Discounted Cash Flow Theory under Conservation Constraints

1. Free Cash Flow to Firm (FCFF)

Let $\text{NOPAT}_t$ denote net operating profit after tax in year $t$, and let $\Delta IC_t$ denote the change in invested capital. Free cash flow to the firm (FCFF) is defined as

$$\text{FCFF}_t = \text{NOPAT}_t - \Delta IC_t.$$

The accounting conservation view decomposes $\Delta IC_t$ into net working capital and net fixed assets:

$$\Delta IC_t = \Delta NWC_t + \Delta NFA_t.$$

Both components are observable from the statement of financial position, making FCFF a fully reconciled measure aligned with the 15-constraint accounting core.

2. Enterprise Value Decomposition

Discounted cash flow valuation computes enterprise value (EV) as the present value of projected FCFF:

$$EV = \sum_{t=1}^{T} \frac{\text{FCFF}_t}{(1 + WACC)^t} + \frac{\text{TV}_T}{(1 + WACC)^T},$$

where $WACC$ is the weaverage cost of capital, and $\text{TV}_T$ is the terminal value at horizon $T$.

In the conservation framework, each projected FCFF must satisfy the accounting constraints. Forecasts that breach $\Delta IC_t = \text{Reinvestment}_t$ or the growth-reinvestment identity introduce violations in $EV$ that our feasibility solvers detect.

3. Conservation-Consistent Terminal Value

Analysts often apply terminal multiples such as $EV/\text{EBITDA}$. Under steady-state assumptions with constant ROIC, tax rate $\tau$, and growth $g$, we enforce a consistent relationship between the multiple and the DCF parameters:

$$\frac{EV}{\text{EBITDA}} \approx \frac{(1-\tau) \times \frac{\text{EBIT}}{\text{EBITDA}} \times \left(1 - \frac{g}{\text{ROIC}}\right)}{WACC - g}.$$

Derivation sketch:

In steady state, $\text{FCFF} = \text{NOPAT} - g \times IC$ with $IC = \text{NOPAT}/\text{ROIC}$.
Substitute $\text{NOPAT} = (1-\tau) \times \text{EBIT}$.
Express $\text{FCFF}$ as $\text{EBITDA}$ less normalized reinvestment and taxes.
Solve for $EV/\text{EBITDA}$ given $EV = \text{FCFF} / (WACC - g)$.

The constraint requires coherence between terminal growth, capital intensity, and return on capital. Violations are reported as $\delta^*_{\text{terminal}} = | \text{Observed multiple} - \text{DCF-implied multiple} |$.

4. Feasibility Bounds

To keep valuations physically plausible, we enforce:

$WACC > g$ — ensures convergence of the present value series.
$g \leq \text{ROIC}$ — implied by the growth-reinvestment identity.
$g \geq 0$ for mature firms, or explicit justification for negative terminal growth.
$\tau \in [0, 0.35]$ in developed markets (empirical tax bounds).

Violations trigger deterministic error codes in the DCF validator.

5. Sensitivity Framework

We analyze robustness to small perturbations $\Delta g$, $\Delta WACC$, and $\Delta \text{ROIC}$:

$$\frac{\partial EV}{\partial g} = \frac{\text{FCFF}_T + \partial \text{FCFF}_T/\partial g}{(WACC - g)^2},$$

and similarly for other parameters. Our implementation produces tornado plots and scenario tables for auditability.

6. Implementation Notes

The finance extension uses the same constraint matrix interface as the accounting core, augmenting it with terminal value conservation rows.
All formulas accept vectorized inputs for consistent batch evaluation across thousands of forecast scenarios.
Sensitivity routines support Sobol’ sampling for robust Monte Carlo analysis when uncertainty distributions are supplied.

Citations

Koller, T., Goedhart, M., & Wessels, D. (2020). Valuation (7th ed.). McKinsey & Company.
Damodaran, A. (2012). Investment Valuation (3rd ed.). Wiley.