DCF — unlevered free cash flow (dcf.fcff)¶

Values the operating business by discounting free cash flows to the firm (FCFF) at the weighted average cost of capital (WACC) over one to three explicit stages, adds a terminal value, and moves from the value of operations to enterprise value and equity value. Every choice that changes the value is an explicit input: the stages and how their flows are obtained, the terms of the flows and of the rates, the timing, the terminal technique and when it is discounted.

Method version: 3. Version 3 replaced the single list of flows of version 2 with stages, made the timing and the terminal technique required inputs, and renamed the version 2 enterprise_value (present values only) to value_of_operations; the version 2 examples give the same numbers in version 3.

Formula¶

The explicit horizon N is the sum of the periods of the stages; one period is one year. The flows of each stage are:

A generated stage starts from the last flow of the previous stage, or from the base flow \(CF_0\) when it is the first stage. In a linear decline, \(g_{from}\) is the growth before the stage and \(g_{to}\) is reached in the last period: "from 10% to 3% over 5 years" gives 8.6%, 7.2%, 5.8%, 4.4% and 3.0%.

Under mid-period timing the flow of period k is discounted half a period earlier, \(DF_k = DF_{k-1} (1 + r_k)^{-0.5}\). The terminal value \(TV_N = CF_{N+1} / (r_{TV} - g)\) is computed by tv.gordon and discounted at the horizon, \(DF_{TV} = DF_N\), or, under mid-period timing when chosen, half a period earlier, \(DF_{TV} = DF_N (1 + r_N)^{0.5}\); the half-period shift uses the rate of period N, \(r_N\), not the terminal rate.

Present values are computed by dividing each amount by its compounding factor \(\prod_{i \le k} (1 + r_i)\) (under mid-period timing, \(\prod_{i < k} (1 + r_i) \, (1 + r_k)^{0.5}\)), which is the same quantity as multiplying by \(DF_k\) but keeps every significant digit at long horizons; the discount factors are reported as computed by tvm.present_value.

Example¶

A plan with three given years, then four years in which growth declines in linear increments from 12% to 3%, then stable growth of 3%:

import fairmark as fm

r = fm.dcf.fcff(
    [
        fm.dcf.given([100, 112, 125]),
        fm.dcf.linear_decline(4, from_growth="0.12", to_growth="0.03"),
    ],
    cash_flow_terms="nominal",
    discount_rate_terms="nominal",
    wacc="0.095",
    timing="end_of_period",
    terminal=fm.dcf.gordon("0.03", first_flow="last_flow_grown"),
    net_financial_position={"gross_debt": 500, "cash": 120},
    non_operating_assets=[{"label": "Land held for sale", "value": 60}],
)
print([row["growth"] for row in r.periods][3:])  # 0.0975, 0.075, 0.0525, 0.03
print(r.value_of_operations)    # 1985.96014516242602644247722...
print(r.enterprise_value)       # value of operations + 60
print(r.equity_value)           # enterprise value − (500 − 120)
print(r.terminal_value_weight)  # share of the value of operations from the terminal value
assert r.enterprise_value - r.value_of_operations == 60

With rates per period, give wacc_per_period (one rate per period of the explicit horizon) and terminal_wacc instead of wacc. With mid-period timing, state when the terminal value is discounted:

r = fm.dcf.fcff(
    [fm.dcf.constant_growth(3, "0.10"), fm.dcf.linear_decline(3, "0.10", "0.025")],
    base_flow=80,
    cash_flow_terms="real",
    discount_rate_terms="real",
    wacc_per_period=["0.11", "0.11", "0.105", "0.10", "0.095", "0.09"],
    terminal_wacc="0.09",
    timing="mid_period",
    terminal_discount_time="horizon",
    terminal=fm.dcf.gordon("0.025", first_flow="last_flow_grown"),
)
print(r.terminal["discount_time"])  # 6

Inputs¶

Outputs¶

Conventions (explicit, reported in explain.conventions)¶

Warnings are reported in explain.conventions and never refuse the input: a growth applied to a negative flow; a growth of the last explicit period that differs from the perpetual growth of the terminal value; under mid-period timing, a Gordon terminal value discounted at the horizon N, where practice commonly discounts it at N − 0.5; a negative equity value, which is reported as computed and never floored at zero. A high terminal_value_weight signals that the valuation depends mostly on long-term assumptions.

Errors¶

Validation errors name the field by its JSON pointer, for example /stages/1/flows, and state the fix; errors raised while reading the input (a missing or unknown field, a wrong type) report the line and column of the input instead.

Accuracy¶

Discount factors are computed by tvm.present_value (tvm::period_discount_factors): at end of period by exact divisions, at mid period with a decimal square root. Present values divide by the compounding factor, so they keep their significant digits at long horizons: measured against an independent 50-digit computation, the value of operations agrees to about \(10^{-24}\) relative or better, at end and at mid period, also over 300 to 1,000 periods. The golden cases from the reference are checked to \(10^{-20}\).

Verified against¶

• Damodaran, Investment Valuation, 2^nd edition, chapter 15, Illustration 15.3 (Amgen, three-stage FCFF with costs of capital changing every year): present value of the explicit period 8,327, value of the operating assets 39,161, equity value 40,867 and terminal value 81,364 are matched within 2 (terminal value: 20) of the published figures, which come from rounded inputs; the golden files decompose the differences.
• An independent 50-digit implementation, scripts/reference_dcf.py, for every golden case in tests/golden/dcf.fcff/, and properties: a constant growth stage equals the same flows given; a linear decline with equal growths equals constant growth; discount factors equal those of tvm.present_value.
• numpy-financial NPV of the explicit flows plus the Gordon value, in binary floating point (agreement of conventions, not of digits).