4. (Bond matrix ) The cash matching and other problems can be conveniently represented in matrix form. Suppose there are m bonds We define for each bond j its associated yearly cash flow stream (column) vector e,, which is n-dimensional. The yearly obligations are likewise represented by the n-dimensional vector y We can stack the

e, vectors side by side to form the columns of a bond matrix C Finally we let p and x be m-dimensional column vectors The cash matching problem can be expressed as minimize p' x subject to Cx 2 y X0.

(a) Identify C, y, p, and x in Table 5.3

(b) Show that if all bonds are priced according to a common term structure of interest rates, there is a vector v satisfying C'v=p What are the components of v?

(c) Suppose b is a vector whose components represent obligations in each period Show that a portfolio x meeting these obligations exactly satisfies Cx = b

(d) With x and y defined as before, show that the price of the portfolio x is vb. Interpret this result