Let's say we have the following situation: an asset (mortgage) with fixed payments, a prepayment & oas models to run through, and calculations for duration, convexity, and price, based on them. The asset is priced at par. We never sell the asset, i.e., hold it until it prepays completely or matures.
We are funding the asset with a blend of liabilities that have the same duration and convexity as the asset, and a minimized time-weighted initial cost. All liabilities also trade at par initially.
The question is: what exactly does this immunization guarantee from income point of view? Are we locking in the initial delta in asset ytm - liabilities blend ytm and immunizing that spread vs parallel changes in the yield curve?