CHAPTER 12

CAPITAL BUDGETING UNDER CERTAINTY

1. Shareholder wealth maximization occurs when projects are chosen with a rate of return greater than the market-determined rate of return or the cost of capital of the firm. Stated in other words, profit maximization occurs when the firm’s marginal costs equal its marginal revenue. Shareholder wealth maximization and profit maximization are synonymous with maximizing the value of the firm.

The relevant cash flows to examine are the expected cash inflows from the project over time and the initial and continuing outflows or costs associated with the project. The future cash flows that are incremental to the project or that accrue to the firm only as a result of the specific project in question should be examined. In addition, any decrease in cash flows to the company caused by the project in question, e.g., the tax depreciation benefit when old equipment is replaced by equipment, must be considered as well.

Accounting regulations attempt to adjust cash flows over several periods; e.g., the expense of an asset is depreciated over several time periods. Hence, project costs associated with the project are matched with the project revenues in the periods in which those revenues are expected to accrue. Because this time frame may span the life of the project, accounting for cash flows at a given point in time for capital budgeting purposes is accomplished by using economic cash flows. Economic cash flows are calculated as they occur to the firm.

2.a. The incremental after-tax operating cash flow (net cash inflow) in a given

period t is defined as the after tax incremental operating cash flow, (1 – τt)COt

plus the incremental depreciation tax subsidy, τtdept, where COt represents the

pretax incremental operating cash flow, τt represents the marginal tax rate, and

dept represents the incremental depreciation. Incremental means the change in

a particular item as a result of the firm undertaking the project.

b. A mutually exclusive project is one whose acceptance precludes the

acceptance of another project while an independent project can be accepted

even other projects are also accepted.

c. The opportunity cost of a new investment is the yield that is given up to

invest in the new project. The cost of capital is the rate of return required by

an investor to invest in a project which means that another investment must be

given up. Thus, the cost of capital is the investor's opportunity cost.

d. The accounting rate of return (ARR) method is the ratio of the investment's

annual net income after taxes to either total outlay or average outlay. (See

page 462 of text.)

e. The reinvestment rate is that rate at which cash flows from a project are

reinvested.

f. The profitability index, often referred to as the benefit/cost ratio, is defined as

the present value of (incremental) net cash inflows over the life of the project

divided by the initial outlay for the project.

g. Capital rationing refers to the scenario where a firm sets limits on the amount

of funds it will spend on fixed assets due to the large number of available

(profitable) investments.

3. In estimating cash flows without debt, the following equation expresses the

necessary tax adjustments:

CF = ICFBT (1 – T) + TD,where ICFBT = incremental CF before tax,

T = corporate tax rate, D = incremental annual change in depreciation (DEP) (i.e., DEP of new－

DEP of old).

Included in the CF, then, are the increase in income due to the new project, adjustments to the CF for additional taxes due to the increased income and due to the difference in DEP per year.

With debt financing we must also adjust the increased interest payments per year, net of tax savings. The first and last year CF will include investment, set-up costs, and salvage value, respectively.

4. Professor Pinches suggests the following process:a. Identification of areas of opportunity

b. Development of information

c. Selection of the best alternative or courses of action to be implemented, and

d. Control or feedback of the degree of success or failure of both the project and

the decision process itself

5. Both the IRR and the NPV have advantages relative to the ARR and the PBP in that they account for the time value of money. A disadvantage of both the IRR and the NPV is that they do not explicitly account for the risk of the project. Furthermore, the IRR and NPV can give conflicting accept/reject decisions

6. Linear programming can be used to solve the capital rationing problem by formulating an objective function which maximizes the NPV of the set of projects selected subject to constraints representing the scarcity of resources. For a detailed discussion of capital rationing with linear programming, see pp.476-480

7. a) The payback method calculates the time period required for the firm to recover the cost of its investment. It is the point in time at which the cumulative net cash

flow from the project equals the initial investment. The annual cash inflows are estimated for the project’s life. The cumulative time period is calculated at the point where the sum of annual cash inflows equals the initial investment. The time period is the payback period.

b) The accounting rate-of-return method averages the after-tax profit from an investment for every period over the initial outlay.

,

where APt = after tax profit in period t I = initial investment N = the life of the project.

c) The internal rate-of-return is that discount rate which equates the discounted cash flows from a project to the initial investment cost. Thus, IRR must be solved iteratively as follows:

,

where CFt = cash flow (positive or negative) in period t I = initial investment, N = the life of the project.

The IRR is then compared with the cost of capital of the firm to determine whether the project will return benefits greater than its cost.

d) The net present value of a project is computed by discounting the cash flows to the present by the appropriate cost of capital. A net present value greater than one means that undertaking the project will increase the value of the firm.

where k = the appropriate discount rate.

e) The profitability index is calculated by dividing the discounted cash flows by the initial investment to arrive at the present value per dollar of outlay:

.

The project should be undertaken if the PI is greater than 1.

The net present value, profitability index, and internal rate-of-return methods are theoretically and empirically more acceptable than the accounting rate-of-return and net payback period methods in that they explicitly consider the cost of capital and the time value of money.

8.

The net present value method assumes reinvestment of intermediate funds at a single discount rate. In contrast, the IRR method assumes that intermediate net cash flows are reinvested to earn a rate equal to the internal rate of return. The NPV and IRR methods may lead to similar accept/reject decisions; however, in some instances the two methods will provide different rankings for capital investment projects. Projects result in different rankings if the cost of one project is greater than that of another or if the timing of cash flow differs among projects. Moving along the horizontal axis to larger discount rates in the above diagram, note that the NPV of each project falls. In addition, the NPV of the project with larger future cash flows will be affected more severely by the increasing discount rate and thus the NPV of project A decreases more rapidly than that of project B. The points at which NPV = 0 are the projects’ IRR’s.

The ranking derived from the NPV differs from that of the IRR if the discount rate used in the NPV calculation is below point C or 7.1%. Thus the conflict

between the NPV and IRR rankings depends on whether the discount rate used by the firm is less than the cross-over rate (point C) or the rate that equates the projects NPV’s.

9. a) Multiple rates of return can occur using IRR if the cash flows change sign. When this occurs, there will be a new root solution to the IRR problem. It is also possible for the IRR solution to be imaginary. Multiple or imaginary roots make interpretation of the capital budgeting solution difficult.

b) IRR calculations use the assumption that the shareholder can reinvest the cash flows at the IRR. This causes two problems: (1) it divorces the discounting process from the cost of capital and (2) it implies that reinvestment rates are contingent on individual projects. In a world of certainty where each project has the same risk, it is illogical to assume that shareholders can invest funds for instance at both 14% and 25% depending on the IRR of the two projects.

c) When groups of projects are considered, IRR can lead to conflicting capital budgeting solutions, depending on what combination of projects are considered. Ideally, a decision rule should allow the manager to choose projects independently of one another. With this in mind, the IRR’s legitimacy is weakened.

Although these arguments are valid, the viability of using the IRR method should be evaluated only within the corporate or business context. The IRR may be the most appropriate return measure depending upon the firm’s specific policies, goals, and constraints. For example, in considering two mutually exclusive projects, firm policy may set a planning horizon of 4 to 5 years, regardless of what the length of the project’s useful life is. Given this constraint, the IRR is most appropriate. Although there are drawbacks implicit in the assumptions of IRR in real-world problems, IRR is often used in conjunction with other capital budgeting methods so that these assumptions are somewhat counterbalanced.

10. Inflation has a major impact upon all financial decisions of the firm mainly because tax deductibility of depreciation charges is based upon historical costs. Nelson goes on to isolate 5 areas in which inflation has impact on capital budgeting decisions:

a) The optimal level of capital investment will typically decrease as the rate of

inflation increases. The higher the rate of inflation, the larger the discount rate, and thus the lower the marginal present value for successive dollars invested.

b) High rates of inflation result in lower capital/labor ratios and thus influence the firm’s choice of technology. The “price” of labor does not depend on the rate of inflation, whereas the “price” of investment does depend on inflation. Thus the level of inflation corresponds to different price ratios between labor and capital, and influences the chosen amount of each factor.

c) Inflation’s impact on mutually exclusive projects deals with depreciation in the sense that depreciation rates are based on historical costs and the distribution of these charges over the life span of the projects determines the net present value.

d) When inflation is high, projects with shorter life spans will be favored over those with longer expected lives because those with shorter life spans will have their depreciation costs restated in current dollars more frequently as they are replaced.

e) This proposition relates to those projects whose lifetime is influenced by managerial decisions about replacement. In these cases, higher inflation will work against current replacement. This occurs because the present value of the replacement is directly influenced by the effect of inflation on future tax savings from depreciation. Thus, although with low inflation an investment may be replaced because of high operating costs, with high inflation the present value of the old project may be higher than that of the replacement project.

11. Practitioners face the problem of managing uncertainty when performing capital budgeting. Tuttle and Litzenberger suggest that returns on investment projects can be made risk-equivalent to the firm’s cost of equity capital by financing the projects with the proper amount of borrowing or lending. Assuming a competitive market with many small, risk-averse investors, the price of the firm’s common stock (its market trade-off point) is a function of its expected rate of return and estimated standard error of return. With perfect correlation of projected returns, acceptance criteria are obvious with two exceptions. These occur when the expected IRR of project i exceeds the firm’s equity capitalization rate, but the standard error of the estimate of the project’s IRR exceeds the firm’s standard error; or when IRRi < Ri, but Si < Sf. Then capital budgeting decision is not so clear. In these instances, the return is made “risk-equivalent” by neutralizing the

risk inherent in the project through a specific debt-equity ratio or through long-term borrowing and lending. The risk-adjusted rate of return equals the yield to maturity of the firms’ long-term debt plus the financing mix factor multiplied by the expected IRR less the yield to maturity. This then becomes the discount rate that reflects the risk-adjusted cost of financing the project, allowing for net present value analysis in the risk-equivalent context.

If the perfect correlation assumption is removed, the new cost of financing is determined through the financing ratio that equates the estimated standard error of returns after acceptance of the project with the error calculated before acceptance.

In markets dominated by large institutional investors, systematic risk is the relevant risk measure. In this framework, the relevant discount rate can be determined to make projects “risk-equivalent.”

Semi-variance is also used as a risk measure in capital budgeting because many managers are concerned only with downside risk. Probability of excess returns is not considered “risk” in this case.

The two methods most commonly used to quantify risk in this context are the certainty equivalent approach and the risk-adjusted discount rate approach.

Hastie feels the major problems for practitioners include capital rationing, judgment errors in estimation, and the failure of the financial analyst to be objective or realistic. Capital rationing may be caused by limits on borrowing; but theory does not offer a method of ranking projects with different risks, strategic purposes, and the quality of analytic support. Errors in judgment in estimating uncertain future profits may come from excessive pessimism or optimism or just bad judgment in general.

In dealing with these problems, overall corporate strategy should be clear and communicated to all involved planners and analysts. The analytic techniques employed should be well understood by all who work with them and should generate the relevant information a firm uses in its decision making. Perhaps academicians should place more attention on the non-quantitative aspects of the problem. That is, the project definition, estimation of CF, and implementation and review stages should be considered more by theoreticians.

Although in theory a project’s coavariance with other projects is the relevant risk measure, consideration must be given to non-quantifiable factors such as human dedication to the project. Such factors are important in determining risk associated with projects in real-world applications.

12. Linear programming can be used in capital rationing when the objective of the firm is to be maximized or minimized and the constraints limiting the firm’s actions are linear functions of the decision variables involved. The first step is to model the problem in linear programming form. This is done by a) identifying the controllable decision variables and b) defining the objective to be maximized or minimized and representing it as a linear function of the controllable decision variables. This is followed by defining the constraints and expressing them as linear equations or inequalities of the decision variables. The following assumptions are made:(1) the solution values of the decision variables are divisible, and(2) the constant coefficients are assumed known and deterministic. If these

assumptions are not valid, integer and stochastic programming can be used.

13. Traditional NPV techniques may not be appropriate to select a project from mutually exclusive investment alternatives, if the projects have different lives. The reason is that a short-lived project can be replicated more quickly than a long-lived project. In order to compare projects with different lives, we can compute NPV of infinite replications of the investment project. The adjustment to compensate for projects with unequal lives can be carried out using either equation (12.12) or (12.13). Besides this infinite replication method the manager can also use the finite replication method. To employ the latter approach the following formula should be used:

Definitions and interpretation for this formula’s components can be found on page 473-474.

14. NPV in break-even analysis uses the following model:

where NPV(k) = net present value of the project discounted at the cost-of-capita1 rate, k.

R(t) = the stream of cash revenues at time t,C(t) = the stream of cash outlays at time t,

t = the investment time horizon,p = the continuously compounded discount rate, equal to log (1 + k).

Once the NPV of various projects is determined, the values are plotted against units sold. Various discount rates are used to determine various breakeven plots.

Curves are drawn on the assumptions that total nonrecurring costs are (R + I) = $900mm, average ρ = 15.5, 3 products/month are produced and sold, development and initial investment phase = 42 months, corporate tax rate = 50%, and that costs will always have positive revenues. Discount rates are an important factor in dynamic break-even analysis. Here the break-even sales levels are 287, 360 and 510 units. In this model, finance theory, accounting information and mathematical tools are incorporated to make the project decision analysis more acceptable.

15. a) NPV(A) = ($4,351)(5.650)=$24,853.15 – 25,000 = $(146.85) NPV(B) = ($6,990)(3.605)=$25,198.95 – 25,000 = $ 198.95

b) NPB(A) = ($4,351)(5.216)=$22,694.82 – 25,000 =$2,305.18) NPV(B) = ($6,990)(3.443)=$24,066.47 – 25,000 = $(933.43)

c) NPV(A) = ($4,351)(6.145)=$26,736.89 – 25,000 =$1,736.89 NPV(B) = ($6,990)(3.791)=$26,499.09 – 25,000 =$1,499.09

Since the initial cost for both projects is $25,000, in Cash (a), Project B should be accepted, in Cash (b) neither project should be accepted, and in Cash (c) project A should be accepted.

(d) Using the information from (a), (b), and (c), students can graph the NPV profiles following the example given on page 467of the text.

16.

Project A

Project B

Project C

A is the best project.

17.

Project A: PBP = 2 years

Project B: PBP = 2 + (40/60) = 2 years

Project C: PBP = 100/120 = years

C is the best project if the payback method is adopted.

18.

Using the trial and error method, the IRR’s of projects A and B are given as

follows:

rA = 53.63%

rB = 27.09%

19.

a.

b. Infinite Replication

20. Whether project A or B is better depends on whether the appropriate discount

rate is less than or greater than the IRR.

For example:

Project A’s IRR

IRRA = 10%

Project B’s IRR

IRRB = 10%

CASE 1: If the discount rate is 8%, which is less than the IRR of 10%, then the

projects are both acceptable.

Since the NPV of B is less than the NPV of A, project A is the better of the two.

CASE 2

If the discount rate is 12% which is larger than the IRR of 10%, then the projects

are unacceptable.

NPVA = –5.33

NPVB = –5.15

NPVA < NPVB but both projects have negative) NPV’s and are therefore

unacceptable.

21.

ZZZ should reject this project, because the NPV is negative.

22.

a. Initial Net Cash Outlay

Cost of System $200,000Installation Expenses 50,000

Net Cash Outlay $250,000

b. Operating Cash Flows

Net Operating Cash Flow = Ct = CFt(1 – T) + (Td)

Depreciation: $250,000/5 = $50,000 per year

Savings in Operating Expenses

Salaries (10 × $15,000) $150,000Production Delays 8,000Lost Sales 12,000Timely Billing 3,000

$173,000

Additional Expenses:Salaries of Specialists $ 80,000Maintenance Expenses 12,000

$92,000Incremental Cash flow (173,000 – 92,000) = $81,000

Net Operating Cash flow = $81,000 (1 – .40) + (.40)(50,000)

= 48,600 + 20,000 = $68,600

c.

d. IRR (r):

A value of r = 11.5% results in the following NPV:

= 68,600 (3.6499) – 250,000

= 250,383.14 – 250,000

= $383.14

The actual IRR of 11.56% is slightly higher than 11.5%. However, the IRR is

still less than the required rate of return of 12%. Thus, the project is

unacceptable.

e.

The PI is less than 1.0. Thus, the project should be rejected.

f. Payback Period = 3 Years

g. Terminal Cash Flow:

Sale Price (t = 5) $25,000Tax on gains @ 40 % 10,000

Net TCF % $15,000

= 68,600(3.6045) + 15,000(0.5674) – 250,000

= 247,289.28 + 8511 – 250,000

= $5,800.28

Project is acceptable.

h. Depreciation = (250,000 – 20,000) / 5 = $46,000 per year

Operating Cash Flows = 81,000 (1 – .4) + (.40) (46,000) = $67,000

Terminal Cash Flow:

Book Value $ 20,000

Market Value 0

Loss $20,000

Tax Credit from loss on sale of asset (@ 40%) $8,000

Thus, the terminal value = $8,000.

NPV = 67,000 (3.6048) + 8000 (0.5674) – 250,000

= $241521.60 + 4539.20 – 250,000

= –$3939.20

The NPV is negative. The project should be rejected.

23.

a. Initial Net Cash Outlay:

Cashflow from Sale of Old Vehicle:

Book Value $ 30,000

Market Value 14,000

Tax Credit on Loss

(30,000 – 14,000)(.40) 6.400

Total Inflow $20,400

Initial Outlay

Cost of New Truck $90,000

Cash Flow From Sale of Old Truck $20,400

Net Outlay $69,600

Depreciation on new = $90,000/5 = $18,000 per year

Depreciation on old = $30,000/3 = $10,000 per year

b. Operating and Terminal Cash flows:

Savings:

Maintenance $ 8,000

Breakdowns 15,000

Total Savings $23,000

Net Annual Operating Cash Flow = CFt(1 – T) + Td

= 23,000(1 – .40) + (.40)(8,000)

= 13,800 + 3,200

= $17,000

Terminal Cash Flow:

Sale Price $20,000

Tax on gain @ 40% 8,000

$12,000

c.

d. The IRR is approximately 11.1%.

e.

The project is acceptable because:

NPV > 0

IRR > required rate of return

PI > 1.0

f. Payback Period = 4.0941 (assuming the terminal value of $12,000 is not

considered)

24.

NPV = (PI – 1.0) I

= (2 – 1)(5000)

= $5000

25.

Initial Cash Outlay :Land $ 40,000Building 80,000Equipment 20,000

Total Outlay $140,000

b. Operating and Terminal Cash flows:

Depreciation = (80,000 + 20,000)/10 = $10,000 per year

Cash flow from Operations:

Annual Revenues $ 90,000Variable Costs 27,000Fixed Costs 25,000

Annual Cash Flow $ 38,000

Annual Operating Cash Flow = CFt(1 – T) + dT

= 38,000(1 – .40) + (10,000)(.40)

= $22,800 + 4,000

= $26,800

Terminal Cash Flow:Sale Value of land and Building $95,000 $95,000

Book Value 40,000Gain on Sale $55,000

Tax on Gain (@ 40%) 22,000Net Terminal Cash Flow $73,000

c.

= 26,800(5.2161) + 73,000(0.2697) – 140,000

= $159,479.58 – 140,000

= $19,479.58

The IRR is approximately 17% as shown below:

The project is acceptable based on the NPV, IRR, and PI.

d. This would not affect the initial cash outlay. However, the operating cashflows

would be reduced by $13,000(1 – .40) = $78,000 resulting in annual operating

cash flows of $19,000. The terminal value cash flow would not change either.

NPV = 19,000(5.2161) + 73,000(0.2697) – 140,000

= $118,794 – 140,000

= –$21,206

IRR equals approximately 10.66 %.

The project should be rejected.

e. No. The allocation of existing overhead to the new facility is an accounting

adjustment only.

26.

Alpha (A) Gamma (B)

Initial Outlay $105,000 $ 90,000

Depreciation 105,000/10= $10,500

90,000/10= $9,000

Cash flow/year –$10,500 –$ 9,000

Annual Net Cash Flows:

Project A: = –10,500(1 – .40) + (.40)(10,500) = –$2,100

Project B: = –9,000(1 –.40) + (.40)( 9,000) = –$1,800

NPVB = –1800(6.1446) – 90,000

= –$101,060.28

Project Gamma should be selected because it has a lower negative NPV.