Introduction Distributions Example
CORPFIN 2503 – Business Data Analytics: Monte Carlo simulation
Week 8: September 13th, 2021
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Introduction
Distributions
Introduction
Distributions
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Introduction Distributions Example
Introduction
Future is uncertain; thus, analysts and managers use several techniques to analyze future events or outcomes:
• Sensitivity analysis shows how the (many) different values of one uncertain variable affect another variable:
• e.g., how different values of oil price affect Qantas profit
• Scenario analysis shows how (a few) different values for several uncertain variables affect another variable:
• e.g., best / most likely / worst case scenarios for BHP profit taking into account commodity prices, exchange rates, and interest rates
• Simulation is the synthesis of sensitivity and scenario analysis: many scenarios for many different values of several uncertain variables
• Monte Carlo simulation is sophisticated simulation.
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Introduction Distributions Example
Monte Carlo simulation
1. We need to identify factors that affect a variable of interest (e.g., stock prices, int. rates, exchange rates).
2. Then we determine the distribution of the factors as well as correlation coefficients between the factors.
3. We generate randomly, e.g. 1,000 values for these factors. ⇒ So we have 1,000 scenarios.
4. For each scenario, we calculate the value of the variable of interest.
5. We sort the obtained values in the ascending order.
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Introduction Distributions Example
Monte Carlo simulation II
The distribution of outcomes can be used to:
1. compute the expected, worst, and best outcomes
• e.g., best / most likely / worst case scenarios for BHP profit taking into account commodity prices, exchange rates, and interest rates
2. calculate the probability of a particular outcome (if the number of scenarios is relatively small)
3. calculate the probability that the outcome is greater or lower than a particular value
• e.g., what is the probability that BHP profit will be greater than AUD10 billion?
4. find the lowest 1st and 5th percentiles of the outcome (for value at risk (VaR)):
• e.g., if VaR on an asset is $100m at a one-week, 95% confidence level, it means that there is a only a 5% chance that the value of the asset will drop more than $100m over any 1 week.
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Introduction Distributions Example
Distributions
In this lecture, we will consider the following distributions: 1. normal
2. uniform.
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Introduction Distributions
Normal distribution
Normal distribution is defined by 2 properties: • mean
• standard deviation.
̌ius CORPFIN 2503, Week 8
Introduction
Distributions
Uniform distribution
Uniform distribution is defined by 2 properties:
• minimum value
• maximum value.
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Introduction Distributions Example
Let’s use Monte Carlo simulation to predict net income of the following firm:
• Sales follow normal distribution: N (100, 20)
• Fixed costs follow uniform distribution: U(20,22)
• Variable costs depend on sales: Sales×U(0.4,0.5)
• A firm has debt of 100; interest rate follows normal distribution: N(0.02,0.005)
• Income tax rate follows uniform distribution: U(0.25,0.30).
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Introduction Distributions
Example II
Net income using expected (i.e., average) values:
Variable Value
Sales 100 – Fixed costs –21
Calculation
– Variable costs
– Interest payments
0.4+0.5 × 100 2
32 Net income 23.2
100 × 0.02
100−21−45−2 0.25+0.3 × 32
Net income before taxes
–Taxes –8.8
32 − 8.64 Monte Carlo simulation will reveal how this value is impacted by
uncertainty of the future.
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Introduction Distributions Example
Example III
First, we need to generate 5 random variables:
data work.monte_carlo;
do scenario = 1 to 1000;
sales = rand(“Normal”, 100, 20);
fixed_costs = rand(“Uniform”, 20, 22);
variable_costs_ratio = rand(“Uniform”, 0.4, 0.5);
interest_rate = rand(“Normal”, 0.02, 0.005);
income_tax_rate = rand(“Uniform”, 0.25, 0.30);
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Introduction Distributions Example
Notes for the SAS code on the previous slide
SCENARIO is an index (or counter) variable. It is set to iterate from 1 to 1,000:
do scenario = 1 to 1000;
Now the loop has been setup; thus, we need to specify what should happen during each iteration.
OUTPUT option instructs SAS to record each iteration in our dataset.
END statement closes off the DO loop.
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Introduction Distributions Example
Example IV: Sample
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Introduction Distributions Example
Example IV
Let’s look at the statistical properties of these variables:
proc means data=work.monte_carlo n mean std min p1 p5
p25 median p75 p95 p99 max maxdec=3;
var sales fixed_costs variable_costs_ratio interest_rate
income_tax_rate;
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Introduction Distributions Example
̌ius CORPFIN 2503, Week 8
Introduction Distributions
Example VI
Let’s look at their histograms:
proc sgplot data=work.monte_carlo;
histogram fixed_costs;
proc sgplot data=work.monte_carlo;
histogram variable_costs_ratio;
proc sgplot data=work.monte_carlo;
histogram income_tax_rate;
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Introduction Distributions Example
Example VII
proc sgplot data=work.monte_carlo noautolegend;
histogram sales;
density sales;
density sales / type=kernel;
keylegend / location=inside position=topright across=1
proc sgplot data=work.monte_carlo noautolegend;
histogram interest_rate;
density interest_rate;
density interest_rate / type=kernel;
keylegend / location=inside position=topright across=1
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Introduction Distributions Example
Histogram for FIXED_COSTS
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Introduction Distributions Example
Histogram for VARIABLE_COSTS_RATIO
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Introduction Distributions Example
Histogram for INCOME_TAX_RATE
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Introduction Distributions Example
Histogram for SALES
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Introduction Distributions Example
Histogram for INTEREST_RATE
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Introduction Distributions
Example VIII
Let’s compute net income for each scenario:
data work.monte_carlo;
set work.monte_carlo;
net_income= (sales – fixed_costs
– sales*variable_costs_ratio
– 100*interest_rate)*(1-income_tax_rate);
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Introduction Distributions Example
Example IX
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Introduction Distributions Example
Now let’s look at the statistical properties of NET_INCOME:
proc means data=work.monte_carlo n mean std min p1 p5 p25
median p75 p95 p99 max maxdec=3;
var net_income;
proc sgplot data=work.monte_carlo noautolegend;
histogram net_income;
density net_income;
density net_income / type=kernel;
keylegend / location=inside position=topright across=1
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Introduction Distributions Example
Descriptive statistics of NET_INCOME
Net income using expected (i.e., average) values is 23.2; mean and median values from Monte Carlo simulation are also around 23.
Min and max values are –4.265 and 54.726.
5th percentile value is 9.110 =⇒ we are 95% confident that net income will be higher than 9.110.
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Introduction Distributions Example
Histogram of NET_INCOME
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Introduction Distributions Example
What if we increase the number of scenarios?
The key difference: when the sample size increases, the range becomes wider.
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Introduction Distributions Example
What if we increase the number of scenarios? II
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