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Risk Management & Real Options VI. Diversification Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05

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2 September 2004 © Scholtes 2004Page 2 Have a go at diversification Example courtesy of Sam Savage: Which of the following portfolios minimizes the probability of loosing money? 100% safe 90% safe, 10% risky 20% safe, 80% risky None of the above Safe investmentProbabilityOutcome 40%-10 60%50 Risky investmentProbabilityOutcome 60%-10 40%80 Expected value= 26

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2 September 2004 © Scholtes 2004Page 3 What does diversification mean? Instead of investing all your money in one uncertain payoff, invest it in several ones Rolling 1 die: 1:6 chance of each number between 1 and 6 Rolling 2 dice: (sum of two random numbers) ̵ 2 can only occur as 1+1 1:36 chance ̵ 7 occurs in every row and has a 6:36=1:6 chance 123456 1234567 2345678 3456789 45678910 56789 11 6789101112 Each cell has 1:36 chance

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2 September 2004 © Scholtes 2004Page 4 Towards the normal curve… 23456789101112 # cells in the matrix Extremes (low or high) numbers can only occur if BOTH dice show the same extreme (low or high) numbers Numbers in the middle have a higher chance of being realised

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2 September 2004 © Scholtes 2004Page 5 Driver for diversification: The central limit theorem Central limit theorem: If we aggregate many independent random effects, the aggregated random variable looks like a normal random variable. Mathematically: The sum of n independent random variables is approximately normal Mean of this normal = sum of the mean of the individual r.v.’s Variance of this normal = sum of the variances of the individual r.v.’s If we take the average of many independent variables with the same variance, then the shape of the average becomes more and more normal but also narrower and narrower as we increase the number of random variables

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2 September 2004 © Scholtes 2004Page 6 Important Caveat: Statistical Dependence The statistical independence of the variables you aggregate is crucial for diversification to work. Statistical independence of two variables means, for all practical matters, that there is no common driving source shared by the variables E.g. Volumes of two oil wells can be thought of as independent random variables, whilst the revenues generated by two oil wells are dependent (with oil price being the common driver of the latter) Market uncertainties are often a driver of statistical dependence Exchange rate risk, fashion, oil price, etc. Cannot be diversified away! Uncertainties that are specific to one company are often called “private risks” Can be dealt with through diversification

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2 September 2004 © Scholtes 2004Page 7 Correlation For which value of p are the two gambles statistically independent? What if p=0? If I know that “safe” is higher than expected, then “risky” MUST be lower than expected Negative correlation One variable gives complete knowledge about the other correlation coefficient = -1 What if p=0.4? If I know that “safe” is 50 then ̵ 80 is more likely for “risky” (POSITIVE CORRELATION) ̵ But there is still a 1/3 chance that I am wrong (Correlation coefficient = 1 – chance of being wrong=66%) Risky Payoff80-10 Safe50p0.6-p0.6 -100.4-pp0.4 0.6

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2 September 2004 © Scholtes 2004Page 8 Correlation Which of the following portfolios minimizes the probability of loosing money? 100% safe 90% safe, 10% risky 20% safe, 80% risky None of the above Risky Payoff80-10 Safe500.40.20.6 -1000.4 0.6

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2 September 2004 © Scholtes 2004Page 9 Portfolio optimization Problem: Choose a portfolio of investments under budget constraints Chose which wells to drill if you have an exploration budget of $200 M Problem: Each portfolio has a shape as its value – some portfolios will be optimal in some scenarios, others will be optimal in other scenarios BUT: Need to make a decision before scenario is observed Can’t we go with the portfolio with maximal expectation, assuming that the size of the portfolio will take care of the risk (diversification argument)? Yes if the portfolio is large and the investments are independent The latter is unlikely; e.g. wells depend on the movement of the oil price Need to understand “residual risk” of portfolio Want to maximise the “value” and minimise the “risk”

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2 September 2004 © Scholtes 2004Page 10 Portfolio optimization Numbers have natural rank (maximal, minimal, etc), shapes don’t Boil shapes into numbers Measures for “value” e.g. expected value Many possible measures for “risk”, e.g. Variance (Expected squared deviation from expected value) “Semi-variance” (Expected (squared) deviation below the expected value) Probability of loosing money 10% value at risk Two objectives: Maximise value and minimize risk

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2 September 2004 © Scholtes 2004Page 11 Portfolio optimization Which portfolios would you choose?

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2 September 2004 © Scholtes 2004Page 12 Portfolio optimisation - practicality Fix return expectation and minimise risk, subject to return at least at expectation and cost of portfolio not exceeding budget Alternative: fix risk level and maximise return subject to portfolio does not exceed fixed risk level and cost of portfolio does not exceed budget

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2 September 2004 © Scholtes 2004Page 13 Portfolio optimisation - technicalities Typically a huge number of possible portfolios e.g. if you have 20 wells to choose from, the number of possible portfolios is 2 20 (>1,000,000) Can use Excel “solver” to solve moderate size problems (e.g. choose from 10-20 wells) see Decision Making with Insight for details on “solver” and other optimization software

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2 September 2004 © Scholtes 2004Page 14 Where are we? I. Introduction II. The forecast is always wrong I. The industry valuation standard: Net Present Value II. Sensitivity analysis III. The system value is a shape I. Value profiles and value-at-risk charts II. SKILL: Using a shape calculator III. CASE: Overbooking at EasyBeds IV. Developing valuation models I. Easybeds revisited V. Designing a system means sculpting its value shape I. CASE: Designing a Parking Garage I II. The flaw of averages: Effects of system constraints VI. Coping with uncertainty I: Diversification I. The central limit theorem II. The effect of statistical dependence III. Optimising a portfolio

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