A simple, accurate formula for the duration of a portfolio of bonds under a non-parallel shift of a non-flat yield curve

It is well known that the various formulas for the duration of a vanilla bond give inaccurate results. Their accuracy can be improved by the addition of extra elements, such as convexity or duration vectors. But the results remain inaccurate. A recent paper proposed a new formula for the duration of a portfolio of vanilla bonds. The formula gives a precise, accurate value for any parallel shift in a flat yield curve, without the need for auxiliary concepts. The analysis is performed in the complex plane, and uses all possible interest rates that solve the time value of money equation. In this paper, the analysis is reworked to produce a second, 'complex' formula that is more general. It copes with any non-flat yield curve and any non-parallel shift in the curve, and it is simpler and easier to prove. Some insights and puzzles presented by the new analysis are discussed.