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P1: OSO c02 JWST022-Promislow October 13, 2010 13:12 Printer Name: Yet to Come VALUES AND ACTUARIAL EQUIVALENCE 13 For such a discount function, the rate of interest i k is a constant i = v −1 − 1, and the rate of discount dk is a constant d = 1 − v. The discount function is therefore conveniently given by simply stating a single parameter, which is usually taken as i, the constant rate of interest. So, for example, if we want to know how much we will accumulate at time n from an investment of 1 at time 0, this is just v(n, 0) = (1 + i)n , which is the usual starting point for the subject of compound interest in elementary texts.
For example if c = (2, 2), and d = (3, 7, 4, 1) then (c, d) = (2, 2, 3, 7, 4, 1, 0, . . , 0). ) For any number r and integer k we will write (rk ) to denote the vector consisting of k entries of r . For example (13 , 25 ) = (1, 1, 1, 2, 2, 2, 2, 2, 0, . . , 0). We let ei denote the standard ith basic vector, i = 0, 1, 2, . . , N , that is, a vector with an entry of 1 in position i and zeros elsewhere. 8 Regular pattern cashflows Before the introduction of computers and calculators, finding values of cashflows could involve a lengthy computation.
Assuming the normal case where entries in the vector k c are nonpositive as they represent repayments, the negative of these values are nonnegative, and multiplying by the higher values of v and summing will lead to a higher reserve. The penalty represents the excess value of the reserve as calculated by an up-to-date discount function, over the original value. ) As a second example, suppose you purchase a bond, which means that you are in the position of the lender. The nature of a bond is that this debt is assignable to another party by trading in the bond market.