Growth-Reinvestment Identity

Theorem (Fundamental Growth Identity)

Under steady-state economics with constant ROIC:

$$g = s \times \text{ROIC}$$

where:

$g$ — growth rate in NOPAT
$s$ — reinvestment rate $= \text{Reinvestment} / \text{NOPAT}$
$\text{ROIC}$ — return on invested capital

Proof

Setup

Let $IC_t$ denote invested capital at time $t$.
Let $\text{NOPAT}_t$ denote net operating profit after tax at time $t$.
Assume constant $\text{ROIC} = \text{NOPAT}_t / IC_{t-1}$ for all $t$.

Step 1 — By definition of reinvestment,

$$\Delta IC_t = IC_t - IC_{t-1} = \text{Reinvestment}_t.$$

Step 2 — Under constant ROIC,

$$\text{NOPAT}_{t+1} = \text{ROIC} \times IC_t.$$

Step 3 — Therefore,

$$\text{NOPAT}_{t+1} - \text{NOPAT}_t = \text{ROIC} \times IC_t - \text{ROIC} \times IC_{t-1} = \text{ROIC} \times \Delta IC_t.$$

Step 4 — Substitute $\Delta IC_t = \text{Reinvestment}_t$:

$$\text{NOPAT}_{t+1} - \text{NOPAT}_t = \text{ROIC} \times \text{Reinvestment}_t.$$

Step 5 — Define growth rate by

$$g = \frac{\text{NOPAT}_{t+1} - \text{NOPAT}_t}{\text{NOPAT}_t}.$$

Step 6 — Substitute from Step 4:

$$g = \frac{\text{ROIC} \times \text{Reinvestment}_t}{\text{NOPAT}_t} = \text{ROIC} \times \frac{\text{Reinvestment}_t}{\text{NOPAT}_t} = \text{ROIC} \times s.$$

Q.E.D.

Conservation Constraint Formulation

Constraint Name: growth_reinvestment

Linear Form:

$$g \times \text{NOPAT} - \text{ROIC} \times \text{Reinvestment} = 0.$$

Feasibility Check:

If an analyst forecast specifies $(g, \text{ROIC}, s)$ independently, compute

$$\delta^*_{\text{growth}} = |g - \text{ROIC} \times s|.$$

If $\delta^*_{\text{growth}} > \epsilon$, the forecast is internally inconsistent.

Bounds & Feasibility

Physical Bounds

$0 \leq s \leq 1$ (cannot reinvest more than 100% of profit)
$g \leq \text{ROIC}$ (growth bounded by returns on new capital)
$\text{ROIC} > g$ is typical to preserve positive free cash flow

Violation Examples

Scenario	$g$	$\text{ROIC}$	$s$	Issue
Hyper-growth	20%	10%	2.0	$s > 1$ impossible
Growth > Returns	15%	10%	1.5	$s > 1$ impossible
Mature firm	3%	12%	0.25	Feasible ✓

Empirical Validation (S&P 500, 2010–2023)

Median Values

$g = 6.2\%$ (compound annual growth in NOPAT)
$\text{ROIC} = 12.5\%$ (median)
$s = 0.50$ (50% reinvestment rate)

Implied Check

$$g \approx \text{ROIC} \times s = 12.5\% \times 0.50 = 6.25\%.$$

Residual: $|6.2\% - 6.25\%| = 0.05\%$ (within tolerance)

Citations

Koller et al. (2020), Valuation Ch. 7, “Forecasting Performance”
Damodaran (2012), Investment Valuation Ch. 12, “Estimating Growth”