {
    "normal": 50.55,
    "g_and_h": 58.74,
    "severe": 57.02
   },
   "likelihood_ratio_vs_normal": 16.38,
   "aic": {
    "normal": -97.11,
    "g_and_h": -109.49
   }
  },
  "us_equity_small_cap": {
   "method": "not distinguishable from log-normal",
   "chosen": null,
   "maximum_likelihood": {
    "g": -0.083,
    "h": 0.07
   },
   "quantile_estimate": {
    "g": -0.204,
    "h": 0.16
   },
   "years": [
    1926,
    2024
   ],
   "n": 99,
   "log_return_mean": 0.1135,
   "log_return_volatility": 0.3218,
   "skewness": -0.244,
   "excess_kurtosis": 0.381,
   "moment_interval_95": {
    "skewness": [
     -0.702,
     0.237
    ],
    "excess_kurtosis": [
     -0.244,
     1.265
    ]
   },
   "candidate_implied_moments": null,
   "severe_candidate": {
    "g": -0.083,
    "h": 0.07
   },
   "log_likelihood": {
    "normal": -27.73,
    "g_and_h": -26.66,
    "severe": -26.66
   },
   "likelihood_ratio_vs_normal": 2.13,
   "aic": {
    "normal": 59.46,
    "g_and_h": 61.33
   }
  },
  "bonds": {
   "method": "quantile (Hoaglin), likelihood flat up to the kurtosis bound, rejected: implied skewness +4.63 and excess kurtosis +163.6 outside the sample's 95% bootstrap intervals",
   "chosen": null,
   "maximum_likelihood": {
    "g": 0.264,
    "h": 0.24
   },
   "quantile_estimate": {
    "g": 0.423,
    "h": 0.157
   },
   "years": [
    1926,
    2024
   ],
   "n": 99,
   "log_return_mean": 0.0445,
   "log_return_volatility": 0.0739,
   "skewness": 0.327,
   "excess_kurtosis": 1.443,
   "moment_interval_95": {
    "skewness": [
     -0.437,
     1.107
    ],
    "excess_kurtosis": [
     0.071,
     2.959
    ]
   },
   "candidate_implied_moments": {
    "skewness": 4.626,
    "excess_kurtosis": 163.561
   },
   "severe_candidate": {
    "g": 0.264,
    "h": 0.24
   },
   "log_likelihood": {
    "normal": 117.99,
    "g_and_h": 124.53,
    "severe": 124.53
   },
   "likelihood_ratio_vs_normal": 13.08,
   "aic": {
    "normal": -231.98,
    "g_and_h": -241.07
   }
  },
  "long_treasuries": {
   "method": "quantile estimate beyond the finite-kurtosis bound",
   "chosen": null,
   "maximum_likelihood": {
    "g": 0.254,
    "h": 0.24
   },
   "quantile_estimate": {
    "g": 0.428,
    "h": 0.273
   },
   "years": [
    1928,
    2024
   ],
   "n": 97,
   "log_return_mean": 0.0445,
   "log_return_volatility": 0.0967,
   "skewness": 0.065,
   "excess_kurtosis": 3.156,
   "moment_interval_95": {
    "skewness": [
     -1.114,
     1.365
    ],
    "excess_kurtosis": [
     0.171,
     5.568
    ]
   },
   "candidate_implied_moments": null,
   "severe_candidate": {
    "g": 0.254,
    "h": 0.24
   },
   "log_likelihood": {
    "normal": 89.47,
    "g_and_h": 101.05,
    "severe": 101.05
   },
   "likelihood_ratio_vs_normal": 23.15,
   "aic": {
    "normal": -174.95,
    "g_and_h": -194.1
   }
  }
 },
 "shapes": {
  "equity": {
   "source": "fitted 1871-2024 (maximum likelihood)",
   "historical": {
    "g": -0.268,
    "h": 0.0
   },
   "severe": {
    "g": -0.3,
    "h": 0.08
   }
  },
  "bonds": {
   "source": "log-normal (quantile (Hoaglin), likelihood flat up to the kurtosis bound, rejected: implied skewness +4.63 and excess kurtosis +163.6 outside the sample's 95% bootstrap intervals)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "us_equity_small_cap": {
   "source": "borrowed from equity (not distinguishable from log-normal)",
   "historical": {
    "g": -0.268,
    "h": 0.0
   },
   "severe": {
    "g": -0.3,
    "h": 0.08
   }
  },
  "long_treasuries": {
   "source": "borrowed from bonds (quantile estimate beyond the finite-kurtosis bound)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "intl_equity_developed": {
   "source": "borrowed from equity",
   "historical": {
    "g": -0.268,
    "h": 0.0
   },
   "severe": {
    "g": -0.3,
    "h": 0.08
   }
  },
  "emerging_markets": {
   "source": "borrowed from equity",
   "historical": {
    "g": -0.268,
    "h": 0.0
   },
   "severe": {
    "g": -0.3,
    "h": 0.08
   }
  },
  "reits": {
   "source": "borrowed from equity",
   "historical": {
    "g": -0.268,
    "h": 0.0
   },
   "severe": {
    "g": -0.3,
    "h": 0.08
   }
  },
  "tips": {
   "source": "borrowed from bonds",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "intl_bonds": {
   "source": "borrowed from bonds",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "cash": {
   "source": "log-normal (no long-history relative)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "gold": {
   "source": "log-normal (no long-history relative)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "commodities": {
   "source": "log-normal (no long-history relative)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  },
  "managed_futures": {
   "source": "log-normal (no long-history relative)",
   "historical": {
    "g": 0.0,
    "h": 0.0
   },
   "severe": {
    "g": 0.0,
    "h": 0.0
   }
  }
 },
 "us_stock_tail_frequencies": {
  "history": {
   "-10%": 0.1234,
   "-20%": 0.0455,
   "-30%": 0.0195,
   "-40%": 0.0065
  },
  "history_years": {
   "-10%": 19,
   "-20%": 7,
   "-30%": 3,
   "-40%": 1
  },
  "none": {
   "-10%": 0.1346,
   "-20%": 0.0377,
   "-30%": 0.0055,
   "-40%": 0.0003
  },
  "historical": {
   "-10%": 0.1319,
   "-20%": 0.0537,
   "-30%": 0.0179,
   "-40%": 0.0047
  },
  "severe": {
   "-10%": 0.1137,
   "-20%": 0.0497,
   "-30%": 0.0207,
   "-40%": 0.0083
  },
  "previous_stretch_2_0": {
   "-10%": 0.137,
   "-20%": 0.0763,
   "-30%": 0.035,
   "-40%": 0.0121
  }
 },
 "market_turbulence": {
  "window_years": 30,
  "full_record_stock_volatility": 0.1748,
  "presets": {
   "calmer": {
    "era": [
     1871,
     1900
    ],
    "stock_volatility": 0.131,
    "multiplier": 0.749,
    "us_stock_tail_frequencies": {
     "-10%": 0.0813,
     "-20%": 0.0229,
     "-30%": 0.005,
     "-40%": 0.0008
    }
   },
   "historical": {
    "era": [
     1871,
     2024
    ],
    "stock_volatility": 0.1748,
    "multiplier": 1.0,
    "us_stock_tail_frequencies": {
     "-10%": 0.1319,
     "-20%": 0.0537,
     "-30%": 0.0179,
     "-40%": 0.0047
    }
   },
   "rougher": {
    "era": [
     1927,
     1956
    ],
    "stock_volatility": 0.2455,
    "multiplier": 1.404,
    "us_stock_tail_frequencies": {
     "-10%": 0.1949,
     "-20%": 0.1072,
     "-30%": 0.0514,
     "-40%": 0.0208
    }
   }
  }
 }
}