FREQUENCY(range of values, range of bins)
FREQUENCY(range of values, range of bins)
Use the mouse to select Cells C85:C107, type =FREQUENCY(B70:GS70,B85:B107), and press Ctrl/ Shift/Enter. To be sure that all 200 NPVs have been counted, enter =SUM(C85:C107) in Cell C108. The result should be 200 in Cell C108.
To calculate the cumulative percent of values for the NPVs, enter =SUM($C$85:C85)/200 in Cell D85, copy the entry to D86:D107, and format the values as percents. The values for cumulative frequency should
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 467
run from 0% in D85 to 100% in D107. One way to create the downside risk curve is to plot the cumulative frequencies in Cells D85:D107 against the NPV bin values in Cells B85:B107.
Recall that the analysis of the skewness and kurtosis indicated the 200 values of NPV closely followed a normal distribution. We will test this in our downside risk curve for NPV by including a plot of the line for the cumulative percentages of a normal distribution with the average value and standard deviation shown in Cells B77 and B79 for the 200 NPV values. To calculate the values on this curve for the bin values, enter =NORMDIST(B85,$B$77,$B$79,TRUE) in Cell E85 and copy to E86:E107.
To create the downside risk chart for NPV shown at the right of Figure 15-9, highlight the range B85:B107, press and hold down the Ctrl key, highlight the range D85:E107, and release the Ctrl key. Click on the Chart Wizard button, select XY Scatter chart, and proceed as before in the Scenario section. The result is the downside risk chart at the right of Figure 15-9. Values in Cells D85:D107 are shown as solid points, and values in Cells E85:E107 have been plotted as a smooth curve. The downside risk curves and the values calculated indicate that there’s a probability of 20.5 percent that the investment will fail to break even by the end of five years.
Notice that the points in Figure 15-9 follow the curve closely. This, of course, is because our analysis of the skewness and kurtosis showed that the NPV values closely follow a normal distribution. This is not always the case. When the results of the iterations are highly skewed or are significantly more peaked or flatter than a normal curve, the downside risk curve should be plotted from the values for the iterations. But it does not hurt to test a normal distribution and satisfy yourself whether or not a normal distribution is justified. In many cases, as here, the values do follow a normal distribution fairly closely.
Downside Risk Chart for Modified Internal Rates of Return: Figure 15-10 shows a downside risk curve for the investment’s modified internal rate of return at the end of five years. The chart is prepared in the same manner as the downside risk chart for NPV.
Figure15-10
Downside Risk Curve for Modified Internal Rate of Return at End of Five Years
KL 109
ALADDIN GAMES: DOWNSIDE RISK CURVE FOR MIRR
110 MIRR Counts Cum.Freq. NormProb. 111
Will Be Less) 70%
The probability the investment 131 19.50%
WNSIDE RISK (Probability MIRR
will fail to earn an MIRR of 13.0% 132 20.00%
is about 20.5%
MODIFIED INTERNAL RATE OF RETURN, MIRR 136
Sum 200
468 ❧ Corporate Financial Analysis with Microsoft Excel ®
Risk Curve for Years to Break Even: Figure 15-11 shows a risk curve for the probability that the investment will take more than a specified number of years to break even. Note the differences between this curve and those for NPV and MIRR (Figures 15-9 and 15-10). The “downside” is that the years to break even will be more than the specified number of years to break even rather than less, as with the values of NPV and MIRR. Therefore, the cumulative frequency percentages in Cells D139:D159 have been converted to downside risks in Cells E139:E159 by subtracting their values from 100 percent. The curve ends to 20.5 percent for five years rather than zero percent because there’s a 20.5 percent chance the investment will fail to break even in five years, which is the duration of the analysis of the financial investment. It is not possible to determine a normal curve for the years to break even because, lacking values beyond five years, we do not know the average or standard deviation of the distribution of years to break even. Instead, we have inserted a second-order trend line through the points we do have. Reading from the chart or interpolating between values in the table gives a value of 4.53 years for the point at which there is a 50 percent chance the years to break even will be more or will be less.
Figure15-11
Risk Curve for Years to Break Even
B C D E F G H I J KL 137
ALADDIN GAMES: RISK CURVE FOR YEARS TO BREAK EVEN
en Counts Cum.Freq.
ears to
wnside
138 Y Break Ev
Do Risk
Will Be More
ears to Break Ev Y
The probability is 50% that it will
take more than 4.53 years for the investment to break even.
RISK (Probability
YEARS TO BREAK EVEN 160
Optimization of Wholesale Selling Price
Although values for the product’s total market, the cost of the investment, and the unit variable cost are beyond the ability of the company to determine exactly at the time the investment is to be made, the wholesale selling price is a variable the company can control. The goal is to select a selling price that maximizes the value of the investment. To determine the optimum selling price, use Excel’s Solver tool with the settings shown in Figure 15-12. The result, $4.65/set, is almost the same as determined for the optimum selling price for the most probable scenario in Chapter 14.
(Continued)
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 469
Figure15-12
Excel’s Solver Tool with Settings to Find Selling Price for Maximum NPV
Sensitivity Analysis
Figure 15-13 is a one-variable input table for analyzing the sensitivity of NPV and other payoffs to the wholesale selling price.
Figure15-13
One-Variable Input Table to Analyze the Sensitivity of Average Net Present Value and Other Financial Measures to the Unit Selling Price
(N.B.Thevalue0.00inCellG149fortheminimumyearstobreakevenatasellingpriceof $5.25/unitisanunfortunateresultofaspreadsheetlimitation.TochangetheentryinCellG149 tona,copytheRangeB140:H149andpasteitbackwithPasteSpecial/Values.Thenchangethe entryinCellG149to“na.”)
B C D E F G H 161
ALADDIN GAMES: SENSITIVITY TO UNIT WHOLESALE PRICE
Minimum
Probabililty
for Failing Wholesale
Unit
Years to
to Break 162
Market
Average
Average
Average
Break
Even 163 164
Price
Share
NPV
IRR
MIRR
Even
(Continued)
470 ® ❧ Corporate Financial Analysis with Microsoft Excel
To create the table shown in Figure 15-13, enter a series of selling prices in Cells B164:B173. Make the following entries in Row 163 to transfer values from the main body of the spreadsheet:
CellB163=B17
TransfersvaluesfromCellsB164:B1173
CellC163=B18
Transfersvaluesformarketshare
CellD163=B77 TransfersvaluesforaverageNPVatendof5years CellE163=C77
TransfersvaluesforaverageIRRatendof5years CellF163=D77
TransfersvaluesforaverageMIRRatendof5years CellG163=E76
Transfersvaluesforminimumyearstobreakeven CellH163=E82
Transfersvaluesforprobabilityforfailingtobreakeven To avoid confusion, the entries in Row 143 have been hidden by using “;;;” (i.e, three semicolons) to
custom format them. To complete the table, drag the mouse to select the Range B163:H173. Use Data/Table to access the Table dialog box and enter B17 for the column input value, as shown in Figure 15-14.
Figure15-14
Table Dialog Box with Entry for One-Variable Input Table
The tabular results in Figure 15-13 are used to create the charts shown in Figures 15-15 and 15-16. These show the sensitivity of the average net present value and the probability for breaking even in five years to whole- sale prices in the range from $4.25 to $5.00/unit.
Figure15-15
Sensitivity of Average NPV to Unit Wholesale Price
GE NPV $2,000
VERA A $0
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 471
Figure15-16
Sensitivity of Average Net Present Value and the Probability for Failing to Break Even in Five Years to the Whole Price
OBABILITY FOR F BREAK EVEN IN 5 20% PR
WHOLESALE PRICE, $/SET
Figure 15-15 shows that the NPV is reasonably close to its maximum so long as the selling price is within the range $4.50 to $4.75/unit, but that it drops off rather sharply below $4.50 and above $4.75/unit. Figure 15-16 shows similar behavior for the probability of breaking even within five years. In other words, choosing the best wholesale price is a critical decision. This type of curve is sometimes referred to as a “bath tub curve.” It has a relatively flat bottom with steep sides. So long as one sits near the bottom, there is not much change from the low point. But sitting on one of the steep sides exposes one to the danger of falling overboard.
An advantage of Monte Carlo simulation is that models of great complexity can be created. In the following case study, for example, the total market increases at first, reaches a maximum, and then falls off in the manner followed by many high-tech products with short lifetimes. The example also includes decreases in unit variable cost and selling price from the product’s initial values. Such reductions are typi- cal as production costs decline with the learning curve effect and as selling prices are dropped to maintain market share in the face of competition and consumer preference for newer products.
472 ❧ Corporate Financial Analysis with Microsoft Excel ®
Case Study: Allegro Products The chief financial officer (CFO) of Allegro Products has been asked to analyze the returns and risks on an
investment to manufacture a new product. The CFO will use four years for the financial analysis period, 12 percent for the discount and reinvestment rates, and 38 percent for the income tax rate. The total of selling costs and general and administrative (G&A) expenses are estimated to be 20 percent of sales revenue.
As a result of its market research, the firm’s marketing division has forecast the total, industry-wide demand for the type of product being considered will be 400,000 units during the first year of the product’s introduc- tion and will increase to 600,000 units during the second year; following that, the total market will drop to 500,000 units during the third year and 250,000 units during the fourth. The standard errors for the forecasts, as percentages of the total market, are 10, 11, 13, and 15 percent for the first, second, third, and fourth years, respectively.
Allegro’s share of the total market will depend on how much the company charges for its product. The marketing division estimates that at a selling price of $30.00/unit, the company’s share of the total market would
be 25 percent. They also estimate that increasing the selling price would reduce the market share according to the relation
where MS = market share (percent) and SP = selling price ($/unit). Thus, at a selling price of $31/unit, the predicted market share would be 24 percent; at a selling price of $32/unit, the predicted market share would
be 21 percent, and so forth. Actual market share would be normally distributed above the value predicted, with a standard deviation of 2 percent; that is, for a predicted market share of 24 percent, the range for one standard deviation about the predicted value would be from 22 to 26 percent. In order to retain the same percentage market share in the second, third, and fourth years as achieved in the first year, marketing analysts estimate that each year they will have to drop the selling price by 10 percent of the average selling price for the preceding year.
The firm’s industrial engineers estimate the required capital investment in equipment will most probably be $3.3 million, with a minimum of $2.5 million and a maximum of $4.5 million. The equipment will be depreciated to zero salvage value by straight-line depreciation over four years. Allegro’s industrial engineers also estimate that the variable cost of producing the product will be between $6.80/unit and $8.00/unit, with any value in that range equally likely. As a result of the “learning curve” effect, they expect that the average unit cost will decrease 10 percent each year.
Would you recommend Allegro to make the investment? Justify your recommendation. Solution: Figure 15-17 is the upper and lower portions of the spreadsheet solution for a selling price of
$30.58/unit, which is the optimum value for maximizing the average NPV. Data values are shown in the upper portion of Figure 15-17 and are italicized. Detailed results for the first 6 of 200 iterations are shown below the data section in Rows 19 to 82. For convenience, a summary of important results from 200 iterations is shown in the upper-right corner of Figure 15-17.
Total Market: The total market (in units) for each of the four years is simulated by using Excel’s random number generator for a normal distribution. For Year 1, the mean value is the data value for the forecast total market (in units) in Cell C12 and the standard deviation is the value calculated in Cell C14 by the entry =C12*C13. Two hundred values are inserted into Cells C20:GT20 by executing the random number genera- tor for the normal distribution with a mean of 400,000 and a standard deviation of 40,000 (the values in Cells C12 and C14). Note: To use the random number generator, it is necessary to specify the values (400,000 and 40,000) rather than the cell identities (C12 and C14).
For Years 2, 3, and 4, the random number generator is used with the mean values in Cells D12, E12, and F12 and the standard deviations in Cells D14, E14, and F14. The outputs are placed in Cells C21:GT21, C22:GT22, and C23:GT23.
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 473
Figure15-17
Rows 1 to 61 of Spreadsheet Solution for Allegro Products
1 Case Study: ALLEGRO PRODUCTS
2 Capital investment: Minimum (MIN)
Summary of Results for 200 Iterations
3 Most probable (MP)
Minimum
Average Maximum
4 Maximum (MAX)
5 Ratio, (MP-MIN)/(MAX-MIN)
6 Unit variable cost, year 1
Minimum
Period for analysis, years
4 Simulation Averages
Market, units 8 Annual cost decrease, pct
7 Maximum
Depreciation method
Straight line
0 Year 1 400,032 9 Selling & G&A expenses, pct of sales
Salvage value
Year 2 599,501 10 Discount and reinvestment rates
Income tax rate
Year 3 500,872 11 Year
Year 4 251,971 12 Total market forecast, units
Market Share 24.80% 13 Standard forecast error
Unit Var Cost $7.447 14 Standard forecast error, units
Investmt RN 0.5088 15 Unit selling price, years 1 to 4
Investment $3,448,541 16 Expected market share, pct
Selling price has been optimized.
Note how well simulation averages meet expectations. 19 Iteration Number
17 Std. deviation of market share, pct
18 Annual price decrease, pct
20 Total market, units
267,055 253,947 24 Random number for market share
1.1984 1.7331 25 Market share
27.06% 28.13% 26 Units sold
72,258 71,427 30 Sales receipts
1,610,968 $ 1,592,453 34 Unit variable cost
$7.166 $7.162 35 Cost of goods sold
377,459 $ 372,920 39 Gross profit
1,233,509 $ 1,219,533 43 Selling & G&A expenses
322,194 $ 318,491 47 Gross profit less selling
1,738,960 $ 1,996,253 48 and G&A expenses
911,316 $ 901,042 51 Investment random number
2,785,305 $ 3,102,048 53 Annual depreciation
696,326 $ 775,512 54 Taxable income
214,989 $ 125,530 58 Income tax, @ 38%
57 Year 4
Year 1
59 Year 2
60 Year 3
61 Year 4
474 ❧ Corporate Financial Analysis with Microsoft Excel ®
Figure15-17
Rows 62 to 82 of Spreadsheet Solution for Allegro Products (Continued)
A B C D E F G H 62 After-tax cash flow
829,620 $ 853,341 67 Net present value
$1,488,605 $1,214,938 71 Internal rate of return
36.66% 31.05% 72 Modified internal rate of return
24.65% 21.65% 73 Break-even point, years
74 Summary of Results for 200 Iterations Years to
Break Even
79 Std. Dev.
82 Probability for failing to break even in 4 years
Unit Selling Price: Start with an arbitrary value, such as $31.00/unit, for the Year 1 selling price in Cell C15. After completing the spreadsheet with the selected arbitrary value, we will use a one-variable input table to evaluate results for a range of selling prices, and we will use Solver to locate the optimum value. (To provide the results shown in Figures 15-17 and 15-18, the trial selling price for the first year has been replaced by the optimum value of $30.58/unit.)
In order to hold the expected market share constant for four years, Allegro plans to drop the selling price by 10 percent each year from the preceding year’s selling price (Cell C18). Selling prices for Years 2, 3, and 4 are calculated by entering =C15*(1-$C$18) in Cell D15 and copying it to E15:F15.
Expected Market Share: The expected market share is a function of the selling price selected, as defined by the equation given in the problem statement. Its value in Year 1 is calculated by entering =0.25-0.01*(C15- 30)^2 in Cell C16. Note that the market share will remain the same for all four years as a result of the reduction in selling price from its original value.
Actual Market Share: Values for the actual market shares are simulated in Cells C25:GT25. Because the marketing division does not know the relationship between selling price and market share exactly, the equation that forecasts market share as a function of price is subject to error.
The actual market share each year is expected to follow a normal distribution with a mean equal to the expected market share (Cell C16) and a standard deviation equal to 2 percent of the expected market share (Cell C17). To simulate the actual market share, we will add the product of the standard deviation multiplied by a random number that is normally distributed about a mean of zero and a standard deviation of one to the expected market share. The series of normally distributed random numbers is generated in Cells C24:GT24 by using the random number generator for a normal distribution with a mean of 0 and a standard deviation of
1. The actual market shares are then simulated in Row 25 by entering =$C$16+C24*$C$17 in Cell C25 and copying the entry to D25:GT25.
(Continued)
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 475
Units Sold: The number of units sold is the product of the total market for the product (Row 20 to 23) and Allegro’s share of the market (Row 25). The total market changes from year to year, whereas Allegro believes that by dropping its prices each year, it can maintain a constant market share equal to its first-year value. The units sold in each of the four years is simulated by entering =C20*C$25 in Cell C26 and copying the entry to C26:GT29. (Note the placement of the dollar sign in the entry.)
Sales Receipts: Sales receipts are the products of the units sold (Rows 26 to 29) and selling price (Cells C15 to F15). They are calculated by the following entries:
Cell
Entry
Copy to
C30
=C26*$C15
D30:GT30
C31
=C27*$D15
D31:GT31
C32
=C28*$E15
D32:GT32
C33
=C29*$F15
D33:GT33
Unit Variable Cost: Values for the unit variable cost of units sold during the first year can be simulated by using the random number generator for a uniform distribution between the values of $6.80 and $8.00 in Cells C6 and C7. The values are placed in Cells C34:GT34.
Cost of Goods Sold: The total cost of goods sold in any year equals the product of the number of units sold (Rows 25 to 28) and the unit variable cost. Recall that as a result of the “learning curve” effect, the unit variable cost drops each year by 10 percent of the cost the year before. Values for the four years are calculated by the following entries:
Cell
Entry
Copy to
C35
C26*C34
D35:GT35
C36
C27*C34*(1-$C$18)
D36:GT36
C37
C28*C34*(1-$C$18)^2
D37:GT37
C38
C29*C34*(1-$C$18)^3
D38:GT38
Gross Profit: Gross profit is the difference between sales receipts and the cost of goods sold. It is calculated in Rows 39 to 42 by entering =C30-C35 in Cell C39 and copying it to C39:GT42.
Selling and G&A Expenses: These expenses are estimated as 20 percent of the sales receipts (Cell C9). They are calculated in Rows 43 to 44 by entering =$C$9*C30 in Cell C43 and copying it to C43:GT46.
Gross Profit Less Selling and G&A Expenses: These are the difference between gross profit (Rows 39 to 42) and selling and G&A expenses (Rows 43 to 46). They are calculated by entering =C39-C43 in Cell C47 and copying it to C47:GT50.
Investment: Allegro’s capital investment to produce the new product has a most probable value (C3) and a range from a minimum to a maximum value (Cells C2 and C4). The general form of a triangular distribution is shown in Figure 15-5.
Using equations 15.1 and 15.2 on a spreadsheet can be implemented by the following steps: (1) Enter values for MIN, MP, and MAX in Cells C2, C3, and C4; (2) compute the ratio (MP-MIN)/(MAX-MIN) by the entry =(C3-C2)/(C4-C2) in Cell C5; (3) use Excel’s Random Number generator to generate a uniform series of random numbers between zero and one in Cells C51:GT51; and (4) enter the following expression in Cell C52 and copy it to D52:GT52:
=IF(C51<$C$5,$C$2+SQRT(C51*($C$3-$C$2)*($C$4-$C$2)), $C$4-SQRT((1-C51)*($C$4-$C$3)*($C$4-$C$2))
(Continued)
476 ❧ Corporate Financial Analysis with Microsoft Excel ®
Annual Depreciation: The capital investment is depreciated by the straight-line method to zero salvage value over a four-year life (Cells F8 and F6). Annual depreciation (Row 53) is therefore the same each year and equals one-fourth the capital investment (Row 52). To calculate annual depreciation, enter =(C52-$F$8)/$F$6 or =SLN(C52,$F$8,$F$6) in Cell 53 and copy it to D53:GT53.
Taxable Income: Depreciation is a deductible expense that reduces a firm’s taxable income. Taxable income is calculated by subtracting annual depreciation (Row 53) from the difference “gross profits less selling and G&A expenses” (Rows 47 to 50). To calculate taxable incomes, enter =C47-C$53 in Cell C54 and copy it to C54:GT57.
Income Tax: Income tax is the product of taxable income times tax rate. Allegro’s tax rate is given as 38 percent (Cell F9). To calculate income tax, enter =$F$9*C54 in Cell C58 and copy it to C58:GT61. After-Tax Cash Flow: The capital investment creates an after-tax cash outflow at time zero (Row 62). To calculate this, enter =-C52 in Cell C62 and copy it to D62:GT62. After-tax cash inflows for Years 1 to 4 are what is left of the “gross profit less selling and G&A expenses” (Rows 47 to 50) after paying income taxes (Rows 58 to 61). To calculate these, enter =C47-C58 in Cell C63 and copy it to C63:GT66.
Net Present Value: The net present value (NPV) of the investment is the future cash flows generated by it (Rows 63 to 66) discounted back to their present value less the capital investment. The future values are discounted at the given discount rate of money (Cell C10).
Use Excel’s NPV function to calculate the net present values of the investment at the end of each year. To do this, enter =NPV($C$10,C$63:C63)+C$62 in Cell C67 and copy it to D67:GT70. Internal Rate of Return at End of Four Years: Use Excel’s IRR function to calculate the internal rate of return (ROR). To do this, enter =IRR(C62:C66,0.1) in Cell C71 and copy it to D71:GT71. Modified Internal Rate of Return at End of Four Years: Use Excels MIRR function to calculate the modi- fied internal rate of return (MIRR). To do this, enter MIRR(C62:C66,$C$10,$C$10) in Cell C72 and copy it to D72:GT72.
Note that this calculation assumes that the rate of interest for reinvesting future cash flows is the same as the discount rate in Cell C10. Enter the correct rate if it is different. Break-Even Point, Years: To calculate the break-even point, enter the following in Cell C73 and copy it to D73:GT73: =IF(C68>0,1-C67/(C68-C67),IF(C69>0,2-C68/(C69-C68),IF(C70>0,3-C69/(C70-C69),”failed”))). Note that the investment fails to break even for the first iteration (Cell C73). Summary of Results for 200 Iterations: Use Excel’s MIN, AVERAGE, and MAX functions to calculate the
minimum, average, and maximum values of the NPV, IRR, and MIRR for the 200 iterations. The entries for NPV are =MIN(C70:GT70) in Cell C76, =AVERAGE(C70:GT70) in Cell C77, and =MAX(C70:GT70) in Cell C78. Similar entries are made for IRR and MIRR and for the minimum value of the number of years to break even. Note that we cannot calculate average or maximum values for the number of years to break even because the investment fails to break even on some of the iterations.
For convenience, the minimum, average, and maximum values of NPV and MIRR are transferred to Cells F4:H5 at the top of Figure 15-17. Rows 79, 80, and 81 show the values for several statistical measures of the distributions of NPV, IRR, and MIRR values. For NPV, the standard deviation is calculated entering =STDEV(C70:GT70) in Cell C79, the skewness is calculated by entering =SKEW(C70:GT70) in Cell C80, and the kurtosis is calculated by entering =KURT(C70:GT70) in Cell C81. Similar calculations are made for the distributions of IRR and MIRR values.
The probability for the investment’s failing to break even in four years is calculated by entering =(200- COUNT(C73:GT73))/200 in Cell F82. Downside Risk Analysis: Figure 15-18 shows the results of the downside risk analysis. Frequency distribu- tions for the NPV and MIRR values have been determined in the same manner as before—that is, by setting up bins that cover the range from slightly below the minimum values to slightly above the maximum values, using Excel’s FREQUENCY command to count the values in each bin, and converting the cumulative frequencies to the downside percentages. The downside percentages are shown as solid points on the charts at the bottom of Figure 15-18.
(Continued)
Capital Budgeting: Risk Analysis with Monte Carlo Simulation ❧ 477
Figure15-18
Downside Risk Analysis for Net Present Value and Modified Internal Rate of Return
NPV Bin Frequency
Percent
Normal Dist.
MIRR Bin
Frequency
Percent Normal Dist. –$800,000
Frequency Distribution, NPV Frequency Distribution, MIRR
Downside Risk Analysis
WNSIDE PR
WNSIDE RISK
(Continued)
478 ❧ Corporate Financial Analysis with Microsoft Excel ®
The normal distribution curves are created by using Excel’s NORMDIST function and the values for the averages and standard deviations of the 200 values of NPV and MIRR to calculate the cumulative probabilities for the bin values. These calculations are made with the following entries:
=NORMDIST(A86,$C$77,$C$79,TRUE)
D87:D100
H86
=NORMDIST(E86,$E$77,$E$79,TRUE)
H87:H108
Note that the normal distribution curve is a close approximation to the trend of the calculated values (plot- ted as points). This is because the distributions of the values are close to being normally distributed, as indicated by the values calculated earlier for skewness and kurtosis. This is not always the case. The values for the normal distribution are plotted as curves without points on the charts at the bottom of Figure 15-18.
Sensitivity Analysis: Figure 15-19 shows the impact of first-year selling prices on the expected market share, average NPV, average MIRR, minimum number of years to break even, and the probability that the investment will fail to break even by the end of four years. The table of results was created by using a one-variable input table with the following entries in Row 141:
G141 Entry
Cell B141
F82 Conclusions and Recommendation: The results show that the optimum selling price is located at about
$30.58/unit. At the optimum selling price, the probability for failing to break even is about 9 percent, and there is a 50:50 chance for doing better or worse than a net present value of approximately $680,000 and a modified internal rate of return of approximately 17.2 percent. Allegro Products should make the investment.