A loan of P0 which accrues an interest i per period is to be paid of with a unif

Question

A loan of P0 which accrues an interest i per period is to be paid of with a uniform gradient annuity with initial payment A0 and gradient G over n periods. Thus, the change in the amount of principal is governed by Pn = Pn-1(1 + i) - A0 - (n - 1)G (1) Find expressions for P1, P2, P3 and P4 as functions of i, P0, A0 and G. From these results determine/guess/infer an expression for Pn as functions of i, P0, A0 and G. Show that the expression you guessed is correct by showing that it satises eq. 1. If the loan is paid of in n payments, derive an equation for P0 as a function of A0, G and n.

Explanation / Answer

P = A(P/A, i, n) To transfer a given Future value into an Annuity or Annual amount I would write A = F(A/F, i, n) To GET the Gradient FROM an Annual amount I would write: G = A(G/A, i, n) To GET the Present value OF a (given) Future value, I write: P = F(P/F, i, n) Note also that sometimes the reference manual does not give a column, such as for G/A. In that case: G = A(G/A, i, n) = A / (A/G, i, n) Also, you may have to piece together what you want: F = G(F/G, i, n) (but F/G is not in the reference manual) = G(F/A, i, n)(A/G, i, n) (which are both in the manual.)

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