1. Risk Analysis Simulate a scenario of possible input values that could occur and observe key financial impacts Pick many different input scenarios according to their likelihood of occurring Record and summarize the key impacts observed to measure financial risks

2. 5 Steps of Risk Analysis Build a spreadsheet model that has dynamic relationships between input assumptions and key outputs Perform sensitivity analysis to identify the key inputs that have the most potential impact on the key outputs Quantify the possible values for the key uncertain inputs by specifying probability distributions Run a simulation to pick scenarios from the input probability distributions and record observed output results Summarize recorded output results to measure risks and likelihood of different outcomes

3. Random Number Generator (RNG) =Rand() function in Excel Randomly generates a number between 0 and 1. This number represents a cumulative probability P(X) between 0% and 100%. We want to program Excel to identify the input value X such that the probability that this value X or a lower value will occur is equal to the generated cumulative probability P(X). For example, a formula that refers to Rand()=.5 would tell you the input assumption value X where 50% of the input assumption values are smaller and 50% of possible input assumption values are larger than X. A formula with Rand()=.9 would tell you the input assumption value X where 90% of the input assumption values are smaller than X.

4. Prob Normal (u, σ) P(X≤u)=.5 P(u- σ ≤x ≤u+ σ )=.65 σ u- σ uu+ σ x

5. Normal Input Distributions =Norminv(rand(), mean, standard deviation) For a specified mean and standard deviation, this formula looks up the value for the input distribution that results in rand()% of the assumption values being smaller than the returned value. The Normal distribution is a continuous distribution

6. Continuous –vs- Discrete Distributions In discrete distributions, the values generated for a random variable must be from a finite distinct set of individual values. In continuous distributions, the values generated for a random variable are specified from a set of uninterrupted values over a range; an infinite number of values is possible

7. Uniform (a, b) a u=a+(b-a) b X 2 Prob P(X ≤ u)=. 5

8. Uniform Distribution =a + (b-a)*rand() Where a is the smallest value that could occur, b is the largest value Values between a and b are assumed to be equally likely to occur Values are assumed to be continuous and not discrete

9. Discrete Distribution P(x) X P(x) 10.2 11.3 12.4 13.1 10 11 12 13 x

10. .2 1.0/0.9.5 11 10 13 12 Cumulative Probability Distribution X P(x) 10.2 11.3 12.4 13.1 P(X≤x).2.5.9 1.0

11. Discrete Distributions Set up a table where the first two columns contain the cumulative probability range for a value and the third column contains the respective discrete values. Make sure the first column starts at 0 Lower Prob Upper Prob Discrete Value 0.210.2.511.5.912.91.013

12. Vlookup formula for the Discrete Distribution Use the Excel function: =vlookup(rand(),table,3) This function will look up the rand() number in column 1 of the table, identify the row that represents the correct cumulative probability, and look up the value associated with that probability in column 3. See Vlookup Excel Tutorials in Blackboard for logic of the Vlookup formula

13. Alternative Histogram Approach Set up the bins you would like in the first column. Make sure that the first bin value is lower than the minimum result and the last bin value is greater than the maximum result Format column 2 to display frequency results for the bins in column 1. Highlight the array of cells in column 2 that should hold the results. Use the array function : =frequency(column with simulation output data, column with bin values) Enter function by pressing [Ctrl][Shift][Enter] instead of the [Enter] key