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Lecture 4: Introduction to Risk and the normal distribution

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Title: Lecture 4: Introduction to Risk and the normal distribution


1
Lecture 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

2
Using 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.

3
An 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..

4
Working 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

6
Probability 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

7
The 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

8
Picture of the standard normal distribution
9
Normal distribution with mean178 and sd6.7
10
What 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

11
Computing 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

12
Facts 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

13
Examples of use of normal distribution
  • Aerosol nozzle flow rates
  • Heights
  • Number of heads when a coin is tossed ten times
    (use _at_risk)

14
Other 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

15
Lecture 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

16
Monte 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

17
Sources 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

18
Probability 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

19
Five common distributions
  • Normal
  • Uniform
  • Discrete
  • Triang
  • PERT
  • Use _at_risk to see what these are like

20
An 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

21
More examples
  • CashBudget (see attached spreadsheet)
  • Organic farming project
  • How have I worked out the sds of the yields?
  • Any other examples ?

22
Uses 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

23
Interpreting 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

24
Lecture 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

25
Statistical 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?

26
Statistical 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

27
Example 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?

28
See 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?

29
Example of a decision with uncertainties
30
Methods 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.

32
Attitudes 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)

33
Questions on the assignment
34
Lecture 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

35
Must 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

36
Independence 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)

37
Dealing 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.

38
Assessing 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

39
Interpreting the answers
  • Histogram
  • Mean and sd
  • Cumulative graphs see next slide
  • (See organic farming question)

40
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41
Uses 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?

42
The 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?

43
The 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?

44
Questions?
  • On Monte Carlo or Multi-criteria analysis, or the
    assignment .
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