Linear combinations of cash flows. We consider cash flow vectors over T time periods, with a positive entry meaning a payment received, and negative meaning a payment made. A (unit) single period loan, at time period t, is the T-vector lt that corresponds to a payment received of $1 in period t and a payment made of $(1 + r) in period t + 1, with all other payments zero. Here r > 0 is the interest rate (over one period). Let c be a $1 T − 1 period loan, starting at period 1. This means that $1 is received in period 1, $(1 + r) T −1 is paid in period T, and all other payments (i.e., c2, . . . , cT −1) are zero. Express c as a linear combination of single period loans.
