Price-to-Book Valuation

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/** Reference calculations for D18-F03 Relative Valuation. */

type Row = Record<string, unknown>;

function numberValue(name: string, value: unknown, options: {positive?: boolean; nonnegative?: boolean} = {}): number {
  if (typeof value !== "number" || !Number.isFinite(value)) throw new TypeError(`${name} must be a finite number`);
  if (options.positive && value <= 0) throw new RangeError(`${name} must be greater than zero`);
  if (options.nonnegative && value < 0) throw new RangeError(`${name} must be nonnegative`);
  return value;
}

function record(name: string, value: unknown): Row {
  if (!value || typeof value !== "object" || Array.isArray(value)) throw new TypeError(`${name} must be an object`);
  return value as Row;
}

function rows(name: string, value: unknown, minimum = 3): Row[] {
  if (!Array.isArray(value) || value.length < minimum) throw new RangeError(`${name} must contain at least ${minimum} rows`);
  const seen = new Set<string>();
  return value.map((raw, index) => {
    const item = record(`${name}[${index}]`, raw);
    if (typeof item.id !== "string" || !item.id.trim()) throw new RangeError(`${name}[${index}].id must be a nonempty string`);
    if (seen.has(item.id)) throw new RangeError(`${name} ids must be unique`);
    seen.add(item.id);
    return item;
  });
}

function median(values: number[]): number {
  const ordered = [...values].sort((a, b) => a - b);
  const middle = Math.floor(ordered.length / 2);
  return ordered.length % 2 ? ordered[middle] : (ordered[middle - 1] + ordered[middle]) / 2;
}

function summary(values: number[]): Record<string, number> {
  const ordered = [...values].sort((a, b) => a - b);
  return {minimum: ordered[0], median: median(ordered), maximum: ordered.at(-1)!, range: ordered.at(-1)! - ordered[0]};
}

function premiumDiscount(implied: number | null, current: unknown, label = "target.current_price"): number | null {
  if (implied === null || current === undefined || current === null) return null;
  return implied / numberValue(label, current, {positive: true}) - 1;
}

function peComparable(inputs: Row): Row {
  const target = record("target", inputs.target);
  const targetEps = numberValue("target.earnings_per_share", target.earnings_per_share, {positive: true});
  const peerMultiples = rows("peers", inputs.peers).map((peer, index) => {
    const price = numberValue(`peers[${index}].price`, peer.price, {positive: true});
    const eps = numberValue(`peers[${index}].earnings_per_share`, peer.earnings_per_share, {positive: true});
    return {id: peer.id, price, earnings_per_share: eps, pe_ratio: price / eps};
  });
  const multipleSummary = summary(peerMultiples.map(row => row.pe_ratio));
  const selected = multipleSummary.median;
  const implied = selected * targetEps;
  return {model: "median-positive-eps-peer-pe", peer_count: peerMultiples.length, peer_multiples: peerMultiples,
    multiple_summary: multipleSummary, selected_multiple: selected, target_earnings_per_share: targetEps,
    implied_price: implied, premium_discount_to_current: premiumDiscount(implied, target.current_price), state: "valuation-complete"};
}

function evEbitda(inputs: Row): Row {
  const target = record("target", inputs.target);
  const ebitda = numberValue("target.ebitda", target.ebitda, {positive: true});
  const shares = numberValue("target.shares_outstanding", target.shares_outstanding, {positive: true});
  const netDebt = numberValue("target.net_debt", target.net_debt ?? 0);
  const preferred = numberValue("target.preferred_equity", target.preferred_equity ?? 0, {nonnegative: true});
  const nci = numberValue("target.noncontrolling_interest", target.noncontrolling_interest ?? 0, {nonnegative: true});
  const nonoperating = numberValue("target.nonoperating_assets", target.nonoperating_assets ?? 0, {nonnegative: true});
  const peerMultiples = rows("peers", inputs.peers).map((peer, index) => {
    const enterpriseValue = numberValue(`peers[${index}].enterprise_value`, peer.enterprise_value, {positive: true});
    const peerEbitda = numberValue(`peers[${index}].ebitda`, peer.ebitda, {positive: true});
    return {id: peer.id, enterprise_value: enterpriseValue, ebitda: peerEbitda, ev_ebitda: enterpriseValue / peerEbitda};
  });
  const multipleSummary = summary(peerMultiples.map(row => row.ev_ebitda));
  const selected = multipleSummary.median;
  const impliedEnterprise = selected * ebitda;
  const impliedEquity = impliedEnterprise - netDebt - preferred - nci + nonoperating;
  const impliedPrice = impliedEquity > 0 ? impliedEquity / shares : null;
  return {model: "median-positive-ebitda-peer-ev-ebitda", peer_count: peerMultiples.length, peer_multiples: peerMultiples,
    multiple_summary: multipleSummary, selected_multiple: selected, target_ebitda: ebitda,
    implied_enterprise_value: impliedEnterprise, equity_bridge: {net_debt: netDebt, preferred_equity: preferred,
      noncontrolling_interest: nci, nonoperating_assets: nonoperating}, implied_equity_value: impliedEquity,
    shares_outstanding: shares, implied_price: impliedPrice,
    premium_discount_to_current: premiumDiscount(impliedPrice, target.current_price),
    state: impliedPrice === null ? "nonpositive-equity-bridge" : "valuation-complete"};
}

function priceToBook(inputs: Row): Row {
  const target = record("target", inputs.target);
  const targetBvps = numberValue("target.book_value_per_share", target.book_value_per_share, {positive: true});
  const peerMultiples = rows("peers", inputs.peers).map((peer, index) => {
    const price = numberValue(`peers[${index}].price`, peer.price, {positive: true});
    const bvps = numberValue(`peers[${index}].book_value_per_share`, peer.book_value_per_share, {positive: true});
    return {id: peer.id, price, book_value_per_share: bvps, price_to_book: price / bvps};
  });
  const multipleSummary = summary(peerMultiples.map(row => row.price_to_book));
  const selected = multipleSummary.median;
  const implied = selected * targetBvps;
  return {model: "median-positive-book-value-peer-pb", peer_count: peerMultiples.length, peer_multiples: peerMultiples,
    multiple_summary: multipleSummary, selected_multiple: selected, target_book_value_per_share: targetBvps,
    implied_price: implied, premium_discount_to_current: premiumDiscount(implied, target.current_price), state: "valuation-complete"};
}

function peg(inputs: Row): Row {
  const target = record("target", inputs.target);
  const forwardPe = numberValue("target.forward_pe", target.forward_pe, {positive: true});
  const growth = numberValue("target.expected_eps_growth_percent", target.expected_eps_growth_percent, {positive: true});
  const peerRatios = rows("peers", inputs.peers).map((peer, index) => {
    const peerPe = numberValue(`peers[${index}].forward_pe`, peer.forward_pe, {positive: true});
    const peerGrowth = numberValue(`peers[${index}].expected_eps_growth_percent`, peer.expected_eps_growth_percent, {positive: true});
    return {id: peer.id, forward_pe: peerPe, expected_eps_growth_percent: peerGrowth, peg_ratio: peerPe / peerGrowth};
  });
  const pegSummary = summary(peerRatios.map(row => row.peg_ratio));
  const targetPeg = forwardPe / growth;
  const impliedPe = pegSummary.median * growth;
  const forwardEps = target.forward_earnings_per_share === undefined || target.forward_earnings_per_share === null
    ? null : numberValue("target.forward_earnings_per_share", target.forward_earnings_per_share, {positive: true});
  const impliedPrice = forwardEps === null ? null : impliedPe * forwardEps;
  return {model: "forward-pe-divided-by-growth-percentage-points", growth_unit: "percentage-points",
    peer_count: peerRatios.length, peer_ratios: peerRatios, peg_summary: pegSummary, target_forward_pe: forwardPe,
    target_growth_percent: growth, target_peg_ratio: targetPeg, selected_peer_peg: pegSummary.median,
    relative_peg: targetPeg / pegSummary.median, implied_forward_pe: impliedPe,
    forward_earnings_per_share: forwardEps, implied_price: impliedPrice,
    premium_discount_to_current: premiumDiscount(impliedPrice, target.current_price),
    state: impliedPrice === null ? "ratio-only" : "valuation-complete"};
}

function solve(matrix: number[][], vector: number[]): number[] {
  const n = vector.length;
  const augmented = matrix.map((row, index) => [...row, vector[index]]);
  for (let column = 0; column < n; column += 1) {
    let pivot = column;
    for (let row = column + 1; row < n; row += 1) if (Math.abs(augmented[row][column]) > Math.abs(augmented[pivot][column])) pivot = row;
    if (Math.abs(augmented[pivot][column]) <= 1e-12) throw new RangeError("regression design matrix is rank deficient");
    [augmented[column], augmented[pivot]] = [augmented[pivot], augmented[column]];
    const scale = augmented[column][column];
    augmented[column] = augmented[column].map(value => value / scale);
    for (let row = 0; row < n; row += 1) {
      if (row === column) continue;
      const factor = augmented[row][column];
      augmented[row] = augmented[row].map((value, index) => value - factor * augmented[column][index]);
    }
  }
  return augmented.map(row => row.at(-1)!);
}

function features(name: string, row: Row): number[] {
  return [1, numberValue(`${name}.expected_growth_percent`, row.expected_growth_percent) / 100,
    numberValue(`${name}.return_on_equity_percent`, row.return_on_equity_percent) / 100,
    numberValue(`${name}.net_debt_to_ebitda`, row.net_debt_to_ebitda)];
}

function regression(inputs: Row): Row {
  const peers = rows("peers", inputs.peers, 6);
  const target = record("target", inputs.target);
  const xRows: number[][] = [];
  const y: number[] = [];
  const multiples: number[] = [];
  peers.forEach((peer, index) => {
    const multiple = numberValue(`peers[${index}].multiple`, peer.multiple, {positive: true});
    xRows.push(features(`peers[${index}]`, peer)); y.push(Math.log(multiple)); multiples.push(multiple);
  });
  const width = xRows[0].length;
  const xtx = Array.from({length: width}, (_, i) => Array.from({length: width}, (_, j) => xRows.reduce((sum, row) => sum + row[i] * row[j], 0)));
  const xty = Array.from({length: width}, (_, i) => xRows.reduce((sum, row, index) => sum + row[i] * y[index], 0));
  const beta = solve(xtx, xty);
  const fittedLogs = xRows.map(row => row.reduce((sum, value, index) => sum + value * beta[index], 0));
  const mean = y.reduce((a, b) => a + b, 0) / y.length;
  const sse = y.reduce((sum, value, index) => sum + (value - fittedLogs[index]) ** 2, 0);
  const sst = y.reduce((sum, value) => sum + (value - mean) ** 2, 0);
  const rSquared = sst === 0 ? (sse === 0 ? 1 : 0) : 1 - sse / sst;
  const fittedPeers = peers.map((peer, index) => ({id: peer.id, observed_multiple: multiples[index],
    fitted_multiple: Math.exp(fittedLogs[index]), log_residual: y[index] - fittedLogs[index]}));
  const targetX = features("target", target);
  const predictedLog = targetX.reduce((sum, value, index) => sum + value * beta[index], 0);
  const impliedMultiple = Math.exp(predictedLog);
  const metric = numberValue("target.valuation_metric", target.valuation_metric, {positive: true});
  const impliedValue = impliedMultiple * metric;
  return {model: "ols-log-multiple-on-growth-roe-and-leverage", peer_count: peers.length,
    feature_order: ["intercept", "expected_growth_decimal", "return_on_equity_decimal", "net_debt_to_ebitda"],
    coefficients: {intercept: beta[0], expected_growth_decimal: beta[1], return_on_equity_decimal: beta[2], net_debt_to_ebitda: beta[3]},
    r_squared_log_space: rSquared, fitted_peers: fittedPeers, target_predicted_log_multiple: predictedLog,
    implied_multiple: impliedMultiple, target_valuation_metric: metric, implied_value: impliedValue,
    premium_discount_to_current: premiumDiscount(impliedValue, target.current_value, "target.current_value"),
    smearing_correction: "not-applied", state: "valuation-complete"};
}

export function calculate(topicId: string, rawInputs: unknown): Row {
  const inputs = record("inputs", rawInputs);
  if (topicId === "D18-F03-A01") return peComparable(inputs);
  if (topicId === "D18-F03-A02") return evEbitda(inputs);
  if (topicId === "D18-F03-A03") return priceToBook(inputs);
  if (topicId === "D18-F03-A04") return peg(inputs);
  if (topicId === "D18-F03-A05") return regression(inputs);
  throw new RangeError(`unsupported topic_id: ${topicId}`);
}