© 2007 Pearson Education Financial Analysis Supplement J.

© 2007 Pearson Education

Financial Analysis

Supplement JSupplement J


Future Value of an Investment

F = PF = P(1 +(1 + r r))nn

wherewhere

FF == future value of the investment at the future value of the investment at the end ofend of n n periodsperiods

PP == amount invested at the beginning, amount invested at the beginning, called the principalcalled the principal

rr == periodic interest rateperiodic interest raterr == number of time periods for which the number of time periods for which the

interest compoundsinterest compounds

The value of an investment at the end of the period over which interest is compounded.


Application J.1

Future Value of a $500 Investment in 5 Years

500(1 + .06)5 = 500(1.338) = $669.11


Present Value of a Future Amount

wherewhere

FF == future value of the investment at the future value of the investment at the end ofend of n n periodsperiods

PP == amount invested at the beginning, amount invested at the beginning, called the principalcalled the principal

rr == periodic interest rate (discount rate)periodic interest rate (discount rate)rr == number of time periods for which the number of time periods for which the

interest compoundsinterest compounds

P P ==FF

(1 +(1 + r r))nn

The amount that must be invested now to accumulate to a certain amount in the future at a specific interest rate.


Application J.2

Present Value of $500 Received in Five Years

500/1.338 =

$373.63


Present Value Factors

P P = = = = FFFF

(1 +(1 + r r))nn

11(1 +(1 + r r))nn

11(1 +(1 + r r))nn

= present value factor (or pf)= present value factor (or pf)


Present Value Factors (pf)

Present Value Factors for a Single PaymentNumber of Interest Rate (r)

Periods

(n) 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.14

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9259 0.9091 0.8929 0.8772

2 0.9803 0.9612 0.9426 0.9246 0.9070 0.8900 0.8573 0.8264 0.7972 0.7695

3 0.9706 0.9423 0.9151 0.8890 0.8638 0.8396 0.7938 0.7513 0.7118 0.6750

4 0.9610 0.9238 0.8885 0.8548 0.8227 0.7921 0.7350 0.6830 0.6355 0.5921

5 0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.6806 0.6209 0.5674 0.4194

6 0.9420 0.8880 0.8375 0.7903 0.7462 0.7050 0.6302 0.5645 0.5066 0.4556

7 0.9327 0.8706 0.8131 0.7599 0.7107 0.6651 0.5835 0.5132 0.4523 0.3996

8 0.9235 0.8635 0.7894 0.7307 0.6768 0.6274 0.5403 0.4665 0.4039 0.3506

9 0.9143 0.8368 0.7664 0.7026 0.6446 0.5919 0.5002 0.4241 0.3606 0.3075

10 0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.4632 0.3855 0.3220 0.2697


Present Value Factor (pf) for Application J.2


Application J.2 using the pf Factor


Annuities

P P = + + …= + + …FF(1 +(1 + r r))nn

FF(1 +(1 + r r))n+1n+1

oror P = A P = A (af)(af)

where where PP = present value of an investment = present value of an investment AA = amount of the annuity received each year = amount of the annuity received each year af = present value factor for an annuity af = present value factor for an annuity

A series of payments on a fixed amount for a specified number of years.


Present Value Factors (af)

Present Value Factors of an AnnuityPresent Value Factors of an AnnuityNumber of Interest Rate (r)

Periods

(n) 0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.10 0.12 0.14

1 0.9901 0.9804 0.9709 0.9615 0.9524 0.9434 0.9259 0.9091 0.8929 0.8772

2 1.9704 1.9416 1.9135 1.8861 1.8594 1.8334 1.7833 1.7355 1.6901 1.6467

3 2.9410 2.8839 2.8286 2.7751 2.7732 2.6730 2.5771 2.4869 2.4018 2.3216

4 3.9020 3.8077 3.7171 3.6299 3.5460 3.4651 3.3121 3.1699 3.0373 2.9137

5 4.8534 4.7135 4.5797 4.4518 4.3295 4.2124 3.9927 3.7908 3.6048 3.4331

6 5.7955 5.6014 5.4172 5.2421 5.0757 4.9173 4.6229 4.3553 4.1114 3.8887

7 6.7282 6.4720 6.2303 6.0021 5.7864 5.5824 5.2064 4.8684 4.5638 4.2883

8 7.6517 7.3255 7.0197 6.7327 6.4632 6.2098 5.7466 5.3349 4.9676 4.6389

9 8.5660 8.1622 7.7861 7.4353 7.1078 6.8017 6.2469 5.7590 5.3282 4.9464

10 9.4713 8.9826 8.3302 8.1109 7.7217 7.3601 6.7201 6.1446 5.6502 5.2161


Present Value Factor (af) for Application J.3

Interest Rate (r)(n) 0.06 0.08 0.10 0.12 0.14

1 0.9434 0.9259 0.9091 0.8929 0.8772

2 1.8334 1.7833 1.7355 1.6901 1.6467

3 2.6730 2.5771 2.4869 2.4018 2.3216

4 3.4651 3.3121 3.1699 3.0373 2.9137

5 4.2124 3.9927 3.7908 3.6048 3.4331


Application J.3

P = A (af)

A = $500 for 5 years at 6%

af = 4.2124 (from table)

P = 500(4.2124) = $2,106.20

Present Value of a $500 Annuity for 5 Years


Straight-Line Depreciation

D D ==I I – S– S

nn

wherewhere

DD = annual depreciation= annual depreciationII = amount of investment= amount of investmentSS = salvage value= salvage valuenn = number of years of project’s life= number of years of project’s life


Modified Accelerated Cost Recovery System (MACRS)

3-year class:3-year class: tools and equipment used in researchtools and equipment used in research

5-year class:5-year class: autos, copiers, and computersautos, copiers, and computers

7-year class:7-year class: industrial equipment and office furnitureindustrial equipment and office furniture

10-year class:10-year class: longer-life equipmentlonger-life equipment


Modified Accelerated Cost Recovery System (MACRS)

3-year class:3-year class: tools and equipment used in researchtools and equipment used in research

5-year class:5-year class: autos, copiers, and computersautos, copiers, and computers

7-year class:7-year class: industrial equipment and office furnitureindustrial equipment and office furniture

10-year class:10-year class: longer-life equipmentlonger-life equipment

Class of InvestmentYear 3-Year 5-Year 7-Year 10-Year

1 33.33 20.00 14.29 10.002 44.45 32.00 24.49 18.003 14.81 19.20 17.49 14.404 7.41 11.52 12.49 11.525 11.52 8.93 9.226 5.76 8.93 7.377 8.93 6.558 4.45 6.559 6.55

10 6.5511 3.29

100.0% 100.0% 100.0% 100.0%

Modified ACRS Depreciation AllowancesModified ACRS Depreciation Allowances


Example J.1Calculating After-Tax Cash Flows

YEARYEARITEMITEM 20082008 20092009 20102010 2011 2011 20122012 20132013 20142014

Initial InformationInitial InformationAnnual demand (salads)Annual demand (salads) 11,00011,000 11,00011,000 11,00011,000 11,00011,000 11,00011,000InvestmentInvestment $16,000$16,000Interest (discount) rateInterest (discount) rate 0.140.14

Cash FlowsCash FlowsRevenueRevenue $38,500$38,500 $38,500$38,500 $38,500$38,500 $38,500$38,500 $38,500$38,500Expenses: Variable costsExpenses: Variable costs 22,00022,000 22,00022,000 22,00022,000 22,00022,000 22,00022,000Expenses: Fixed costsExpenses: Fixed costs 8,0008,000 8,0008,000 8,0008,000 8,0008,000 8,0008,000Depreciation (D)Depreciation (D) 3,2003,200 5,1205,120 3,0723,072 1,8431,843 1,8431,843 922922

Pretax incomePretax income $5,300$5,300 $3,380$3,380 $5,428$5,428 $6,657$6,657 $6,657$6,657 – $922– $922Taxes (40%)Taxes (40%) 2,1202,120 1,3521,352 2,1712,171 2,6632,663 2,6632,663 – 369– 369

Net operating income (NOI)Net operating income (NOI) $3,180$3,180 $2,208$2,208 $3,257$3,257 $3,994$3,994 $3,994$3,994 – $533– $533

Total cash flow (NOI + D)Total cash flow (NOI + D) $6,380$6,380 $7,148$7,148 $6,329$6,329 $5,837$5,837 $5,837$5,837 $369$369

Local restaurant considering the addition of a salad bar:


Example J.2 Calculating NPV

2009:2009: $$ 6,380(0.8772)6,380(0.8772) == $$ 5,5975,5972010:2010: $$ 7,148(0.7695)7,148(0.7695) == $$ 5,5005,5002011:2011: $$ 6,329(0.6750)6,329(0.6750) == $$ 4,2724,2722012:2012: $$ 5,837(0.5921)5,837(0.5921) == $$ 3,4563,4562013:2013: $$ 5,837(0.5194)5,837(0.5194) == $$ 3,0323,0322014:2014: $$ 369(0.4556)369(0.4556) == $$ 168168

NPV = ($5,597 + $5,500 + $4,272 + $3,456 + $3,032 + $168) NPV = ($5,597 + $5,500 + $4,272 + $3,456 + $3,032 + $168) –– $16,000 $16,000NPV = $6,024NPV = $6,024


Example J.2 Calculating IRR

2009:2009: $$ 6,380(0.8772)6,380(0.8772) == $$ 5,5975,5972010:2010: $$ 7,148(0.7695)7,148(0.7695) == $$ 5,5005,5002011:2011: $$ 6,329(0.6750)6,329(0.6750) == $$ 4,2724,2722012:2012: $$ 5,837(0.5921)5,837(0.5921) == $$ 3,4563,4562013:2013: $$ 5,837(0.5194)5,837(0.5194) == $$ 3,0323,0322014:2014: $$ 369(0.4556)369(0.4556) == $$ 168168

NPV = ($5,597 + $5,500 + $4,272 + $3,456 + $3,032 + $168) NPV = ($5,597 + $5,500 + $4,272 + $3,456 + $3,032 + $168) –– $16,000 $16,000NPV = $6,024NPV = $6,024

IRR by Trial and Error

Discount Rate NPV

14% $ 6,02518% $ 4,09222% $ 2,42526% $ 97730% – $ 199

28% $ 322


Example J.2 Calculating Payback Period

YEARYEARITEMITEM 20012001 20022002 20032003 20042004 20052005 20062006 20072007

Initial InformationInitial InformationAnnual demand (salads)Annual demand (salads) 11,00011,000 11,00011,000 11,00011,000 11,00011,000 11,00011,000InvestmentInvestment $16,000$16,000Interest (discount) rateInterest (discount) rate 0.140.14

Cash FlowsCash FlowsRevenueRevenue $38,500$38,500 $38,500$38,500 $38,500$38,500 $38,500$38,500 $38,500$38,500Expenses: Variable costsExpenses: Variable costs 22,00022,000 22,00022,000 22,00022,000 22,00022,000 22,00022,000Expenses: Fixed costsExpenses: Fixed costs 8,0008,000 8,0008,000 8,0008,000 8,0008,000 8,0008,000Depreciation (D)Depreciation (D) 3,2003,200 5,1205,120 3,0723,072 1,8431,843 1,8431,843 922922

Pretax incomePretax income $5,300$5,300 $3,380$3,380 $5,428$5,428 $6,657$6,657 $6,657$6,657 – $922– $922Taxes (40%)Taxes (40%) 2,1202,120 1,3521,352 2,1712,171 2,6632,663 2,6632,663 – 369– 369

Net operating income (NOI)Net operating income (NOI) $3,180$3,180 $2,208$2,208 $3,257$3,257 $3,994$3,994 $3,994$3,994 – $533– $533

Total cash flow (NOI + D)Total cash flow (NOI + D) $6,380$6,380 $7,148$7,148 $6,329$6,329 $5,837$5,837 $5,837$5,837 $369$369

Payback Period

Add after-tax cash flows to get as close as possible to without exceeding the initial investment ($16,000)

$6,380 + $7,148 = $13,528 (2009 and 2010)

$16,000 – $13,528 = $2,472 (remainder for 2010)

$2,472/$6,329 = 0.39 (portion of 2010 required)

Payback Period = 2.39 years


OM Explorer Financial Analysis Solver

Salad Bar example:


NPV for ProjectApplication J.4

Year 1: $500Year 2: $650Year 3: $900

The discount rate is 12%, and the initial investment is $1,550, so the project’s NPV is:

Present value of investment (Year 0): ($1,550.00)

Present value of Year 1 cash flow: 446.40



Project NPV: $ 55.20


IRR for ProjectApplication J.5

Discount Rate

NPV

10% $500 (0.9091) + $650 (0.8264) + $900 (0.7513) = $117.88

12% $500 (0.8929) + $650 (0.7972) + $900 (0.7188)

= $ 55.20

14% $500 (0.8772) + $650 (0.7695) + $900 (0.6750) = ($ 3.72)


Payback Period for ProjectApplication J.6

Payback for year 1 = $500

Payback for years 1 and 2 = $500 + $650 = $1,150

Proportion of year 3 = ($1,550 $1,150) / $900 = 0.44 year

Payback periods for project = 2.44 years

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