Decision and Risk Analysis

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

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

6. Cumulative functions of standard distributions Cumulative function Distribution 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. Rj+1 = MOD(a Rj +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 Min 2,000 Max: 20,500 St.d.: 2900 Assuming correlation of -0.8 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 Cumulative function RiskCumul(50,300,{125,150,190},{0.25,0.5,0.75})

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 Cumulative function RiskCumul(0.3,0.6,{0.36,0.4,0.45},{0.25,0.5,0.75})

17. Dynatron - Simulaton Results

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

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

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

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