Single liability immunization problem

In some cases, why and how the immunization portfolio can prevail even with non-parallel yield curve movements such as an upward and steepening shift , an upward and flattening shift, a downward and steepening shift , or a downward and flattening shift?

How can the yield curve shifts above results in the same 100 bp increase in the cash flow yield and each shift in the yield curve produces virtually the same reduction in the portfolio’s market value?

what does cash flow yield means?

That’s why it’s called Portfolio Immunization, because in theory the changes in assets and liabilities will offset each other. The duration of the assets exactly match the duration of the liabilities so to put it as simply as possible when liabilities go up, assets go up and when assets go down, liabilities go down.

That only works for parallel movements.

To answer the question: it’s a combination of yield (IRR, YTM) and market value change. For example in a curve steepening scenario and a barbell portfolio (short + long) vs. an intermediate bullet liability, two things happen. One, PVA will be lower than PVL, which isn’t a problem per se. If the increase in yield of assets is greater than yield of liabilities, the hedge may still be sufficient to immunize liabilities. Lower PVA & higher IRR may result in sufficient FVA to match FVL.

PVA = present value of assets, FVA = future value of assets.