Machinery's Handbook, 31st Edition
Cash Flow
147
i + − ( ) log log( 1) A A iP
n
=
Example: If an annuity of $200 is to be paid for 10 years, what is the present amount of money that needs to be deposited if the interest is 5%? Here, A = 200, i = 0.05, n = 10: P 200 1+0.05 ( ) 10 – 1 0.05 1 + 0.05 ( ) 10 = -------------------- $1,544.35 = The annuity a principal P drawing interest at the rate i will give for a period of n years is A P i 1 i + ( ) n 1 i + ( ) n – 1 = -------------- Example: A sum of $10,000 is placed at 4%. What is the amount of the annuity payable for 20 years out of this sum: Here, P = 10000, i = 0.04, n = 20: A 100000.04 1 + 0.04 ( ) 20 1+0.04 ( ) 20 – 1 -------------------- $735.82 = = Sinking Funds.— Amortization is the extinction of debt, usually by means of a sinking fund . The sinking fund is created by a fixed investment A placed each year at compound interest for a term of years n , and is therefore an annuity of sufficient size to produce at the end of the term of years the amount F necessary for the repayment of the principal of the debt, or to provide a definite sum for other purposes. Then, F A 1 i + ( ) n – 1 i = -------------- and A F i 1 i + ( ) n – 1 = -------------- Example: If $2,000 is invested annually for 10 years at 4% compound interest, as a sink ing fund, what would be the total amount of the fund at the expiration of the term? Here, A = 2000, n = 10, i = 0.04: F 2000 1+0.04 ( ) 10 – 1 0.04 ------------------- $24,012.21 = = Cash Flow Diagrams.— The following conventions are used to standardize cash flow di - agrams. The horizontal (time) axis (as seen in several patterns in Table 1 and Table 2) is marked off in equal increments, representing periods, up to the duration of the project. Receipts are represented by arrows directed upwards and disbursements are represented by arrows directed downwards. Arrow length is proportional to magnitude of cash flow. In the following, A = uniform annuity, i = interest rate, and n = number of payments or periods. Table 1. Cash Flow Patterns P -pattern P = present value A single amount P occurring at the beginning of n years. P represents Present amount.
P t =0
F t = n
A single amount F occurring at the end of n years. F represents Future amount. Equal amounts A occurring at the end of each of n years. A represents Annual amount. G is increasing by an equal amount for each time increment (e.g., month) over the period of life n . G represents Gradient amount.
F -pattern F = future value
A each
A -pattern A = annual value
t = 1
t = n
G -pattern G = uniform gradient of expense
G 2 G
( n −1) G
t = 2
t = n
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