Title: Lecture 4: Introduction to Risk and the normal distribution
1Lecture 4 Introduction to _at_Risk and the normal
distribution
- Tree diagrams
- _at_Risk
- The normal distribution
- Michael Wood.
- Printed or viewed on 23 November 2009
2Using probabilities to work out further
probabilities
- Mathematical probability theory
- Tree diagrams are a simple aspect of this. Well
look at a very simple example only. - The normal distribution is a widely used result
from probability theory - Monte Carlo simulation
- Using software like _at_Risk. Well focus on this
because it is easier and much more powerful.
3An example
- Probability of Heads when coin tossed
- Three coins tossed
- Prob of 0 Heads
- Prob of 1 Heads
- Prob of 2 Heads
- Prob of 3 Heads
- We can work out these probabilities
- by a tree diagram
- by doing it lots of times
- By computer simulation using _at_risk
- Well now do all three..
4Working out probabilities by
- Tree diagram
- Draw it!
- However, for more realistic problems this
approach can get very complicated! - Doing it lots of times
- Do it!
- Only possible for simple things like tossing
coins! - Computer simulation using _at_risk
- Often very powerful and flexible, and fairly easy
5_at_Risk
- See handout
- Just an introduction now more later
- See seminar exercise and handout (especially the
4 steps) - Use RiskDiscrete function to model 1 coin
- Sum heads from three coins
- Mark as output
- Simulate lots of experiments
- Look at results
6Probability distributions
- Suppose weve got a variable like
- Earnings, aerosol nozzle flow rate, height,
number of heads if a coin is tossed ten times - Often want to know frequency of different values
- With lots of empirical data we could draw a
histogram - Sometimes we can work out the pattern using
probability theory. This is called a probability
distribution because it tells us how the
probabilities are distributed which values are
more likely etc. - Many of these eg binomial, Poisson, negative
exponential distributions. Most important is the
normal distribution
7The normal distribution
- Commonly occurring symmetrical bell shaped
distribution - Heights
- Aerosol nozzle flow rates
- Coin tosses and babies
- Complex mathematical formula or use computer
- Maths based on assumption that variable depends
on a large number of small independent factors
(formula is ) - Not all distributions are normal (eg marital
problems), but many are so its widely used
8Picture of the standard normal distribution
9Normal distribution with mean178 and sd6.7
10What does this diagram mean?
- Just to help you see whats going on. Not
intended as a scale drawing! - Think of diagram as very detailed histogram
- Probs/frequencies represented by areas
- Dont worry about the vertical scale
- Horizontal scale is sds above the mean
- Eg if mean 30, sd5, then 2 represents 40, -2
is 20
11Computing the normal distribution
- Need mean and sd
- _at_risk use function RiskNormal
- (Excel use function Normdist)
- Eg probability of man being taller than 1.80m
- Use facts on the next slide
12Facts about the normal distribution
- Whatever the mean and sd the following are always
(roughly) true - 68 within 1 sd of the mean
- 95 within 2 sds of the mean
- 99.8 within 3 sds of the mean
- Now shade in the area between -1 and 1 on the
diagram and label it with the appropriate
percentage. Does it look right? - Now do the same with the area to the right of 2
13Examples of use of normal distribution
- Aerosol nozzle flow rates
- Heights
- Number of heads when a coin is tossed ten times
(use _at_risk)
14Other probability distributions
- For example
- Poisson
- Binomial
- t distribution (a bit like the normal
distribution widely mentioned in research, but
the maths is very complex) - PERT (which you will meet with _at_risk)
- And lots more
- We will meet a few in _at_risk
15Lecture 5 Introduction to Monte Carlo simulation
- Useful for analysis of complicated probability
models - Easy models can be dealt with by a tree diagram,
but this gets too difficult even for simple
models like the café problem - Basic idea is to simulate lots of possible
futures - See handout on Monte Carlo simulation and _at_risk
16Monte Carlo simulation with _at_Risk
- Ordinary (deterministic) spreadsheet to estimate
profit / return etc - Decide where main uncertainties are
- Assess appropriate probability distributions
- Paste in _at_risk distribution. Useful ones
uniform, discrete, normal, triang, PERT - Mark outputs
- Run simulation. Look at the data this shows you
lots of possible futures. - Interpret results
17Sources of probability information
- Subjective expert opinion. Ideally ask several
questions to cross-check answers. (See attached
article on Elicitation for more detailed
suggestions.) - Past data
- Past data adjusted by expert opinion
- Equally likely arguments
18Probability distributions
- For a few discrete (separate) values use
RiskDiscrete - Occasionally, the theory of the distribution will
tell you which distribution to use eg Normal,
Poisson, binomial, negative exponential - Otherwise use data or expert opinion to guess the
likely shape of the distribution and base
decision on this - Remember useful distributions are normal,
uniform, discrete, PERT
19Five common distributions
- Normal
- Uniform
- Discrete
- Triang
- PERT
- Use _at_risk to see what these are like
20An example
- Suppose you are given 100,000 to set up a
business - Decide what the business is
- Now use Excel to set up a simple cash forecast
for the end of the first and second years - Now put some uncertainties into this model
- First we will simulate it with dice
- Then we will use _at_risk
- Then we will think about what the answer means
21More examples
- CashBudget (see attached spreadsheet)
- Organic farming project
- How have I worked out the sds of the yields?
- Any other examples ?
22Uses of Monte Carlo simulation
- Many uses. Eg
- Investment appraisal
- Value at risk (banking)
- Finance
- .
- Normally you want to compare two
options/strategies or do a what-if analysis
23Interpreting the results of a simulation
- Plotting graphs to show the risk profile of each
strategy histogram or cumulative (latter useful
to check stochastic dominance CR GW, 194-5) - Mean and sd as a summary of each alternative
(especially in finance) - Estimate probability of achieving target (_at_Risk
detailed statistics) - Do a simulation of the difference between
alternatives - Need to think about what users want to know
24Lecture 6 - Risk and Uncertainty
- Statistical expectations (EMV if working in
money) - Basic concepts for analysing risk and uncertainty
- The assignment
- Michael Wood.
- Printed or viewed on 23 November 2009
25Statistical expectations
- Example 1 a builder earns 200 a day in fine
weather and 50 a day in wet weather. One day in
four is wet. - What are his expected daily earning?
- Can he expect to earn this next Monday?
26Statistical expectations
- The expected value or expectation is the long
term average taking probabilities into account.
When talking about money EMV - The mean of an _at_risk simulation is the average of
lots of possible futures ie the expected values
27Example 2 a choice
- Imagine you had the choice of
- A 1000 (definitely), or
- B A 50 chance of 2000 and a 50 chance of 0
- C A 99 chance of 2000 and a 1 chance of
losing 10000 - D A 99 chance of 1 and a 1 chance of 50,000
- Which would you choose? Why?
- What is the EMV of each?
28See spreadsheet choice.xls
- Use _at_risk to simulate this
- Not really necessary but useful to see how _at_risk
works - Note that the EMV is the same as the mean of the
simulated data - The standard deviation gives a measure of the
risk - Do you think its a good measure of the risk?
29Example of a decision with uncertainties
30Methods of choosing the best option despite
risk/uncertainty
- Eliminate dominated strategies (always worth
starting with this) - Maximin
- (Maximax)
- Greatest expected payoff (but dont forget risk)
- Work out probability distributions and then let
the decision maker use judgment - Other possibilities see handout
- For complex situations Monte Carlo simulation
(with _at_Risk) can help with the last three of these
31- An organisation has 10000 to invest. Two
alternatives are being considered. The first,
safe, alternative would guarantee them 11 000 at
the end of one year the second, risky,
alternative involves backing a bid to find oil in
the region. If this succeeds they are confident
their investment will be worth 100 000 after a
year, but if it fails they will lose their 10
000. They estimate that the chance of finding oil
is 20. - (a) find out what they should do to maximise the
expected value of their investment after a year - (b) find out what they should do if they adopt
the maximin criterion - (c) find out what they should do if they adopt
the maximax criterion - (d) decide what you would advise them to do if
the organisation was small and would be in severe
difficulties if the 10 000 were to be lost - (e) decide what you would advise them to do if
the organisation were a large multi-national
corporation which makes decisions of this kind
regularly.
32Attitudes to risk
- Decision makers may be
- Risk neutral
- Risk averse
- Risk seeking
- Can you think of examples of each?
- What would each type of decision maker do in the
choice example above? - A common measure of risk (especially in finance)
is the standard deviation of the possible
outcomes. This measures how variable they are
(e.g. choice example above and Exercise 5)
33Questions on the assignment
34Lecture 9 More on Monte Carlo simulation and the
assignment
- The problem of variables which are not
independent - Assessing probability distributions
- Interpreting the answers from a simulation
- Uses of _at_risk value at risk, investment
appraisal, finance, cash flow forecasts. - The assignment
- Printed on 25 October 2006
35Must check
- Probabilities, costs etc are realistic
- Use data whenever possible, and show how you used
it - See article on elicitation for getting views from
an expert
36Independence of probabilities
- To see the problem, work out the probability of
getting rain three days in a row in Portsmouth
(the probability of rain on one day is 0.25
according to an expertDr French) - If not independent can set correlations see
handout. - Example returns from shares in two firms may not
be independent because (see Exercise)
37Dealing with variables which are not independent
- Example 1 FourShares.xls
- Can we assume returns from different shares are
uncorrelated? Probably not because - If we assume a correlation we can get _at_risk to
take account of it when generating random
values of the variables (see _at_Risk handout and
Q3) - What difference do you expect this to make to the
answers? - Example 2 Organic farming
- There is likely to be a correlation between
yields because of the weather. This correlation
is built into the model by means of the weather
scenario variable.
38Assessing probability distributions
- Take care this is important!
- See Lecture 6 slides
- Important distributions are discrete, uniform,
normal, triang, Pert - Pert / triang useful if you can get an assessment
of min, most likely and max from expert - Look at attached Halfway houses data
- See cash flow example too
39Interpreting the answers
- Histogram
- Mean and sd
- Cumulative graphs see next slide
- (See organic farming question)
40(No Transcript)
41Uses of Monte Carlo simulation
- Can analyse uncertainties in any spreadsheet
model. - Eg Bruces cash flow model
- Decide whats uncertain (sales on row 131?)
- Decide on outputs
- How does the Monte Carlo answer compare with the
deterministic answer?
42The assignment
- Sources of information any questions?
- A lot of reading on probabilities etc. Also
article on Elicitation in Significance (June
2005) - Evaluating the model (deciding how valuable or
useful it is). - How to do this?
- Assumptions
- Eg variables uncorrelated how do we check this?
43The assignment
- Dont forget that the soft (non-numerical)
issues are often very important eg - Have you got the value tree right
- Have you omitted anything from the model
- Have you got a realistic short list of options?
- Has your decision maker used the swing weighting
process correctly?
44Questions?
- On Monte Carlo or Multi-criteria analysis, or the
assignment .