《独立开发者的商业指南·收入与财务工程》2.1 单位经济的四个数：ARPA、毛利、CAC、回本

单客户月毛利 = $77.00 × 84.2% = $64.83
回本期 = $140.00 ÷ $64.83 = 2.16 → 2.2 个月
LTV = $64.83 ÷ 1.5% = $4,322.00
LTV/CAC = $4,322.00 ÷ $140.00 = 30.87 → 30.9

// file: src/unit/economics.ts
// tier: L1
// verify: node --test test/unit/economics.test.ts

import { type Cents } from "../core/money.ts";
import { type Bps } from "../core/rate.ts";
import { type Result, fail, ok } from "../core/result.ts";

export type UnitEconomicsInput = {
  mrrCents: Cents;
  customers: number;
  grossMarginBps: Bps;
  marketingCents: Cents;
  newCustomers: number;
  monthlyChurnBps: Bps;
};

export type UnitEconomics = {
  arpaCents: Cents;
  grossProfitPerCustomerCents: Cents;
  cacCents: Cents;
  paybackMonths: number;
  ltvCents: Cents;
  ltvCacRatioBps: Bps;
};

function isCents(value: number): boolean {
  return Number.isInteger(value) && value >= 0;
}

function isBps(value: number): boolean {
  return Number.isInteger(value) && value >= 0 && value <= 10000;
}

/** Monthly average revenue per account, in cents. */
export function arpa(mrrCents: Cents, customers: number): Result<Cents> {
  if (!isCents(mrrCents)) {
    return fail("MRR must be a non-negative integer number of cents");
  }
  if (!Number.isInteger(customers) || customers <= 0) {
    return fail("customers must be a positive integer");
  }
  return ok(Math.round(mrrCents / customers));
}

/** Gross profit contributed by one customer in one month, in cents. */
export function grossProfitPerCustomer(
  arpaCents: Cents,
  grossMarginBps: Bps,
): Result<Cents> {
  if (!isCents(arpaCents)) {
    return fail("ARPA must be a non-negative integer number of cents");
  }
  if (!isBps(grossMarginBps)) {
    return fail("gross margin must be an integer between 0 and 10000 bps");
  }
  return ok(Math.round((arpaCents * grossMarginBps) / 10000));
}

/** Customer acquisition cost: acquisition spend divided by new customers. */
export function cac(
  marketingCents: Cents,
  newCustomers: number,
): Result<Cents> {
  if (!isCents(marketingCents)) {
    return fail("marketing spend must be a non-negative integer number of cents");
  }
  if (!Number.isInteger(newCustomers) || newCustomers <= 0) {
    return fail("new customers must be a positive integer");
  }
  return ok(Math.round(marketingCents / newCustomers));
}

/**
 * Months of gross profit needed to recover CAC, rounded to one decimal.
 * Returns 0 when CAC is zero.
 */
export function paybackMonths(
  cacCents: Cents,
  monthlyGrossProfitCents: Cents,
): Result<number> {
  if (!isCents(cacCents)) {
    return fail("CAC must be a non-negative integer number of cents");
  }
  if (!isCents(monthlyGrossProfitCents)) {
    return fail("monthly gross profit must be a non-negative integer number of cents");
  }
  if (cacCents === 0) {
    return ok(0);
  }
  if (monthlyGrossProfitCents === 0) {
    return fail("monthly gross profit must be positive when CAC is positive");
  }
  return ok(Math.round((cacCents / monthlyGrossProfitCents) * 10) / 10);
}

/**
 * Gross-profit LTV under a constant monthly churn rate.
 * The formula is monthly gross profit divided by monthly churn.
 */
export function ltvFromGrossProfit(
  monthlyGrossProfitCents: Cents,
  monthlyChurnBps: Bps,
): Result<Cents> {
  if (!isCents(monthlyGrossProfitCents)) {
    return fail("monthly gross profit must be a non-negative integer number of cents");
  }
  if (!Number.isInteger(monthlyChurnBps) || monthlyChurnBps <= 0 || monthlyChurnBps > 10000) {
    return fail("monthly churn must be an integer between 1 and 10000 bps");
  }
  return ok(
    Math.round((monthlyGrossProfitCents * 10000) / monthlyChurnBps),
  );
}

/** LTV divided by CAC, expressed in basis points, so 3.0x is 30000 bps. */
export function ltvCacRatio(ltvCents: Cents, cacCents: Cents): Result<Bps> {
  if (!isCents(ltvCents)) {
    return fail("LTV must be a non-negative integer number of cents");
  }
  if (!isCents(cacCents) || cacCents === 0) {
    return fail("CAC must be a positive integer number of cents");
  }
  return ok(Math.round((ltvCents * 10000) / cacCents));
}

/** Compute the four numbers and the two headline ratios in one pass. */
export function buildUnitEconomics(
  input: UnitEconomicsInput,
): Result<UnitEconomics> {
  const arpaResult = arpa(input.mrrCents, input.customers);
  if (!arpaResult.ok) {
    return arpaResult;
  }
  const grossProfitResult = grossProfitPerCustomer(
    arpaResult.value,
    input.grossMarginBps,
  );
  if (!grossProfitResult.ok) {
    return grossProfitResult;
  }
  const cacResult = cac(input.marketingCents, input.newCustomers);
  if (!cacResult.ok) {
    return cacResult;
  }
  const paybackResult = paybackMonths(cacResult.value, grossProfitResult.value);
  if (!paybackResult.ok) {
    return paybackResult;
  }
  const ltvResult = ltvFromGrossProfit(
    grossProfitResult.value,
    input.monthlyChurnBps,
  );
  if (!ltvResult.ok) {
    return ltvResult;
  }
  const ratioResult = ltvCacRatio(ltvResult.value, cacResult.value);
  if (!ratioResult.ok) {
    return ratioResult;
  }
  return ok({
    arpaCents: arpaResult.value,
    grossProfitPerCustomerCents: grossProfitResult.value,
    cacCents: cacResult.value,
    paybackMonths: paybackResult.value,
    ltvCents: ltvResult.value,
    ltvCacRatioBps: ratioResult.value,
  });
}