23. Given a portfolio of 3 bonds with market values in $millions of: 200, 450, and 350, and respective durations of 3.5, 5.0, and 8:5.

(a) Calculate the duration of the portfolio, where D ¼

P xiDi=

P xi and xi denotes the amount invested in bond i.

(b) Find the trade in R3 that changes the portfolio duration to 4:0 that has the lowest transaction fee, assuming that this fee is proportional to the market value bought and sold, and that all final positions must be long. (Hint: See (3.47), but note that while the constraint P

xi ¼ 0 allows you to analytically consider this a problem in R2, because x3 ¼ x1  x2, the norm minimization in R2 will not work in general.)

(c) Repeat part

(b) but now with a duration target of 6:5, and where final positions can be long or short.

(d) Achieve the same objective in part (c), but adding the constraint that the investment policy maximum for any bond is 462 on a long or short basis.