Machinery's Handbook, 31st Edition
146 Cash Flow Example: For a nominal per annum rate of 12%, with monthly compounding, the effec tive per annum rate is: 1 0.12 12 ⁄ + ( ) 12 – 1 0.1268 12.7% effective annual interest rate = = = EIR Example: Same as before but with quarterly compounding: 1 0.12 4 ⁄ + ( ) 4 – 1 0.1255 12.6% effective annual interest rate = = = EIR Cash Flow and Equivalence Cash flow refers to the sum of money received or disbursed, as shown in a project’s financial report. Due to the time value of money, the timing of cash flows over a project’s lifespan plays a vital role in project success. Engineering economics problems involve four patterns of cash flow, both separately and in combination. Two cash flow patterns are equivalent if they have the same value at a particular time. Present Value and Discount.— The present value or present worth P of a given future or final sum F is the amount P that, when placed at interest i for a given time n , will produce the given amount F . Simple interest: P F ni F ni = + = + − 1 1 1 ( ) Compound interest: P F i F n n i = + = + − ( ) ( ) 1 1 The true discount D is the difference between F and P : D = F - P . Example: Find the present value and discount of $500 due in six months at 6% simple interest. Here, F = 500; n = 6 ⁄ 12 = 0.5 year; i = 0.06. Then, P = 500/(1 + 0.5 × 0.06) = $485.44. The discount is D = 500 – 485.44 = $14.56. Example: Find the sum that, placed at 5% compound interest, will in three years produce $5,000. Here, F = 5000, i = 0.05, n = 3. Then, P 5000 1+0.05 ( ) 3 = -------------- $4,319.19 = Annuities.— An annuity is a fixed sum paid at regular (uniform) intervals. In the formulas that follow, yearly payments are assumed. When a situation entails a monthly payment, such as rent, the interest rate and duration are modified as needed. It is customary to cal - culate annuities on the basis of compound interest. If an annuity A is to be paid out for n consecutive years, the interest rate being i , then the present value P of the annuity is P A 1 i + ( ) n – 1 i 1 i + ( ) n = -------------- If at the beginning of each year a sum A is set aside at an interest rate i , the total value F of the sum set aside, with interest, at the end of n years, will be F A 1 i + ( ) 1 i + ( ) n – 1 [ ] i = ------------------------ If at the end of each year a sum A is set aside at an interest rate i , then the total value F of the principal, with interest, at the end of n years will be F A 1 i + ( ) n – 1 i = -------------- If a principal P is increased or decreased by a sum A at the end of each year, then the value of the principal after n years will be F P 1 i + ( ) n A 1 i + ( ) n – 1 i -------------- ± = If the sum A by which the principal P is decreased each year is greater than the total yearly interest on the principal, then the principal, with the accumulated interest, will be entirely used up in n years:
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