Question :

Reading 9 :What does it mean to be risk averse or risk neutral? How important is it to evaluate risk and to evaluate how sensitive to risk any business, personal, or academic decision you have made or are making? (Attachement Reading_9.pdf)

Reading 10 :What is the essence of Decision Theory? How can the concept of decision theory be used in your business, personal or academic activities?

Part of Reading 10: Short Decision Theory - Computer Leasing Case (File 20Ar reference only) 1 Decision Theory is another important tool available to us when we are facing making a decision under uncertain conditions (called states of nature). Decision Theory Three Step Process Assume you head-up a computer processing venture that wants to enter the political market providing surveying information to various politicians and political action groups in Washington, DC. You are faced with three possible decisions. You may lease a large computer, a medium computer or a small computer. Which do you do? First Step: Construct a Payoff Table. Remember a payoff table is in terms of profit. This means that there has been sales forecast for each intersection between the state of nature and the decision facing the business. These sales forecasts begins with anticipated sales, less cost of sales, less overhead and other expenses to arrive at the bottom line profit or loss expected under each separate conditions. Many assumptions have been made concerning each profit or loss stated in the payoff table. Possible Decisions d1 - lease the large computer d2 - lease the medium computer d3 - lease the small computer States of Nature (How well our services will be received in the market place) s1 - high acceptance of your product s2 - low acceptance of your product A state of nature is simply what might happen in the market place when you place your offer mix (product and price) before your target market. Payoff Table 1 States of Nature Decisions s1 - high s2 - low d1 - lease large $200,000 -$20,000 d2 - lease medium $150,000 $20,000 $100,000 $60,000 d3 - lease small 1 Part of Reading 10: Short Decision Theory - Computer Leasing Case (File 20Ar reference only) 2 If we knew the state of nature would be high, then we would have no problem with the decision. We would select d1. This would maximize our profit at $200,000. It is the most optimistic decision and is called the maxi-max decision. If we knew the state of nature would be low, then we would have no problem with the decision. We would select d3. This also maximizes our profit under this state of nature. This is the most pessimistic or most conservative decision and is noted as the maxi-min decision. The difficulty is that we do not know the state of nature with any reasonable degree of certainty, so what decision is best for us given we do not know the reaction of the target market to our offer mix? Step Two: Construct and Opportunity Lost Table. This table is a statement about the money we could have made if we had made the maxi-max or the maxi-min decision, so those values will be the benchmark from which we measure the profit we lost. Opportunity Lost Table 2 States of Nature Decisions s1 - high s2 - low Zero - 0 d1 - lease large $80,000 d2 - lease medium $ 50,000 $40,000 d3 - lease small $100,000 Zero - 0 Step Three: Next we construct the maximum opportunity lost table. This means we select from Table 2, those values related to each possible decision that are the maximum profit we lost, because of making each decision. Maximum Opportunity Lost Table 3 Decisions Max Opportunity Lost d1 - lease large $ 80,000 d2 - lease medium $ 50,000 d3 - lease small $100,000 Finally you select the minimum of the maximum, or you select d2. Decision d2 - lease a medium computer - is the decision which causes you to lose the least profit. 2 Reading 9: Risk Assessment & Sensitivity Analysis (File019r reference only) 1 Risk Assessment & Sensitivity Analysis Assessment: If uncertain quantities exist, potential performance cannot be effectively estimated. The probability of an exact forecasted number occurring is very small if not zero. Possibilities occur within a range. An optimistic forecast can be made, but that may not be the best. A pessimistic forecast can be made, but that too may not be the best. These are, of course, the extremes. Thinking should not be limited to the extremes. Forecasting should consider other alternatives, which can and do often happen. An example will be followed to show how sensitive to risk decision is. The technique is referred to as sensitivity analysis. This analysis can be shown in a visual portrait of the sensitivity which is called a risk profile. From the risk profile it can been seen if the negative possibilities are more prone to occur then the positive possibilities. An Example: Let's assume we manufacture breakfast cereal. Since our target market is children, we are faced with making a decision about either including a trinket with our 28 ounce of cereal or not including the trinket. We decide to look at the risk of our decision. Some of the key factors of the case are listed below: Key Factors: We are a children's breakfast cereal manufacturer. We want to consider placing a Holiday trinket in our 28 ounce boxes. They are to be Included 2 months prior to the Holiday Season which would make the insertion in October and November. Our focus group, the Children's Preference Panel, gave the trinket good reception in our preliminary study. Expectations: Based on prior history, we expect a 12% increase in sales by including the holiday trinket. However, the increase in sales might range for +8% to + 14%. Without the trinket, the sales forecast is 6.5 million boxes plus or minus 2%. Assumption: Production can match demand. The Trinket Cost is $0.10 each and one is inserted into each box. One potential problem has arisen. There may be problems with the trinket manufacturer meeting our production schedule, so two alternative suppliers have been contacted and one will provide the trinket for 0.01 cents more ($0.11 total) and the other will provide the trinket for 0.015 Reading 9: Risk Assessment & Sensitivity Analysis 2 (File019r reference only) cents ($0.115 total) per unit. It is believed that the alternate suppliers have an equal chance of delivering the product in a timely manner. The artwork displaying the trinket will cost $125,000. However, the artwork cost could increase 20% or decrease 5% depending on the layout and final artwork. The unit contribution to gross margin for the 28-ounce cereal is $1.48 per box. The decision facing our company is: Do we insert the holiday trinket or not? Sensitivity Analysis: Let's look at the three possible outcomes - best, worst and base case results. Optimistic Scenario: (All un-certainities result in Favorable results). Pessimistic Scenario: (All un-certainities result in Unfavorable results). Best Estimate (Base Case) Scenario: (This is the base case and uses what could be described as the best guess or best estimate approach. Our benchmark will begin with the base case and then develop the other two outcomes (optimistic and pessimistic) using these results.) We can make our analysis based on total cost but such detail is not necessary. We are interested in the change in profit not the total profit. To accomplish this we will work with Gross Margin but more specifically, we will use incremental gross margin. Table 1: Increase in Incremental Gross Margin: Best Estimate (Base Case) Item Forecasted Error in Base Volume Volume Increase in Sales in Percent Unit Cost Per Trinket Artwork Cost Variance Base Case Volume in Boxes Plus: Increase Volume (12% of 6,500,000) Total Volume in Boxes Values 0% 12.0% $0.10 0% 6,500,000 780,000 7,280,000 Increase in Contribution ($1.48 times 780,000) $1,154,400 Less: Trinket Cost ($0.10 times 7,280,000 boxes) $ 728,000 Less: Artwork Base Case Estimate $ 125,000 Incremental Gross Margin $ 301,400 Reading 9: Risk Assessment & Sensitivity Analysis 3 (File019r reference only) From the base case analysis, we would easily conclude that we will insert the trinket, since the promotion is expected to yield an increase in incremental gross margin of $301,400. However, this analysis is does not consider any risk associated with the base case value. By using a base case (best guess) we have not consider possible variances in the various factors and costs? We better consider them. Let's look at both the optimistic and pessimistic outcomes again using the incremental gross margin approach. By looking at both, the expected income or loss will be bracketed. Sensitivity analysis is just like firing for effect in artillery situations - one long and one short with the third right on target. The estimates using the base case can be adjusted based on the possible variations given in the case. See the information listed in the example section above. For example, the forecast was estimated at 6.5 million boxes, but could vary 2% up or down, so let's consider those possibilities. The optimistic outcome would forecast +2%. The pessimistic outcome would forecast -2%. Table 2: Increase in Contribution Based on the Optimistic and Pessimistic Scenarios. Item Forecast Error In Base Volume Volume Increase Percentage - Total Unit Cost Per Trinket Artwork Cost Variance Optimistic +2.0% +14.0% $0.10 -5.0% Pessimistic -2.0% +8.0% $0.115 +20.0% Adjusted Base Volume - Boxes Plus: Increased Volume Total Volume in Boxes 6,630,000 928,200 7,558,200 6,370,000 509,600 6,879,600 Increase in Gross Margin Contribution ($1.48) $1,373,736 $ 754,208 Less: Trinket Cost $ 755,820 $ 791,154 Less: Artwork Costs $ 118,750 $ 150,000 Incremental Gross Margin $ 499,166 -$186,946 Loss Calculations: Volume: The base volume from Table 1 of 6,500,000 is multiplied by 1.02 (2% increase) to get 6,630,000, the most optimistic adjusted base volume. For the pessimistic volume, the base volume of 6,500,000 is multiplied by 0.98 (2% decrease) to get an adjusted base volume of 6,370,000. Increase in Volume: The optimistic increase is the result of multiplying the optimistic 6,630,000 times 0.14 (14.0% increase) to get an additional 928,200 Reading 9: Risk Assessment & Sensitivity Analysis (File019r reference only) 4 boxes. The pessimistic increase is the result of multiplying the pessimistic 6,370,000 times 0.08 (8% increase) to get an additional 509,600 boxes. Increase in Gross Margin Contribution: If you multiply $1.48 (given in the case) times each of the volume increases just calculated you will get the values shown as the optimistic of $1,373,736 ($1.48 times 928,200) and the pessimistic of $754,208 ($1.48 times 509,600). Trinket Cost: The trinket cost is set at $0.10 times 7,558,200 for the most optimistic cost of $755,820. For the most pessimistic cost the trinket cost is set at the upper limit of $0.115 times 6,879,600 or $791,154. Artwork Cost: The artwork reflects a decrease of 5% for the most optimistic view, which is $125,000 times 0.95 (5% decrease) or $118,750. If the cost of artwork increases by 20%, this value is expressed as the most pessimistic cost of $150,000 ($125,000 times 1.20). In calculating the most optimistic and the most pessimistic outcomes, you are bracketing the upper and lower possibilities. What you expect to happen will fall somewhere between those two numbers, since you rarely have all negative things happen or all positive things happen. As a summary, you believe the best possible outcome will yield a contribution to gross margin of $499,166, but you also believe that the contribution cannot fall below a negative $186,946. Your best estimate (base case) is a contribution to gross margin of $301,400. There are four variables that affect the outcome of the contribution margin - forecasted error in base volume, volume increase, unit cost and artwork. There is a fifth, which is price, but for now we will leave that out of the equation. Since we have bracketed the maximum gain or loss (optimistic and pessimistic), we may want to conduct sensitivity analysis. We will hold three of the variables constant and determine the impact on contribution margin for the fourth. This means we will have four, one-at-a-time sensitivity analysis charts. We will first vary volume and hold the other three constant (base case values). The base case is our benchmark in any sensitivity analysis. This is shown in Table 3 below. Reading 9: Risk Assessment & Sensitivity Analysis 5 (File019r reference only) Table 3: One-at-a-Time Sensitivity Analysis - Volume Changes Item Forecast error in base volume Volume increase percentage Unit Cost per Trinket Artwork Cost Variance Lowest Value 0% 8% $.10 0% Highest Value 0% 14% $.10 0% Base Volume (Boxes) Volume Increase Total Volume (Boxes) 6,500,000 520,000 7,020,000 6,500,000 910,000 7,410,000 Increase in Contribution ($1.48) $769,600 Less: Trinket Costs $702,000 Less: Artwork Costs $125,000 Incremental Gross Margin $1,348,600 $ 741,000 $ 125,000 -$57,400 Loss $ 480,800 To verify the calculations in Table 3 will be left to you, but they follow the same pattern detailed in Table 2 above. What you need to do is continue this one-at-a-time sensitivity analysis for the other three variables. Create a blank table similar to the one given below and begin the process. The answers will immediately follow the blank table given below, but try not to access them until you make an attempt. Table 4: One-at-a-Time Sensitivity Analysis - Base Volume Changes Item Forecast error in base volume Volume increase percentage Unit Cost per Trinket Artwork Cost Variance Base Volume (Boxes) Volume Increase Total Volume (Boxes) Increase in Contribution ($1.48) Less: Trinket Costs Less: Artwork Costs Incremental Gross Margin Lowest Value Highest Value Reading 9: Risk Assessment & Sensitivity Analysis 6 (File019r reference only) Table 5 would be One-at-a-Time Sensitivity Analysis - Unit Cost Per Trinket. Table 6 would be One-at-a-Time Sensitivity Analysis - Art Cost Variance. The correct answers are shown in the completed tables below, but again do not look at them until you have attempted them. Answers: Table 4: One-at-a-Time Sensitivity Analysis - Base Volume Changes . Item Lowest Value Highest Value Forecast error in base volume -2.0% +2.0% Volume increase percentage 12.0% 12.0% Unit Cost per Trinket $0.10 $0.10 Artwork Cost Variance 0.0% 0.0% Base Volume (Boxes) Volume Increase Total Volume (Boxes) 6,370,000 764,400 7,134,400 6,630,000 795,600 7,425,600 Increase in Contribution ($1.48) Less: Trinket Costs Less: Artwork Costs $1,131,312 $ 713,440 $ 125,000 $1,177,488 $ 742,560 $ 125,000 Incremental Gross Margin $ 292,872 $ 309,928 Table 5: One-at-a-Time Sensitivity Analysis - Unit Cost Per Trinket Item Forecast error in base volume Volume increase percentage Unit Cost per Trinket Artwork Cost Variance Lowest Value Highest Value 0.0% 0.0% 12.0% 12.0% $0.10 $.115 0.0% 0.0% Base Volume (Boxes) Volume Increase Total Volume (Boxes) 6,500,000 780,000 7,280,000 6,500,000 780,000 7,280,000 Increase in Contribution ($1.48) $1,154,400 Less: Trinket Costs $ 728,000 Less: Artwork Costs $ 125,000 $1,154,400 $ 837,200 $ 125,000 Incremental Gross Margin $ 192,200 $ 301,400 Reading 9: Risk Assessment & Sensitivity Analysis 7 (File019r reference only) Table 6: One-at-a-Time Sensitivity Analysis - Art Work Changes Item Forecast error in base volume Volume increase percentage Unit Cost per Trinket Artwork Cost Variance Lowest Value Highest Value 0% 0% 12.0% 12.0% $0.10 $0.10 -5.0% +20.0% Base Volume (Boxes) Volume Increase Total Volume (Boxes) 6,500,000 780,000 7,280,000 6,500,000 780,000 7,280,000 Increase in Contribution ($1.48) $1,154,400 Less: Trinket Costs $ 728,000 Less: Artwork Costs $ 118,750 $1,154,400 $ 728,000 $ 150,000 Incremental Gross Margin $ 276,400 $ 307,650 Recap of Results - Incremental Gross Margin Expected Results $301,400 Optimistic (Everything Goes Well) $499,166 Pessimistic (Everything Goes Bad) -$186,946 Sensitivity Analysis: Sensitivity Analysis Vary Volume Unit Cost Per Trinket Error in Base Forecast Art Work Variance Lowest -$57,400 $301,400 $292,872 $307,650 Highest $480,800 $192,200 $309,928 $276,400 The incremental gross margin can vary from a loss of $186,946 to a profit of $499,166, but the expected result is a profit increase of $301,400. The optimistic and pessimistic reflect a bracket between which all the sensitivity values must fall. Notice that they do. Sensitivity analysis analyzes the results as if only one of the four variables changes with the other three remaining constant. Using Sensitivity Analysis the incremental gross profits may vary between a loss of $57,400 to a maximum profit of $480,800. Volume is the most sensitive, so particular attention must be paid to it. Reading 9: Risk Assessment & Sensitivity Analysis (File019r reference only) 8 Other Considerations: While not a part of the original example, what about price changes? Is it possible for the company to increase overall prices based on the insertion of the trinket? Just using the base case volume of 6,500,000 plus the 780,000 or total expected sales of 7,280,000 we can look at what a small $0.01 change in price would do to incremental gross margin (add $72,800 for each 1 cent increase in price). While this may not be a method chosen by the manufacturer, it is worth consideration. A 3 cent increase would cut off any downside risk and all values would be positive. One other question you might want to answer is the total gross margin return on sales. You would then want to match this gross margin percentage to the normal gross margin percentages. Unfortunately there is not enough information in the case to allow us to make this calculation without making some assumption about the sales price of each box of cereal. Let's make a couple of assumptions so you can see the direction of this idea. If we assume that the cereal sells for $3.50 per box, then the sales using the best estimate approach and a volume of 7,280,000 boxes would be $25,480,000. The total gross margin at $1.48 per box is $10,774,400. This percentage is 42.28% ($10,774,400 $25,480,000). What we are adding to the gross margin using the most optimistic calculation is $499,000 (see Table 2). The change in our gross margin is now $10,774,400 + $499,000 = $11,273,400. The gross margin increases to 44.24% from 42.28% (approximately a 5% positive change). If everything bad happens and the most pessimistic view prevails, the change in the gross margin is from $10,774,400 to $10,587,400. Now the gross margin drops to 41.55% from 42.28%. This is less than a 2% negative change. Clearly the downside risk is less than the upside potential, but either way the decision is not going to make or break the company. Non-Quantifiable Considerations: Another idea is worthy of consideration although it is not quantifiable. How many new customers would be attracted to the product based on the added value of the trinket? Additionally even after the trinket promotion was over, how many of these new customer would be permanently attracted to the product and continue to repurchase it trinket or not?. Breakeven Analysis: One very important value to know is the breakeven volume increase necessary to breakeven on the incremental volume. This can best be done algebraically. Let's reason to the volume increase necessary to breakeven in percentage which then could be expressed in number of boxes if desired. Often the term used to describe this is threshold analysis. Reading 9: Risk Assessment & Sensitivity Analysis 9 (File019r reference only) First we will let X = volume increase percentage. relationships as follows. Next we can express $1.48 times 6,500,000 boxes is the base case contribution margin. We are, however, interested in the total contribution margin increase. This can be expressed as follows: ($1.48 times 6,500,000 boxes times X) is the total contribution margin increase. This reduces to 9,620,000 X. The trinket cost is $0.10. This times the base case box volume of 6,500,000 yields the base case cost of the trinkets ($0.10 times 6,500,000). However we need to express this cost in terms of the base case volume and the increased breakeven volume. Thus the expression becomes [$0.10 times (6,500,000 + 6,500,000 times X)] This ultimately reduces to [650,000 + 650,000X] The last cost to be considered is the artwork cost of $125,000 which is fixed. Remember X is the volume increase in percentage. We now can combine these three expressions into an algebraic formula, which we must set to Zero since we are looking for breakeven volume in percentage form. 9,620,000X less [650,000 + 650,000X] less $125,000 = 0 Simplifying this equation we have the following: 9,620,000X - 650,000X - 650,000 - 125,000 = 0 8,970,000 X - 775,000 = 0 8,970,000X = 775,000 X = 775,000 8,970,000 X = 0.0864 or 8.64%. Our volume must increase by 8.64% to 561,600 boxes to breakeven on inserting the trinket. Of course, we have made this breakeven calculation assuming $0.10 cost for the trinket and $125,000 for the artwork. If either of those two variables increase, then the breakeven percentage will increase. Given the relationships established with the algebraic equation, changing either cost would not make the additional calculations difficult. The breakeven can be displayed in tabular format as follows: Reading 9: Risk Assessment & Sensitivity Analysis 10 (File019r reference only) Table 7: Breakeven Using the Base Case Item Forecast Error in Base Volume Volume Increase Percentage Unit Cost Per Trinket Artwork Cost Variance Breakeven 0.0% 8.64% $0.10 0.0% Base Volume (Boxes) Volume Increase Total Volume 6,500,000 561,600 7,061,600 Increase in Contribution (Income) $831,168 Trinket Costs (Expense) $706,160 Artwork Costs (Expense) $125,000 Incremental Gross Margin (Profit) Zero* *The $8.00 is due to rounding and is essentially zero. Diagrams: Often times it may be important to graphically display the results in either a Tornado Diagrams or in a Histogram or Bar Chart. The Tornado Diagram is usually done by making use of the drawing bar. The horizontal line shows the sensitivity values from negative on the left to the positive values on the right. Usually a textbook will have pictures of Tornado Diagrams. The name of this diagram is associated with the shape of the chart. Everything is developed from the Base Case incremental value. The incremental gross margin forms the horizontal axis. The other lines show the resulting incremental gross margin for the maximum and minimum values for the four variables. Go to your textbook and look at the index for this graphic display. We will take a look at plotting risk on a Histogram or Bar Chart in a later lecture. Usually a graphic display conveys the idea of downside risk and upside potential. If the values are to the right of the base case you have greater upside potential than downside risk. The corollary is also true. Probabilities: As you assess the decisions to be made in this example, it is clear you could develop a probability distribution for each of the variables or unknown quantities. One quick example will explain the meaning. We will not go into much detail on Reading 9: Risk Assessment & Sensitivity Analysis (File019r reference only) 11 these distributions, but probabilities are the next logical step in assessing the outcomes expressed in the sensitivity analysis approach. . Through this process, you have developed the following probabilities. Implicit in this approach is the belief that the other two alternate suppliers have an equal chance of delivering the trinket on time. The three suppliers become our sample space - all of the possible outcomes under consideration. The probability of any sample space must total 1. Assume that we have determined the probability of the original supplier delivering the product on time is 0.70. If the other two suppliers have a probability that is equally likely, we must then subtract the 0.7 from the 1.00 and we get a combined probability of 0.3. Next we must divide the 0.3 by 2, since they are equally likely to occur. This means the probability of occurrence for each of the alternate suppliers is 0.15 each. X Unit Cost f(X) or P (X) Probability $0.10 0.70 $0.110 0.15 $0.115 0.15 Total 1.00 This is considered to be a triangular probability distribution and the outcome would lean heavily on the unit cost of the trinket staying at $0.10 (70% chance) with the other two being equally likely. If the original supplier does not appear to be on time, then selecting the supplier with a cost of $0.11 will cut off some of the downside risk (sensitivity analysis was done using $0.115). Uncertainties with Many Potential Outcomes (Probability Distributions): There are four uncertain variables in the trinket example. The Unit Cost of the Trinket. (Graphic Representation) The Base Volume Forecast Error. (Uniform Distribution) The Range of Volume Increase Including the Trinket. (Triangular Distribution) The Artwork Cost Variance. (Triangular Distribution) Reading 9: Risk Assessment & Sensitivity Analysis (File019r reference only) 12 Working with each of these distributions would be the next most important decision process. If we are able to determine a reasonable probability of each of the uncertain quantities, we can enrich our possibility of making the correct decision. For now we will not visit these calculations, but it is important you recognize we can fine tune our decision process by working the probabilities of the uncertain quantities. By applying probabilities, we might be able to determine the expected value of the random variable for this particular distribution. Deriving the Probability Distribution for Performance: In the example we used earlier, the introduction of the trinkets using the Best Estimate Approach yielded an incremental gross margin of $301,400. The range of values between the most optimistic outcome and the most pessimistic outcome was between a profit of $499,166 and a loss of $186,946. This is a swing of $686,112. The actual outcome will be somewhere between those two values. The decision depends on the probability distribution of the incremental gross margin. If the probability distribution is weighted to the high end (more positive outcomes may happen), the decision may be to introduce the trinket. If the probability distribution is weighted to the low end (more negative outcomes may happen), the decision may be to NOT introduce the trinket