1. Risk Analysis Simulate a scenario of possible input values that could occur and observe key impacts Pick many input scenarios according to their likelihood of occurring Record and summarize the key impacts observed to measure 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 (0 and 100%). Think of this number as the cumulative probability of seeing a value for the input assumption that would be smaller than the number associated with this probability. For example, a formula that uses Rand()=.5 would tell you the input assumption value where 50% of the input assumption values would be smaller than the number calculated. A formula with Rand()=.9 would tell you the input assumption value where 90% of the input assumption values would be smaller than the number calculated.

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 will look up the value for the input distribution that will result in rand()% of the assumption values being smaller than the returned value x.

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

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

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

9. .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

10. Discrete Distributions Set up a table where the first column contains the cumulative probability for a value, the second column contains the discrete values that could occur. Use the Excel function: =vlookup(rand(),table,2) 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 2.

11. 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