def get_content():
return [
pl.Section([
pl.SubSection([
'You are an analyst trying to decide whether it is worth it to launch a new product line selling '
't-shirts. The success of the t-shirt business will depend on the state of the economy. '
'You can purchase a machine which costs \\$10,000,000 at $t=0$, then each year you will earn '
'the profit per unit (revenue minus variable cost), multiplied by the output. The machine will '
'last for 10 years. Should this project be undertaken? Will it be successful in all scenarios?',
],
title='Problem Statement'),
pl.SubSection([
pl.Center(
lt.Tabular([
lt.TopRule(),
lt.ValuesTable.from_list_of_lists([[
'State of Economy', 'Probability', 'Price',
'Output', 'Interest Rate'
]]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Expansion', '20%', '15', '1000000', '7%'],
['Normal', '70%', '12', '500000', '5%'],
['Recession', '10%', '10', '200000', '3%'],
], ),
lt.BottomRule(),
],
align='l|cccc'))
],
title='Scenarios'),
],
title='Capital Budgeting')
]
def get_content():
return [
pl.Section([
pl.SubSection([
'Create a model with two stocks, A and B. Both start at price 100. Generate stock prices for '
'100 periods. Do this by drawing returns from normal distributions defined with the inputs '
'below. Then apply the returns to the prior prices to get the price in each period. Create a '
"portfolio of the two stocks, by taking the weighted-average of the stock's returns, then applying "
"that return to a third Portfolio series starting at 100. Graph "
"the two stocks and porfolio performance over time. Calculate the mean and standard deviation "
"of the generated returns for the two stocks and the portfolio."
],
title='Problem Statement'),
pl.SubSection([
pl.Center(
lt.Tabular([
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Input', 'Default Value']]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Stock A Weight', '60%'],
['Stock A Mean Return', '10%'],
['Stock A Return Standard Deviation', '30%'],
['Stock B Mean Return', '5%'],
['Stock B Return Standard Deviation', '10%'],
], ),
lt.BottomRule(),
],
align='l|c'))
],
title='Inputs'),
],
title='Stock Portfolio')
]
def get_content():
return [
pl.Section([
pl.SubSection([
'Create a model of stock returns and correlations. The asset returns should be based on the '
'capital asset pricing model (CAPM), which states:',
pl.Equation(str_eq=r'r_s = r_f + \beta (r_m - r_f) + \epsilon',
inline=False),
pl.UnorderedList([
'$r_e$: Return on stock',
'$r_f$: Return on risk free asset',
'$r_m$: Return on the market portfolio',
r'$\beta$: Covariance between the market portfolio and the stock',
r'$\epsilon$: Idiosyncratic return, (random, normally distributed with mean 0)'
]), 'Your model should accept the number of assets, '
'the average idiosyncratic standard deviation across the assets, the standard deviation of the '
'idiosyncratic standard deviation across the assets, the market average return, the '
'market standard deviation, the risk-free rate, and the number of periods as the inputs. You should randomly draw '
"each asset's standard deviation of idiosyncratic returns from a normal distribution based on the "
"inputs. You should also randomly draw the stock's beta from a uniform distribution between 0 and 2. "
"Then draw the market returns from a normal distribution with its mean and standard deviation. Then "
"the return for each asset in each time period will be determined by drawing its idiosyncratic return "
"for that period from its normal distribution, then calculating the CAPM formula. After all the assets "
"returns are generated, calculate the correlation between all the assets. Your model should be updating "
"the number of assets and number of periods merely by changing the inputs."
],
title='Problem Definition'),
pl.SubSection([
pl.Center(
lt.Tabular([
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Input', 'Default Value']]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Number of Assets', '3'],
['Number of Periods', '20'],
[
'Average Idiosyncratic Standard Deviation',
'20%'
],
[
'Standard Deviation of Idiosyncratic Return Standard Deviation',
'10%'
],
['Market Average Return', '7%'],
['Market Standard Deviation', '15%'],
['Risk-Free Rate', '2%'],
], ),
lt.BottomRule(),
],
align='l|c'))
],
title='Model Inputs'),
],
title='Generating Asset Returns using CAPM')
]
def get_content():
jinja_templates_path = os.path.sep.join(['pltemplates', 'projects', 'p4'])
jinja_env = pl.JinjaEnvironment(
loader=FileSystemLoader(jinja_templates_path))
return [
pl.Section([
pl.SubSection([
Project4ProblemModel(template_path='prob_definition.j2',
environment=jinja_env),
],
title='Problem Definition'),
pl.SubSection([
Project4ProblemModel(template_path='notes.j2',
environment=jinja_env),
],
title='Notes'),
pl.SubSection([
Project4ProblemModel(template_path='bonus.j2',
environment=jinja_env),
],
title='Bonus')
],
title='Overview'),
pl.Section([
pl.SubSection([
Project4ProblemModel(template_path='submission.j2',
environment=jinja_env),
],
title='Submission'),
pl.SubSection([
pl.Center(
lt.Tabular([
pl.MultiColumnLabel('Grading Breakdown', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Category', 'Percentage']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists(
[['Model Accuracy', '60%'],
['Model Readability', '30%'],
['Model Formatting', '10%'], ['Bonus', '5%']]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[['Total Possible', '105%']]),
lt.BottomRule()
],
align='l|c'))
],
title='Grading'),
],
title='Submission & Grading')
]
def get_content():
jinja_templates_path = os.path.sep.join(
['pltemplates', 'practice', 'python_retirement'])
jinja_env = pl.JinjaEnvironment(
loader=FileSystemLoader(jinja_templates_path))
return [
pl.Section([
pl.SubSection([
PythonRetirementPracticeProblemModel(
template_path='prob_definition.j2', environment=jinja_env),
],
title='Problem Definition'),
pl.SubSection([
pl.Center(
lt.Tabular([
lt.TopRule(),
lt.ValuesTable.from_list_of_lists([[
'Input',
'Default Value',
]]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Starting Salary', '\$50,000'],
['Salary Growth', '3%'],
['Mid-Salary Cutoff', r'\$80,000'],
['High-Salary Cutoff', r'\$120,000'],
['Low Savings Rate', '10%'],
['Mid Savings Rate', '25%'],
['High Savings Rate', '40%'],
['Interest Rate', '5%'],
['Desired Cash', r'\$1,500,000'],
], ),
lt.BottomRule(),
],
align='l|cc'))
],
title='Inputs'),
pl.SubSection([
"""
The final answer with the default inputs should be 37 years to retirement. Try hard to get
there working from scratch. If you are very stuck, then try taking the Dynamic Salary
Retirement model and modifying it. If you are still stuck, then check the provided Jupyter
notebook solution. If you have a lot of trouble with this, please see me in office hours or
after class, as your first project will be similar but a bit more difficult.
"""
],
title='Solution')
],
title=
'Capital Budgeting Probabilities with Monte Carlo Simulation'
)
]
width=0.5), ),
case_solutions_str,
pl.UnorderedList(case_solutions_list),
],
title='Selected Solutions'),
pl.SubSection([
pl.Center(
lt.Tabular([
lt.MultiColumnLabel('Grading Breakdown', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Category', 'Percentage']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists([
['Model Accuracy', '60%'],
['Model Readability', '20%'],
['Model Formatting', '10%'],
['Following the Template', '10%'], ['Bonus', '5%']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[['Total Possible', '105%']]),
lt.BottomRule()
],
align=align))
],
title='Grading'),
],
title='Submission & Grading')
]
def get_content():
random.seed(1000)
lecture = get_probability_lecture()
scenario_excel_lab = get_scenario_analysis_excel_lab_lecture().to_pyexlatex()
scenario_python_lab = get_scenario_analysis_python_lab_lecture().to_pyexlatex()
randomness_excel_lab = get_randomness_excel_lab_lecture().to_pyexlatex()
randomness_python_lab = get_randomness_python_lab_lecture().to_pyexlatex()
random_stock_lab = get_random_stock_model_lab_lecture().to_pyexlatex()
full_model_internal_randomness_lab = get_extend_model_internal_randomness_lab_lecture().to_pyexlatex()
appendix_frames = [
lecture.pyexlatex_resources_frame,
scenario_excel_lab.appendix_frames(),
scenario_python_lab.appendix_frames(),
randomness_excel_lab.appendix_frames(),
randomness_python_lab.appendix_frames(),
random_stock_lab.appendix_frames(),
full_model_internal_randomness_lab.appendix_frames(),
]
df_mono = pl.Monospace('DataFrame')
next_slide = lp.Overlay([lp.NextWithIncrement()])
with_previous = lp.Overlay([lp.NextWithoutIncrement()])
rand_mono = pl.Monospace('=RAND')
rand_between_mono = pl.Monospace('=RANDBETWEEN')
norm_inv_mono = pl.Monospace('=NORM.INV')
excel_random_normal_example = pl.Monospace('=NORM.INV(RAND(), 10, 1)')
random_module_mono = pl.Monospace('random')
py_rand_mono = pl.Monospace('random.random')
py_rand_uniform_mono = pl.Monospace('random.uniform')
py_rand_norm_mono = pl.Monospace('random.normalvariate')
py_random_link = Hyperlink('https://docs.python.org/3.7/library/random.html#real-valued-distributions',
'(and other distributions)')
py_random_normal_example = pl.Monospace('random.normalvariate(10, 1)')
random_seed_example = pl.Monospace('random.seed(0)')
next_slide = lp.Overlay([lp.NextWithIncrement()])
n_iter = pl.Equation(str_eq='n_{iter}')
df_mono = pl.Monospace('DataFrame')
df_std = pl.Monospace('df.std()')
df_mean = pl.Monospace('df.mean()')
random_choices_mono = pl.Monospace('random.choices')
random_choices_example = pl.Monospace("random.choices(['Recession', 'Normal', 'Expansion'], [0.3, 0.5, 0.2])")
return [
pl.Section(
[
lp.TwoColumnGraphicDimRevealFrame(
[
'So far everything in our models has been deterministic',
'Further, we have not explored any scenarios in our models, we have taken the base case as '
'the only case',
'Unfortunately, the real world is very random. Many possible scenarios could occur.'
],
[
images_path('dice.jpg'),
],
title='Why Model Probability'
),
lp.DimRevealListFrame(
[
'There are a few ways we can gain a richer understanding of the modeled situation by '
'incorporating probability',
f'The simplest is {pl.Bold("scenario modeling")}, in which different situations are defined with probabilities, '
'and the result of the model is the expected value across the cases.',
['Another is', pl.Bold('internal randomness'),
'where randomness is incorporated directly within '
'the model logic'],
['Finally,', pl.Bold("Monte Carlo simulation"),
'treats the model as deterministic but externally varies the '
'inputs to get a distribution of outputs.']
],
title='How to Bring Probability In'
),
],
title='Motivation for Probability Modeling',
short_title='Intro',
),
pl.Section(
[
lp.Frame(
[
pl.UnorderedList([
lp.DimAndRevealListItems([
['When something is measured numerically, it can be either a', pl.Bold('discrete'),
'variable, or a', pl.Bold('continuous'), 'variable.'],
])
]),
lp.Block(
[
pl.Equation(str_eq=r'x \in \{x_1, x_2, ... x_n\}', inline=False),
pl.VSpace(-0.4),
pl.UnorderedList([
[pl.Equation(str_eq=r'\{x_1, x_2, ... x_n\}:'), 'A specific set of values'],
])
],
title='Discrete Variables'
),
lp.Block(
[
pl.Equation(str_eq=r'x \in \mathbb{R} \text{ or } [a, b]', inline=False),
pl.VSpace(-0.4),
pl.UnorderedList([
[pl.Equation(str_eq='\mathbb{R}:'), 'All real numbers'],
[pl.Equation(str_eq='[a, b]:'), 'Some interval between two values or infinity'],
])
],
title='Continuous Variables'
),
],
title='Math Review: Discrete and Continuous Variables'
),
lp.Frame(
[
pl.TextSize(-3),
pl.UnorderedList([
lp.DimAndRevealListItems([
[pl.Bold('Expected value'), 'is the average outcome over repeated trials'],
"It is generally useful to get a single output from multiple possible cases",
], dim_earlier_items=False)
]),
lp.Block(
[
pl.Equation(str_eq=r'E[x] = \sum_{i=1}^{N} p_i x_i', inline=False),
pl.VSpace(-0.4),
pl.UnorderedList([
[pl.Equation(str_eq=r'E[x]:'), 'Expected value for', pl.Equation(str_eq='x')],
[pl.Equation(str_eq=r'x_i:'), 'A specific value for', pl.Equation(str_eq='x')],
[pl.Equation(str_eq=r'p_i:'), 'The probability associated with value',
pl.Equation(str_eq='x_i')],
[pl.Equation(str_eq=r'N:'), 'The total number of possible values of',
pl.Equation(str_eq='x')],
])
],
title='Discrete Variables'
),
lp.Block(
[
pl.Equation(str_eq=r'E[x] = \frac{1}{N} \sum_{i=1}^{N} x_i', inline=False),
pl.VSpace(-0.4),
pl.UnorderedList([
[pl.Equation(str_eq=r'N:'), 'The number of samples collected for',
pl.Equation(str_eq='x')],
])
],
title='Continuous Variables'
),
],
title='Math Review: Expected Value'
),
lp.GraphicFrame(
images_path('different-variance-plot.pdf'),
title='Math Review: Variance in One Picture'
),
lp.Frame(
[
pl.TextSize(-2),
pl.UnorderedList([
lp.DimAndRevealListItems([
[pl.Bold('Variance'), 'and', pl.Bold('standard deviation'),
'are measures of the dispersion '
'of values of a random variable.'],
'Variance is the real quantity of interest, but standard deviation is easier to understand '
'because it has the same units as the variable, while variance has units squared'
], dim_earlier_items=False),
]),
lp.Block(
[
EquationWithVariableDefinitions(
r'Var[x] = \sigma^2 = \frac{1}{N - 1} \sum_{i=1}^{N} (x_i - \mu)^2',
[
[pl.Equation(str_eq=r'N:'), 'Number of samples of', pl.Equation(str_eq=r'x')],
[pl.Equation(str_eq=r'\mu:'), 'Sample mean'],
]
),
],
title='Variance of a Continuous Variable'
),
lp.Block(
[
EquationWithVariableDefinitions(
r'\sigma = \sqrt{Var[x]}',
[
[pl.Equation(str_eq=r'\sigma:'), 'Standard deviation'],
],
space_adjustment=-0.5
),
],
title='Standard Deviation'
),
],
title='Math Review: Variance and Standard Deviation'
),
lp.TwoColumnGraphicDimRevealFrame(
[
['A', pl.Bold('probability distribution'),
'represents the probabilities of different values of '
'a variable'],
'For discrete variables, this is simply a mapping of possible values to probabilities, e.g. for a coin '
'toss, heads = 50% and tails = 50%',
'For continuous variables, a continuous distribution is needed, such as the normal distribution',
],
graphics=[
images_path('normal-distribution.png'),
],
title='Math Review: Probability Distributions'
),
lp.TwoColumnGraphicDimRevealFrame(
[
pl.TextSize(-2),
["You've probably heard of the", pl.Bold('normal distribution'),
'as it is very commonly used because it occurs a lot in nature'],
['It is so common because of the', pl.Bold('central limit theorem'), 'which says that '
'averages of variables will follow a normal distribution, regardless of the distribution of the '
'variable itself'],
'This has many applications. For example, we can view the investment rate as an average across '
'individual investment returns, and so it will be normally distributed.',
],
graphics=[
images_path('normal-distribution-percentages.png'),
],
title='Math Review: Normal Distribution'
),
lp.Frame(
[
pl.TextSize(-3),
pl.UnorderedList([
lp.DimAndRevealListItems([
'We want to extend our retirement model to say that the investment return is not constant.',
'We can treat the interest rate as either a discrete (specific values) or a continuous '
'(range of values, more realistic) variable'
], dim_earlier_items=False),
]),
lp.Block(
[
pl.Center(
[
lt.Tabular(
[
lt.ValuesTable.from_list_of_lists([
['Interest Rate', 'Probability']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['2%', '30%'],
['5%', '50%'],
['7%', '20%'],
]),
],
align='cc'
)
]
)
],
title='As a Discrete Variable'
),
lp.Block(
[
pl.Equation(str_eq=r'r_i \sim N(\mu, \sigma^2)', inline=False),
pl.VSpace(-0.5),
pl.UnorderedList([
[pl.Equation(str_eq=r'N:'), 'Normal distribution'],
[pl.Equation(str_eq=r'\mu:'), 'Interest rate mean'],
[pl.Equation(str_eq=r'\sigma:'), 'Interest rate standard deviation'],
])
],
title='As a Continuous Variable'
),
],
title='A Non-Constant Interest Rate'
),
],
title='Mathematical Tools for Probability Modeling',
short_title='Math Review'
),
pl.Section(
[
lp.Frame(
[
pl.TextSize(-1),
lp.Block(
[
pl.Center(
[
lt.Tabular(
[
lt.ValuesTable.from_list_of_lists([
['State of Economy', 'Interest Rate', 'Savings Rate', 'Probability']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Recession', '2%', '35%', '30%'],
['Normal', '5%', '30%', '50%'],
['Expansion', '7%', '25%', '20%'],
]),
],
align='l|ccc'
)
]
)
],
title='Interest Rate Scenarios'
),
pl.UnorderedList([
lp.DimAndRevealListItems([
['In scenario modeling, different cases for model parameters are chosen. Several '
'parameters may be altered at once in a given case.'],
"Here we are making the different cases the state of the economy. When the economy is doing "
"poorly, the individual earns a lower return, but also saves more because they don't want to "
"overspend at a bad time",
"When the economy does well, the individual earns a higher return, but also spends more"
])
]),
],
title='Scenario Modeling'
),
lp.DimRevealListFrame(
[
['We can implement scenario modeling', pl.Bold('internal'), 'or', pl.Bold('external'),
'to our model'],
['With an internal implementation, the cases are built', pl.Underline('into the model logic'),
'itself, '
'and model logic also takes the expected value of the case outputs. The inputs of the model',
'are now the cases and probabilities.'],
['With an external implementation, the', pl.Underline('model logic is left unchanged,'),
'instead the '
'model is run separately with each case, then the expected value is calculated across the outputs '
'from the multiple model runs.'],
],
title='Implementing Scenario Modeling'
),
lp.Frame(
[
pl.Center(
[
lt.Tabular(
[
lt.ValuesTable.from_list_of_lists([
[pl.Bold('Internal'), pl.Bold('External')]
]),
lt.MidRule(),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Original model is now an old version',
'Original model can still be used normally'],
]),
# TODO [#14]: each row should come one per slide, but need to allow overlays in lt items
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Model runs exactly as before',
'Getting full results of model requires running the model multiple times and '
'aggregating output']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Model complexity has increased', 'Model complexity unchanged']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Complexity to run model is unchanged',
'Complexity to run model has increased']
]),
],
align='L{5cm}|R{5cm}'
)
]
)
],
title='Internal or External Scenario Analysis?'
),
lp.DimRevealListFrame(
[
'For internal scenario analysis, set up a table of the cases and probabilities. Then calculate the '
'expected value of these cases for each model parameter. Then use the expected value as the new '
'model parameter.',
'For external scenario analysis, a data table is useful. Create the data table of outputs for each case '
'and another table of case probabilities, then combine them to produce the expected value of '
'the output.',
'If you are trying to change more than two inputs at once in external scenario '
'analysis, this becomes more '
'challenging but you can assign a number to each set of inputs and have the model look up the '
'inputs based on the case number, using the case number as the data table input.'
],
title='Scenario Analysis in Excel'
),
InClassExampleFrame(
[
'I will now go through adding external scenario analysis to the Dynamic Salary Retirement Model '
'in Excel',
'The completed exercise on the course site as "Dynamic Salary Retirement Model Sensitivity.xlsx"',
],
title='Scenario Analysis in Excel',
block_title='Adding Scenario Analysis to the Dynamic Retirement Excel Model'
),
scenario_excel_lab.presentation_frames(),
lp.DimRevealListFrame(
[
['For internal scenario analysis, set up a', df_mono,
'or dictionary of the cases and probabilities. Then calculate '
'the expected value of these cases for each model parameter. Then use the expected value as the new '
'model parameter.'],
'For external scenario analysis, just call your model function with each input case, collect the '
'results, and combine them to produce the expected value of the output.'
],
title='Scenario Analysis in Python'
),
InClassExampleFrame(
[
'I will now go through adding external scenario analysis to the Dynamic Salary Retirement Model '
'in Python',
'he completed exercise on the course site as "Dynamic Salary Retirement Model Scenario.ipynb"',
],
title='Scenario Analysis in Python',
block_title='Adding Scenario Analysis to the Dynamic Retirement Python Model'
),
scenario_python_lab.presentation_frames(),
],
title='Scenario Modeling'
),
pl.Section(
[
lp.DimRevealListFrame(
[
["Using the technique of", pl.Bold('internal randomness,'),
'something random is added internally to the model'],
'Instead of taking a fixed input, random values for that variable are drawn',
'This technique can be used with both discrete and continuous variables'
],
title='What is Internal Randomness?'
),
lp.GraphicFrame(
internal_randomness_graphic(),
title='Internal Randomness in One Picture'
),
lp.DimRevealListFrame(
[
'Internal randomness makes sense when the random behavior is integral to your model',
'If you are just trying to see how changing inputs affects outputs, or trying to get confidence intervals for outputs, '
'an external method such as sensitivity analysis or Monte Carlo simulation would make more sense.',
'For example, if we want to allow investment returns to vary in our retirement model, an external method fits well because '
'the core model itself is deterministic',
'If instead we were modeling a portfolio, we might use internal randomness to get the returns for each asset.'
],
title='Should I Use Internal Randomness?'
),
lp.DimRevealListFrame(
[
'Similarly to our discussion of internal vs. external sensitivity analysis, internal randomness keeps '
'operational complexity (how to run the model) low, but increases model complexity.',
'The main drawback of internal randomness is that the same set of inputs will give different outputs each time the model is run',
'While this is the desired behavior, it can make it difficult to determine whether everything is working.'
],
title='Internal Randomness Advantages and Pitfalls'
),
lp.TwoColumnGraphicDimRevealFrame(
[
'Instead of taking the input as fixed, draw it from a distribution',
'We need to define a distribution for each input we want to randomize. This will typically be a normal distribution, and then '
'we just need to give it a reasonable mean and standard deviation',
'Put the most reasonable or usual value as the mean. Then think about the probabilities of the normal distribution relative '
'to standard deviation to set it'
],
graphics=[
images_path('normal-distribution-percentages.png'),
],
title='Internal Randomness with Continuous Variables'
),
lp.DimRevealListFrame(
[
['The main functions for randomness in Excel are', rand_mono, 'and', rand_between_mono],
'The latter gives a random number between two numbers, while the former gives a random number '
'between 0 and 1. Both of these draw from a uniform distribution (every number equally likely)',
['Meanwhile, the', norm_inv_mono,
'function gives the value for a certain normal distribution at a certain probability (it is not random)'],
'We can combine these two functions to draw random numbers from a normal distribution',
[excel_random_normal_example,
'would draw a number from a normal distribution with mean 10 and standard deviation 1'],
],
title='Internal Randomness with Continuous Variables in Excel'
),
InClassExampleFrame(
[
'I will now go through generating random continuous variables '
'in Excel',
'The completed exercise on the course site is called "Generating Random Numbers.xlsx"',
'We will focus only on the "Continuous" sheet for now',
],
title='Example for Continuous Random Variables in Excel',
block_title='Generating Random Numbers from Normal Distributions in Excel'
),
randomness_excel_lab.presentation_frames(),
lp.DimRevealListFrame(
[
['In Python, we have the built-in', random_module_mono, 'module'],
['It has functions analagous to those in Excel:', py_rand_mono, 'works like', rand_mono,
'and', py_rand_uniform_mono, 'works like', rand_between_mono],
['Drawing numbers from a normal distribution', py_random_link, 'is easier: just one function',
py_rand_norm_mono],
[py_random_normal_example,
'would draw a number from a normal distribution with mean 10 and standard deviation 1']
],
title='Internal Randomness with Continuous Variables in Python'
),
InClassExampleFrame(
[
'I will now go through generating random continuous variables '
'in Python',
'The completed exercise on the course site is called "Generating Random Numbers.ipynb"',
'We will focus only on the "Continuous" section for now',
],
title='Example for Continuous Random Variables in Python',
block_title='Generating Random Numbers from Normal Distributions in Python'
),
randomness_python_lab.presentation_frames(),
lp.DimRevealListFrame(
[
'We can also build randomness into the model for discrete variables',
"With discrete variables, our distribution is just a table of probabilities for the different values",
'To pick a random value for a discrete variable, first add another column to your table which has the '
'cumulative sum of the prior probabilties, and then another column which is that column plus the '
'current probability',
'Then generate a random number between 0 and 1 from a uniform distribution',
'If the generated number is between the probability and the cumulative sum of prior probabilities, choose that case'
],
title='Internal Randomness with Discrete Variables'
),
lp.Frame(
[
pl.TextSize(-1),
lp.Block(
[
pl.Center(
[
lt.Tabular(
[
lt.ValuesTable.from_list_of_lists([
['State of Economy', 'Interest Rate', 'Probability', 'Begin Range',
'End Range']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists([
['Recession', '2%', '30%', '0%', '30%'],
['Normal', '5%', '50%', '30%', '80%'],
['Expansion', '7%', '20%', '80%', '100%'],
]),
],
align='L{2cm}|cccc'
)
]
)
],
title='Interest Rate Scenarios'
),
pl.UnorderedList([
pl.TextSize(-2),
lp.DimAndRevealListItems([
'The Begin Range column is calculated as the cumulative sum of prior probabilities',
'The End Range column is calculated as Begin Range + Probability',
"Generate a random number between 0 and 1. If it is between the begin and end range, "
"that is the selected value",
"If it's 0.15, it's a recession. If it's 0.45, it's a normal period. If it's 0.94, it's "
"an expansion period."
], vertical_fill=True)
]),
],
title='An Example of Internal Randomness with Discrete Variables'
),
lp.DimRevealListFrame(
[
'The steps in the preceeding slides need to be carried out manually in Excel',
['In Python, there is a built-in function which is doing all of this in the background,',
random_choices_mono],
['Simply do', random_choices_example, 'to yield the exact same result for the prior example']
],
title='Random Discrete Variables in Python'
),
InClassExampleFrame(
[
'I will now go through generating random discrete variables '
'in both Excel and Python',
'We will be continuing with the same Excel workbook and Jupyter notebook from before, '
'"Generating Random Numbers.xlsx" and "Generating Random Numbers.ipynb"',
'We will focus only on the "Discrete" sheet/section now',
],
title='Example for Discrete Random Variables in Excel and Python',
block_title='Generating Random Numbers from Discrete Distributions in Excel and Python'
),
random_stock_lab.presentation_frames(),
InClassExampleFrame(
[
'I will now add internal randomness with discrete variables to '
'both the Excel and Python Dynamic Salary Retirement models to simulate economic conditions '
'changing year by year',
'The completed models on the course site are called '
'"Dynamic Salary Retirement Model Internal Randomness.xlsx" and '
'"Dynamic Salary Retirement Model Internal Randomness.ipynb"',
],
title='Adding Internal Randomness to Excel and Python Models',
block_title='Extending the Dynamic Salary Retirement Model with Internal Randomness'
),
full_model_internal_randomness_lab.presentation_frames(),
],
title='Internal Randomness'
),
pl.PresentationAppendix(appendix_frames),
]
def get_content():
align_c = lt.ColumnAlignment('c')
align_l = lt.ColumnAlignment('l')
align = lt.ColumnsAlignment([align_l, align_c])
n_phones = 100000
price_scrap = 50000
price_phone = 500
cogs_phone = 250
price_machine_adv = 1000000
n_life = 10
n_machines = 5
d_1 = 100000
g_d = 0.2
max_year = 20
interest = 0.05
pmm = PhoneManufacturingModel(n_phones, price_scrap, price_phone, n_life,
n_machines, d_1, g_d, max_year, interest)
scipy_mono = pl.Monospace('scipy')
scipy_minimize_link = 'https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.minimize_scalar.html#' \
'scipy-optimize-minimize-scalar'
scipy_minimize_mono = pl.Monospace('scipy.optimize.minimize_scalar')
possible_elasicity_str = ', '.join([
f'($E = {ec[0]}$, $d_c = {ec[1]}$)' for ec in ELASTICITY_CONSTANT_CASES
])
possible_elasicity_str = ', '.join([
f'($E = {ec[0]}$, $d_c = {ec[1]}$)' for ec in ELASTICITY_CONSTANT_CASES
])
possible_elasicity_str = '[' + possible_elasicity_str + ']'
return [
pl.Section([
pl.SubSection([
'You work for a new startup that is trying to manufacture phones. You are tasked with building '
'a model which will help determine how many machines to invest in and how much to spend on '
'marketing. Each machine produces $n_{output}$ phones per year. Each phone sells '
r'for \$$p_{phone}$ and costs \$$c_{phone}$ in variable costs to produce. '
'After $n_{life}$ years, the machine can no longer produce output, but may be scrapped for '
r'\$$p_{scrap}$. The machine will not be replaced, so you may end up with zero total output '
r'before your model time period ends. '
'Equity investment is limited, so in each year you can spend $c_{machine}$ to either buy a machine or buy '
'advertisements. In the first year you must buy a machine. Any other machine purchases must '
'be made one after another (advertising can only begin after machine buying is done). '
'Demand for your phones starts at '
'$d_1$. Each time you advertise, demand increases by $g_d$%. The prevailing market interest '
'rate is $r$.'
],
title='The Problem'),
pl.SubSection([
pl.UnorderedList([
'You may limit your model to 20 years and a maximum of 5 machines if it is helpful.',
'For simplicity, assume that $c_{machine}$ is paid in every year, '
'even after all machines have shut down.',
'Ensure that you can change the inputs and the outputs change as expected.',
'For simplicity, assume that fractional phones can be sold, you do not '
'need to round the quantity transacted.'
])
],
title='Notes'),
pl.SubSection([
pl.SubSubSection([
pl.UnorderedList([
'$n_{output}$: Number of phones per machine per year',
'$n_{machines}$: Number of machines purchased',
'$n_{life}$: Number of years for which the machine produces phones',
'$p_{phone}$: Price per phone',
'$p_{scrap}$: Scrap value of machine',
'$c_{machine}$: Price per machine or advertising year',
'$c_{phone}$: Variable cost per phone',
'$d_1$: Quantity of phones demanded in the first year',
'$g_d$: Percentage growth in demand for each advertisement',
'$r$: Interest rate earned on investments'
])
],
title='Inputs'),
pl.SubSubSection([
pl.UnorderedList([
'Cash flows in each year, up to 20 years',
'PV of cash flows, years 1 - 20',
])
],
title='Outputs')
],
title='The Model'),
pl.SubSection([
"It is unrealistic to assume that price and demand are unrelated. To extend the model, "
"we can introduce a relationship between price and demand, given by the following equation: ",
pl.Equation(str_eq=r'd_1 = d_c - Ep_{phone}', inline=False),
pl.UnorderedList([
f'{pl.Equation(str_eq="E")}: Price elasticity of demand',
f'{pl.Equation(str_eq="d_c")}: Demand constant'
]),
[
f"For elasticities and constants {possible_elasicity_str} "
f"({len(ELASTICITY_CONSTANT_CASES)} total cases), and taking the other "
"model inputs in the ",
pl.NameRef('check-work'),
' section, determine the optimal price for each '
'elasticity, that is the price which maximizes the NPV.'
],
pl.SubSubSection([
pl.UnorderedList([
'$d_1$ is no longer an input, but an output.',
'This bonus requires optimization, which we have not yet covered in class.',
'In Excel, you can use Solver.',
[
f'In Python, the {scipy_mono} package provides optimization tools. You will '
f'probably want to use:',
],
pl.UnorderedList([
Hyperlink(scipy_minimize_link,
scipy_minimize_mono),
"You will need to write a function which accepts price and returns NPV, "
"with other model inputs fixed.",
pl.UnorderedList([[
'Depending on how you set this up,',
Hyperlink(
'https://www.learnpython.org/en/Partial_functions',
'functools.partial'),
'may be helpful for this.'
]]),
"It will actually need to return negative NPV, as the optimizer only minimizes, "
"but we want maximum NPV.",
[
'No answers to check your work are given for this bonus. The',
pl.NameRef('check-work'),
'section only applies to without the bonus.'
]
])
])
],
title='Notes')
],
title='Bonus Problem'),
],
title='Overview'),
pl.Section([
'You must start from "Project 1 Template.xlsx". Ensure that you reference all inputs from '
'the Inputs/Outputs tab. Also ensure that all outputs are referenced back to the Inputs/Outputs tab. '
'Do not change any locations of the inputs or outputs. '
'The final submission is your Excel workbook.'
],
title='Excel Exercise'),
pl.Section([[
'You must start from "Project 1 Template.ipynb". '
'I should be able to run all the '
'cells and get the output of your model at the bottom. ',
[
'You should not change the name of the',
pl.Monospace('ModelInputs'), 'class or the',
pl.Monospace('model_data'), 'variable.'
], 'You need to define',
pl.Monospace('cash_flows'), 'as your output cash flows (numbers, '
'not formatted), and ',
pl.Monospace('npv'),
'as your NPV (number, not formatted). When you '
'show your final outputs in the notebook, then they should be formatted.'
]],
title='Python Exercise'),
pl.Section([
Center(
lt.Tabular([
lt.MultiColumnLabel('Grading Breakdown', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Category', 'Percentage']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists([[
'Model Accuracy', '60%'
], ['Model Readability', '20%'], [
'Model Formatting', '10%'
], ['Following the Template', '10%'], ['Bonus', '5%']]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[['Total Possible', '105%']]),
lt.BottomRule()
],
align=align))
],
title='Grading'),
pl.Section(
[
'If you pass the following inputs (to the basic model, not bonus model): ',
pl.UnorderedList([
f'{pl.Equation(str_eq="n_{output}")}: {n_phones:,.0f}',
f'{pl.Equation(str_eq="p_{scrap}")}: \${price_scrap:,.0f}',
f'{pl.Equation(str_eq="p_{phone}")}: \${price_phone:,.0f}',
f'{pl.Equation(str_eq="c_{machine}")}: \${price_machine_adv:,.0f}',
f'{pl.Equation(str_eq="c_{phone}")}: \${cogs_phone:,.0f}',
f'{pl.Equation(str_eq="n_{life}")}: {n_life}',
f'{pl.Equation(str_eq="n_{machines}")}: {n_machines}',
f'{pl.Equation(str_eq="d_1")}: {d_1:,.0f}',
f'{pl.Equation(str_eq="g_d")}: {g_d:.0%}',
f'{pl.Equation(str_eq="r")}: {interest:.0%}',
]),
'You should get the following result:',
# TODO [#10]: replace project 1 result using notebook executor
"""
Cash Flows:
Year 1: \$24,000,000
Year 2: \$24,000,000
Year 3: \$24,000,000
Year 4: \$24,000,000
Year 5: \$24,000,000
Year 6: \$29,000,000
Year 7: \$35,000,000
Year 8: \$42,200,000
Year 9: \$50,840,000
Year 10: \$61,208,000
Year 11: \$73,699,600
Year 12: \$74,050,000
Year 13: \$49,050,000
Year 14: \$24,050,000
Year 15: \$-950,000
Year 16: \$-1,000,000
Year 17: \$-1,000,000
Year 18: \$-1,000,000
Year 19: \$-1,000,000
Year 20: \$-1,000,000
NPV: \$369,276,542
"""
],
title='Check your Work',
label='check-work')
]
def get_content():
jinja_templates_path = os.path.sep.join(['pltemplates', 'projects', 'p3'])
jinja_env = pl.JinjaEnvironment(
loader=FileSystemLoader(jinja_templates_path))
with open(ANSWERS_OUTPUT_PATH, 'r') as f:
answers_json = json.load(f)
answers_dict = answers_json[0]
beta_std = answers_dict['beta_std']
mkt_ret_std = answers_dict['mkt_ret_std']
bond_price_std = answers_dict['bond_price_std']
tax_rate_std = answers_dict['tax_rate_std']
bond_years = answers_dict['bond_years']
bond_coupon = answers_dict['bond_coupon']
bond_price = answers_dict['bond_price']
bond_par = answers_dict['bond_par']
risk_free = answers_dict['risk_free']
price = answers_dict['price']
shares_outstanding = answers_dict['shares_outstanding']
libor_rate = answers_dict['libor_rate']
inputs_table = pl.Center(
lt.Tabular([
pl.MultiColumnLabel('Baseline Inputs', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists([['Variable', 'Baseline Value']
]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists([
['Market Bond Maturity (Years)', f'{bond_years:.0f}'],
['Market Bond Coupon', f'{bond_coupon:.2%}'],
['Market Bond Price', rf'\${bond_price:.2f}'],
['Market Bond Par Value', rf'\${bond_par:.2f}'],
['Risk Free Rate', f'{risk_free:.2%}'],
['Stock Price', rf'\${price:.2f}'],
['Shares Outstanding', f'{shares_outstanding:,.0f}'],
['LIBOR Rate', f'{libor_rate:.2%}'],
]),
lt.BottomRule()
],
align='l|c'))
stdev_table = pl.Center(
lt.Tabular([
pl.MultiColumnLabel('Standard Deviations', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Variable', 'Standard Deviation']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists([
[r'$\beta$', beta_std],
['Market Return', f'{mkt_ret_std:.0%}'],
['Walmart Bond Market Price', rf'\${bond_price_std}'],
['Tax Rate', f'{tax_rate_std:.0%}'],
]),
lt.BottomRule()
],
align='l|c'))
return [
pl.Section([
Project3ProblemModel(template_path='prob_definition.j2',
environment=jinja_env),
Project3ProblemModel(template_path='notes.j2',
environment=jinja_env),
Project3ProblemModel(template_path='bonus.j2',
environment=jinja_env),
pl.SubSection([inputs_table],
title='Baseline Model Inputs',
label='baseline-inputs'),
pl.SubSection(
[stdev_table], title='Monte Carlo Inputs', label='mc-inputs'),
],
title='Overview'),
pl.Section([
Project3ProblemModel(template_path='submission.j2',
environment=jinja_env),
pl.SubSection([
'Selected solutions with the baseline inputs:',
pl.UnorderedList([
f'WACC: {answers_dict["wacc"]:.2%}',
rf'MV Debt: \${answers_dict["mv_debt"] / 1000000000:.1f} billion',
f'Cost of Equity: {answers_dict["coe"]:.2%}',
f'Pre-Tax Cost of Debt: {answers_dict["pretax_cost_of_debt"]:.2%}'
])
],
title='Solutions'),
pl.SubSection([
pl.Center(
lt.Tabular([
pl.MultiColumnLabel('Grading Breakdown', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Category', 'Percentage']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists([
['Model Accuracy', '60%'],
['Model Readability', '20%'],
['Model Formatting', '10%'],
['Following the Template', '10%'], ['Bonus', '5%']
]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[['Total Possible', '105%']]),
lt.BottomRule()
],
align='l|c'))
],
title='Grading'),
],
title='Submission & Grading')
]
def get_content():
return [
pl.Section(
[
pl.SubSection(
[
'You are a financial analyst for an aircraft manufacturer. Your company is trying to decide '
'how large its next line of planes should be. Developing a line of planes takes a one-time '
'research cost, then each plane has a cost to manufacture afterwards. The larger the plane, '
'the higher the research and manufacture costs, but also the more revenue per plane. The '
'larger planes are more risky because if the economy goes poorly, not many airlines will want '
'to invest in such a large plane, but if the economy goes well, airlines will be rushing to the '
'larger planes to fit rising demand.',
'',
"""
The research cost will be paid at $t=0$. For simplicity, assume that all planes are manufactured
at $t=1$ and sold at $t=2$. The interest rate is given in the below table.
Find the expected NPV and standard deviation of NPV for each plane. Which plane has the lowest
chance of a negative NPV? Which has the highest chance of a positive NPV? Visualize the range
of possible NPVs for each plane, as well as the probability of acheiving different NPV levels
(probability table). For the mid-size plane, which has a larger impact on the NPV, an
additional plane sold or a decrease in the interest rate by 1%? In your opinion, which
plane should the manufacturer create?
"""
],
title='Problem Definition'
),
pl.SubSection(
[
pl.Center(
lt.Tabular(
[
lt.TopRule(),
lt.ValuesTable.from_list_of_lists([[
'Plane', 'Research Cost', 'Manufacture Cost', 'Sale Price', 'Expected Unit Sales', 'Stdev Unit Sales'
]]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[
['Super Size', r'\$100,000,000', r'\$10,000,000', r'\$11,500,000', '200', '120'],
['Large', r'\$50,000,000', r'\$5,000,000', r'\$5,600,000', '400', '50'],
['Mid-size', r'\$25,000,000', r'\$3,000,000', r'\$3,350,000', '500', '20'],
],
),
lt.BottomRule(),
],
align='l|ccccc'
)
)
],
title='Possible Planes'
),
pl.SubSection(
[
pl.Center(
lt.Tabular(
[
lt.TopRule(),
lt.ValuesTable.from_list_of_lists([[
'Input', 'Default Value', 'Default Stdev'
]]),
lt.MidRule(),
lt.ValuesTable.from_list_of_lists(
[
['Interest Rate', '7%', '4%'],
],
),
lt.BottomRule(),
],
align='l|cc'
)
)
],
title='Other Inputs'
)
],
title='Capital Budgeting Probabilities with Monte Carlo Simulation'
)
]
def get_content():
random.seed(1000)
ev_bet = (999999 / 1000000) * 1 + (1 / 1000000) * (-750001)
xlwings_mono = pl.Monospace('xlwings')
pd_mono = pl.Monospace('pandas')
quickstart_mono = pl.Monospace('quickstart')
read_from_excel_example = pl.Python("""
my_value = xw.Range("G11").value # single value
# all values in cell range
my_value = xw.Range("G11:F13").value
# expands cell range down and left getting all values
my_values = xw.Range("G11").expand().value
""")
write_to_excel_example = pl.Python("""
xw.Range("G11").value = 10
xw.Range("G11").value = [10, 11] # horizontal
xw.Range("G11").value = [[10], [11]] # vertical
xw.Range("G11").value = [[10, 11], [12, 13]] # table
""")
ball_options = ['fill', 'circle', 'inner sep=8pt']
blue_ball_options = ball_options + ['blue']
red_ball_options = ball_options + ['red']
def rand_pos():
return random.randrange(-150, 150) / 100
blue_nodes = [
lg.Node(None, (rand_pos(), rand_pos()), options=blue_ball_options)
for _ in range(10)
]
red_nodes = [
lg.Node(None, (rand_pos(), rand_pos()), options=red_ball_options)
for _ in range(10)
]
red_blue_ball_graphic = lg.TikZPicture([
lg.Rectangle(5, 5, shape_options=['blue', 'thick']), *blue_nodes,
*red_nodes
])
lecture = get_monte_carlo_lecture()
intro_mc_python_lab = get_intro_monte_carlo_lab_lecture().to_pyexlatex()
mc_python_lab = get_python_retirement_monte_carlo_lab_lecture(
).to_pyexlatex()
mc_excel_lab = get_excel_retirement_monte_carlo_lab_lecture().to_pyexlatex(
)
return [
pl.Section([
lp.
TwoColumnGraphicDimRevealFrame([
[
pl.Bold('Monte Carlo Simulation'),
'is a technique which allows understanding the probability '
'of acheiving certain outputs from a model.'
],
'This gives the modeler a greater understanding of the likelihood of different outputs, rather '
'than relying on a single number',
],
graphics=[
images_path(
'random-numbers.jpg')
],
title=
'What is Monte Carlo Simulation?'),
lp.DimRevealListFrame([
r'Imagine you have a one-time opportunity to place a bet for \$1. ',
r'If you win the bet, you will receive \$2. If you lose the bet, you will lose \$750,000. '
r'You cannot avoid the payment by declaring bankruptcy.',
r'The odds of winning the bet are 999,999/1,000,000. In 1/1,000,000 you lose the \$750,000.',
fr'The expected profit from the bet is \${ev_bet:.2f}. Should you take it? Depends on your '
fr'risk tolerance.',
'Therefore not only the expected outcome matters, but also what other outcomes may occur and '
'their probabilities.'
],
title='Why Use Monte Carlo Simulation?'),
lp.GraphicFrame(explore_parameters_graphic(),
title='Monte Carlo Simulation in One Picture'),
lp.DimRevealListFrame([
'Monte Carlo simulation is carried out similarly to external scenario analysis.',
'The main difference is that we manually picked specific cases for the inputs with scenario '
'analysis.',
'In Monte Carlo simulation, we assign distributions to the inputs, and input values are drawn '
'from the distributions for each run of the model',
'Finally, we can fit a probability distribution to the outputs to be able to talk about the '
'chance of a certain outcome occurring'
],
title=
'Basic Process for Monte Carlo Simulation')
],
title='Introduction'),
pl.Section(
[
lp.DimRevealListFrame([
'Monte Carlo simulation can be applied to any model',
'It is generally easier to run them in Python than in Excel.',
"With pure Excel, you're either going to VBA or hacking something with data tables",
'In Python, just loop for N iterations, each time drawing inputs, running the model, and collecting '
'outputs.',
[
'We will start with a pure Python model, then move to using',
xlwings_mono, 'to add Monte Carlo '
'simulations to our Excel models.'
],
],
title=
'Running Monte Carlo Simulations - Python or Excel?'
),
lp.Frame([
lp.Block([
r'You have \$1,000 now and need to pay \$1,050 in one year. You have available to you '
r'two assets: a risk free asset that returns 3%, and a stock that returns 10% with a '
r'20% standard deviation. How much should you invest in the two assets to maximize '
r'your probability of having at least \$1,050 in one year?'
],
title='An Investment Problem'),
pl.VFill(),
pl.UnorderedList([
lp.DimAndRevealListItems([
'We must first construct the basic model which gets the portfolio value for given '
'returns',
'Then draw values of the stock return from a normal distribution, and run the model '
'many times and visualize the outputs. ',
'Then repeat this process with each weight to determine the best weight.'
])
])
],
title='An Example Application'),
InClassExampleFrame([
'Go to the course site and download the Jupyter notebook "MC Investment Returns.ipynb" from '
'Monte Carlo Examples',
'I will go through this example notebook to solve the problem from the prior slide.'
],
title='Simluating Portfolio Values',
block_title=
'Example for Simulating Portfolio Values'),
pl.TextSize(-2),
intro_mc_python_lab.presentation_frames(),
pl.TextSize(0),
],
title='Running a First Monte Carlo Simulation',
short_title='Run MC',
),
pl.Section(
[
lp.Frame([
pl.TextSize(-2), 'For the model given by:',
pl.Equation(str_eq='y = f(X)', inline=False),
pl.Equation(str_eq='X = [x_1, x_2, ..., x_n]',
inline=False),
pl.UnorderedList([[
pl.Equation(str_eq='y:'), 'Model output'
], [pl.Equation(str_eq='X:'), 'Model input matrix'],
[
pl.Equation(str_eq='x_i:'),
'Value of $i$th $x$ variable'
]]),
'To run $N$ Monte Carlo simulations, follow the following steps:',
pl.OrderedList(
[[
'Assign a probability distribution for each',
pl.Equation(str_eq='x_i')
],
[
'For each',
pl.Equation(str_eq='x_i'),
'randomly pick a value from its probability distribution. Store them as',
pl.Equation(str_eq='X_j')
],
[
'Repeat the previous step $N$ times, yielding',
pl.Equation(str_eq='[X_1, X_2, ..., X_N]')
],
[
'For each',
pl.Equation(str_eq='X_j'), 'calculate',
pl.Equation(str_eq='y_j = f(X_j)')
],
[
'Store the values of',
pl.Equation(str_eq='X_j'), 'mapped to',
pl.Equation(str_eq='y_j')
],
[
'Visualize and analyze',
pl.Equation(str_eq='y_j'), 'versus',
pl.Equation(str_eq='X_j')
]])
],
title='Monte Carlo Simulation Process'),
lp.DimRevealListFrame([
'There are a multitude of outputs we can get from a Monte Carlo simulation. We saw a few '
'already in the example.',
[
pl.Bold('Outcome probability distributions'),
'are the main output. We saw this with two '
'approaches in the example, a',
pl.Underline('histogram'), 'and a',
pl.Underline('probability table.')
],
[
'We also examined the',
pl.Bold('probability of a certain outcome'),
'in whether we reached '
'the desired cash.'
],
[
'The last main output is examining the',
pl.Bold('relationship between inputs and outputs.'),
'for which common approaches include',
pl.Underline('scatter plots'), 'and',
pl.Underline('regressions.')
]
],
title=
'Outputs from Monte Carlo Simulation'),
lp.TwoColumnGraphicDimRevealFrame(
[
pl.TextSize(-3),
'The outcome probability distribution represents the chance of receiving different '
'outcomes from your model.',
'There are two main ways to visualize a probability distribution: a plot and a table.',
[
'The plot, usually a',
pl.Underline('histogram'), 'or',
pl.Underline('KDE'),
'gives a high-level overview of the probabilities and can uncover any non-normal '
'features of the distribution.'
],
[
'The probability table represents the chance of receiving the given value or '
'lower.'
],
'The Value at Risk (VaR) is a common measure calculated in the industry, and it represents '
'the probability of losing at least a certain amount. This would be a subset of this analysis '
'and so this analysis can be used to calculate VaR',
],
graphics=[
images_path('outcome-probability-distribution.png'),
lt.Tabular([
pl.MultiColumnLabel('Probability Table', span=2),
lt.TopRule(),
lt.ValuesTable.from_list_of_lists(
[['Probability', 'Value']]),
lt.TableLineSegment(0, 1),
lt.ValuesTable.from_list_of_lists(
[['25%', '1020'], ['50%', '1039'],
['75%', '1053']]),
lt.BottomRule()
],
align='c|c')
],
title='Outcome Probability Distributions',
graphics_on_right=False,
),
lp.TwoColumnGraphicDimRevealFrame(
[
'Imagine a box which contains red and blue balls. You do not know in advance how many there '
'are of each color.',
'You want to estimate the probability of getting a blue ball when pulling a ball from the box.',
'To evaluate this, you grab a ball, write down its color, and put it back, 1,000 times.',
'You pull a blue ball in 350 out of the 1,000 trials. What is the probability of getting blue?'
],
graphics=[red_blue_ball_graphic
],
title='Probability of a Certain Outcome - A Simple Example'
),
lp.DimRevealListFrame([
'We followed the same logic when estimating the probability of receiving our desired cash '
'in the investment example.',
pl.Equation(
str_eq=
fr'p = \frac{{{pl.Text("Count of positive outcomes")}}}{{{pl.Text("Count of trials")}}}'
),
[
'For the balls example, this is simply',
pl.Equation(str_eq=r'p = \frac{350}{1000} = 0.35'),
],
[
'In the investment example, we used', pd_mono,
'to check for each trial, whether it was a '
'positive outcome (made it a 1) or not (made it a 0). Then the sum is the count of '
'positive outcomes and so the mean is the probability.'
],
],
title=
'Probability of a Certain Outcome, Formally'
),
lp.DimRevealListFrame([
'Monte Carlo simulation can also provide a more comprehensive look at the relationship between '
'inputs and outputs.',
'While sensitivity analysis can be used to estimate the relationship between an input and '
'output, it is usually done with other inputs at their base case',
'The values of inputs may affect how other inputs affect the output. E.g. for the retirement '
'model, an increase in interest rate increases wealth more if the initial salary was higher.',
'As all the inputs change each time, you can get a more realistic view of the relationship, e.g. '
'some trials with a higher interest rate will have high salary and some will have low salary.'
],
title=
'Why Monte Carlo Simulations Help Understand Inputs vs. Outputs'
),
lp.TwoColumnGraphicDimRevealFrame(
[
pl.TextSize(-1),
'A scatter plot is a simple way to visualize the relationship between two variables',
'If there is a relationship, you will see some defined pattern in the points. This may be '
'somewhat of an upward or downward line (linear relationship) or some other shape such '
'as a U (non-linear relationship).',
'If there is no relationship, then there will just be a random cloud of points (lower '
'plot) or a horizontal line.'
],
graphics=[
images_path('scatter-plot-line.png'),
images_path('scatter-plot-no-relationship.png')
],
graphics_on_right=False,
title=
'Visualizing the Relationship between Inputs and Outputs'),
lp.TwoColumnGraphicDimRevealFrame([
pl.TextSize(-2),
'The scatter plots help give a broad understanding of the relationship but do not answer the '
'question, how much will my output change if my input changes? E.g. if I earn 10,000 more '
'for a starting salary, how much sooner can I retire?',
'Simply increasing the input in your model and checking the output is not enough, because it '
'does not take into account how all the other variables may be changing.',
'Multivariate regression is a general tool which is good at answering these kinds of questions, '
'while taking into account all the changing inputs.'
],
graphics=[
images_path(
'excel-multivariate-reg.png'
)
],
title=
'Numerically Analyzing the Relationships'
),
lp.DimRevealListFrame([
pl.TextSize(-1),
'The coefficient in a multivariate regression represents how much the outcome variable '
'changes with a one unit change in the input variable.',
'E.g. a coefficient of -.0002 on starting salary in explaining years to retirement would mean '
r'that a \$1 increase in starting salary is associated with a decrease in years to retirement by .0002 years, or '
r'a \$10,000 increase in starting salary is associated with a decrease in years to retirement by 2 years.',
'All interpretations are "all else constant", meaning that it does not consider relationships '
'between the inputs. E.g. if starting salary is higher because of a good economy, and interest '
'rates are also higher due to the good economy, the starting salary coefficient is not taking '
'into account the increase in interest rates.',
'Be careful about units. If you use decimals for percentages, you will need to multiply or '
'divide by 100 to get the effect in percentages.'
],
title='How to use Multivariate Regression'
),
InClassExampleFrame(
[
'I will now go through adding a Monte Carlo simulation to the Dynamic Salary Retirement '
'Model in Python',
'The completed example is on the course site in '
'Monte Carlo Examples',
],
title='Adding Monte Carlo Simulation to a Formal Model',
block_title='Dynamic Salary Retirement with Monte Carlo'),
mc_python_lab.presentation_frames(),
],
title='A More Formal Treatment of Monte Carlo Simulation',
short_title='Formal MC',
),
pl.Section([
lp.DimRevealListFrame([
'In pure Excel, it is much more difficult to run a Monte Carlo Simulation',
'Without going to VBA, typically the only way is to use a data table',
'A data table can be used in situations where you only want to have one or two inputs '
'varying at once. Just generate the random inputs and use them as the axes of the data table',
'If you want to vary more than two inputs, VBA or Python would be required',
'There are also add-ons that accomplish this but they are usually not free'
],
title=
"How is it Different Running MC in Excel?"),
lp.DimRevealListFrame([
'The process for Monte Carlo Simulation which works for any number of variables is '
'very similar to what we were doing in Python.',
'We are still just changing the inputs, running the model, and storing the outputs from each run',
[
'Using', xlwings_mono,
'from Python code we can change and retrieve the values of cells'
],
'This allows us to change inputs, run the model, and store outputs, just as in Python, but running our Excel model.',
'We can either analyze the outputs in Python or output them back to Excel for analysis'
],
title=
'Monte Carlo in Excel with More than Two Variables'
),
InClassExampleFrame([
'Go to the course site and download the "Dynamic Salary Retirement Model.xlsx" and '
'"Excel Monte Carlo.ipynb" from the Monte Carlo Examples',
'Open up the Jupyter notebook and follow along with me',
'The completed Excel model is also there in case you lose track. Visualizations '
'were added after running the Jupyter notebook on the original Excel model.',
],
title='Monte Carlo Excel Retirement Model',
block_title=
f'Using {xlwings_mono} to Run Monte Carlo Simulations'
),
mc_excel_lab.presentation_frames(),
],
title='Monte Carlo Simulation in Excel',
short_title='Excel MC'),
pl.PresentationAppendix([
lecture.pyexlatex_resources_frame,
intro_mc_python_lab.appendix_frames(),
mc_python_lab.appendix_frames(),
mc_excel_lab.appendix_frames(),
])
]
