Decision TechnologyDecision TechnologyModeling, Software and ApplicationsModeling, Software and Applications

Matthew J. LiberatoreMatthew J. Liberatore

Robert L. NydickRobert L. Nydick

John Wiley & Sons, Inc. John Wiley & Sons, Inc.

Financial Simulation Financial Simulation Using @RiskUsing @Risk

CASH FLOW FOR GMGM is trying to estimate the cash flows from a new car GM is trying to estimate the cash flows from a new car

that will sell for 5 years. During the current year that will sell for 5 years. During the current year (year 0) a fixed development cost of $1.4 billion is (year 0) a fixed development cost of $1.4 billion is incurred and is depreciated on a straight-line basis incurred and is depreciated on a straight-line basis over the next 5 years. over the next 5 years.

Year 1 unit sales follow a triangular distribution with a Year 1 unit sales follow a triangular distribution with a worst case of 100,000 units, most likely case of worst case of 100,000 units, most likely case of 150,000, and best case of 170,000. 150,000, and best case of 170,000.

Sales during years 2-5 are assumed to decay at the same Sales during years 2-5 are assumed to decay at the same rate each year. This annual decay rate follows a rate each year. This annual decay rate follows a triangular distribution with a best case of 5%, most triangular distribution with a best case of 5%, most likely case of 8%, and a worst case of 10%.likely case of 8%, and a worst case of 10%.

Each year a car sells for $15,000.Each year a car sells for $15,000.

CASH FLOW FOR GMDuring year 1 each car sold incurs a variable During year 1 each car sold incurs a variable

cost of $10,000. Variable cost increases 4% cost of $10,000. Variable cost increases 4% per year.per year.

The tax rate is 40% and the cash flows are The tax rate is 40% and the cash flows are discounted at 15% per year.discounted at 15% per year.

Assume that all cash flows occur at the end of Assume that all cash flows occur at the end of the year.the year.

Estimate the NPV of the cash flows from the Estimate the NPV of the cash flows from the new car.new car.

What fraction of the time will the new model What fraction of the time will the new model add value?add value?

CASH FLOW FOR GMAfter making cell D22 an output cell and After making cell D22 an output cell and

running for 1000 simulations, we obtain the running for 1000 simulations, we obtain the output for this problem.output for this problem.

The NPV is $43 million.The NPV is $43 million.

After viewing the histogram of profits, we see After viewing the histogram of profits, we see that there is a 32% chance that the project that there is a 32% chance that the project will have negative cash flows.will have negative cash flows.

CASH FLOW FOR LILLYIn the car business a new model usually sees reduced In the car business a new model usually sees reduced

sales every year. A new drug; however, sees sales every year. A new drug; however, sees increased sales the first few years followed by increased sales the first few years followed by reduced sales. Consider the following problem.reduced sales. Consider the following problem.

Lilly is producing a new drug that will be sold for 10 Lilly is producing a new drug that will be sold for 10 years. Year 1 unit sales are assumed to follow a years. Year 1 unit sales are assumed to follow a triangular distribution with worst case 100,000 triangular distribution with worst case 100,000 units, most likely case 150,000 units, and best case units, most likely case 150,000 units, and best case 170,000. 170,000.

The year 0 fixed cost is $2.1 billion and is depreciated The year 0 fixed cost is $2.1 billion and is depreciated on a 10-year straight line basis.on a 10-year straight line basis.

Sales are equally likely to increase for 3,4,5, or 6 years Sales are equally likely to increase for 3,4,5, or 6 years with the average percentage increase during those with the average percentage increase during those years following a triangular distribution with worst years following a triangular distribution with worst case 6%, most likely case 9%, and best case 11%.case 6%, most likely case 9%, and best case 11%.

CASH FLOW FOR LILLYDuring the remainder of the life of the drug unit sales During the remainder of the life of the drug unit sales

will decrease according to a triangular distribution will decrease according to a triangular distribution with best case 8%, most likely 12% and worst case with best case 8%, most likely 12% and worst case 18%.18%.

During each year a unit sells for $15,000.During each year a unit sells for $15,000.

Year 1 variable cost is $10,000.Year 1 variable cost is $10,000.

The unit variable cost of producing the drug increases at The unit variable cost of producing the drug increases at 4% per year.4% per year.

Estimate the mean NPV of the cash flows.Estimate the mean NPV of the cash flows.

What is the probability that the drug will add value?What is the probability that the drug will add value?

What source of uncertainty is the most important driver What source of uncertainty is the most important driver of NPV?of NPV?

CASH FLOW FOR LILLYWe use Autoconvergence to determine the We use Autoconvergence to determine the

number of iterations for @Risk to run. Select number of iterations for @Risk to run. Select Iterations Auto in the Simulation Setting Iterations Auto in the Simulation Setting window and change to 1%. window and change to 1%.

This will make @Risk keep running iterations This will make @Risk keep running iterations until during the last 100 iterations the mean, until during the last 100 iterations the mean, standard deviation, and selected other standard deviation, and selected other statistics change by 1% or less.statistics change by 1% or less.

The histogram of NPV shows the probability of The histogram of NPV shows the probability of having a negative value.having a negative value.

The mean NPV can also be found in the The mean NPV can also be found in the Summary Statistics.Summary Statistics.

CASH FLOW FOR LILLYA tornado graph can be used to determine key A tornado graph can be used to determine key

drivers of NPV.drivers of NPV.

Select Simulation Settings and Sampling dialog Select Simulation Settings and Sampling dialog box, select All for Collect Distribution box, select All for Collect Distribution Samples.Samples.

From the output Results window, select the From the output Results window, select the Graph selected items button and Tornado Graph selected items button and Tornado Graph option.Graph option.

The results show that the three key drivers for The results show that the three key drivers for NPV are: Unit Sales (89%), length of growth NPV are: Unit Sales (89%), length of growth (40%), and growth rate (18%).(40%), and growth rate (18%).

CASH FLOW FOR LILLYThis means that Year 1 Unit sales has a .89 This means that Year 1 Unit sales has a .89

correlation with NPV, length of growth has a correlation with NPV, length of growth has a .4 correlation, and annual growth has a .18 .4 correlation, and annual growth has a .18 correlation.correlation.

The uncertainty about Year 1 Unit sales and The uncertainty about Year 1 Unit sales and length of growth are very important for length of growth are very important for determining NPV.determining NPV.

If you spent money to reduce the uncertainty of If you spent money to reduce the uncertainty of one of the four sources of randomness, what one of the four sources of randomness, what cell would you change? Re-run the cell would you change? Re-run the simulation after you reduce the uncertainty simulation after you reduce the uncertainty of that cell to show the improvement in of that cell to show the improvement in NPV.NPV.

MANAGERIAL ISSUESWe studied how GM and Lilly might make a We studied how GM and Lilly might make a

decision about whether to introduce a new decision about whether to introduce a new car or a new drug, respectively.car or a new drug, respectively.

How does your company make these types of How does your company make these types of decisions?decisions?

What information does our approach provide What information does our approach provide that other approaches lack?that other approaches lack?

OPTIMAL CONSTRUCTION BIDA firm is considering bidding on a construction A firm is considering bidding on a construction

contract. If they win the bid, the cost to complete contract. If they win the bid, the cost to complete the project is uncertain but normally distributed the project is uncertain but normally distributed with a mean of $25,000 and a standard deviation of with a mean of $25,000 and a standard deviation of $3,000.$3,000.

It costs $1,000 to prepare the bid. It costs $1,000 to prepare the bid.

There are 6 potential competitors and it is estimated that There are 6 potential competitors and it is estimated that there is a 50% chance that each competitor will bid there is a 50% chance that each competitor will bid on the project (independent probabilities).on the project (independent probabilities).

If a competitor places a bid, their bid follows a normal If a competitor places a bid, their bid follows a normal distribution with a mean of $50,000 and a standard distribution with a mean of $50,000 and a standard deviation of $10,000. deviation of $10,000.

OPTIMAL CONSTRUCTION BIDAssume that our bid must be in multiples of Assume that our bid must be in multiples of

$1000 and we are considering bids from $1000 and we are considering bids from $30,000 to $60,000.$30,000 to $60,000.

What should be bid to maximize expected What should be bid to maximize expected profit?profit?

Assume that the low bid always wins.Assume that the low bid always wins.

Run the simulation for 1000 iterations.Run the simulation for 1000 iterations.

How will your optimal bid change if there are 12 How will your optimal bid change if there are 12 competitors?competitors?

Why is the best bid less when there are 12 Why is the best bid less when there are 12 competitors compared to 6?competitors compared to 6?

OPTIMAL CONSTRUCTION BIDIn order to solve this problem, you will need to In order to solve this problem, you will need to

determine how many competitors will bid. determine how many competitors will bid.

To accomplish this you will use the To accomplish this you will use the =riskbinomial(n,p) function. This will =riskbinomial(n,p) function. This will simulate the number of successes in n trials simulate the number of successes in n trials each of which has a probability of success p.each of which has a probability of success p.

For example, =riskbinomial(100,0.9) will For example, =riskbinomial(100,0.9) will simulate how many free throw attempts a simulate how many free throw attempts a player makes out of 100 if her probability of player makes out of 100 if her probability of making any individual free throw is 90%.making any individual free throw is 90%.

NEW DRUG DEVELOPMENTThe pharmaceutical business deals with a very The pharmaceutical business deals with a very

high degree of uncertainty. Over 90% of all high degree of uncertainty. Over 90% of all products under development fail to come to products under development fail to come to market resulting in large losses.market resulting in large losses.

Products that do come to market can earn Products that do come to market can earn multibillion profits annually for 10-15 years. multibillion profits annually for 10-15 years.

Eli Daisy wants to determine whether a new Eli Daisy wants to determine whether a new drug is worth developing.drug is worth developing.

NEW DRUG DEVELOPMENTBefore coming to market, the new drug must go Before coming to market, the new drug must go

through the following stages:through the following stages: Initial R&DInitial R&D Preclinical TestingPreclinical Testing Testing I (first phase clinical trials)Testing I (first phase clinical trials) Testing II (second phase clinical trials)Testing II (second phase clinical trials)Only after all drug stages succeed can the drug Only after all drug stages succeed can the drug

be sold.be sold.If the drug fails at any stage, then development If the drug fails at any stage, then development

is terminated.is terminated.A success at any stage leads us to pursue the A success at any stage leads us to pursue the

next stage.next stage.

NEW DRUG DEVELOPMENTWe would like to determine the risk adjusted We would like to determine the risk adjusted

NPV (15% discount rate per year).NPV (15% discount rate per year).

We would also like to determine the key drivers We would also like to determine the key drivers for the drug’s profitability.for the drug’s profitability.

For each stage we will model the cost, For each stage we will model the cost, probability of success, and time required to probability of success, and time required to complete the stage with a triangular random complete the stage with a triangular random variable. variable.

We also model the profit earned if the drug We also model the profit earned if the drug makes it to market as a triangular random makes it to market as a triangular random variable.variable.

NEW DRUG DEVELOPMENTThe following costs (in millions) and time estimates (in The following costs (in millions) and time estimates (in

years) for each stage have been determined:years) for each stage have been determined:StageStage BestBest WorstWorst Most LikelyMost LikelyInitial R&DInitial R&D CostCost 5050 120120 7070Init R&D TimeInit R&D Time 33 77 44Preclin Test CostPreclin Test Cost 1010 3030 1515Prec Test TimePrec Test Time .5.5 33 11Testing I CostTesting I Cost 350350 600600 480480Testing I TimeTesting I Time 33 66 44Testing II CostTesting II Cost 35003500 60006000 42004200Testing II TimeTesting II Time 33 66 44ProfitProfit 60000600001400014000 1800018000

NEW DRUG DEVELOPMENTThe following probabilities for each stage have The following probabilities for each stage have

also been determined.also been determined.

StageStage WorstWorst Most LikelyMost Likely BestBest

Init R&DInit R&D .2.2 .35.35 .42.42

Precl TestPrecl Test .3.3 .5.5 .6.6

Testing ITesting I .4.4 .5.5 .6.6

Testing IITesting II .7.7 .9.9 .96.96

We assume that these random variables are We assume that these random variables are independent.independent.

Retirement Investing A 25 year old person wants to invest for 40 years until

retirement. She plans to invest $1000 at the beginning of each of

the next 40 years. Each year, she plans to put fixed percentages – the

same each year – of this $1000 in stocks, bonds and T-bills. However, she is not sure which percentage to use.

She does have historical annual returns from stocks, bond and T-bills from 1946-2001.

Retirement.xlsThis file contains the historical data for the

stocks, bond and T-bills.This file also includes inflation factors for

these years.For example, for 1993 the annual returns

for stocks, bonds, and T-bills were 9.99%, 18.24% and 2.90%, and the inflation rate was 2.75%.

SolutionWe must decide how to use the historical returns

and inflation factors to generate future values of these quantities.

We will use a “scenario” approach.Each historical year is a possible scenario, where

the scenario specifies the returns and inflation factor for that year.

Then for any future year, we randomly choose one of these scenarios, using RISKDISCRETE function.

Solution -- continued

Intuitively more recent scenarios ought to have a larger chance of being chosen.

A weight is given to each scenario, starting with 1 for 2001. A “damping factor” is multiplied by the weight from the next year.

To change weights to probabilities, divide each weight by the sum of all the weights. We use a damping factor of 0.98.

Developing the Model The model can be developed as follows.

1. Inputs. Enter the data in the shaded regions. These include the historical returns and inflation factors, the alternative sets of investment weights we plan to test, and other inputs.

2. Weights. The investment weights we will use for the model are in row 10 to 12. We do this with a RISKSIMTABLE and VLOOKUP combination. Specifically, enter the formulas =RISKSIMTABLE({1,2,3}) in cell A16 and =VLOOKUP($A$16,$A$10,$D$12,2) in cell B16 and copy the latter to the cells C16 and D16.

Developing the Model3. Probabilities. Enter value 1 in cell F75. Then enter the

formula =$B$4*F75 in cell F74 and copy it up to cell F20. Sum these values with the SUM function in cell F76. Then to convert them to probabilities, enter the formula =F21/$F$76 in cell G20 and copy it down to cell G75.

4. Scenarios. We want to simulate 40 scenarios in columns K through O, one for each year of Sally’s investing. To do this, enter the formulas =RISKDISCRETE($A$20:$A$75, $G$20:$G$75) and =1+VLOOKUP($K20, $A$20:$A$75,L$18) in cells K20 and L20, and then copy this latter formula to the range M20:O20.

Developing the Model5. Beginning, ending cash. The bookkeeping part begins by entering the

formula =B5 in cell J20 for the initial investment. Then enter the formulas =J20*SUMPRODUCT($B$16:$D$16,L20:N20) and =$B$5+P20 in cells P20 and J21 for ending cash in the first year and beginning cash in the second year. The former shows how the beginning cash grows in a given year. The latter implies that Sally reinvests her previous money, plus she invests a new $1000. Copy these formulas down column invests a new $1000. Copy these formulas down column J and P.

6. Deflators. We eventually want to deflate future dollars to today’s dollars. The proper way to do this is to calculate deflators. Do this by entering the formula =1/O20 in cell Q20. Then enter the formula Q20/O21 in cells Q21 and copy it down.

Developing the Model Final cash. Calculate the final value in today’s dollars in cell K15

with the formula =P59*Q59

Then designate this cell as an @RISK output cell. Note that multiplying by the deflator for year 40 is similar to taking an NPV. The only difference is that the inflation rates differ through the 40 years, whereas NPV calculations typically involve the same discount rate each year.

The number of iterations is 1000 and the number of simulations is 3. The results show that the simulation which invests the most heavily

in stocks is easily the winner.

@Risk Results -- continued

The histogram for simulation 1 indicates a lot of variability – and skewness – in the distribution of final cash.

A useful concept we might introduce here is value at risk (VAR). It is defined as the 5th percentile of a distribution and is often the value investors worry about.

@Risk Results -- continued

You also could run this simulation with other investment weights, both for the 40-year horizon and for shorter time horizons such as 10 or 15 years.

Even though the stock strategy appears to be best for a long horizon, it might not fare as well for a shorter horizon. You could also try getting newer data, after 2001.

Winning at Craps We would like to use simulation to estimate the

probability of winning a single game of craps.1. Simulate tosses. Simulate the results of 40 tosses in the

range B5:D44 by entering the formula =RISKDUNIFORM({1, 2, 3, 4, 5, 6}) in cells B5 and C5 and the formula =SUM(B5:C5) in cell D5. Then copy these to the range B6:D44. RISKDUNIFORM takes a list of numbers and randomly selects one of the numbers from the list, where each number has equal probability. This step is like rolling two die (columns B and C) and summing the results (column D).

Developing the Model2. First toss outcome. Determine the outcome of the first toss with the

formulas =IF(OR(D5=7,D5=11),1,0), =IF(OR(D5=2,D5=3,D5=12),1,0) and =IF(AND(E5=0,F5=0),”Yes”,”No”) in cells E5 and F5, and G5. Similarly, the OR condition in cell F5 checks whether he loses right away. In cell G5, we use the AND condition to check whether both cells E5 and F5 are 0, in which case the game continues. Otherwise, the game is over.

3. Outcomes of other tosses. Assuming the game continues beyond the first toss, Joe’s point is the value in cell D5. Then we are waiting for a toss to have the value in D5 or 7, whichever occurs first. To implement this logic, enter the formulas =IF(OR(G5=“No”,G5=“”), “”,IF(D6=$D$5,1,0)), =IF(OR(G5=“No”,G5=“”), “”,IF(D6=7,1,0)) and =IF(OR(G5=“No”,G5=“”), “”,IF(AND(E6=0,F6=0),”Yes”,”No”)) in cells E6, F6, and G6. Then copy these to the range E7:G44.

Developing the Model4. Game outcomes. We keep track of two aspects of the

game in @Risk output cells, whether Joe wins or loses and how many tosses are required. To find these, enter the formula =SUM(E5:E44) and =COUNT(E5:E44) in cells B47 and B48. Note that both functions, SUM and COUNT, ignore blank cells.

5. Simulation summary. Although we get various summary measures in the @RISK Results window when we run the simulation, it is useful to see some key summary measures right on the model sheet.

Developing the ModelTo get these, enter the formula =RISKMEAN(B47) in cell B50 and copy it to cell B51. As the labels indicate, the RISKMEAN in cell B50, being an average of 0’s and 1’s, is just the fraction of iterations where Joe wins. The average in cell B51 is the average number of tosses until the game’s outcome is determined.

@Risk ResultsAfter running @RISK, we obtain the summary results

in cells B50 and B51 of the simulation.Our main interest is in the average in cell B50 because

it represents our best estimate of the probability of winning, 0.494. (A probability argument can be used to show that the exact probability of winning in craps is 0.493.)

We also see that the average number of tosses needed to determine the outcome of a game was 3.395. (The maximum number of tosses ever needed was 29.)

March Madness We can simulate the NCAA basketball tournament to estimate the probability

that each teams wins. We use 2005 tournament pairings and each team’s Sagarin’s nationally

syndicated rating. The Sagarin’s ratings can be used to estimate the probability that a team wins

any particular game. Those results can be used to play out the tournament. Suppose team A plays team B and Sagarin’s ratings for these teams are 85

and 78. Sagarin predicts that the point differential will be the difference between the ratings.

We assume the actual point differential is normally distributed with mean equal to Sagarin’s prediction (7) with a standard deviation for each game of 10.

March-Madness-Men2005.xlsWe will only outline the simulation model.The file contains the full details.The entire simulation is on a single Model sheet.

Columns A to C list team indexes, team names, and Sagarin ratings.

Winners from one round are automatically carried over to the next round with appropriate formulas.

Portions of the Model sheet appear on the next slides.

Developing the Model We now describe the model.

1. Teams and ratings. We first enter the teams and their ratings, as shown.

2. Simulate rounds. Jumping ahead to the fourth-round simulation shown on the previous slide, we capture the winners from the previous round 3 and then simulate the games in round 4. The key formulas are in columns N and O. For example, the formulas in cells N126 and O126 are =VLOOKUP(L126,LTable,3)-VLOOKUP(L127,LTable,3) and =RISKNORMAL(N126,10) The first of these looks up the ratings of the two teams involved and subtracts them to get the predicted point spread. The second formula simulates a point spread with the predicted point spread as its mean.

Developing the Model Outputs. As shown by the boxed-in cells in Figure 12.44, we

designate seven cells as @RISK output cells: the index of the winner, the indexes of the two finalists, and the indexes of the four semifinalists (the Final Four teams). However, the results we really want are tallies, such as the number of iterations where North Carolina (or any other team) wins the tournament. This takes some planning.

In the @RISK Reports dialog box, if we check the Output Data option, we get a sheet called Outputs Data Report that lists the values of all @RISK output cells for each of the iterations. (We used 5000 iterations.)

Developing the ModelAfter we have these, we can use COUNTIF functions to tally the number of wins (or finalist or semifinalist appearances) for each team, right in the original Model sheet. Some of these tallies appear on the next slide. For example, the formula in cell U5 is =COUNTIF(‘Outputs Data Report’!$I$8:$I$5007,S5)

Developing the Model In this case, the range I8:I5007 of the Outputs Data

Report contains the indexes of the 5000 winners, so this formula simply counts the number of these that are index 1.

As you can see, the top-rated team in the Chicago region, Illinois, won the tournament in 1014 of the 5000 iterations and reached the Final Four almost half of the time.

In contrast, the lowly rated Fairleigh Dickinson did not make the Final Four in any of the 5000 iterations.

@Risk ResultsThis simulation came out somewhat differently

than in previous years. The two teams predicted by the simulation to

win most often, North Caroline and Illinois, actually made it to the finals, and the higher rated of these two teams, North Carolina, actually won the championship.

Simulations of sporting events are not always this accurate!

SUMMARY In this chapter, we:In this chapter, we:

Showed how @Risk could be used for Showed how @Risk could be used for financial simulation and additional financial simulation and additional applications.applications.

Specific areas covered include: cash flow Specific areas covered include: cash flow models, a bidding problem, retirment models, a bidding problem, retirment planning, a game of chance, and a NCAA planning, a game of chance, and a NCAA basketball tournament simulation.basketball tournament simulation.

COPYRIGHTCopyright Copyright 2003 Matthew J. Liberatore and Robert L. Nydick. All rights reserved. 2003 Matthew J. Liberatore and Robert L. Nydick. All rights reserved.

Reproduction or translation of this work beyond that named in Section 117 of the Reproduction or translation of this work beyond that named in Section 117 of the United States Copyright Act without the express written consent of the copyright United States Copyright Act without the express written consent of the copyright owners is unlawful. Requests for further information should be addressed to owners is unlawful. Requests for further information should be addressed to Matthew J. Liberatore and Robert L. Nydick. Adopters of the textbook are granted Matthew J. Liberatore and Robert L. Nydick. Adopters of the textbook are granted permission to make back-up copies for their own use only, to make copies for permission to make back-up copies for their own use only, to make copies for distribution to students of the course the textbook is used in, and to modify this distribution to students of the course the textbook is used in, and to modify this material to best suit their instructional needs. Under no circumstances can copies be material to best suit their instructional needs. Under no circumstances can copies be made for resale. Matthew J. Liberatore and Robert L. Nydick assume no made for resale. Matthew J. Liberatore and Robert L. Nydick assume no responsibility for errors, omissions, or damages, caused by the use of these programs responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.or from the use of the information contained herein.