First picture after the question is page 22 and the second two are page 93
Cash flows and single period loans. As a practical example, we consider cash ows over so periods, with positive entries meaning income or cash in and negative entries meaning payments or cash out. We dene the singleperiod loan cash ow vectors as Ol 1 (1 + 'r) ' Untfl lg: i=1,...,TL1, where 'r 2 0 is the perperiod interest rate. The cash ow lg represents a loan of $1 in period '11, which is paid back in period i+ 1 with interest 'r. (The subscripts on the zero vectors above give their dimensions.) Scaling It; changes the loan amount; scaling It: by a negative coefcient converts it into a loan to another entity (which is paid back in period *5 + 1 with interest). The vectors 61, ll, . . . ,ln_1 are a basis. (The rst vector 61 represents income of $1 in period 1.) To see this, we show that they are linearly independent. Suppose that le]. +132l1 + ' ' ' + Bnlnl = 0. We can express this as 51 + 32 133 (1 + T)52 f = 0. Jan _ (1 + T)n1 (1 + r)n The last entry is (1 + rm\" = 0, which implies that B\" = 0 (since 1 + 'r > 0). Using n = 0, the second to last entry becomes (1 + 7') 13141 = 0, so we conclude that nd = 0. Continuing this way we nd that n_2, . . . \"82 are all zero. The rst entry of the equation above, 51 + 52 = 0, then implies 51 = 0. We conclude that the vectors 61,!1, . . . ,l.n_1 are linearly independent, and therefore a basis. This means that any cash ow nvector c can be expressed as a linear combi nation of (ta, replicated by) an initial payment and one period loans: 6 = 0:161 + 0:211 + . ' ' + anln_1. It is possible to work out what the coefcients are (see exercise 5.3). The most interesting one is the rst coeicient 1+r+ +(1+r)\"1' 11=Cl+ which is exactly the net present value (N PV) of the cash ow, with interest rate 7'. Thus we see that any cash ow can be replicated as an income in period 1 equal to its net present value, plus a linear combination of oneperiod loans at interest rate r. . Discounted total. Let c be an n-vector representing a cash flow, with ci the cash received (when c; > 0) in period i. Let d be the n-vector defined as d = (1, 1/(1 +r), ...,1/(1+r)"-1), where r 2 0 is an interest rate. Then d c = c1 + c2/(1 +r)+ ... + cn/(1+r)n-1 is the discounted total of the cash flow, i. e., its net present value (NPV), with interest rate r.is the supervisor right! Did the intern make a mistake! Give a ve brief explanation. 5.3 Replicating a cash flow with single-period loans. We continue the example described on page 93. Let c be any n-vector representing a cash flow over n periods. Find the coefficients Q1, . .., On of c in its expansion in the basis e1, l1, ..., In-1, i. e., c = ale1 + a2l1 + . . . + anln-1. Verify that on is the net present value (NPV) of the cash flow c, defined on page 22, with interest rate r. Hint. Use the same type of argument that was used to show that e1, l1, ..., In-1 are linearly independent. Your method will find an first, then an-1, and so on
