Monte Carlo Simulation A technique that helps modelers examine the consequences of continuous risk Most risks in real world generate hundreds of possible outcomes Provides fuller picture of the risk in an asset or investment by considering Different input assumptions & scenarios Likelihood of inputs & scenarios occurring

5 Steps of Monte Carlo Simulation Build a spreadsheet model that has dynamic relationships between input assumptions and key outputs Perform sensitivity analysis to identify the key uncertain inputs that have the most potential impact on the key outputs Quantify possible values for the key uncertain inputs by specifying probability distributions Simulate numerous scenarios from the input probability distributions and record output results Summarize recorded output results to measure risks and likelihood of different outcomes

Random Number Generator (RNG) =Rand() function in Excel Randomly generates a number between 0 and 1 Used to represent a cumulative probability P(X) between 0% and 100% Will be used to identify the input value X such that the probability that value X or a lower value occurs is equal to P(X) For example, Rand()=.5 Used to find input assumption value X where 50% of the input assumption values are smaller and 50% of possible input assumption values are larger than X. Rand()=.9 Used to find input assumption value X where 90% of the input assumption values are smaller than X.

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

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

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

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

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

Probability Distributions in Analytic Solver Select cell that represents an uncertain input assumption On the Analytic Solver Platform ribbon, click Distributions in the Simulation Model group Pick a Common continuous distribution or a Discrete distribution Enter values for its parameters in dialog box and click Save

Generating Values for Random Variables Analytic Solver will insert: =PsiDistribution(parameters) function into the input assumption cell from which values will be randomly selected Pressing [F9] Recalc key will simulate next set of values in model Double-clicking on the input cell will open the dialog box for further editing or review This function can be copied to other cells

Specifying Input Distributions Practice in Class Exercise 2!

Tracking Output Measures (KPI’s) in Analytic Solver Select cell that represents an output measure On the Analytic Solver Platform ribbon, click Results in the Simulation Model group Click Output and In Cell option This will append the PsiOutput() function to the formula in the cell which triggers ASP to record the cell’s value during the simulation

Tracking Output Statistics in Worksheet with Analytic Solver Select cell that represents an output measure On the Analytic Solver Platform ribbon, click Results in the Simulation Model group Click Statistic and desired statistic option (e.g. Mean) Click on the blank cell where you want this statistic entered =PsiStatistic(OutputMeasure) function will be entered in the cell and calculated after the simulation is run

Simulation Settings Select number of trials (e.g. runs, scenarios) to be performed: On the Analytic Solver Platform ribbon, click Options in the Options group Set Trials per Simulation to desired number on Simulation tab. You can also check Use Correlations if you want to correlate input distributions

Run Simulation On the Analytic Solver Platform ribbon, click Interactive in the Simulate button for the Solve Action group The simulation will be run and a dialog box will pop up with the distributions for your output measures. The requested statistics will also become available in worksheet. You can click on any chart to see statistics and interactive graph.

Correlated Input Variables On the Analytic Solver Platform ribbon, click Correlations in the Simulation Model group Select Matrices and move random variables that you want to correlate to right side of task bar. Click on the square that has a scatter plot for two random variables and move the correlation bar to define the relationship between the two variables. Save in blank cells in worksheet and then run simulation.