1 uncertain numbers and diversification

Judge Business School

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THE LANGUAGE OF UNCERTAINTY I

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

Today

2. What happens when uncertain numbers are combined in plans?

PLUS

Introduction to Monte Carlo simulation in Excel


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”


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


Example:
A histogram of uncertain revenues

What is the chance that revenues will be between $9M and $10M?


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

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

Target probabilities…

What is the chance of achieving a target of $8M?


Turning a histogram into a target curve

Add up probabilities (stack up blue bars) to the left


Turning a histogram into a target curve

Probability of missing a $8M target is about 45%
Probability of achieving the target is about 55%


Bar chart is often turned into a curve: the target curve

The target curve is also known as the “cumulative distribution function”


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%

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”


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

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

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

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”

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

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)

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

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?

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…


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


The wheel of fortune uncertainty has a
flat shape between 0 and 1M


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


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,…?


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


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


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

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”


Recall the four basic questions

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)?


Crucial Concept # 2
When uncertainties are combined the shape
goes up in the middle

+

=

So what? – Here is your choice….

In both cases revenues range between 0 and $1M
In both cases the expected revenue is $0.5M


Which shape would you prefer if
your were sure to get fired
if you achieve less than $200k?


What is the chance of meeting a 200k target?


Which shape would you prefer if
your contract promises a huge bonus
if you achieve at least $800k?


Why did the shape go up in the middle?

(Picture courtesy of Analycorp.com )

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


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

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

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

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

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


Group work

Perform Group Activity Session 5


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