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This is all off the top of my head, but how about this: $$PEGY = \frac{PE}{EG+DY}$$ $$PE = \frac{P}{E_n}$$ $$EG_{forward} = \frac{E_{n+1} - E_n}{E_n}$$ $$DY = \frac{D}{E_n}$$ $$PEGY = \frac{\frac{P}{E_n}}{\frac{E_{n+1} - E_n}{E_n} + \frac{D}{E_n}} = \frac{P}{E_n} \frac{E_n}{E_{n+1} - E_n + D}$$ $$PEGY = \frac{P}{\Delta E + D}$$ And ...
Based on what I know for a leacture on Arrow Debrue Security assume that we live in a world with only 2 states and a Security $S$ State 1 - Inflation > 1% State 2 - Inflation < 1% And In State 1 security $S$ pays 1 i.e the payout fraction is 100% In State 2 security $S$ pays 0 i.e the payout fraction is 0% This collection of states ...