Sensitivity analysis in capital budgeting using crystal ball
This article illustrates how Crystal Ball software can be used in capital budgeting analysis to achieve a better indication of project results. We illustrate how sensitivity analysis and Monte Carlo simulation modeling, such as incorporated in Crystal Ball software, can be used to analyze and measure risk in capital budgeting. Using Monte Carlo simulation overcomes the limitations present in using just spreadsheet modeling. In Crystal Ball, a range of possible values for each uncertain input variable is described within the spreadsheet. The Monte Carlo simulation process then computes up to 1,000 possible forecasts (i.e. net present values), based on the input information and creates a probability distribution for the forecasts. The expected outcome (i.e net present value) is identified as the mean of all possible forecasts, which can be quite different from the outcome using static input variables. The techniques presented in this paper are considered to be especially useful to corporate financial managers.
Traditional capital budgeting analysis requires that investment decisions be dependent upon the results of discounted cash flow analysis. If the resulting net present value (NPV) from the discounted cash flows is positive, the project should be accepted. If the NPV is negative, then the project should be rejected. Cash flow estimates in most cases are forecasts of the unknown future. In many cases, the cash flows are assumed to follow a straight-line growth curve. Forecasts can be constructed using historical data and fitting these data to a time-series model or a regression analysis. However, whatever the method of obtaining these forecasts, they are only point estimates of an uncertain future.
Performing a discounted cash flow analysis on static cash flow estimates provides an accurate value of the project assuming all the future cash flows are known with certainty, that is, there exists zero volatility around the forecast values. However, in reality, business and economic conditions are difficult to forecast. Industry and product risk can be appropriately treated in cash flow analysis by adjusted the discount rate upwards to account for higher risk. However, uncertainty associated with cost and price variability is more difficult to account for.
Uncertainty means that actual cash flows may follow a different path than projected cash flows or they may fluctuate around the expected values. The uncertainty or volatility inherent in cash flow projections can be captured through sensitivity analysis. Sensitivity analysis refers to the attempt by financial managers to determine the implications of unknown events (Brealy et. al., 2001). Bodie and Merton (2000) suggest that sensitivity analysis in capital budgeting consists of testing whether the project will still be worthwhile even if some of the underlying variables turn out to be different from that projected.
A classic example of sensitivity analysis involves computing the NPVs of projected cash flows for a possible range of values taken on by an input variable (for example, future demand or future prices for the firm’s products). This procedure can be accomplished by using a spreadsheet software package, such as Excel. One can then modify the input variables and observe the affect on the net present value of the project. The use of spreadsheet modeling for sensitivity analysis, however, has limitations Exploring an entire range of possible outcomes is next to impossible, especially if there is more than one uncertain input variable. Thus the amount of risk associated with the underlying uncertainty of the input variables is difficult to quantify, at best, when using spreadsheet modeling.
The focus of this paper, and its contribution to the literature, is to illustrate how Monte Carlo simulation modeling, such as incorporated in Crystal Ball software can be used to analyze value and risk in capital budgeting. Using Monte Carlo simulation overcomes the limitations present in using just spreadsheet modeling. In Crystal Ball, a range of possible values for each uncertain input variable is described within the spreadsheet. The Monte Carlo simulation process then computes up to 1,000 possible forecasts (i.e. NPV), based on the input information and creates a probability distribution for the forecast. The expected forecast for the project is identified as the mean of all possible forecasts (i.e. NPVs), which can be quite different from the NPV using static input variables. Further, Crystal Ball can identify the likelihood of desired outcomes, which is very useful in valuing capital budgeting projects.
This paper presents a hypothetical capital budgeting replacement project. Two mutually exclusive alternatives are presented. One major feature of many replacement proposals is the cost savings that arise from the use of more efficient machinery or equipment. Another reason to replace equipment may be the reduction of variability of actual costs from projected amounts. If a company is replacing equipment decisions, regarding whether or not to replace and when to replace, have to be made. In analyzing a replacement decision the risk associated with the uncertainty of certain costs may be an important consideration in these decisions.
Our hypothetical company is faced with two alternatives. One alternative is to purchase a new machine to replace the existing equipment, right away. The other alternative is to wait two years, before purchasing the equipment. In this example, using static cash flows, the decision to purchase the equipment right away has a lower NPV than the NPV of the decision to wait two years. However the uncertainty the cash flows associated with the decision to purchase the equipment later will be much higher. Crystal Ball software can be used to get an indication of a more accurate measure of the NPVs and the associated risk for each of the decisions.
Replacement projects have two initial cash flows that have to be taken into consideration: the cost of the new equipment and the proceeds from sale of the old equipment. The proceeds from disposal of the old equipment should be adjusted for the tax effect – the difference between the sales price and the book value of the equipment is the gain or loss, and will create either a tax liability or a tax credit that has to be taken into consideration in the cash flow analysis. Other incremental cash flows that may arise from replacement analysis include changes in depreciation tax shields, projected disposal values of new equipment, and tax write-offs (expenses) of disposed equipment.
MACRS depreciation is used in this example. We assume the hypothetical replacement equipment has a 7-year tax life. We also utilize the new IRS ruling on depreciation for qualified property. The new rule allows a depreciation deduction of 30 percent of the adjusted basis of the qualified property purchased and placed in service after September 10, 2001 and before September 11, 2004. Due to the September 11, 2001 attacks, Congress moved quickly to encourage new investment by passing new depreciation rules. Qualified property is tangible property depreciated under MACRS with a recovery period of 20 years or less. The 30% depreciation deduction is taken in the year the asset is placed into service. The MACRS depreciation is then based on the adjusted balance (purchase price minus the 30 percent depreciation). The total allowable depreciation is not affected by the ruling. However, the higher depreciation tax shield in the first year of operating the new qualified property (and lower in the subsequent years) increases the value of the project.
We use nominal cash flows for all of our revenue and cost estimates, thus we adjust for inflation and real growth. The expected growth rate of revenues is 20%. After three years the expected real growth rate is expected to be constant at 7% for the remaining project life. We assume an inflation rate of 2.3% that is a five-year (1998-2002) average of the change in the US Consumer Price Index. An effective tax rate of 25% is assumed for the project life. Revenues for the current year are assumed to be $97.0 million.
We assume that our hypothetical company is considering the purchase of a replacement machine. If the new machine is purchased immediately, the purchase price would be $490.8 million. However the equipment must be purchased within two years, since the old equipment will be unable to handle the increased production at that time. At the end of two years the purchase price is expected to be $530.0 million, an 8% increase in price.
We assume a salvage value equal to book value, for the new equipment, at the end of the project life under both alternatives. The old machine has a book value of $54.3 million, which is being depreciated on a straight-line basis over the remaining life of three years. The current salvage value of the old machine is $40,000. The old machine will have a book value of $18.1 million at the end of two years. The expected salvage value of the old machine at that time is $21.1 million. We assume that the new property qualifies for the 30% tax depreciation benefit under both alternatives.
Equipment maintenance, production, and labor costs have been increasing, as a result of declining efficiency of the current equipment. Maintenance and production costs have averaged 30% of gross revenues. However operations managers believe that these costs will increase to 35% of gross revenues if the current equipment is used. Purchase of the new equipment will result in maintenance and production costs being 15% of revenues. The new replacement equipment comes with a contract that sets maintenance costs to 8% of gross revenues. It is expected that other production costs will be 7% of gross revenues if the new equipment is used.
In the past labor costs have averaged 35% of gross revenues. Because of inefficiencies of the old equipment, labor costs are expected to increase to 40% of the company’s gross revenues for the next two years, if the current equipment is used. However, it is believed that use of the new replacement equipment would result in labor costs consistently being 35% of gross revenues.
Several of the project assumptions are surrounded with some uncertainty. The input variables of the analysis which are uncertain include the annual real growth rate, expected maintenance and production costs, expected labor costs, and the purchase price of the new equipment in two years. The probability distribution of expected annual real growth rate of revenues for the next three years is expected to be a normal distribution in the range of 18% to 22%, with the expected growth rate equal to 20%. Operations managers believe that maintenance and other production costs could be as high as 50% of revenues or as low as 35%. Operations managers also believe labor costs could be as high as 55% and as low as 30%. The price of the equipment has fluctuated and it is possible that the equipment price could be as high as $570.6 million or as low as $510.4 million.
Static Discounted Cash Flow Analysis
Table 1 illustrates the discounted cash flow analyses for the first alternative; purchase the new replacement equipment now. Table 1, Panel A, presents the project assumptions inflation, real growth rates, and MACRS depreciation rates, as defined above. Table 1, Panel B illustrates the cash outlay on June 30, 2002 for the new replacement equipment. We assume the project is placed in service on June 30, thus the 30% depreciation deduction is allowed in the fiscal year ending June 30, 2002. The MACRS depreciation basis and net outlay for the replacement equipment, after the 30% depreciation deduction is -$343.6 million. The tax benefit from disposal of the old equipment is $3.6 million. This was computed by multiplying the effective tax rate (25%) times the loss, which is the difference between the disposal value and the book value of the old equipment ($40 million and $54.3 million, respectively). The total net cash outlay on June 30, 2002 is $300.0 million.
Table 1, Panel C, illustrates the operating cash flows for the next six years for the alternative to purchase the new replacement equipment now. Forecasted gross revenues are computed based on current revenues of $97 million using an inflation rate of 2.3%. The real growth rate is expected be 20% for the next three years and 7% for the following three years. Maintenance and production costs are 15% of gross revenues. Labor costs are 35% of grow revenues. The depreciation tax shields are also presented in Table 1, Panel C. The depreciation tax shield is calculated as the effective tax rate times the difference between the depreciation on the new equipment and the depreciation on the old equipment, as required in replacement analysis. Table 1, Panel D presents the total annual free cash flows and the Net Present Value of $ 11.0 million, using a discount rate of 12%.
Traditional discounted cash flow analysis favors the decision to wait. The net present value of the decision to wait is $22.5 million versus $11.0 million for purchasing the equipment now. Table 2 illustrates the discounted cash flow analysis of purchasing the equipment later on June 30, 2004. Table 2, Panel A, presents the expected maintenance and production, and labor costs as a percent of revenues. Table 2, Panel B presents the annual inflation rates, the expected annual real growth rates, and the MACRS depreciation rates. Note the MACRS depreciation starts in the year 2005, since the equipment is purchased at the end of 2004 under the decision to wait.
Table 2, Panel C, illustrates the cash outlay on June 30, 2004 for the new replacement equipment under the decision to purchase the equipment later. The purchase price in 2004 is expected to be $530 million, and increase of 8% over the 2002 price. We assume the project is placed in service on June 30, 2004, thus the 30% depreciation deduction is allowed in the fiscal year ending June 30, 2004. The MACRS depreciation basis and net outlay for the replacement equipment, after the 30% depreciation deduction is -$371.0 million. The net proceeds from the sale of the old equipment in 2004 is $20.4 million, which is the disposal price of $21.1 million less the tax, $.75 million, on the gain. This was computed by multiplying the effective tax rate (25%) times the difference between the disposal value and the book value of the old equipment ($21.1 and $18.1 million, respectively). The total net cash outlay on June 30, 2004 is $349.9 million.
Table 2, Panel D, illustrates the operating cash flows for the next six years for the alternative to wait two years to purchase the new replacement equipment. Forecasted gross revenues are computed based on current revenues of $97 million using an inflation rate of 2.3%. The real growth rate is expected to be 20% for the next three years and 7% for the following three years. Maintenance and production costs are expected to be 35% of gross revenues. Labor costs are 40% of gross revenues. The depreciation tax shields are also presented in Table 2, Panel D. The depreciation tax shield in the first two years is calculated as the effective tax rate times the depreciation on the old equipment. The depreciation tax shield for 2005 is computed as the effective tax rate times the difference between the depreciation on the new equipment and the remaining depreciation on the old equipment. Table 2, Panel E presents the total annual free cash flows and the Net Present Value of $22.5 million, using a discount rate of 12%.
The highlighted boxes in Tables 1 and 2 represent the uncertain input variables and the forecast variable (NPV), which have been identified in Crystal Ball Software. As illustrated in the next section, the results are somewhat different when sensitivity analysis and Crystal Ball software are introduced into the analysis, versus the results obtained from using static cash flow analysis.
Monte Carlo Simulation and Crystal Ball
Table 3 illustrates the assumptions (uncertainty descriptions) for each of the uncertain variables in the analysis. The only uncertain variable in the sensitivity analysis for the decision to purchase the equipment now is the real growth rate of revenues. Thus the risk associated with the projected cash flows is minimized under this alternative.
The uncertain variables for the alternative to purchase the equipment in two years are maintenance and production costs in on the current equipment, labor costs on the current equipment, and purchase price of the new equipment in two years, as well as the real growth rate of revenues. The probability distributions for each of the uncertain variables are assigned within the Crystal Ball software. A normal probability distribution is appropriate for growth rates (Mun, 2002), and is used in this example for the real growth rate of revenues. A triangular probability distribution is often used for costs or prices. With this distribution minimum, maximum and likeliest amounts are set. The distribution forms a triangle and is popular to use when limited data is available. A triangular distribution is used for maintenance and production costs, labor costs, and purchase price of equipment in 2004.
Table 4 illustrates a tornado diagram for the real growth rate. The entire volatility of the cash flows is attributed to this variable, under the first alternative. Table 5 presents the tornado diagram for the four uncertain variables in the second alternative. Tornado diagrams are used in sensitivity analysis to help determine the impact the uncertainty of certain input variables can have on outcomes. The tornado diagram illustrates the swing between the maximum and minimum forecast values for each variable, placing the variable that causes the largest swing at the top and the variable that causes the smallest swing at the bottom. The bars next to each variable represent the forecast value range across the variable tested. The direction of the bars next to each variable represent the direction of the relationship between the variables and the forecast.
Table 6 illustrates the Crystal Ball forecast of the NPV for the decision to purchase the equipment now. From Table 6, Panel A, the range of possible NPVs based on the real growth forecast is $ zero to $22 million with 99.4% certainty. As can be seen from the frequency distribution and statistics analysis, in Table 6, Panel B, the mean (expected) NPV is still $11 million, with a standard deviation of $4 million.
Table 7 illustrates the Crystal Ball forecast of the NPV for the decision to wait two years to purchase the equipment. From Table 7, Panel A, the entire range of values that the NPV can take is ($29 million) to $40 million. Using Monte Carlo simulation, the expected NPV under the alternative to wait to years to purchase the equipment if $9 million, with a standard deviation of $12 million (see Table 7, Panel B). In Table 7, Panel C reveals that the bottom two deciles of NPV are expected to take on a negative value. In Crystal Ball, one can identify the probability of achieving a certain amount of the forecast variable (NPV). Table 8, Panel A, illustrates this concept by showing that there is 80.1% certainty that the NPV will fall within the range of $0 to $40 million (a positive NPV). Thus we have 19.9% certainty that the NPV will be negative under the second alternative. Table 8, Panel repeats the forecast distribution from Table 7, Panel B, bust also pictorially shows the likelihood of a positive NPV.
These results indicate that the decision over when to purchase the equipment would be different after examination of the sensitivity analysis. The Crystal Ball simulation will give more realistic numbers on the capital budgeting analysis than the traditional NPV calculation since it takes into consideration the sensitivity of the forecast variables to the uncertainty or range of values assigned to the input variables. In this example, using Crystal Ball permits us to more accurately assess the NPV and quantify the risk associated the two alternatives.
By: Nancy Beneda, Ph.D.
Petter Gokstad is a Graduate Assistant for the Department of Finance, University of North Dakota, Box 7096 Grand Forks, ND 58202-7096, USA Dsokstad@hotmaiLcom
Nancy Beneda, Ph.D. is an Assistant Professor for the Department of Finance, University of North Dakota, Box 7096, Grand Forks, ND 58202-7096, USA nancy.beneda(a),und.nodak.edu
Copyright Credit Research Foundation Third Quarter 2003
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