19. Prove for a portfolio of fixed income securities with price function given by Pi Xn...

19. Prove for a portfolio of fixed income securities with price function given by PðiÞ ¼

Xn j¼1 PjðiÞ

that the duration and convexity of the portfolio, assuming PðiÞ00, and PjðiÞ00 for all j, is given by DðiÞ ¼

Xn j¼1 wjDjðiÞ; CðiÞ ¼

Xn j¼1 wjCjðiÞ;

where wj ¼ Pj ðiÞ

PðiÞ

, and hence Pn j¼1 wj ¼ 1.

Remark 9.166 It is important to note that you will not need to make an assumption about the signs of fPjðiÞg to prove this result. So this result applies equally well to long positions, PjðiÞ > 0, short positions, PjðiÞ < 0, or a mixed portfolio of longs and shorts.