Colum

If you reallocate more to the pension equity, the firm risk has increased and its stock will be more volatile. So with all else being equal, equity beta would increase.

I think what you are remembering from the book is the chart that showed the required change in the capital structure needed to OFFSET the changes in pension assets to maintain the equity beta.

For instance, if they rebalance to more equity in the pension, the company would need to delever, ie pay down debt or raise equity to maintain the equity beta at the same level.

Vice versa if they allocated more pension assets to fixed income, the firm risk would decrease and equity beta would fall. If they wanted to maintain the equity beta, they should add more risk to the operating balance sheet, ie buyback stock or issue more debt.