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Risk Management & Real Options VI. Diversification Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05 Have a go at diversification Example courtesy of Sam Savage: Safe investment Probability Outcome 40% -10 60% 50 Expected value= 26 Risky investment Probability Outcome 60% -10 40% 80 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 2 September 2004 © Scholtes 2004 Page 2 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 September 2004 ̵ 2 can only occur as 1+1 1:36 chance ̵ 7 occurs in every row and has a 6:36=1:6 chance 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 © Scholtes 2004 Each cell has 1:36 chance Page 3 Towards the normal curve… # cells in the matrix 2 3 4 5 6 7 8 9 10 11 12 • 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 2 September 2004 © Scholtes 2004 Page 4 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 X 1 ... X n Xn X1 X1 V ( X1) V ( X1) V( ) V ( ) ... V ( ) nV ( ) n 2 n n n n n n 2 September 2004 © Scholtes 2004 Page 5 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 • Market uncertainties are often a driver of statistical dependence • • 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) 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 2 September 2004 © Scholtes 2004 Page 6 Correlation Risky Safe 80 -10 50 p 0.6-p 0.6 -10 0.4-p p 0.4 0.4 0.6 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 • • Payoff 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 ̵ ̵ 2 September 2004 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%) © Scholtes 2004 Page 7 Correlation Risky Payoff 80 -10 50 0.4 0.2 0.6 -10 0 0.4 0.4 0.4 0.6 Safe 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 2 September 2004 © Scholtes 2004 Page 8 Portfolio optimization Problem: Choose a portfolio of investments under budget constraints • Problem: Each portfolio has a shape as its value – some portfolios will be optimal in some scenarios, others will be optimal in other scenarios • Chose which wells to drill if you have an exploration budget of $200 M 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” 2 September 2004 © Scholtes 2004 Page 9 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 2 September 2004 © Scholtes 2004 Page 10 Portfolio optimization Return Risk-Return Profile Risk Which portfolios would you choose? 2 September 2004 © Scholtes 2004 Page 11 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 2 September 2004 © Scholtes 2004 Page 12 Portfolio optimisation - technicalities Return Risk-return profile Risk Typically a huge number of possible portfolios e.g. if you have 20 wells to choose from, the number of possible portfolios is 220 (>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 2 September 2004 © Scholtes 2004 Page 13 Where are we? I. II. III. IV. V. VI. Introduction The forecast is always wrong I. The industry valuation standard: Net Present Value II. Sensitivity analysis 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 Developing valuation models I. Easybeds revisited Designing a system means sculpting its value shape I. CASE: Designing a Parking Garage I II. The flaw of averages: Effects of system constraints Coping with uncertainty I: Diversification I. The central limit theorem II. The effect of statistical dependence III. Optimising a portfolio 2 September 2004 © Scholtes 2004 Page 14