APPENDIX I
Relationship between Observed Price-to-Earnings (“P/E”) Ratios and Nominal Interest Rates
We start by repeating equation (9.11) and then carrying out a first-order Taylor Series differential, holding the unleveraged discount rate constant.
Carrying out the partial differentiation from (9.11) as required in (I.1), we obtain:
We can cancel and rearrange terms, after which we can divide through by Δi, which produces:
As the changes in P/E ratio and interest rate, i, approach zero in the limit, equation (I.3) can be changed from a differential into an exact first derivative:
From inspection, it can be seen that when P/E ratios are large and when leverage is large, that is, φ is small, the P/E ratio will be most significantly, positively, related to the change in nominal yields.
