Energy Project Financing : Resources and Strategies for Success

218 Energy Project Financing: Resources and Strategies for Success

Figure A-4. Combined uniform series and gradient series cash flow

Tables of (P|G,i,n) are provided in Appendix 4A. An algebraic expression

can also be derived for the (P|G,i,n) factor, which expresses P in terms of

G, i, and n. The derivation of this formula is omitted here, but the result-

ing expression is shown in the summary table (Table A-6) at the end of

this section.

It is not uncommon to encounter a cash flow series that is the sum of

a uniform series and a gradient series. Figure A-4 illustrates such a series.

The uniform component of this series has a value of 1000 and the gradi-

ent series has a value of 500. By convention the first element of a gradient

series has a zero value. Therefore, in Figure A-4, both the uniform series

and the gradient series have length four (n = 4). Like the uniform series

factor, the “P” calculated by a (P|G,i,n) factor is located one period before

the first element of the series (which is the zero element for a gradient

series).

Example 12

Assume you wish to make the series of withdrawals illustrated in

Figure A-4 from an account which pays 15%/yr. How much money would

you have to deposit today such that the account is depleted at the time of

the last withdrawal?

This problem is best solved by recognizing that the cash flows are a

combination of a uniform series of value 1000 and length 4 (starting

at time = 1), plus a gradient series of size 500 and length 4 (starting at

time = 1).

P = A * (P|A,15%,4) + G * (P|G,15%,4)

P = 1000 * (2.8550) + 500 * (3.7864)

P = 2855.00 + 1893.20 = $4748.20

Occasionally it is useful to convert a gradient series to an equivalent