1 Cash Flow Cash flow is the sum of money recorded as receipts or disbursements in a project s financial records. A cash flow diagram presents the flow of cash as arrows on a time line scaled to the magnitude of the cash flow, where expenses are down arrows and receipts are up arrows. Year-end convention ~ expenses occurring during the year are assumed to occur at the end of the year. Example (FEIM): A mechanical device will cost $20,000 when purchased. Maintenance will cost $1000 per year. The device will generate revenues of $5000 per year for 5 years. The salvage value is $
2 Discount Factors and Equivalence 4-2a1 Present Worth (P): present amount at t = 0 Future Worth (F): equivalent future amount at t = n of any present amount at t = 0 Annual Amount (A): uniform amount that repeats at the end of each year for n years Uniform Gradient Amount (G): uniform gradient amount that repeats at the end of each year, starting at the end of the second year and stopping at the end of year n.
3 Discount Factors and Equivalence 4-2a2 NOTE: To save time, use the calculated factor table provided in the NCEES FE Handbook.
4 Discount Factors and Equivalence 4-2b Example (FEIM): How much should be put in an investment with a 10% effective annual rate today to have $10,000 in five years? Using the formula in the factor conversion table, P = F(1 + i) n = ($10,000)( ) 5 = $6209 Or using the factor table for 10%, P = F(P/F, i%, n) = ($10,000)(0.6209) = $6209
5 Discount Factors and Equivalence 4-2c Example (FEIM): What factor will convert a gradient cash flow ending at t = 8 to a future value? The effective interest rate is 10%. The F/G conversion is not given in the factor table. However, there are different ways to get the factor using the factors that are in the table. For example, or (F/G,i%,8) = (P/G,10%,8)(F/P,10%,8) = ( )(2.1436) = (F/G,i%,8) = (F/A,10%,8)(A/G,10%,8) = ( )(3.0045) = NOTE: The answers arrived at using the formula versus the factor table turn out to be slightly different. On economics problems, one should not worry about getting the exact answer.
6 Nonannual Compounding 4-3 Effective Annual Interest Rate An interest rate that is compounded more than once in a year is converted from a compound nominal rate to an annual effective rate. Effective Interest Rate Per Period Effective Annual Interest Rate Example (FEIM): A savings and loan offers a 5.25% rate per annum compound daily over 365 days per year. What is the effective annual rate? " i e = 1+ r % $ ' # m& m " (1= % $ ' # 365 & 365 (1= !
7 Discount Factors for Continuous Compounding 4-4 The formulas for continuous compounding are the same formulas in the factor conversion table with the limit taken as the number of periods, n, goes to infinity.
8 Comparison of Alternatives 4-5a Present Worth When alternatives do the same job and have the same lifetimes, compare them by converting each to its cash value today. The superior alternative will have the highest present worth. Example (EIT8):
9 Comparison of Alternatives 4-5b1 Capitalized Costs Used for a project with infinite life that has repeating expenses every year. Compare alternatives by calculating the capitalized costs (i.e., the amount of money needed to pay the start-up cost and to yield enough interest to pay the annual cost without touching the principal). NOTE: The factor conversion for a project with no end is the limit of the P/A factor as the number of periods, n, goes to infinity.
10 Comparison of Alternatives 4-5b2 Example (EIT8):
11 Comparison of Alternatives 4-5c Annual Cost When alternatives do the same job but have different lives, compare the cost per year of each alternative. The alternatives are assumed to be replaced at the end of their lives by identical alternatives. The initial costs are assumed to be borrowed at the start and repaid evenly during the life of the alternative. Example (EIT8):
12 Comparison of Alternatives 4-5d Cost-Benefit Analysis Project is considered acceptable if B C 0 or B/C 1. Example (FEIM): The initial cost of a proposed project is $40M, the capitalized perpetual annual cost is $12M, the capitalized benefit is $49M, and the residual value is $0. Should the project be undertaken? B = $49M, C = $40M + $12M + $0 B C = $49M $52M = $3M < 0 The project should not be undertaken.
13 Comparison of Alternatives 4-5e Rate of Return on an Investment (ROI) The ROI must exceed the minimum attractive rate of return (MARR). The rate of return is calculated by finding an interest rate that makes the present worth zero. Often this must be done by trial and error.
14 Depreciation 4-6a Straight Line Depreciation The depreciation per year is the cost minus the salvage value divided by the years of life.
15 Depreciation 4-6b Accelerated Cost Recovery System (ACRS) The depreciation per year is the cost times the ACRS factor (see the table in the NCEES Handbook). Salvage value is not considered.
16 Depreciation 4-6c Example (FEIM): An asset is purchased that costs $9000. It has a 10-year life and a salvage value of $200. Find the straight-line depreciation and ACRS depreciation for 3 years. Straight-line depreciation/year $9000 " $200 = 10 = $880 / yr ACRS depreciation First year ($9000)(0.1) = $ 900 Second year ($9000)(0.18)! = $1620 Third year ($9000)(0.144) = $1296
17 Depreciation 4-6d Book Value The assumed value of the asset after j years. The book value (BV j ) is the initial cost minus the sum of the depreciations out to the j th year. Example (FEIM): What is the book value of the asset in the previous example after 3 years using straight-line depreciation? Using ACRS depreciation? Straight-line depreciation $9000 (3)($800) = $6360 ACRS depreciation $9000 $900 $1620 $1296 = $5184
18 Tax Considerations 4-7a Expenses and depreciation are deductible, revenues are taxed. Example (EIT8):
19 Tax Considerations 4-7b Tax Credit A one-time benefit from a purchase that is subtracted from income taxes. Example (EIT8):
20 Tax Considerations 4-7c Gain or loss on the sale of an asset: If an asset has been depreciated and then is sold for more than the book value, the difference is taxed.
21 Bonds 4-8 Bond value is the present worth of payments over the life of the bond. Bond yield is the equivalent interest rate of the bond compared to the bond cost. Example (EIT8):
22 Break-Even Analysis 4-9 Calculating when revenue is equal to cost, or when one alternative is equal to another if both depend on some variable. Example (FEIM): How many kilometers must a car be driven per year for leasing and buying to cost the same? Use 10% interest and year-end cost. Leasing: $0.15 per kilometer Buying: $5000 purchase cost, 3-year life, salvage $1200, $0.04 per kilometer for gas and oil, $500 per year for insurance EUAC (leasing) = $0.15x, where x is kilometers driven EUAC (buying) = $0.04x + $500 + ($5k)(A/P,10%,3) ($1.2k)(A/F,10%,3) = $0.04x + $500 + ($5k)(0.4021) ($1.2k)(0.3021) = $0.04x + $2148 Setting EUAC (leasing) = EUAC (buying) and solving for x $0.15x = $0.04x + $2148 x = 19,527 km must be driven to break even
24 Additional Examples 4-11a Example 1 (FEIM): What is the uninflated present worth of $2000 in 2 years if the average inflation rate is 6% and i is 10%? d = i + f + if = (0.06)(0.10) = P = ($2000)(P/F,16.6%, 2) = ($2000)(1 + d) n = ($2000)( ) 2 = $1471
25 Additional Examples 4-11b Example 2 (FEIM): It costs $75 per year to maintain a cemetery plot. If the interest rate is 6.0%, how much must be set aside to pay for maintenance on each plot without touching the principal? (A) $1150 (B) $1200 (C) $1250 (D) $1300 P = ($75)(P/A,6%, ) = ($75)(1/0.06) = $1250 Therefore, (C) is correct.
26 Additional Examples 4-11c Example 3 (FEIM): It costs $1000 for hand tools and $1.50 labor per unit to manufacture a product. Another alternative is to manufacture the product by an automated process that costs $15,000, with a $0.50 per-unit cost. With an annual production rate of 5000 units, how long will it take to reach the break-even point? (A) 2.0 yr (B) 2.8 yr (C) 3.6 yr (D) never Cumulative cost (hand tools) = $ $1.50x, where x is the number of units. Cumulative cost (automated) = $15,000 + $0.50x Set cumulative costs equal and solve for x. $ $1.50x = $15,000 + $0.50x $1x = $14,000 x = 14,000 units t break-even = x/production rate = 14,000/5000 = 2.8 yr Therefore, (B) is correct.
27 Additional Examples 4-11d Example 4 (FEIM): A loan of $10,000 is made today at an interest rate of 15%, and the first payment of $3000 is made 4 years later. The amount that is still due on the loan after the first payment is most nearly (A) $7000 (B) $8050 (C) $8500 (D) $14,500 loan due = ($10k)(F/P,15%,4) $3000 = ($10k)( ) 4 $3000 = ($10k)(1.7490) $3000 = $14,490 ($14,500) Therefore, (D) is correct.
28 Additional Examples 4-11e Example 5 (FEIM): A machine is purchased for $1000 and has a useful life of 12 years. At the end of 12 years, the salvage value is $130. By straight-line depreciation, what is the book value of the machine at the end of 8 years? (A) $290 (B) $330 (C) $420 (D) $580 BV = $1000 ($1000 $130)(8/12) = $1000 $580 = $420 Therefore, (C) is correct.
29 Additional Examples 4-11f Example 6 (FEIM): The maintenance cost for an investment is $2000 per year for the first 10 years and $1000 per year thereafter. The investment has infinite life. With a 10% interest rate, the present worth of the annual disbursement is most nearly (A) $10,000 (B) $16,000 (C) $20,000 (D) $24,000 The costs or benefits for a cash flow that repeat should be broken into different benefits and costs that all start or finish at the time of interest. Take the $2000 cost that repeats for 10 years and break it into two $1000 costs to have one $1000 cost that goes on infinitely and one $1000 cost that goes on for 10 years. P = ($1000)(P/A,10%,10) + ($1000)(P/A,10%, ) = ($1000)(6.1446) + ($1000)(1/0.10) = $ $10,000 = $16,144.6 ($16,000) Therefore, (B) is correct.
30 Additional Examples 4-11g Example 7 (FEIM): With an interest rate of 8% compounded semiannually, the value of a $1000 investment after 5 years is most nearly (A) $1400 (B) $1470 (C) $1480 (D) $1800 i e = (1 + r/m) m 1= ( /2) 2 1 = F = ($1000)(F/P,8.16%,5) = ($1000)( ) 5 = ($1000)(1.480) = $1480 Therefore, (C) is correct.
31 Additional Examples 4-11h1 Example 8 (FEIM): The following data applies for example problems 8.1 through 8.3. A company is considering the purchase of either machine A or machine B. machine A machine B initial cost $80,000 $100,000 estimated life 20 years 25 years salvage value $20,000 $25,000 other costs $18,000 per year $15,000 per year for the first 15 years $20,000 per year for the next 10 years Example 8.1 (FEIM): The interest rate is 10%, and all cash flows may be treated as end-of-year cash flows. Assume that equivalent annual cost is the value of the constant annuity equal to the total cost of a project. The equivalent annual cost of machine B is most nearly (A) $21,000 (B) $21,500 (C) $23,000 (D) $26,500
32 Additional Examples 4-11h2 The $15,000 cost for 15 years and the $20,000 cost for the next 10 years can be broken into a $20,000 cost for the full 25 years and a $5000 benefit that is present for the first 15 years. The present worth of $20k for 25 years is P($20,25) = ($20k)(P/A,10%,25) = ($20k)(9.0770) = $181.54k The present worth of $5k for 15 years is P($5,15) = ($5k)(P/A,10%,15) = ($5k)(7.6061) = 38.03k P other costs = $ k + $38.03k = $143.51k A other costs = P other costs (A/P, 10%, 25) = ($143.51k)(0.1102) = $15,815 EUAC = ($100k)(A/P,10%,25) ($25k)(A/F,10%,25) + $15,815 = ($100k)(0.1102) ($25k)(0.0102) + $15,815 = $11,020 $255 + $15,815 = $26,610 ($26,500) Therefore, (D) is correct.
33 Additional Examples 4-11h3 Example 8.2 (FEIM): If funds equal to the present worth of the cost of purchasing and using machine A over 20 years were invested at 10% per annum, the value of the investment at the end of 20 years would be most nearly: (A) $548,000 (B) $676,000 (C) $880,000 (D) $1,550,000 P(A) = $80k + ($20k)(P/F,10%,20) ($18k)(P/A,10%,20) = $80k + ($20k)(0.1486) ($18k)(8.5136) = $80k + $2.972k $ k = $230,273 F(A,10%,20) = ($230,273)(F/P,10%,20) = ($230,273)(6.7275) = $1,549,162 ($1,550,000) Therefore, (D) is correct.
34 Additional Examples 4-11h4 Example 8.3 (FEIM): How much money would have to be placed in a sinking fund each year to replace machine B at the end of 25 years if the fund yields 10% annual compound interest and if the first cost of the machine is assumed to increase at a 6% annual compound rate? (Assume the salvage value does not change.) (A) $2030 (B) $2510 (C) $2540 (D) $4110 F = P(F/P,6%,25) salvage value = ($100,000)(4.2919) $25,000 = $404,190 A = F(A/F,10%,25) = ($404,190)(0.0102) = $4123 ($4110) Therefore, (D) is correct.
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