1 uncertain numbers and diversification
- 1. Judge Business School (name) (date) THE LANGUAGE OF UNCERTAINTY I
- 2. Four basic questions for the next 3 sessions 1. What is an uncertain number? 2. What happens when uncertain numbers are combined in our plans? 3. What happens when uncertain numbers in business plans are replaced by projections (aka common practice)? 4. Uncertain numbers can depend on one-another. What effect does that have on our plans? Page 1 Management Science Group
- 3. Today 1. What is an uncertain number? 2. What happens when uncertain numbers are combined in plans? PLUS Introduction to Monte Carlo simulation in Excel Page 2 Management Science Group
- 4. 1. What is an uncertain number? Page 3 Management Science Group
- 5. 1. What is an uncertain number? An uncertain number has a range of possible outcomes To allow for the fact that some outcomes might be considered more likely than others, we use weights to reflect our judgement of these relative likelihoods These weights are called the probabilities of the outcomes (between 0=won’t happen and 1=will happen for sure) Recall the 3 interpretations of “probability” Page 4 Management Science Group
- 6. Representing uncertain numbers HISTOGRAM • An uncertain number is best represented by a bar chart • Bars indicate specific regions within the range of outcomes • Height of bars indicating the probabilities that the outcome will be in the respective region Page 5 Management Science Group
- 7. Example: A histogram of uncertain revenues What is the chance that revenues will be between $9M and $10M? Page 6 Management Science Group
- 8. Exercise How to create a histogram • Your mark on this course is 1. Decide on a sensible range uncertain – create a 2. Create scenarios: Chop range into a histogram few sub-ranges of the SAME SIZE • What is the range for this 3. Draw the histogram: Depict the uncertain number? relative likelihoods of any one of the scenarios as height of a bar in a bar • What is a sensible histogram chart Associated shape for this uncertain probabilities number? Page 7 Scenarios
- 9. Exercise How to create a histogram • Pick another uncertain 1. Decide on a sensible range number that is relevant to 2. Create scenarios: Chop range into a your life and create a few sub-ranges of the SAME SIZE histogram 3. Draw the histogram: Depict the • What would be a reasonable relative likelihoods of any one of the scenarios as height of a bar in a bar range for this uncertain chart number? Associated probabilities • What is a sensible histogram shape for this uncertain number? Page 8 Scenarios
- 10. Target probabilities… What is the chance of achieving a target of $8M? Page 9 Management Science Group
- 11. Turning a histogram into a target curve Add up probabilities (stack up blue bars) to the left Page 10 Management Science Group
- 12. Turning a histogram into a target curve Probability of missing a $8M target is about 45% Probability of achieving the target is about 55% Page 11 Management Science Group
- 13. Bar chart is often turned into a curve: the target curve The target curve is also known as the “cumulative distribution function” Page 12 Management Science Group
- 14. Bar chart is often turned into a curve: the target curve The target curve is also called “cumulative distribution function” Probability of missing a $8M target is ~45% Page 13 Management Science Group
- 15. Crucial Concept # 1 An uncertain number is a shape Stats-Latin: Statisticians call the shape of an uncertain number its “distribution” Statisticians call the target curve the “cumulative distribution function” Page 14 Management Science Group
- 16. Summary statistics are numbers derived from the shape I. The Average Mean = Average The average is the point where the histogram, if it were made of wood, would balances out Not the balance point
- 17. Summary statistics are numbers derived from the shape I. The Average Mean = Average The average is point where the downside (left of the average) “weighs” as much as the upside (right of the average) Too much weight Average = balance point on downside
- 18. Summary statistics are numbers derived from the shape I. The Average Mean = Average • For data: Average = Sum of data divided by the number of data points • Formal definition: 1. multiply all possible realizations of the uncertain number 2. sum up all these probability-weighted possible realizations • Example: average of dice 1*1/6+2*1/6+…+5*1/6+6*1/6=3.5 o Interesting: The average of an uncertain number is not necessarily a possible realisation of this uncertain number, as in the dice example • Approximation from histogram 1. multiply the mid-point of each bin with the height of its bar 2. sum up all these probability-weighted mid-points
- 19. Summary statistics are numbers derived from the shape II. Percentiles xth Percentile: x% of the data are below this value Easily read off target curve (=percentile graph) 60% chance That revenue The target curve (or“cumulative Is below $9M 60th percentile = $9M distribution function”) is also called “percentile curve”
- 20. Summary statistics are numbers derived from the shape III. Quantiles Deciles: chop the y-axis (0%- 100%) into 10 equal intervals Separating points on the x axis are the 10 deciles 10 What’s the “upper decile sales performance?” 4th decile is about $7.5M Chopping the y-axis into a different number intervals lead to other quantiles chopping it into 4 intervals give “quartiles”, what’s the third quartile? chopping it into 5 intervals leads to “quintiles”, what’s the third quintile? chopping it into 100 intervals gives the percentiles
- 21. Summary statistics are numbers derived from the shape II. Median • Median = 50th percentile People often mix up median and mean. Vignette to help you memorize the difference: When Bill Gates enters this lecture room, the median annual income won’t change much – but the mean will increase dramatically • Lower quartile = 25th percentile, upper quartile = 75th percentile • Inter-quartile range = range from lower to upper quartile • Common abbreviations: P10 = lower decile (10th percentile) P90 = upper decile (90th percentile)
- 22. Summary statistics are numbers derived from the shape III. Mode Mode = most likely value (or most likely bin in histogram), if there is one Homework: Make sure you understand the difference between mean, median and mode They are the same only in special cases (e.g. bell-shape distribution) Mode: $6M-$7M
- 23. Data has a shape as well… Open EasyBedsData.xls Use the percentile function in Excel to produce a target curve for Number of enquiries What’s the percentage of days with enquiries between 1000 and 1500?
- 24. Histograms from data… … are a bit trickier to produce Homework: Produce histograms for Number of enquiries Bookings completed Daily Revenue Number of No-shows There are plenty of tutorials on this on the web… Page 23 Management Science Group
- 25. Further practice on histograms I will describe an uncertain number to you Please draw a histogram of this uncertain number You are managing a business unit. Next year’s profit is uncertain. Suppose it can range between £0 and £1M Suppose the uncertainty is resolved by me spinning a wheel of fortune with equally spaced numbers 0,1,…,1M Today, before the wheel of fortune is spun, the revenue is uncertain. What is the shape of this uncertain revenue? Draw a histogram with 5 bins Page 24 Management Science Group
- 26. The wheel of fortune uncertainty has a flat shape between 0 and 1M Page 25 Management Science Group
- 27. The spinner function in @Risk The @Risk software package on your laptop has a spinner function - “=riskuniform(0,1)” and many other shapes If you hit the little button with the picture of the two dice on the Simulation tab, it “wakes up” the uncertain number (which is reset to its average if you hit the button again): Every time you change a cell in the spreadsheet – or hit F9 – it produces a different number in this cell THE EXCEL DEFINITION OF AN UNCERTAIN NUMBER: AN UNCERTAIN NUMBER IS A NUMBER IN A MODEL THAT CHANGES EVERY TIME YOU HIT F9 You can use @Risk to “hit the F9 key” 10,000 times and record the results, so that you can check the shape of the uncertain number Go on the cell with the “=riskuniform(0,1)” formula and hit the “Add Output” button; then increase “Iterations” to 10,000 and hit the “Start Simulation” button Page 26 Management Science Group
- 28. Histogram of 10,000 simulation trials Roughly – but not precisely – 2,000 results in each bin Why don’t we find precisely 2,000 results in each bin? If you rolled 6000 dice, would you expect precisely 1,000 1’s, 2’s,…? Page 27 Management Science Group
- 29. So... An uncertain number is a shape Histogram or target curve An uncertain number cell in Excel is a number that changes when F9 is hit We can use Monte Carlo simulation to check the shape of any uncertain number cell Page 28 Management Science Group
- 30. Practice continued… Now, next year’s uncertain profit of your unit is the sum of the profits of two divisions The revenue for each division ranges between £0 and £0.5M For each division, the uncertainty is resolved by a spin of a wheel of fortune (each division has its own wheel of fortune) Today, before the spinners are spun, the revenues are uncertain. What it the shape of the uncertain revenue of each division? What is the shape of the uncertain revenue of the company as a whole? Draw rough histograms with 5 bars Page 29 Management Science Group
- 31. Monte Carlo Simulation Traditional models: Numbers in numbers out Monte Carlo Simulation models: Shapes in shapes out Monte Carlo works like this: 1. Specify shape for each uncertain input @ Risk: Put a “=riskXXX” fomula in the input cell 2. Specify the output cells of interest to you 3. Specify how often you want to “roll the dice” (the number of Monte Carlo trials) and 4. For each Monte Carlo trial: - Roll the dice for all uncertain input cells (“Press F9”) - Store the realizations of your uncertain inputs and the corresponding calculation of your specified outputs in one row of a “results spreadsheet” 5. Repeat this for the specified number of trials (1000 - 10,000) store many rows of MC records, one for each trial (“F9 hit”) 6. Produce the shape of the output in the MC records spreadsheet Page 30 Management Science Group
- 32. Generating the shape of a 2-division firm Total profit = profit division 1 + profit division 2 Profit division 1 and division 2 are uncertain numbers Generate input shapes Profit division 1: 10,000 trials of =RiskUniform(0,1)*500 Profit division 2: 10,000 trials of another =RiskUniform(0,1)*500 Generate output cell Add the input trials up, trial by trial, to get 10,000 trials of total profit To get a target curve: Right-click on histogram graph, go to “Distribution Format”, choose “Cumulative Ascending” Page 31 Management Science Group
- 33. Recall the four basic questions 1. What is an uncertain number? 2. What happens when uncertain numbers are combined in our plans? 3. Uncertain numbers can depend on one-another. What effect does that have on our plans? 4. What happens when uncertain numbers in business plans are replaced by projections (aka common practice)? Page 32 Management Science Group
- 34. Crucial Concept # 2 When uncertainties are combined the shape goes up in the middle + = Page 33 Management Science Group
- 35. So what? – Here is your choice…. In both cases revenues range between 0 and $1M In both cases the expected revenue is $0.5M Page 34 Management Science Group
- 36. Which shape would you prefer if your were sure to get fired if you achieve less than $200k? Page 35 Management Science Group
- 37. What is the chance of meeting a 200k target? Page 36 Management Science Group
- 38. Which shape would you prefer if your contract promises a huge bonus if you achieve at least $800k? Page 37 Management Science Group
- 39. Why did the shape go up in the middle? (Picture courtesy of Analycorp.com ) Page 38 Management Science Group
- 40. Diversification If you add up uncertain numbers, the shape of the sum is more peaked in the middle than the individual shapes Extreme outcomes become less likely You “buy” a reduction of downside risk and pay for this through a reduction of upside opportunity Diversification works for all kind of shapes, not just flat ones The peaking increases the more shapes you add up Let’s check this with Monte Carlo simulation Page 39 Management Science Group
- 41. The Central Limit Theorem Fundamental mathematical result: When you add up many (independent) uncertain numbers, then the resulting uncertain sum has a bell shape (“normal distribution”) Will discuss the concept of “independent” later Page 40
- 42. Some more statistics-Latin: What’s the “standard deviation”? The 66,95,99% rules KEY: The standard deviation makes only intuitive sense in the case of a bell-shape (normal distribution) If an uncertain number has a bell shape then there is about - 66% chance that the result will be within one standard deviation of the mean -95% chance that the result will be within two standard deviations of the mean - 99.7% chance that the result will be within three standard deviations of the mean
- 43. Target curve for a normal distribution with mean = 2000 and standard deviation = 50 1950 = mean – 1 * stdev = 17th percentile 1900 = mean – 2 * stdev = 2.5th percentile 2050 = mean + 1* stdev = 83rd percentile 2000 = mean + 2* stdev = 97.5d percentile Hence 66% of data is within 1 stdev of mean Hence 95% of data is within 2 stdev of mean
- 44. Key learning points Crucial Concept # 1: Uncertain numbers are shapes - Range of outcomes - Probability of landing in a particular region of the range - Represent uncertain numbers by histograms and / or target charts Crucial Concept #2: When uncertain numbers are combined the shape goes up in the middle - Probability of extreme events (both sides) goes down Crucial Technology: We can calculate with shapes just as with numbers, using Monte Carlo Simulation Page 43
- 45. Individual work Make yourself familiar with @Risk, using steps 1-3 and 4a of the tutorial at http://www.palisade.com/risk/5/tips/en/gs/ Step 1: Quick Start Step 2: Model Step 3: Simulate Step 4.a: Histograms and cumulative curves Page 44 Management Science Group
- 46. Group work Perform Group Activity Session 5 Page 45 Management Science Group