PVA > PVL

How come that in two opposite scenarios PVA is better of than PVL?

"For a large parallel increase in the curve, the immediate decrease in portfolio value will be less than the decrease in the PVL due to the positive convexity effect" but " For a large parallel decrease in the curve, the immediate increase in portfolio value will exceed the increase in the PVL due to the positive convexity effect"

I thought that long-term rates have more weight than short-term weights and yield curve has a parallel upward increase, then increase PVA must be decreasing more than PVL (like it is during steepening or flattening twists). Any thoughts?

I think the concept to focus on here is convexity. Positive convexity makes a portfolio’s value more sensitive to rate decreases and less sensitive to rate increases. From the portfolio value movements mentioned, it means portfolio PVA has more convexity than portfolio PVL.

Not much to do with weights for long-term or short-term rates as you are examining parallel shifts.

It will be easier to show this graphically but I will use numbers to illustrate the point above.

%Change in Price = -Duration x Change in Yield + 0.5 x Convexity x (Change in Yield)^2

Let’s say the duration of your assets and liabilities are both equals to 8.

Let’s assume that the Convexity of Assets = 120 and Convexity of Liabilities = 80.

Scenario 1: Parallel increase of 1% in curve.

% Change in Asset Value = (-6 x 0.01) + (0.5 x 120 x 0.01^2) = -0.054 or -5.4%

% Change in Liability Value = (-6 x 0.01) + (0.5 x 80 x 0.01^2) = -0.056 or -5.6%

"For a large parallel increase in the curve, the immediate decrease in portfolio value will be less than the decrease in the PVL"

Scenario 2: Parallel decrease of 1% in curve.

% Change in Asset Value = (-6 x (-0.01)) + (0.5 x 120 x (-0.01)^2) = +0.066 or +6.6%

% Change in Liability Value = (-6 x (-0.01)) + (0.5 x 80 x (-0.01)^2) = +0.064 or +6.4%

"For a large parallel decrease in the curve, the immediate increase in portfolio value will exceed the increase in the PVL due to the positive convexity effect"