This appendix demonstrates that the price-to-earnings (P/E) ratio of a levered company depends on its unlevered (all-equity) P/E, its cost of debt, and its debt-to-value ratio. When the unlevered P/E is less than 1/*k*_{d} (where *k*_{d} equals the cost of debt), the P/E falls as leverage rises. Conversely, when the unlevered P/E is greater than 1/*k*_{d}, the P/E ratio rises with increased leverage.

In this proof, we assume the company faces no taxes and no distress costs. We do this to avoid modeling the complex relationship between capital structure and enterprise value. Instead, our goal is to show that there is a systematic relationship between the debt-to-value ratio and the P/E.

To determine the relationship between P/E and leverage, we start by defining the unlevered P/E (PE_{u}). When a company is entirely financed with equity, its enterprise value equals its equity value, and its net operating profit less adjusted taxes (NOPLAT) equals its net income:
where*V*_{ENT} = enterprise value NOPLAT_{t+1} = net operating profit less adjusted taxes in year *t* + 1

This equation can be rearranged to solve for the enterprise value, which we will use in the next step:

For a company partially financed with debt, net income (NI) equals ...

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