NPV(project) = PV(with project) - PV(without project)Let's consider an example of this type of calculation.

The net present value is positive. This means that the firm should go ahead with acquisition. Notice that the market value of the computer is not included. It is irrelevant for the upgrading decision. Further note that a number of simplifying assumptions have been made such as a constant discount rate and zero tax rate. Let's be more precise about the capital budgeting decision. First, we need to introduce some notation.

*R_t*= $ cash revenue in time*t**E_t*= $ cash expenses in time*t**TAX_t*= $ taxes in time*t**D_t*= $ depreciation in time*t**T*= $ average and marginal tax rate*I_t*= $ Investment in time*t**S_t*= $ Salvage value in time*t*

X_t= R_t - E_t - TAX_t - I_t + S_tTaxes are defined to be:

Tax_t = T × (R_t - E_t - D_t)Substituting the expression for taxes into the first equation yields:

X_t = (1-T) × (R_t - E_t) + (T × D_t) - I_t + S_tNote that we are making a number of simplifying assumptions about the taxation. In an real world application, one would want to consider (1) carry forward and carry back rules, (2) investment tax credits, (3) sufficiency of taxable income, and (4) special tax circumstances (e.g. mining and petroleum).

X_t = (1 - T) × (R_t - E_t) + (T × D_t) - I_t + S_tIt is possible that

Year 1 Year 2 Year 3 Year 4 Year 5

Cash Revenue 140,000 140,000 140,000 140,000 140,000 Cash Expenses 40,000 40,000 40,000 40,000 40,000 Depreciation 60,000 60,000 60,000 60,000 60,000 Taxes Paid 13,600 13,600 13,600 13,600 13,600 After-tax Cash 86,400 86,400 86,400 86,400 86,400

Calculate the PV of the cash flows:

We could also work this out as:

The PV of the cash flows is $303,889. The initial outlay is $300,000 so the NPV of the project is $3,889.

Year 1 Year 2 Year 3 Year 4 Year 5

Cash Revenue 147,000 154,350 162,068 170,171 178,679 Cash Expenses 42,000 44,100 46,305 48,620 51,051 Depreciation 60,000 60,000 60,000 60,000 60,000 Taxes Paid 15,300 17,085 18,959 20,927 22,954 After-tax Cash 89,700 93,165 96,804 100,624 104,634

Calculate the PV of the cash flows:

The PV of these cash flows is $337,938. With the initial outlay of $300,000, the NPV is $37,938. The project is far more attractive now.

The value of the perpetuity is:

The nominal discount rate is 18%. The nominal cash flows are
growing at a constant rate of 10% per year. Next year's cash
flow is *.4 × 1.1=$.44 * million.
The present value is

This uses the formula of the present value of a growing perpetuity.
If we are assessing the present value (at time *t=0* of some cash flows that grow
at a constant rate *g* and the discount rate is *k*, the
formula is:

Year Sales('000) NWC('000) Capital('000)

0 0 400 20,000 1 5000 500 2 600 500 3 9000 700 4 10,000 700 5 10,000 700 6 10,000 700

Variable costs are 60% of sales, and fixed costs are negligible. The $20 million in production equipment (capital investment CI) will be depreciated straight-line to $0 over 5 years. It will actually be worth $10 million in six years. If 10% is the required return, should Trout proceed. The corporate tax rate is 34%.

Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6

Sales 5000 6000 9000 10000 10000 10000 Costs 3000 3600 5400 6000 6000 6000 Deprec. 4000 4000 4000 4000 4000 0

Operating Cash Flow 2680 2944 3736 4000 4000 2640

NWC 400 100 0 200 0 0 -700 CI 20000 -6600

Capital Required 20400 100 0 200 0 0 -7300

Cash Flows -20400 2580 2944 3536 4000 4000 9940

Notice that I am assuming that Trout can take full advantage of the depreciation tax shield. Notice also that the capital equipment is worth $10 million in year 6. When it is sold, the after tax revenue is $6.6 million. The present value of the future cash flows is $18.261 million. The total initial outlays is $20.4 million. So the NPV of this project is -$2.139 million. Hence it should be rejected.

Division Beta Required Return

Leisure 1.16 .07 + 1.16[.13 - .07] = 13.96% Graphics 1.64 .07 + 1.64[.13 - .07] = 16.84% Paint 0.70 .07 + 0.70[.13 - .07] = 11.20%

Now calculate the present values of the cash flows.

The highest net present value is found with Paint Inc.

$$V_0=\sum_{t=0}^{\infty}{X_t-I_t\over (1+r)^t}$$
where *X_t* represents the gross cash flows from investment projects
and *I_t* represents the costs of the investment projects. All
of the variables are after-tax measures.
Now consider *X_{j,t}* and *I_{j,t}* to be the incremental
after-tax net cash flows associated with candidate project
*j*. The value of the firm with the project is:

The candidate project should be accepted if:

Writing this out in terms of the definitions of the values:

This implies:

This is just the Net Present Value rule for project *j*. Hence,
the NPV rule always provides the correct investment decision.

The IRR is the interest rate that makes the value of the discounted cash flows equal to zero. The project selection rule is:

The rate *R* delivers the net present value. If the discount rate
is higher than *R* and the discounted value is non-negative, then
one would think that the IRR rule is delivering a similar evaluation
as Net Present Value. In fact, IRR can be consistent with NPV.
If the cash flows are *normal* and the term structure is
flat, then the IRR should given the correct evaluation for
independent projects. Consider the graphical representation:

Now let's consider when this rule breaks down:

Period Cash Flow

0 1000 1 -1500

This project can be thought of as borrowing. Now consider solving for the IRR.

We can safely assume that the *IRR* is greater than *R*. The rule suggests that
we should accept this project. Clearly, this is an incorrect
evaluation.

Period Cash Flow

0 -100 1 310 2 -220

Note that the cash flows switch sign from year to year. Now consider solving for the IRR.

There are two solutions to the IRR. At 100%, the IRR tells us to accept the project. At 10%, it is unclear what the IRR is telling us. We need to know what the discount rate for the firm is. If the discount rate is less than 10%, then the IRR is telling us to accept and the NPV is telling us to reject. If the discount rate is greater than 10%, then the NPV is telling us to accept and the IRR is telling us to accept and reject. Now consider a different set of cash flows.

Period Cash Flow

0 100 1 -310 2 220

Note that the cash flows switch sign from year to year. In this case, they are positive, negative and positive. Now solve for the IRR.

The IRR again gives us two solutions. If the firm's discount rate is less than 10%, then the NPV rule tells us to accept and both the IRRs also tell us to accept. If the firm's discount rate is greater than 10% but less than 100%, the NPV rule tells us to reject but we get conflicting advice from the IRR rule.

Period Cash Flow

0 100 1 -200 2 150

To solve for the IRR, set

This is just a quadratic equation. We can solve for the roots with the familiar formula:

where *a* is the coefficient on *x^2* term, *b* is the coefficient
on *x* and *c* is the constant.

We cannot solve the square root of a negative number with a real
number. The solutions to the IRR are *imaginary* numbers.
This does not help us evaluate the worth of the project. We
cannot provide a graphical representation of the differences
between NPV and IRR here unless we use the complex plane.
Note in this particular case that the NPV is greater than zero for
all discount rates.

Period Cash Flow Interest Rate

0 -1000 - 1 80 .20 2 80 .10 3 80 .04 4 80 .04 5 1080 .04

This just represents the cash flows from purchasing a five year bond than pays an 8% coupon. The internal rate of return is 8%. Now let's calculate the net present value of cash flows.

Period Discount Present Value

1 80/1.2 66.67 2 80/(1.2)(1.1) 60.61 3 80/(1.2)(1.1)(1.04) 58.27 4 80/(1.2)(1.1)(1.04)^2 56.03 5 1080/(1.2)(1.1)(1.04)^3 727.36

Present Value of Inflows 968.94 Investment 1000.00

Net Present Value -31.03

In the case of the non-uniform term structure, the IRR is not that meaningful of a measure.

Project Period 0 Period 1 IRR NPV

A -100 200 100% 82 B -10,000 15,000 50% 3636

Notice that the IRR rule tells us to accept project A and the NPV rule tells us to accept project B. The IRR rule can not distinguish between a $1 investment and a $1 million dollar investment.

Project Period 0 Period 1 Period 2 NPV(10%)

A -1000 1100 0 0 B -1000 0 1210 0

The IRR rule will not be able to distinguish between these projects. The IRR is 10%. For any discount rate, below 10% the IRR tells us that both projects are acceptable. But for any discount rate below 10%, project B has a higher NPV than A. Note that project A's cash flows all come in the first year whereas all of project B's cash flows come in the second year. Graphically,

The payback rule says to accept independent projects if

where *N^c* represents the life of the investment project.
Similarly, the payback rule says to accept the mutually
exclusive project that has the smallest *N^**.
The most obvious problem with payback is that it ignores
the time value of money. Cash flows in year 10 are treated
as cash flows today. The rule will not in general give
the correct evaluation of a project.

The payback rule says to accept independent projects if

where *N^c* represents the life of the investment project.
Similarly, the payback rule says to accept the mutually
exclusive project that has the smallest *N^**.
Any project with *N^* < T* has a positive NPV. Let's express
the NPV in terms of payback.

The second term in the NPV is ignored in the payback. This
term could represent very profitable cash flows. If you
are using payback, you may be excluding projects that have
positive net present value.
Now consider mutually exclusive projects. A lower $N^*$ does
*not* imply a higher NPV. Consider the following example.

Project Period 0 Period 1 Period 2 NPV(10%) N*

A -1000 1100 0 0 1 B -1000 0 1300 74.38 2

In this example, the discounted payback tells us to accept project A but project B has a higher NPV.

The rule says to accept projects that have:

Among mutually exclusive projects, accept the project with the highest index which is greater than one. There are no problems in using this rule for independent projects.

which is just another way to express net present value:

There are problems in using the profitability index to evaluate mutually exclusive projects. The index ignores scale. Consider the following example.

Project Period 0 Period 1 PI(10%) NPV(10%)

A -100 200 1.82 82 B -10,000 15,000 1.36 3636

The profitability index suggests that project A is superior to project B. However, the net present value of B far exceeds that of A.

- If the firm is all-equity financed,
*B=0*,*WACC=R_S*, i.e.,*WACC*is the cost of equity; - If the firm is all-debt financed,
*S=0*,*WACC*=R_B(1-T), i.e.,*WACC*is the after-tax cost of debt.

- The project and the firm have the same systematic risk
- The project and the firm have the same debt capacity

2 All-equity value = -10 + --- = 0 0.2Now the PV of tax-shield is:

T × B × R_B Additional effects of debt = ----------- = T × B = $1.7 million R_BHence, the adjusted present value is

APV = 0 + 1.7 = $1.7 millionIn general, the additional effects of debt include:

*floatation costs*,*tax shield from debt*,*effects of subsidized financing*.

- Use WACC if the project is
*close to*scale-enhancing - Use APV if the project is
*far from*scale-enhancing.

**Step 1.** Modeling

**Step 2.** Specifying Distributions for Cash Flows

We do not know the cash flows for sure,
but we can forecast them or make our best guesses.
Assume *CF_1*
and *CF_2* follows normal distributions,
*CF_1* has mean $70 with standard error $7
and *CF_2* has mean $60 and
standard error $12. This implies that
$70 is the expected cash flow next year.
However, due to some uncertain economic conditions,
the actual cash flow next year will be different from
$70. But we believe it has about 68.26% chance to
be in $70 plus or minus $7 and 95.44% chance to
be in $70 plus or minus $14. Or if we take $70
as the forecasting value of *CF_1*,
we have 68.26% chance
of making 10% forecasting error and
95.44% of making 20% error.

**Step 3.** Draw Cash Flows from the Specified Distributions

A computer can be used to easily draw
*CF_1* and *CF_2* from two
normal distributions with means 70
and 60, standard errors
7 and 12 respectively.

**Step 4.** Compute the NPV

With cash flows drawn in Step 3, the NPV
is computed from the model of Step 1.

**Step 5.** Repeat 3 and 4 for a Specified Number of Times

Generally, hundreds or thousands repeated computations of
3 and 4 are required. 10,000 times would be a good choice for
many problems.

**Step 6.** Determine the Distribution of NPV

With the hundreds or thousands repeated computed values of NPV
from Step 5, we are able to determine the distribution
of NPV, in particular the mean and standard error. All this
is what we need for our decision making.

- Burns oil
- Burns either oil or coal

- Option to completely terminate
- Option to stop production temporarily

In short a project with liquidation possibility is worth more than project without possibility of abandonment.

- Change output rate per unit of time.
- Change total length of production run time.

Project with option to expand is worth more than project without possibility of expansion.

Project with option to contract is worth more than project without possibility of contraction.

Project whose operation can be dynamically turned on and off (or switched to two distinct locations) is worth more than the same project without the flexibility to switch. A flexible manufacturing system (FMS) is a good example of this type of option. Other examples--

- Choose plant with high maintenance costs relative to construction costs. Management gains the flexibility to reduce the life of the plant and contract the scale of project by reducing expenditures on maintenance.
- Build plant whose physical life exceeds the expected duration of use (thereby providing the firm with the option of producing more by extending life of project).

- Prices paid for leases would decline
- Some leases would not be purchased at all.

- initial R&D with 20% chance of success
- clinical testing, with 50% chance of success
- ing I, with 40% chance of success
- ing II, with a 90% chance of success.

- Ignoring options can destroy firm value (projects may be rejected that should be accepted)
- Options are to be found in every aspect of the firm's operations
- Options need to be recognized -- and valued properly.

*Merger or Consolidation*- With a
*merger*, one firm absorbs another. The acquiring firm retains its name and identity, but the acquired firm ceases to exist. - With a
*consolidation*, a new firm is created. Both firms involved terminate their previous legal existence.

- With a
*Acquisitions of Stock*- A firm buys another firm's voting stock
in exchange for cash, stock, or other securities.
This is often done by a
*tender offer*, a public offer to buy the stocks directly from shareholders.

- A firm buys another firm's voting stock
in exchange for cash, stock, or other securities.
This is often done by a
*Acquisitions of Assets*- A firm can buy another firm by purchasing the assets of the target firm.

*Merger or Consolidation*- + legally simple.
- + Relatively inexpensive, no transfers of titles necessary.
- - All liabilities assumed, including potential litigation.
- - 2/3 of shareholders (most states) of both firms must approve.
- - Dissenting shareholders can sue to receive their `fair' value (`appraisal rights')

*Acquisitions of Stock*(tender offer)- + No shareholder meetings or votes necessary
- + Bidder can bypass management and go directly to shareholders.
- - Resistance by the target firm's management makes the process costly.
- - Often a minority of shareholders hold out.

*Acquisitions of Assets*- + Only needs 50% of shareholders' approval, thus avoiding dissident minority shareholders.
- - transfers of assets may be costly in legal fees.

*Horizontal Acquisition*- Acquisition of a firm in the same industry as the acquiring firm.

*Vertical Acquisition*- Acquisition of a firm at a different step of production from the acquiring firm. For example, the ill-fated strategy of Kodak acquiring Sterling Drugs.

*Conglomerate Acquisition*- Acquisition of a firm in unrelated business.

*Economies of scale*- Share costly equipment, facilities and personnel, reduce the cost of flotation.

*Acquire valuable technologies and resources*- For example, many oil company acquisitions took place because it was cheaper to buy existing reserves than to explore new ones.

*The target company is undervalued*- The target firm's management may not operating the firm to
its full potential, leaving room for another firm to takeover
and realize the value. Alternatively, the acquiring firm
may have insider information on the target firm
which leads them to believe the firm has a value higher
than the current market value. For example, it is now common
to see `expert' on TV giving estimates of a company's
*break up*value. If this exceeds the company's market value, a takeover specialist could acquire the firm at or somewhat above the current market value, sell it off in pieces, and earn a substantial profit.

- The target firm's management may not operating the firm to
its full potential, leaving room for another firm to takeover
and realize the value. Alternatively, the acquiring firm
may have insider information on the target firm
which leads them to believe the firm has a value higher
than the current market value. For example, it is now common
to see `expert' on TV giving estimates of a company's
*Tax considerations*- A firm with large tax loss carry-forwards may be attractive to another firm that can use the tax benefits. However, IRS may disallow the use of tax loss carry-forwards if no business purpose for the acquisition is demonstrated. Furthermore, under 1986 Tax Reform Act, the carry-forwards is limited.
- Some firms which have unused debt capacity may make them acquisition candidates. The acquiring firm can deduct more interest payments and reduce taxes. For example, this was cited as the logic behind the proposed merger of Hospital Corporation of American and American Hospital Supply in 1985. Insiders said the combined company could increase debt by $1 billion.

*Inefficient management of the target company*- Management could be bad relative to others in the same industry, leading to a horizontal merger. Or, it could be bad in absolute sense, leading to a conglomerate merger. Anybody can come in and do better.

*Market power*- One firm may acquire another to reduce competition. If so, prices can be increased and monopoly rents obtained. However, mergers that reduce competition may be challenged by the US Department of Justice and the Federal Trade Commission.

*Diversification*- A cash rich company may use the cash for acquisitions rather than to pay it out as dividends. A frequent argument for this is that it reduces the investor's risk in the company, thus achieving diversification. However, investors can diversify on their own, likely more easily and cheaply than can the company.

*Pac-man Defense**White Knight**Lockup Defense**Scorched Earth Defense**Golden Parachutes**Poison Pills**Greenmail**Create an Antitrust Problem**Change the state of incorporation**Stalling tactics**Shark repellent charter amendements**Dual class recapitalization*

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