You're saying if interest rates increased and stock price P/E remained the same most funds would allocate less money to bonds?
I don't understand why that would be. If the expected future cash flow of one asset increases (bonds), and remains the same for another (stocks) why would you allocate more money to stocks and away from bonds?
The future cash flow of existing bonds is fixed and it does not increase. If you have a bond that pays a 2% annual coupon you will see it's value drop when interest rates increase. The reason is that you can now get a newly issued bond that pays a higher (say 3%) fixed coupon so your's is worth less.
Your bond's price will drop to say 90% of the notional amount while the new bond will trade at 100% so they will effectively have the same "yield" of 3%.
The future cash flow of any given bond stays the same. But from an asset allocators point of view the cost of the future cash flow from a band decreased, making it a more attractive investment.
I don't understand why that would be. If the expected future cash flow of one asset increases (bonds), and remains the same for another (stocks) why would you allocate more money to stocks and away from bonds?