How fund houses make money: AMC revenue and profit, operating leverage, the value of a SIP customer

"""M06B · How fund houses make money: AMC revenue and profit, operating leverage, the value of a SIP customer.

Exercise stubs. Replace each ``raise NotImplementedError`` with your code, then run the tests in
../tests. See notes.md §6B.5: revenue (6B.10), Proposition 6B.5.1 (profit (6B.11) and its beta to the
market (6B.12), Worked example 6B.6), and Proposition 6B.5.2 (the present value of the fees on a SIP
customer (6B.13), Worked example 6B.7).

Conventions
-----------
* AUM: equity AUM A_e with elasticity β_A = d ln A_e / d ln M to the market level M, and other AUM A_d
  that does not move with M. Fee yields y are annual fractions of average AUM kept by the AMC.
* Costs: a fixed amount F a year plus a share v of revenue. Profit Π = (1 − v) · revenue − F.
* SIP customer (6B.13): monthly rates throughout. Instalment P at the start of each of the first n
  months; holdings grow at g a month (net of fees); the AMC earns y/12 on the end-of-month holding; the
  customer leaves each month with probability λ; cash flows are discounted at k a month.
"""

from __future__ import annotations

import numpy as np


def amc_revenue(fee_yields, avg_aums) -> float:
    """Revenue = Σ_s y_s Ā_s (6B.10)."""
    raise NotImplementedError


def amc_profit(aum_equity: float, aum_other: float, yield_equity: float, yield_other: float, fixed: float,
               var_share: float) -> float:
    """Π = (1 − v)(y_e A_e + y_d A_d) − F (6B.11)."""
    raise NotImplementedError


def profit_at_market(m: float, aum_equity: float, aum_other: float, yield_equity: float, yield_other: float,
                     fixed: float, var_share: float, aum_beta: float = 1.0) -> float:
    """Profit when the market is at m times today's level: equity AUM becomes A_e · m^β_A."""
    raise NotImplementedError


def profit_beta(aum_equity: float, aum_other: float, yield_equity: float, yield_other: float, fixed: float,
                var_share: float, aum_beta: float = 1.0) -> float:
    """β_Π = d ln Π / d ln M = β_A (1 − v) y_e A_e / Π (6B.12); with A_d = 0 it is β_A (1 + F/Π).
    Raise ValueError if Π <= 0."""
    raise NotImplementedError


def monthly_rate(annual: float) -> float:
    """Monthly rate equivalent to an annual compound rate: (1 + annual)^(1/12) − 1."""
    raise NotImplementedError


def monthly_attrition(annual: float) -> float:
    """Monthly leaving probability λ equivalent to losing a fraction ``annual`` of customers a year:
    1 − (1 − annual)^(1/12)."""
    raise NotImplementedError


def pv_fees_sip(P: float, n: int, fee_yield: float, g: float, lam: float, k: float) -> float:
    """PV = y_m P · q/(1 − q) · (1 − a^n)/(1 − a) (6B.13), with y_m = fee_yield/12, a = (1 − λ)/(1 + k)
    and q = (1 + g) a. Raise ValueError unless q < 1."""
    raise NotImplementedError


def pv_fees_sip_simulated(P: float, n: int, fee_yield: float, g: float, lam: float, k: float,
                          n_customers: int, horizon: int, seed: int = 0) -> np.ndarray:
    """Discounted fees from ``n_customers`` simulated customers over ``horizon`` months (the model of
    (6B.13), truncated at the horizon). With rng = np.random.default_rng(seed), each customer's departure
    month is rng.geometric(lam, n_customers): a customer leaving in month L earns fees in months
    1, …, L − 1 and none from month L on. Return the array of per-customer present values."""
    raise NotImplementedError