Risk Management & Real Options III. The value is a shape Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course

2 September 2004 © Scholtes 2004Page 2 An interesting worksheet function… Insert =randbetween(5,15) in a spreadsheet cell and press F9 Every time a different integer between 5 and 15 is generated Each integer has the same chance (=1/16) to appear How would you check this? Simulating the roll of a die =randbetween(1,6) simulates the roll of a die =randbetween(1,6)+randbetween(1,6) simulates the roll of two dice What are the possible values? What’s the distribution of these values? Can we use this tool for management analysis?

2 September 2004 © Scholtes 2004Page 3 The Monty Hall Game Revisited Recall: Contestant chooses one of three closed doors to win the prize behind the door Behind one of the doors is a £30K sports car, behind the other two doors are goats Monty Hall, who knows which door hides the sports car, will now open one of the remaining doors with a goat behind it Now the contestant is asked if she wishes to switch to the other door  Should she switch? LET’S SIMULATE THE GAME

2 September 2004 © Scholtes 2004Page 4 The value shape of the Monty Hall decisions

2 September 2004 © Scholtes 2004Page 5 Don’t over-interpret distributions The fact that one shape is preferable over another does not mean that the corresponding decision always leads to a better outcome What’s the chance that swapping turns out to be the wrong decision? In the presence of uncertainty it is important to distinguish between “good decisions” and “good outcomes” Good decisions may not have good outcomes

2 September 2004 © Scholtes 2004Page 6 Probabilities Objective probability: Assign to a scenario the frequency of its occurrence in the long run Based on “law of averages”: If we repeat a random experiment over and over again and record the number of times a particular scenario occurs, then the percentage of times this scenario occurs converges to some number in the long run Law of nature, not a law of mathematics Basis is the notion of a “repeated random experiment” – flipping a coin, rolling a dice Objective through data recording Subjective probability: Tool for the quantification and communication of uncertainty, even if there is no repeated experiment Most “experiments” in business and economics are never repeated What’s the chance of selling more than 1 Mio cars of a new model? We build mental analogy to repeated experiment, but there is no repeated experiment Why do we work with probabilities? Enable visualisation of uncertainties as shapes, technically called “distributions”

2 September 2004 © Scholtes 2004Page 7 Uncertainties as shapes: Histograms Bin: a range of possible values Bin value: typically the mid-point of the range Chance that a randomly picked value falls in this bin

2 September 2004 © Scholtes 2004Page 8 Uncertainties as shapes: Value-at-risk charts

2 September 2004 © Scholtes 2004Page 9 10% value-at-risk is about 2.5 Mio Uncertainties as shapes: Value-at-risk charts There is a 10% chance of loosing 2.5 Mio or more!

2 September 2004 © Scholtes 2004Page 10 30% chance that the realised value is negative 10% value-at-risk is about 2.5 Mio Uncertainties as shapes: Value-at-risk charts

2 September 2004 © Scholtes 2004Page 11 30% chance that the realised value is negative 20% chance that the realised value is above 7,000,000 10% value-at-risk is about 2.5 Mio Uncertainties as shapes: Value-at-risk charts

2 September 2004 © Scholtes 2004Page 12 The shape calculator: Monte Carlo Simulation Key question: What is the distribution of a bottom line system performance measure? Depends on the distributions of the uncertain inputs to the valuation model (demand, cost, price, etc) Mont Carlo simulation allows us to compute the shape of output measures from the shapes of the uncertain inputs Monte Carlo is for distributions what the calculator is for numbers Can type in “input” shapes and press a button to obtain “output” shapes “Distribution-in-distribution-out” models

2 September 2004 © Scholtes 2004Page 13 Can’t we do this intuitively? Suppose x and y are two uncertain model inputs Both, x and y, can achieve values between 0 and 1 with every number having the same chance of being picked Uniform distribution on [0,1] If x has been picked first and is known, then that information does not change the chances for the y-values “Statistical independence” What’s the shape of the following “output measures”: x+y? x*y? 1/x? y/x? Max{x,y}

2 September 2004 © Scholtes 2004Page 14 Sampling from a distribution What drives MCS?  Can ask your computer to draw a number from a distribution Repeated drawing from a distribution and recording the frequency of occurrences will reproduce the histogram of the distribution Generate one scenario by drawing once from all the distributions of the uncertainties in the model and plugging the numbers in the model Record the performance measure(s) and repeat this MANY times Simulation software helps you to do the book-keeping and to produce nice graphs

2 September 2004 © Scholtes 2004Page 15 A road-map to Monte Carlo Simulation Start with a traditional valuation spreadsheet Logically correct representation of value creation Filled with projections for uncertain values Determine key uncertainty drivers through sensitivity analysis / tornado diagram on uncertainties Determine distribution of these uncertainties (most of the work!) Use data where possible Subjective expertise o/w Perform a Monte Carlo Simulation Use MCS software (e.g. Crystal Ball)

2 September 2004 © Scholtes 2004Page 16 A bit more about the mechanics…

2 September 2004 © Scholtes 2004Page 17 A word about software Professional and Crystal Ball Widespread in financial services Professional version expensive Student versions expire after some months Overkill for most applications Difficult to learn and remember if not used on a regular basis We use Savage’s XLSim (or Sim.xla in the old version) Fine for most practical applications Easy to learn, very close to ordinary Excel Good student version with Decision Making with Insight Inexpensive professional upgrade through web

2 September 2004 © Scholtes 2004Page 18 Key Question Monte Carlo simulation is a “distribution calculator” MCS produces distributions of model outputs (performance measures) from distributions of model inputs Where do we get input distributions from? This is the modelling part  A (rather technical) lecture course in its own right Objective element: use historical / market data where-ever possibly Subjective element: use expert opinions and industry experience Creative challenge: Which industry / market data is relevant to my uncertain inputs and how can I combine this data with subjective information, e.g. expert opinions, to produce sensible input distributions?

2 September 2004 © Scholtes 2004Page 19 Examples of standard distributions: I. Discrete distributions Discrete distribution: Uncertain input can take on finitely many values, each with a certain probability Probabilities add up to 1 Can be generated with XLSim using gen_discrete() Example: Value of a card in Black Jack ̵ 2,3,4,5,6,7,8,9,10 at face value ̵ Jack,King,Queen = 10 ̵ Ace = 11 (or 1 if total above 21) Values 2,3,4,5,6,7,8,9,11 with probability 9/13, value 10 with probability 4/13 Challenge: Set up a an Excel simulation of Black Jack, assuming a very large number of decks and two players with fixed stopping strategies

2 September 2004 © Scholtes 2004Page 20 A histogram is a visualization of a discrete distribution

2 September 2004 © Scholtes 2004Page 21 Examples of standard distributions: II. Uniform distribution All values over a range are equally likely Need only specify the range Gen_Uniform(-1, 4) % 90.0% Histogramfrequency for bins of width = 1

2 September 2004 © Scholtes 2004Page 22 The rand() function Insert =rand() in a spreadsheet cell and press F9 Every time a different decimal number between 0 and 1 is generated The chance for any one number between 0 and 1 is as large as for any other This is a “uniform” random number between 0 and 1 What’s the chance that “=rand()” produces a number below 0.3? above 0.6? between 0.2 and 0.5? What’s the shape of the sum of two (or 3 or 4) uniform random variables? Use XLSim Can you explain this intuitively?

2 September 2004 © Scholtes 2004Page 23 Examples of standard distributions: III. Normal distribution Reason for its prevalence is the Central Limit Theorem: Aggregation of many independent random effects leads to bell-shape Need to specify mean and standard deviation Histogramfrequency for bins of width = 1 Gen_Normal(2, 3) <>95.0% % rule: roughly 95% of sampled values lie within 2 standard deviations of the mean

2 September 2004 © Scholtes 2004Page 24 Sampling from a normal distribution Standard Excel: =norminv(rand(),m,s) samples from a normal distribution with mean m and standard deviation s XLSim: gen_normal(m,s)

2 September 2004 © Scholtes 2004Page 25 Log-normal distributions Shape for product of random variables Multiplicative version of the central limit theorem: The product of many independent random variables is a random variable of the form Y=exp(X), where X is normal Y has so-called log-normal distribution Suppose Y can be interpreted as return for random continuous interest rate X Example: Annual return with random daily returns Prevalent model for stock price returns

2 September 2004 © Scholtes 2004Page 26 Distribution of annual interest rate R=r 1 +r 2 +….+r 365 =normal(5%,30%)

2 September 2004 © Scholtes 2004Page 27 Distribution of return if interest is paid daily and money is re-invested (hold on to stock) Skewedbell-shape

2 September 2004 © Scholtes 2004Page 28 Examples of standard distributions: IV. Triangular distribution User needs to specify lowest, most likely and highest value Gen_Triang(-2, 0, 10) % 90.0%

2 September 2004 © Scholtes 2004Page 29 Sampling from a triangular distribution XLSim: gen_triang(L,M,H) L: lowest value M: most likely value H: highest value Careful: the most likely value is NOT the mean if the triangular distribution is skewed (i.e. if M is not in the middle between L and H) Often easier to communicate than normal Gives similar result to normal but allows for skewness

2 September 2004 © Scholtes 2004Page 30 Using historical data I: Fitting a distribution Begin by inspecting your data Draw a histogram Does the rough shape match any shape you know? Choose from a variety of distributional shapes Some software packages such or Crystal Ball, offer many shapes to choose from Fit the shape to the data Simple cases: Parameters of distribution are easy to estimate from historic data (e.g. historic average and stdev for normal and Crystal Ball do that for you Can use “goodness of fit” measure to compare fit of distributions (chi- squared statistic) Unlikely that you need to know the technical details ̵ If in doubt, ask an expert

2 September 2004 © Scholtes 2004Page 31 When is data fitting appropriate? You need to have an idea about the shape of the distribution E.g. a central limit theorem argument for normal or log-normal distributions Data fitting is particularly appropriate if you have a good reason to assume that the distribution has a particular shape AND only relatively few data points (e.g. 20 rather than 200) If you have plenty of data (e.g. 1000’s of data points) you may as well sample directly from the data…

2 September 2004 © Scholtes 2004Page 32 Using historic data II: Sampling from the data The main challenge for statistics used to be to make an inference on the basis of little data Now we are often overwhelmed with data  Can we use this? Suppose you have 1000 data entries Put them in range b1:b1000 and put numbers 1 to 1000 in range a1:a1000 Use Vlookup(randbetween(1,1000),a1:b1000,2) to sample data Alternative: use gen_resample(b1:b1000) in XLSim

2 September 2004 © Scholtes 2004Page 33 Using historic data II: Sampling from the data Can also use this technique to sample correlated data, e.g. price and demand from the same day Suppose price and demand are in columns b1:c1000 Put numbers 1:1000 in a1:a1000 First sample a number x between, say 1 and 1000 Then use Vlookup(x,a1:c1000,2) to sample the x’s price and vlookup(x,a1:c1000,3) to sample the corresponding demand Alternative: Use gen_ResampleSync function in XLSim

2 September 2004 © Scholtes 2004Page 34 Taking account of trends in sampling To simulate the future from the past, you can use regression Demand at time t = Expected demand at time t + Random error Key assumption: expected demand changes with time (e.g. linear growth) but random error is independent of time (e.g. normal with mean 0 and standard deviation 20) Do a linear regression of your, say daily demand data, giving a relation Expected demand = a+b*week and generate worksheet with past data errors, one for each day: Data error = actual demand – expected demand To sample data for day k (in the future) sample one of the errors and add it to the projected expected value a+b*k Must be careful not to project linear growth too far into the future!

2 September 2004 © Scholtes 2004Page 35 What if you don’t get it right? Garbage-in-garbage-out principle applies to all quantitative analyses Sloppiness is a prevalent reason for garbage-in! Always spell out your assumptions and be prepared to defend them Make sure you have intuitive explanations for your results Always do a careful sensitivity analysis on your distributions Work with different input distributions in the case of MCS if you are not sure about the distribution E.g. use ranges of means or standard deviations and check if the “model story” changes as the parameters change

2 September 2004 © Scholtes 2004Page 36 What if you don’t get it right? Even if you don’t get it right, MCS improves on a projection-based analysis MCS is similar to the way in which shaking a ladder improves your trust in the stability of the ladder You know that you won’t apply the correct forces to the ladder – but you would be silly not to shake it before you climb up The art is to find the right balance between shaking the ladder too little and shaking it too hard

2 September 2004 © Scholtes 2004Page 37 Simulating a process We can use MCS to simulate a process with uncertain inputs Example 1: Customers at a helpline Customers ring up at a helpline, are put in a queue and served successively Shall we open a second line? Example 2: Holding inventory How much inventory do we need to hold if we want to be sure that we can meet customer demand 99% of the time? Worked spreadsheet examples are on the web

2 September 2004 © Scholtes 2004Page 38 Summary MCS Monte Carlo simulations generate the distribution of performance measures from the distributions of inputs Most important: Allows the visualization of risk profiles Histograms and cumulative distribution functions Supported by software such as or Crystal Ball Monte Carlo is a great toy that helps you test your intuition!!!

2 September 2004 © Scholtes 2004Page 39 Group work for today EasyBeds