Equal Risk Contribution: Let the Covariance Matrix Change the Allocation

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/**
 * Pure D14-F02-A02 equal-risk-contribution allocator.
 *
 * This freezes the cyclical coordinate-descent volatility update from
 * Griveau-Billion, Richard, and Roncalli (arXiv:1311.4057, Eq. 7), with equal
 * positive budgets b_i = 1/N. It accepts a supplied covariance matrix only;
 * return estimation, provenance, and market-data repair are out of scope.
 */

export const WEIGHT_SUM_ABSOLUTE_TOLERANCE = 1e-12;
export const RISK_SHARE_TOLERANCE = 1e-10;
export const MAX_SWEEPS = 10_000;

export type EqualRiskContributionErrorCode =
  | "invalid-input"
  | "invalid-covariance"
  | "numerical-range-invalid"
  | "convergence-failure";

export interface EqualRiskContributionResult {
  assetIds: string[];
  weights: number[];
  componentRiskContributions: number[];
  componentRiskShares: number[];
  portfolioVariance: number;
  portfolioVolatility: number;
  sumWeights: number;
  riskShareResidual: number;
  iterations: number;
  method: "equal-risk-contribution";
  solver: "cyclical-coordinate-descent-volatility-eq7";
  covarianceScale: number;
  status: "ok";
}

export class EqualRiskContributionValidationError extends Error {
  readonly code: EqualRiskContributionErrorCode;

  constructor(code: EqualRiskContributionErrorCode, message: string) {
    super(message);
    this.name = "EqualRiskContributionValidationError";
    this.code = code;
  }
}

function invalid(code: EqualRiskContributionErrorCode, message: string): never {
  throw new EqualRiskContributionValidationError(code, message);
}

function neumaierSum(values: readonly number[]): number {
  let total = 0;
  let compensation = 0;
  for (const value of values) {
    const candidate = total + value;
    if (Math.abs(total) >= Math.abs(value)) {
      compensation += (total - candidate) + value;
    } else {
      compensation += (value - candidate) + total;
    }
    total = candidate;
  }
  return total + compensation;
}

function dot(row: readonly number[], vector: readonly number[]): number {
  const products: number[] = [];
  for (let index = 0; index < vector.length; index += 1) {
    products.push(row[index] * vector[index]);
  }
  return neumaierSum(products);
}

function validateIds(assetIds: readonly unknown[]): string[] {
  if (!Array.isArray(assetIds)) {
    invalid("invalid-input", "assetIds must be an ordered array.");
  }
  if (assetIds.length === 0) {
    invalid("invalid-input", "the asset universe must not be empty.");
  }
  const ids: string[] = [];
  for (let index = 0; index < assetIds.length; index += 1) {
    const value = assetIds[index];
    if (typeof value !== "string" || value.trim() === "") {
      invalid("invalid-input", `assetIds[${index}] must be a non-empty string.`);
    }
    ids.push(value);
  }
  if (new Set(ids).size !== ids.length) {
    invalid("invalid-input", "assetIds must be unique.");
  }
  return ids;
}

function validateCovariance(
  covariance: readonly unknown[],
  n: number,
): { matrix: number[][]; scale: number; normalized: number[][] } {
  if (!Array.isArray(covariance)) {
    invalid("invalid-input", "covariance must be a square numeric matrix.");
  }
  if (covariance.length !== n) {
    invalid("invalid-covariance", "covariance must have one row per asset.");
  }
  const matrix: number[][] = [];
  for (let rowIndex = 0; rowIndex < covariance.length; rowIndex += 1) {
    const rawRow = covariance[rowIndex];
    if (!Array.isArray(rawRow)) {
      invalid("invalid-covariance", `covariance[${rowIndex}] must be a numeric row.`);
    }
    if (rawRow.length !== n) {
      invalid("invalid-covariance", "covariance must be square.");
    }
    const row: number[] = [];
    for (let columnIndex = 0; columnIndex < rawRow.length; columnIndex += 1) {
      const value = rawRow[columnIndex];
      if (typeof value !== "number") {
        invalid(
          "invalid-covariance",
          `covariance[${rowIndex}][${columnIndex}] must be numeric data.`,
        );
      }
      if (!Number.isFinite(value)) {
        invalid(
          "numerical-range-invalid",
          `covariance[${rowIndex}][${columnIndex}] is outside the finite numeric range.`,
        );
      }
      row.push(value);
    }
    matrix.push(row);
  }

  for (let i = 0; i < n; i += 1) {
    for (let j = i + 1; j < n; j += 1) {
      if (matrix[i][j] !== matrix[j][i]) {
        invalid(
          "invalid-covariance",
          "covariance must be exactly symmetric; no silent symmetrization is performed.",
        );
      }
    }
    if (!(matrix[i][i] > 0)) {
      invalid("invalid-covariance", "ERC requires a strictly positive covariance diagonal.");
    }
  }

  let scale = 0;
  for (const row of matrix) {
    for (const value of row) {
      if (Math.abs(value) > scale) scale = Math.abs(value);
    }
  }
  if (!Number.isFinite(scale) || scale <= 0) {
    invalid("numerical-range-invalid", "covariance scale is zero or non-finite.");
  }

  const normalized = matrix.map((row) => row.map((value) => value / scale));
  const lower = Array.from({ length: n }, () => Array<number>(n).fill(0));
  for (let i = 0; i < n; i += 1) {
    for (let j = 0; j <= i; j += 1) {
      let remainder = normalized[i][j];
      if (j > 0) {
        const terms: number[] = [];
        for (let k = 0; k < j; k += 1) terms.push(lower[i][k] * lower[j][k]);
        remainder -= neumaierSum(terms);
      }
      if (i === j) {
        if (!Number.isFinite(remainder) || remainder <= 0) {
          invalid("invalid-covariance", "covariance must be strictly positive definite for ERC.");
        }
        lower[i][j] = Math.sqrt(remainder);
      } else {
        const denominator = lower[j][j];
        if (!(denominator > 0)) {
          invalid("invalid-covariance", "covariance Cholesky factor is degenerate.");
        }
        lower[i][j] = remainder / denominator;
      }
    }
  }
  return { matrix, scale, normalized };
}

function riskState(
  covariance: readonly number[][],
  y: readonly number[],
): { product: number[]; quadratic: number; volatility: number } {
  const product = covariance.map((row) => dot(row, y));
  const quadratic = dot(y, product);
  if (!Number.isFinite(quadratic) || quadratic <= 0) {
    invalid(
      "numerical-range-invalid",
      "positive-definite covariance produced a non-positive or non-finite quadratic form.",
    );
  }
  const volatility = Math.sqrt(quadratic);
  if (!Number.isFinite(volatility) || volatility <= 0) {
    invalid("numerical-range-invalid", "portfolio volatility is outside the finite range.");
  }
  return { product, quadratic, volatility };
}

/**
 * Solve equal positive volatility-risk contributions using the frozen CCD
 * positive-root update. The covariance scale is normalized internally.
 */
export function equalRiskContributionWeights(
  assetIds: readonly unknown[],
  covariance: readonly unknown[],
): EqualRiskContributionResult {
  const ids = validateIds(assetIds);
  const validated = validateCovariance(covariance, ids.length);
  const n = ids.length;
  const budget = 1 / n;
  const y = Array<number>(n).fill(1);
  let residual = Number.POSITIVE_INFINITY;
  let iterations = 0;

  for (let sweep = 1; sweep <= MAX_SWEEPS; sweep += 1) {
    for (let i = 0; i < n; i += 1) {
      const row = validated.normalized[i];
      const diagonal = row[i];
      const rowProduct = dot(row, y);
      const other = rowProduct - diagonal * y[i];
      const state = riskState(validated.normalized, y);
      const discriminant = other * other + 4 * diagonal * budget * state.volatility;
      if (!Number.isFinite(discriminant) || discriminant < 0) {
        invalid("numerical-range-invalid", "ERC coordinate discriminant is invalid.");
      }
      const root = Math.sqrt(discriminant);
      const updated = other > 0
        ? (2 * budget * state.volatility) / (root + other)
        : (-other + root) / (2 * diagonal);
      if (!Number.isFinite(updated) || updated <= 0) {
        invalid(
          "numerical-range-invalid",
          "ERC coordinate update left the strictly positive numerical domain.",
        );
      }
      y[i] = updated;
    }

    const state = riskState(validated.normalized, y);
    const shares = y.map((value, index) => (value * state.product[index]) / state.quadratic);
    residual = 0;
    for (const share of shares) {
      const error = Math.abs(share - budget);
      if (error > residual) residual = error;
    }
    iterations = sweep;
    if (!Number.isFinite(residual)) {
      invalid("numerical-range-invalid", "ERC risk-share residual is non-finite.");
    }
    if (residual <= RISK_SHARE_TOLERANCE) break;
  }
  if (residual > RISK_SHARE_TOLERANCE) {
    invalid(
      "convergence-failure",
      `ERC cyclical coordinate descent did not reach residual tolerance after ${MAX_SWEEPS} sweeps.`,
    );
  }

  const ySum = neumaierSum(y);
  if (!Number.isFinite(ySum) || ySum <= 0) {
    invalid("numerical-range-invalid", "ERC solution normalization sum is invalid.");
  }
  const weights = y.map((value) => value / ySum);
  if (weights.some((value) => !Number.isFinite(value) || value <= 0)) {
    invalid("numerical-range-invalid", "ERC weights must remain finite and positive.");
  }
  const sumWeights = neumaierSum(weights);
  if (!Number.isFinite(sumWeights) || Math.abs(sumWeights - 1) > WEIGHT_SUM_ABSOLUTE_TOLERANCE) {
    invalid("numerical-range-invalid", "ERC weights do not satisfy the sum tolerance.");
  }

  // The CCD state is homogeneous: y and c*y have identical risk shares but
  // different absolute variance. Recompute absolute risk from the returned
  // fully-invested weights rather than accidentally reporting the internal y
  // scale as the portfolio's supplied-covariance risk.
  const normalizedWeightProduct = validated.normalized.map((row) => dot(row, weights));
  const normalizedWeightQuadratic = dot(weights, normalizedWeightProduct);
  const normalizedWeightVolatility = Math.sqrt(normalizedWeightQuadratic);
  if (!Number.isFinite(normalizedWeightQuadratic) || normalizedWeightQuadratic <= 0
    || !Number.isFinite(normalizedWeightVolatility) || normalizedWeightVolatility <= 0) {
    invalid("numerical-range-invalid", "normalized ERC portfolio volatility is invalid.");
  }
  const shares = weights.map(
    (weight, index) => (weight * normalizedWeightProduct[index]) / normalizedWeightQuadratic,
  );
  const scaleRoot = Math.sqrt(validated.scale);
  const portfolioVolatility = normalizedWeightVolatility * scaleRoot;
  const portfolioVariance = normalizedWeightQuadratic * validated.scale;
  if (
    !Number.isFinite(scaleRoot)
    || !Number.isFinite(portfolioVolatility)
    || portfolioVolatility <= 0
    || !Number.isFinite(portfolioVariance)
    || portfolioVariance <= 0
  ) {
    invalid("numerical-range-invalid", "restoring the covariance risk scale lost positive risk.");
  }
  const normalizedComponentContributions = weights.map(
    (weight, index) => (weight * normalizedWeightProduct[index]) / normalizedWeightVolatility,
  );
  const componentRiskContributions = normalizedComponentContributions.map((value) => value * scaleRoot);
  if (componentRiskContributions.some((value) => !Number.isFinite(value))) {
    invalid("numerical-range-invalid", "component risk contribution is non-finite.");
  }

  return {
    assetIds: ids,
    weights,
    componentRiskContributions,
    componentRiskShares: shares,
    portfolioVariance,
    portfolioVolatility,
    sumWeights,
    riskShareResidual: residual,
    iterations,
    method: "equal-risk-contribution",
    solver: "cyclical-coordinate-descent-volatility-eq7",
    covarianceScale: validated.scale,
    status: "ok",
  };
}