{
  "checks": {
    "all_fixed_portfolios_fully_invested": true,
    "coexceedance_probability_bounds_0.95": true,
    "coexceedance_probability_bounds_0.99": true,
    "coexceedance_probability_bounds_0.999": true,
    "coexceedance_quadrature_error_0.95": true,
    "coexceedance_quadrature_error_0.99": true,
    "coexceedance_quadrature_error_0.999": true,
    "discrete_atom_var_zero_es_four": true,
    "empirical_es_cash_translation": true,
    "empirical_es_fractional_tail_mass": true,
    "empirical_es_positive_homogeneity": true,
    "large_es_within_six_sampling_errors_equal_weight": true,
    "large_es_within_six_sampling_errors_linear_tilt": true,
    "large_es_within_six_sampling_errors_lower_common_exposure_1000_equal_weight": true,
    "large_scenarios_finite": true,
    "lp_beats_feasible_training_comparators": true,
    "lp_complementarity": true,
    "lp_dual_signs": true,
    "lp_dual_stationarity": true,
    "lp_duality_gap": true,
    "lp_objective_matches_discrete_es": true,
    "lp_primal_feasible": true,
    "mixture_es_within_six_sampling_errors_equal_weight": true,
    "mixture_es_within_six_sampling_errors_minimum_sample_es": true,
    "mixture_es_within_six_sampling_errors_minimum_variance": true,
    "mixture_quantile_root_equal_weight": true,
    "mixture_quantile_root_minimum_sample_es": true,
    "mixture_quantile_root_minimum_variance": true,
    "normal_var_tail_probability": true,
    "population_minimum_variance_satisfies_caps": true,
    "projected_covariance_matches_dense_128": true,
    "projected_covariance_matches_operator": true,
    "scaled_student_variance_one": true,
    "student_es_matches_density_integral": true,
    "student_var_tail_probability": true
  },
  "classification": "synthetic_teaching_example",
  "dependence": {
    "finite_quantile_coexceedance": [
      {
        "gaussian": 0.24378857534349838,
        "max_quadrature_error": 3.00445217289632e-11,
        "quantile_level": 0.95,
        "student_t": 0.3212536865603763
      },
      {
        "gaussian": 0.12939244182646556,
        "max_quadrature_error": 6.787517471771513e-11,
        "quantile_level": 0.99,
        "student_t": 0.25943295659721805
      },
      {
        "gaussian": 0.05425916819456713,
        "max_quadrature_error": 2.840748678041088e-10,
        "quantile_level": 0.999,
        "student_t": 0.22617258137721713
      }
    ],
    "gaussian_tail_coefficient": 0.0,
    "student_tail_coefficient": 0.20703125
  },
  "discrete_example": {
    "alpha": 0.975,
    "es": 3.999999999999997,
    "losses": [
      0,
      10
    ],
    "probabilities": [
      0.99,
      0.01
    ],
    "var": 0.0
  },
  "inputs": {
    "alpha": 0.99,
    "loss_units": "arbitrary additive shocks; one unspecified horizon; no annualisation",
    "pairwise_correlation": 0.5,
    "student_degrees_of_freedom": 5
  },
  "large_universe": {
    "assets": 10000,
    "batch_size": 2000,
    "factors": 50,
    "fixed_portfolios": 3,
    "projected_covariance": [
      [
        0.0272302291484607,
        0.007180629129458101,
        0.02970572369700643
      ],
      [
        0.007180629129458101,
        0.001943514093707128,
        0.007831450498587939
      ],
      [
        0.02970572369700643,
        0.007831450498587939,
        0.03240649504755275
      ]
    ],
    "results": [
      {
        "analytic_es": 0.569112712721655,
        "es_sampling_error_diagnostic": 0.007154018970520728,
        "model_standard_deviation": 0.1650158451436125,
        "portfolio": "equal_weight",
        "sample_es": 0.5703434287730365,
        "sample_var": 0.43488103358835706
      },
      {
        "analytic_es": 0.1520430195530379,
        "es_sampling_error_diagnostic": 0.0019229253377824908,
        "model_standard_deviation": 0.044085304736466636,
        "portfolio": "lower_common_exposure_1000_equal_weight",
        "sample_es": 0.15203506789789406,
        "sample_var": 0.1157794409811713
      },
      {
        "analytic_es": 0.6208528369093135,
        "es_sampling_error_diagnostic": 0.007806778190376224,
        "model_standard_deviation": 0.18001804089466353,
        "portfolio": "linear_tilt",
        "sample_es": 0.6221908660032492,
        "sample_var": 0.47462784752483816
      }
    ],
    "scenarios": 100000,
    "seeds": {
      "common_scale": 20260913,
      "constructed_covariance": 20260911,
      "normal_draws": 20260912
    },
    "storage_bytes": {
      "factor_and_diagonal": 4080000,
      "fixed_weights": 240000,
      "hypothetical_dense_asset_scenarios": 8000000000,
      "one_portfolio_loss_block": 48000,
      "retained_portfolio_losses": 2400000
    }
  },
  "mixture": {
    "centred_event_loadings": [
      4.0,
      0.0,
      1.5
    ],
    "common_normal_loading": 0.2,
    "evaluation_scenarios": 100000,
    "evaluation_seed": 20260909,
    "event_probability": 0.04,
    "optimization": {
      "absolute_duality_gap": 4.440892098500626e-16,
      "complementarity_max_residual": 0.0,
      "dual_sign_violation": -0.0,
      "objective": 1.6829235629375947,
      "primal_max_violation": 4.0314973581701e-15,
      "scenario_constraints": 20000,
      "scenario_matrix_nonzeros": 100000,
      "stationarity_max_residual": 1.9984014443252818e-15,
      "status": 0,
      "variables": 20004
    },
    "population_covariance": [
      [
        0.7768999999999999,
        0.04000000000000001,
        0.2704
      ],
      [
        0.04000000000000001,
        0.9425,
        0.04000000000000001
      ],
      [
        0.2704,
        0.04000000000000001,
        0.5489
      ]
    ],
    "portfolios": [
      {
        "es_sampling_error_diagnostic": 0.011118388332456198,
        "evaluation_es": 2.338056672088381,
        "evaluation_var": 2.0596411421953493,
        "population": {
          "cdf_at_var_error": 0.0,
          "es": 2.329646949115323,
          "standard_deviation": 0.5743692192309752,
          "var": 2.0623006323198494
        },
        "portfolio": "equal_weight",
        "training_es": 2.333877513437333,
        "weights": [
          0.3333333333333333,
          0.3333333333333333,
          0.3333333333333333
        ]
      },
      {
        "es_sampling_error_diagnostic": 0.011608121264831416,
        "evaluation_es": 2.179453121006641,
        "evaluation_var": 1.8992627401038749,
        "population": {
          "cdf_at_var_error": 1.1102230246251565e-16,
          "es": 2.1698974625898533,
          "standard_deviation": 0.5674227070212512,
          "var": 1.8913530468303181
        },
        "portfolio": "minimum_variance",
        "training_es": 2.1710425458778957,
        "weights": [
          0.24393389181295277,
          0.31243050243027676,
          0.4436356057567704
        ]
      },
      {
        "es_sampling_error_diagnostic": 0.010463363359274197,
        "evaluation_es": 1.7285966328255458,
        "evaluation_var": 1.491673373582031,
        "population": {
          "cdf_at_var_error": 0.0,
          "es": 1.7194488640684775,
          "standard_deviation": 0.6148142345647727,
          "var": 1.480623954739633
        },
        "portfolio": "minimum_sample_es",
        "training_es": 1.6829235629375954,
        "weights": [
          0.0,
          0.45500171816944107,
          0.544998281830559
        ]
      }
    ],
    "residual_standard_deviations": [
      0.35,
      0.95,
      0.65
    ],
    "training_scenarios": 20000,
    "training_seed": 20260908,
    "weight_cap": 0.8,
    "training_tail_mass_scenarios": 200,
    "evaluation_tail_mass_scenarios": 1000,
    "weight_cap_nonbinding": true
  },
  "paper": "PSIM-RN-05",
  "runtime": {
    "numpy": "2.3.5",
    "python": "3.13.9",
    "scipy": "1.16.3"
  },
  "scope": "Public methods, invented constants and seeded draws. No market data, proprietary calibration, signals, holdings or internal performance.",
  "student_es_increase_percent": 29.4018594723082,
  "variance_one_models": {
    "normal": {
      "es": 2.665214220345806,
      "var": 2.3263478740408408
    },
    "student_t": {
      "es": 3.448836760047854,
      "var": 2.606463569384323
    }
  }
}