Valuation Consistency Across Methods

1. Clean Surplus Bridge

Clean surplus accounting (Ohlson, 1995) requires

$$BV_{t} = BV_{t-1} + \text{NI}_t - \text{Div}_t,$$
where $BV_t$ is book value of equity. When forecasts violate this bridge the residual income method ceases to match DCF. We embed the bridge in our augmented constraint matrix and track the residual
$$\delta^*_{\text{clean}} = |BV_t - BV_{t-1} - \text{NI}_t + \text{Div}_t|.$$
Any non-zero residual indicates the analyst introduced unmodelled equity issuances or accounting noise that must be reconciled.

2. Implied Multiples from DCF Parameters

Given a DCF forecast with $(g, WACC, \text{ROIC}, \tau)$ and normalized operating metrics, the implied forward EV/EBITDA multiple is

$$m_{\text{DCF}} = \frac{(1-\tau) \times \frac{\text{EBIT}}{\text{EBITDA}} \times (1 - g/\text{ROIC})}{WACC - g}.$$

If an analyst simultaneously asserts a trading multiple $m_{\text{obs}}$, our valuation bridge enforces

$$\delta^*_{\text{multiple}} = |m_{\text{obs}} - m_{\text{DCF}}|.$$
Values above tolerance (default 0.5 turns of EBITDA) trigger an inconsistency warning.

3. Residual Income Cross-Check

Residual income valuation expresses equity value as

$$EV_{\text{RI}} = BV_0 + \sum_{t=1}^{T} \frac{RI_t}{(1 + r)^t},$$
where $RI_t = \text{NI}_t - r \times BV_{t-1}$ and $r$ is the cost of equity. We reconcile residual income to DCF by confirming
$$EV_{\text{DCF}} - EV_{\text{RI}} = 0.$$
The difference is reported as $\delta^*_{\text{ri}}$ and should vanish when clean surplus holds and WACC aligns with the cost of capital used in the residual income model.

4. Feasibility LP for Inconsistent Methods

We frame reconciliation as a linear feasibility problem:

$$\min_{x} \|A_{\text{bridge}} x - b\|_1,$$
where $x$ collects cash flows, dividends, and multiples, while $A_{\text{bridge}}$ encodes the constraints above. The solver proposes minimal adjustments (e.g., tweak $g$ by 30 bps, increase terminal multiple by 0.2 turns) required to restore consistency. Analysts can then apply or reject adjustments with full transparency.

5. Practical Workflow

  1. Input: Forecast FCFF, dividends, terminal multiple, and cost of capital.
  2. Compute: DCF valuation, implied multiple, residual income valuation.
  3. Evaluate: $\delta^*_{\text{growth}}$, $\delta^*_{\text{multiple}}$, $\delta^*_{\text{ri}}$.
  4. Resolve: Use feasibility LP recommendations to reconcile.
  5. Document: Persist rationale in the valuation bridge audit trail.

Citations

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