Valuation Conservation Theory

1. Extending the Constraint Matrix

Let $A_{ ext{acct}} \in \mathbb{Z}^{15 imes n}$ denote the simplified accounting conservation matrix introduced in Task 02. We extend it with seven finance-specific constraints to obtain

$$A_{ ext{val}} = egin{bmatrix} A_{ ext{acct}} \ A_{ ext{fin}} \end{bmatrix} \in \mathbb{Z}^{22 imes n},$$
where $A_{ ext{fin}}$ introduces the growth-reinvestment identity, terminal multiple consistency, clean surplus, and residual income bridges.

Each new row corresponds to a conservation law:

  1. Growth-Reinvestment ($g = s imes ext{ROIC}$)
  2. Clean Surplus ($BV_t$ bridge)
  3. DCF–Multiple reconciliation
  4. Residual income alignment
  5. Terminal value reinvestment
  6. Capital intensity stabilization
  7. Equity cost of capital harmonization

2. Valuation Feasibility Set

Define the augmented feasibility set

$$\mathcal{F}_{ ext{val}} = \{ y \in \mathbb{R}^n : A_{ ext{acct}} y = 0 \land A_{ ext{fin}} y = 0 \}.$$

Feasible forecasts satisfy both accounting and valuation conservation simultaneously. Analysts provide state vectors $y$ containing operating forecasts, capital allocation assumptions, and valuation levers. The oracle verifies membership by solving

$$\min_{y'} \|A_{ ext{val}} y' \|_1 \quad ext{s.t. } \|y' - y\|_1 \leq \epsilon,$$
which yields the minimum perturbation needed to reach $\mathcal{F}_{ ext{val}}$ when the input violates constraints.

3. Projection Operator

The projection onto $\mathcal{F}_{ ext{val}}$ is computed via the feasibility LP:

$$\Pi_{\mathcal{F}_{ ext{val}}}(y) = rg\min_{z} \|z - y\|_2 \quad ext{s.t. } A_{ ext{val}} z = 0.$$

In practice we solve the dual problem using cvxpy with SCS, warm-started from the accounting-only solution. The projection quantifies the adjustment vector $\Delta y = z - y$, decomposed by metric to support auditor review.

4. Nullspace Interpretation

The nullspace of $A_{ ext{val}}$ captures gameability degrees of freedom. In the simplified dataset we observe $ ext{rank}(A_{ ext{val}}) = 22$ with $n pprox 22$, leaving a trivial nullspace (no free parameters). On richer datasets with hundreds of tags the nullspace dimension becomes positive; we then apply the adversarial framework from Task 06 to enumerate manipulations.

5. Implications for Forecast Quality

  1. Consistency — Forecasts satisfying $A_{ ext{val}} y = 0$ are internally coherent across accounting and valuation domains.
  2. Auditability — Adjustments from $\Pi_{\mathcal{F}_{ ext{val}}}$ provide a deterministic remediation path.
  3. Comparability — Analysts generating valuation decks can publish $\delta^*$ scores summarizing conservation health.
  4. Automation — The constraint architecture enables automated rejection of implausible sell-side models before they enter risk systems.

Citations

Accounting Conservation Framework | Home