Page 271 - Analysis, Synthesis and Design of Chemical Processes, Third Edition

F = A(1 + i) n–1 + A(1 + i) n–2 + A(1 + i) n–3 + ... + A
n

2
This equation is a geometric series of the form a, ar, ar ,..., ar n–1 with sum S = F .
n
n

For the present case, a = A; r = 1 + i; n = n. Therefore,

(9.11)

It is important to notice that Equation (9.11) is correct when the annuity starts at the end of the first time
period and not at time zero. In the next section, a shorthand notation is provided that will be useful in CFD
calculations.

9.5.2 Discount Factors

The shorthand notation for the future value of an annuity starts with Equation (9.11). The term F is
n
shortened by simply calling it F, and then dividing through by A yields

n
F/A = [(1 + i) – 1]/i = f(i, n)

This ratio of F/A is a function of i and n—that is, f(i,n). It can be evaluated when both the interest rate, i,
and the time duration, n, are known. The value of f(i,n) is referred to as a discount factor. If either A or F
is known, the remaining unknown can be evaluated.

In general terms, a discount factor is designated as

Discount factor for X/Y = (X/Y, i, n) = f(i, n)

Discount factors represent simple ratios and can be multiplied or divided by each other to give additional
discount factors. For example, assume that we need to know the present worth, P, of an annuity, A—that
is, the discount factor for P/A—but do not have the needed equation. The only available formula
containing the annuity term, A, is the one for F/A derived above. We can eliminate the future value, F, and
introduce the present value, P, by multiplying by the ratio of P/F, from Equation 9.6.
Discount Factor for P/A = (P/A, i, n)

= (F/A, i, n) (P/F, i, n)

Substituting for F/A and P/F gives