Energy Project Financing : Resources and Strategies for Success

216 Energy Project Financing: Resources and Strategies for Success

a uniform series of cash flows. When dealing with uniform series cash

flows, the symbol A represents the amount of each annual cash flow and

n represents the number of cash flows in the series. The factor (P|A,i,n)

is known as the uniform series, esent worth factorpr and is read “to find P

given A at i% for n years.” Tables of (P|A,i,n) are provided in Appendix

4A. An algebraic expression can also be derived for the (P|A,i,n) factor

which expresses P in terms of A, i, and n. The derivation of this formula is

omitted here, but the resulting expression is shown in the summary table

(Table A-6) at the end of this section.

An important observation when using a (P|A,i,n) factor is that the

“P” resulting from the calculation occurs one period prior to the first “A”

cash flow. In our example the first withdrawal (the first “A”) occurred one

year after the deposit (“P”). Restating the example problem above using a

(P|A,i,n) factor, it becomes:

P = A * (P|A,i,n)

P = 2000 * (P|A,9%,4)

P = 2000 * (3.2397) = $6479.40

This result is identical (as expected) to the result using the (P|F,i,n)

factors. In both cases the interpretation of the result is: If we deposit

$6479.40 in an account paying 9%/yr interest, we could make withdraw-

als of $2000 per year for four years, starting one year after the initial de-

posit, to deplete the account at the end of 4 years.

The reciprocal relationship between P and A is symbolized by the

factor (A|P,i,n) and is called the uniform series, capital recovery factor. Tables

of (A|P,i,n) are provided in Appendix 4A, and the algebraic expression

for (A|P,i,n) is shown in Table A-6 at the end of this section. This factor

enables us to determine the amount of the equal annual withdrawals “A”

(starting one year after the deposit) that can be made from an initial de-

posit of “P.”

Example 9

Determine the equal annual withdrawals that can be made for 8

years from an initial deposit of $9000 in an account that pays 12%/yr. The

first withdrawal is to be made one year after the initial deposit.

A = P * (A|P,12%,8)

A = 9000 * (0.2013)