1. Decision and Risk Analysis Financial Modelling & Risk Analysis II Kiriakos Vlahos Spring 2000

2. Session overview Probability distributions for Risk Analysis –Subjective –Regression and Forecasting models –Historic data Resampling Distribution fitting Sampling distributions –Using histograms –The inversion method Correlated random variables Comparing uncertain outcomes –Dynatron case

3. Using regression models in risk analysis Example: Ferric regression model: Cost = 11.75 + 7.93 * (1/Capacity) Standard Error (SE) = 0.98 @RISK formula for cost: Cost = 11.75 + 7.93 * (1/Capacity) + RiskNormal(0,0.98)

4. Using historic data Resampling @RISK funcion RISKDUNIFORM(datarange) At every iteration it picks one of the historic values at random.

5. Historic data - Distribution fitting 1. Historic data 2. Histogram 3.Cumulative function 4. Fit theoretical distribution 5. Then use theoretical distribution in @RISK Use statistical packages for distribution fitting

6. Cumulative functions of standard distributions Distribution function Cumulative function Uniform Triangular Normal

7. Random sampling Probabilistic simulation depends on creating samples of random variables In order to carry out random sampling we need: –a set of random numbers –a distribution or cumulative function for each of the random variables –a mechanism for converting random numbers into samples of the above distributions Tables of random numbers Pseudo random number generators: –e.g. R j+1 = MOD(a R j +c, m) The initial R is the seed Excel RAND() function

8. Inversion method Pick random number between 0 and 1 Read sample

9. Modelling correlated variables Demand = risknormal(100,20) Price = risknormal(100,20) Sales = Demand * Price Assuming correlation of -0.8 Min 2,000 Max: 20,500 St.d.: 2900 Min 5,500 Max: 13,500 St.d.: 1300 Always try to model correlation between random variables

10. Expected value Production = 100 Demand = risknormal(100,20) Sales = min(Production, Demand) If we replace Demand with its expected value then Sales equals 100. But the expected value of Sales is less than 100. In general: i.e. replacing uncertain inputs with their average values does not result in the expected value of the output unless the function is linear.

11. Dynatron Decide about: –The production level of Dynatron toys –the split into super and standard

12. Dynatron - Decision Alternatives Field Sales Representatives Production Manager Gassman

13. Cost Accounting Additional production costs

14. Base case model Profit = Revenue - Inventory cost - Investment cost

15. Dynatron - Demand uncertainty Median demand 150 Minimum 50 and maximum 300 1 in 4 chance that demand is at least 190 3 in 4 chances that demand is at least 125 RiskCumul(50,300,{125,150,190},{0.25,0.5,0.75}) Cumulative function

16. Standard/super split uncertainty % of supers Median 40 % Minimum 30% and maximum 60% 75% chance to be 45% or less 25% to be 36% or less RiskCumul(0.3,0.6,{0.36,0.4,0.45},{0.25,0.5,0.75}) Cumulative function

17. Dynatron - Simulaton Results

18. Comparing risky assets Case 1 B A Profit A>>B Case 2 A B Profit A>>B Case 3 B A Profit A>>B ?

19. Risk-return tradeoff Risk Return Efficient frontier Dominated options

20. Screening risky options Return Cumulative Probability functions 0 1 A B A>>B Return 0 1 B A if area (1) > area (2) (1) (2) then project A >> B Requires risk aversion

21. Dynatron - Simulation Results Cumulative probability distributions

22. Dynatron - Simulation Results Cumulative probability distributions

23. Summary Integrating regression and forecasting models with risk analysis Using historic data in risk analysis Resampling Distribution fitting Sampling distributions –The inversion method Model correlation between random variables! Comparing uncertain outcomes –Screening options –Risk return tradeoff –Risk preferences