Even with a parallel yield curve shift, this causes returns in assets and liaibltiies with same duration to be different.
If the convexity is greater for assets, then shouldnt a drop in rates cause its value to drop more than liabilties?
If we view asset liability management (ALM) as a process that focuses on protecting economic value of equity or share holder value then an ALM framework should focus on insuring interest rate changes should have minimal impact on share holder value (surplus).
To protect surplus or immunize it against interest rate changes two conditions must be met.
Weighted average duration of assets = weighted average duration of liabilities
Weighed average convexity of assets > weighted average convexity of liabilities
In this case, then value of asset should increase. With positive convexity, when rates decline so that prices increase, they increase _ more _ than duration would suggest.
It depends on the surplus if the manager can continue to actively manage. If Asset duration exceed Liability duration if rates fell and if rates increase then asset value will drop more than thel liability value and surplus will decrease. Asset duration and Convexity should be matched if not then surplus might change.