Copyright ©2003 South-Western/Thomson Learning Chapter 10 Capital Budgeting and Risk.

Copyright ©2003 South-Western/Thomson Learning

Chapter 10Capital Budgeting and Risk

Introduction

• This chapter looks at adjusting a project’s risk level when it has more or less than the firm’s average risk level.

Risk

• Project risk– Project risk is the risk that a project will

perform below expectations.– Some of the risk can be diversified away.

• Beta risk– Beta risk depends on the risk of the project

relative to the market-portfolio.– Beta (systematic) risk cannot be diversified

away.

• Capital asset pricing model (CAPM)– The CAPM is used to estimate risk-adjusted

discount rates for capital budgeting.

Information on Risk

• The Society for Risk Analysis (SRA)– http://www.sra.org/index.htm

• Official journal of the SRA is Risk Analysis– http://www.sra.org/journal.htm

Adjusting for Beta Risk in Capital Budgeting

• The beta concept introduced in Chapter 5 for security risk analysis can also be used to determine risk-adjusted discount rates (RADR) for individual capital budgeting projects. This approach is appropriate for a firm whose stock is widely traded and for which there is very little chance of bankruptcy. – The probability of bankruptcy is a function of

total risk, not just systematic risk.

Adjusting for Beta Risk in Capital Budgeting

• Just as the beta (systematic risk) of a portfolio of securities can be computed as the weighted average of the individual security betas, a firm may be considered as a portfolio of assets, each having its own beta. From this perspective, the systematic risk of the firm is simply the weighted average of the systematic risk of the individual assets.

All Equity Case

• The project’s risk-adjusted discount rate is found with the SML equation:

* ( )e f m fk r r r

All Equity Case: Example

• For example, consider the security market line shown in Figure 10.4. The firm has a beta of 1.2 and is financed exclusively with internally generated equity capital. The market risk premium is 7 percent.


• When considering projects of average risk—that is, projects that are highly correlated with the firm’s returns on its existing assets and that have a beta similar to the firm’s beta (1.2)—the firm should use the computed 13.4 percent cost of equity from Figure 10.4.


• When considering projects having estimated beta different from 1.2, it should use an equity discount rate equal to the required rate of return calculated from the security market line.


• For example, if a project’s estimated beta is 1.7 and the risk-free rate is 5 percent, the project’s required rate of return would be 16.9 percent and this would be used as the risk-adjusted discount rate for that project, assuming the project is finance with 100 percent equity.

The Equity and Debt Case

• To better understand the material in this section, we briefly introduce the concept of a weighted (average) cost of capital, which is developed more extensive in Chapter 11.


• At this point, it is only necessary to recognize that the required return on the project discussed in this section reflects the project’s equity return requirement and the debt return requirement for the funds expected to be used to finance the project.


• Consider the example of Vulcan Industries, with a current capital structure consisting of 50 percent debt and 50 percent equity. Vulcan is considering expanding into a new line of business and wants to compute the rate of return that will be required on projects in this area.


• Vulcan has determined that the debt capacity associated with projects in its new business line is such that a capital structure consisting of 40 percent debt and 60 percent common equity is appropriate to finance these new projects.


• Vulcan’s company beta has been estimated to be 1.3, but the Vulcan management does not believe that this beta risk is appropriate for the new business line. Vulcan’s managers must estimate the beta risk appropriate for projects in this new line of business and then determine the risk-adjusted return requirement on these projects.


• Because the beta risk of projects in this new business line is not directly observable, Vulcan’s managers have decided to rely on surrogate market information. They have identified a firm, Olympic Materials, that competes exclusively in the line of business into which Vulcan proposes expanding. The beta of Olympic has been estimated to be 1.5.


• Recall from Chapter 5 that a firm’s beta is computed as the slope of its characteristic line and that actual security returns are used in the computations. Accordingly, a firm’s computed beta is a measure of both its business risk and its financial risk. When a beta is computed for a firm such as Olympic, it reflects both the business and financial risk of that firm.


• To determine the beta associated with Vulcan’s proposed new line of business using the observed beta from another firm (Olympic) that competes exclusively in that business line, it is necessary to convert the observed beta, often called a leveraged beta, l, into an unleveraged, or pure project beta, u.


• This unleveraged beta can then be releveraged to reflect the amount of debt capacity appropriate for this type of project and that will be used by Vulcan to finance it.

The Equity and Debt Case: Summary

• Betas can be observed for firms in the same investment class as the proposed investment.

• These betas can be used to estimate risk-adjusted discount rates.

• A two-step process is used: 1. Calculate an unleveraged beta.

2. Calculate a new leveraged beta to reflect appropriate debt capacity.

Step 1: Calculate an Unleveraged Beta

• Convert the observed, leveraged beta, l, into an unleveraged, or pure project beta, u.

– where u is the unleveraged beta for a project or firm, l is the leveraged beta for a project or firm, B is the market value of the firm’s debt, E is the market value of the firm’s equity, and T is the firm’s marginal tax rate.

1 (1 )( / )l

u T B E

Step 1: Calculate an Unleveraged Beta

• The use of this equation can be illustrated for the Vulcan Material example. The beta, l, for Olympic has been computed to be 1.50. Olympic has a capital structure consisting of 20 percent debt and 80 percent common equity and a tax rate of 35 percent. Substituting these values into the equation yields

1.501.29

1 (1 0.35)(0.25)u

Step 2: Calculate a New Leveraged Beta

• Calculate the new leveraged beta, l, for the proposed capital structure of the new line of business

[1 (1 )( / )]l u T B E


• The unleveraged, or pure project beta for the proposed new line of business of Vulcan is estimated to be 1.29. Vulcan intends to finance this new line of business with a capital structure consisting of 40 percent debt and 60 percent common equity. In addition, Vulcan’s tax rate is 40 percent.


• The equation can be rearranged to compute the leveraged beta associated with this new line of business, given Vulcan’s proposed target capital structure for the project:

1.29[1 (1 0.4)(40 / 60)]

1.81l

Step 2 Continued

• Calculating the required rate of return, ke, based on the new leveraged beta, l:

• Calculate the risk-adjusted required return, ka

*, on the new line of business:

ka* = %debt(ki) + %equity(ke

* )

*e ( )f m f lk r r r

Step 2 Continued

• With a risk-free rate of 5 percent and a market risk premium of 7 percent, the required rate of return on the equity portion of the proposed new line of business is computed from the security market line as

ke* = 5% + 7% 1.81 = 17.7%

Step 2 Continued

• If the after-tax cost of debt, ki, used to finance the new line of business is 8 percent, the risk-adjusted required return, ka

*, on the new line of business, given the proposed capital structure of 40 percent debt and 60 percent equity, is a weighted average of the marginal, after-tax debt and equity costs, or

ka* = 0.4(8%) + 0.6(17.7%) = 13.8%

Step 2 Continued

• Therefore, the risk-adjusted required rate of return on the proposed new line of business for Vulcan is 13.8 percent. This number reflects about the pure project risk and the financial risk associated with the project as Vulcan anticipates financing it.

Risk-Adjusted Net Present Value

• Suppose a company is considering a project whose net investment is $50,000 with expected cash inflows of $10,000 per year for 10 years. As shown in Table 10.2, the project’s NPV is $-1,670 when evaluate at a risk-adjusted discount rate (ka

*) and $6,500 when evaluated at the weighted average cost of capital (Ka).


TABLE 10.2 Calculation of NPV at Weighted Average Cost of Capital and Risk-Adjusted Discount Rate

Weighted Average Cost of Capital (ka = 12%)

Risk-Adjusted Discount Rate (ka*)

Year t Net Cash Flow

Interest Factor

Present Value

Interest Factor

Present Value

0 $-50,000 1.000 $-50,000 1.000 $-50,000 1-10 10,000 5.650 56,500 4.833 48,330 NPV $6,500 $-1,670


• Assuming that the 16 percent RADR figure has been determined correctly by using the security market line with an accurate beta value, the project should not be accepted even though its NPV, calculated using the company’s weighted cost of capital, is positive. This new product project is similar to Project 4 in Figure 10.5.


• The new product project discussed in the previous paragraph has an internal rate of return of about 15 percent, compared to its 16 percent required return. Therefore, the project should be rejected, according to the IRR decision rule.


• When the IRR technique is used, the RADR given by the SML frequently is called the hurdle rate. Some finance practitioners use the term hurdle rate to describe any risk-adjusted discount rate.


• Figure 10.5 illustrates the difference between the use of a single discount rate, the weighted cost of capital, for all projects regardless of risk level and a discount rate based on the security market line for each project. In the example in Figure 10.5, Projects 1, 2, 3, and 4 are being evaluated by the firm.


• Using the weighted cost of capital approach, the firm would adopt Projects 3 and 4. However, if the firm considered the differential levels of systematic risk for the four alternatives, it would accept Projects 1 and 3 and reject Projects 2 and 4.


• In general, the risk-adjusted discount rate approach is considered preferable to the weighted cost of capital approach when the projects under consideration differ significantly in their risk characteristics.

Adjusting for Total Project Risk

• The risk adjustment procedures discussed in this section are appropriate when the firm believes that a project’s total risk is the relevant risk to consider in evaluating the project and when it is assumed that the returns from the project being considered are highly correlated with the returns from the firm as a whole.

Adjusting for Total Project Risk

• Therefore, these methods are appropriate only in the absence of internal firm diversification benefits, which might change the firm’s total risk (or the systematic portion of total risk).

• In addition, total project risk can be measured by calculating the standard deviation and coefficient of variation. These calculations are discussed in Chapter 5.

Analyze Total Project Risk

• NPV-Payback approach

• Simulation approach

• Sensitivity analysis

• Scenario analysis

• Risk-adjusted discount rate approach

• Certainty equivalent approach

NPV-Payback Approach

• Many firms combine net present value (NPV) and with payback (PB) when analyzing project risk. As noted in Chapter 9, the project payback period is the length of time required to recover the net investment. Because cash flow estimates tend to become more uncertain further into the future, applying a payback cutoff point can help reduce this degree of uncertainty.

NPV-Payback Approach

• A project must have a positive NPV and a payback of less than a critical number of years to be acceptable.

• The net present value/payback method is both simple and inexpensive.

NPV-Payback Approach: Weakness

• First, the choice of which payback criterion should be applied is purely subjective and not directly related to the variability of returns from a project. Some investments may have relatively certain cash flows far into the future, whereas others may not. The use of a single payback cutoff point fails to allow for this.

NPV-Payback Approach: Weakness

• Second, some projects are more risky than others during their start-up periods; the payback criterion also fails to recognize this.

• Finally, this approach may cause a firm to reject some actually acceptable projects.

NPV-Payback Approach: Strength

• This approach is helpful when screening investment alternatives, particularly international investments in politically unstable countries and investments in products characterized by rapid technological advances.

NPV-Payback Approach: Strength

• Firms that have difficulty raising external capital and thus are concerned about the timing of internally generated cash flows often find a consideration of a project’s payback period to be useful.

Simulation Approach

• Estimate the probability distribution of each element which influences the CFs of a project.

• Elements:– Number of units sold– Market price– Unit production costs– NINV– Unit selling cost– Project life– Cost of capital

Simulation Approach

• Calculate the NPV using randomly chosen numerical values for the elements.

• Repeat the process until a probability distribution of the NPV can be estimated.

Simulation Approach

• In an actual simulation, the computer program is run a number of different times, using different randomly selected input variables in each instance. Thus, the program can be said to be repeated, or iterated, and each run is termed an iteration. In each iteration, the net present value for the project would be computed accordingly. Figure 10.1 illustrates a typical simulation approach.

Simulation Approach

• The results of these iterations are then used to plot a probability distribution of the project’s net present values and to compute a mean and a standard deviation of returns. This information provides the decision maker with an estimate of a project’s expected returns, as well as its risk. Given this information, it is possible to compute the probability of achieving a net present value that is greater or less than any particular value.

Simulation Approach: Strength

• The simulation approach is a powerful one because it explicitly recognizes all of the interactions among the variables that influence a project’s net present value. It provides both a mean net present value and a standard deviation that can help the decision maker analyze trade-offs between risk and expected return.

Simulation Approach: Weakness

• The simulation approach can take considerable time and effort to gather the information necessary for each of the input variables and to correctly formulate the model. This limits the feasibility of simulation to very large projects.

Simulation Approach: Weakness

• The simulation approach can work only if the values of the input variables are independent of one another. If this is not true, then the simulation must adjust for the dependence among the input variables, making the model even more complexity.

Sensitivity Analysis

• Sensitivity analysis is a procedure that calculates the change in net present value given a change in one of the cash flow elements, such as product price. In other words, a decision maker can determine how sensitive a project’s return is to change in a particular variable.


• Because sensitivity analysis is derived from the simulation approach, it also requires the definition of all relevant variables that influence the net present value of a project.

• The appropriate mathematical relationships between these variables must be defined, too, in order to estimate the cash flow from the project and compute the net present value.


• Rather than dealing with the entire probability distribution for each of the input variables, however, sensitivity analysis allows the decision maker to use only the “best estimate” of each variable to compute the net present value.


• The decision maker can then ask various “what if” questions in which the project’s net present value is recomputed under various conditions.


• Sensitivity analysis can be applied to any variable to determine the effect of changes in one or more of the inputs on a project’s net present value. This process provides the decision maker with a formal mechanism for assessing the possible consequence of various scenarios.


• It is often useful to construct sensitivity curves to summarize the impact of changes in different variables on the net present value of a project. A sensitivity curve has the project’s net present value on the vertical axis and the variable of interest on the horizontal axis. For example, Figure 10.3 shows the sensitivity curves for two variables, sales price and cost of capital for a project.


• The steep slope of the price-NPV curve indicates that the net present value is very sensitive to changes in the price for which the product can be sold.

• In contrast, the relatively flat cost-of-capital-NPV curve indicates that the net present value is not very sensitive to changes in the firm’s cost of capital.


• Similar curves could be constructed for project life, salvage value, units sold, operating costs, and other important variables.


• Spreadsheets, such as Excel, have made the application of sensitivity analysis techniques simple and inexpensive. Once the base case has been defined and entered in the spreadsheet, it is easy to ask hundreds of “what if” questions.

Scenario Analysis

• Scenario analysis considers the impact of simultaneous changes in key variables on the desirability of an investment project.

• When using the scenario analysis technique, the financial analyst might ask the project director to provide various estimates of the project’s expected net present value.

Scenario Analysis

• In addition to what is perceived to be the most-likely scenario, the project director might be asked to provide both optimistic and pessimistic estimates.

Scenario Analysis

• An optimistic (pessimistic) scenario might be defined by the most optimistic (pessimistic) values of each of the most input variables—for example, low (high) development costs, low (high) production costs, high (low) prices, and strong (weak) demand.

Scenario Analysis

• The project manager could also be asked to provide estimates of the probability that the optimistic scenario will result and the probability that the pessimistic scenario will result. With these probability estimates, the financial manager can compute an estimate of the standard deviation of the NPV of the project.

Risk-Adjusted Discount Rate Approach

• An individual project is discounted at a discount rate adjusted to the riskiness of the project instead of discounting all projects at one rate.

• ka* = rf + risk premium

• Calculate the NPV substituting ka* for k in

the formula.


• In the risk-adjusted discount rate approach, net cash flows for each project are discounted at a risk-adjusted rate, ka

* to obtain the NPV:

– The magnitude of ka* depends on the

relationship between the total risk of the individual project and the overall risk of the firm.

*1

NCFNPV NINV

(1 )

nt

tt ak


• When companies assume some amount of risk, it is expected to earn higher returns than those available on risk-free securities. The difference between the risk-free rate and the firm’s required rate of return (cost of capital) is an average risk premium to compensate investors for the fact that the company’s assets are risky.


• This relationship is expressed algebraically as follows:

Θ = ka – rf

– Θ is the average risk premium for the firm.

– rf is the risk-free rate (the yield on U.S. government securities, such as 90-day Treasury bills, is used as the risk-free rate).

– ka is the required rate of return for projects of average risk, that is, the firm’s cost of capital.


• The cash flows from a project having greater than average risk are discounted at a higher rate, ka*—that is, a risk-adjusted discount rate—to reflect the increased riskiness.

Certainty Equivalent Approach

• The certainty equivalent approach adjusts the net cash flows in the numerator of the NPV equation, in contrast to the RADR approach, which involves adjustments to the denominator of the NPV equation.


• A certainty equivalent factor is the ratio of the amount of cash someone would require with certainty at a point in time in order to make him or her indifferent between that certain amount and an amount expected to be received with risk at the same point in time.


• The project is adjusted for risk by converting the expected risky cash flows to their certainty equivalents and then computing the net present value of the project.


• The risk-free rate, rf —not the firm’s cost of capital, k—is used as the discount rate for computing the net present value. This is done because the cost of capital is a risky rate, reflecting the firm’s average risk, and using it would result in a double counting of risk.

• Certainty equivalent factors range from 0 to 1.0. The higher the factor, the more certain the expected cash flows.


• Algebraically, the certainty equivalent factor, t, for the cash flows expected to be received during each time period, t, are expressed as follows:

t = (Certain return)/(Risky return)


• The certainty equivalent factors are used to compute a certainty equivalent net present value as follows:

0 = Certainty equivalent factor associated with the net investment (NINV) at time 0 (Note: 0 =1)n = Expected economic life of the project

t = Certainty equivalent factor associated with the net cash flows (NCF) in each period, t

rf = Risk-free rate

01

NCFNPV -NINV( )

(1 )

nt t

tt fr


TABLE 10.1 Calculation of Certainty Equivalent Net Present Value Year Expected

NCF Certainty

Equivalent Factor ()

Certainty Equivalent Cash Flow

PVIF0.08, t Present Value of

Cash Flows 0 $-10,000 1.0 $-10,000 1.000 $-10,000 1 +5,000 0.9 +4,500 0.926 +4,167 2 +6,000 0.8 +4,800 0.857 +4,114 3 +7,000 0.7 +4,900 0.794 +3,891 4 +4,000 0.6 +2,400 0.735 +1,764 5 +3,000 0.4 +1,200 0.681 +817

Certainty Equivalent NPV +$4,753


• The initial outlay of $10,000 is known with certainty. Hence, the certainty equivalent factor for year 0 is 1.0, and the certainty equivalent cash flow is -$10,000 (= -$10,000*1.0).


• The $5,000 cash inflow in year 1 is viewed as being somewhat risky. Consequently, the decision maker has assigned a certainty equivalent factor yields a certainty equivalent cash flow of $4,500. This means that the decision maker would be indifferent between receiving the promised, risky $5,000 a year from now or receiving $4,500 with certainty at the same time.


• A similar interpretation is given to the certainty equivalent factors and certainty equivalent cash flows for years 2 through 5.

• Note that the certainty equivalent factors decline into the future. This reflects the fact that most cash flows are viewed as being more risky the further into the future they are projected to occur.


• In Table 10.1, we have computed the certainty equivalent net present value for this project assuming an 8 percent risk-free rate. It equals $4,753, and the project therefore is acceptable.


• The certainty equivalent approach of considering risk is viewed as conceptually sound for the following reasons:– The decision maker can adjust separately

each period’s cash flows to account for the specific risk of those cash flows. This normally is not done when the risk-adjusted discount rate approach is applied.


– Decision makers must introduce their own preferences directly into the analysis. Consequently, the certainty equivalent net present value provides an unambiguous basis for making a decision. A positive net present value means that the project is acceptable to that decision maker, and a negative net present value indicates it should be rejected.

Special Elements of Risk When Investing Abroad

• Captive funds

• Foreign government takes over assets

• Exchange rate risk– Risk of inflation

• Uncertain tax rates

Copyright ©2003 South-Western/Thomson Learning Chapter 10 Capital Budgeting and Risk.

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