This notebook contains the second homework for this class, and is due on Friday, February 19th, 2016 at 5:00 p.m.. Please make sure to get started early, and come by the instructors' office hours if you have any questions. Office hours and locations can be found in the course syllabus. IMPORTANT: While it's fine if you talk to other people in class about this homework - and in fact we encourage it! - you are responsible for creating the solutions for this homework on your own, and each student must submit their own homework assignment.
Some links that you may find helpful:
Investing in the stock market can produce tremendous gains over time -- somewhere in the range of 8-10% annual returns on average (see this link for a graph of this trend). However, there is also tremendous volatility in the stock market (i.e., huge change in stock prices), with huge market swings over time. This has happened several times over the past 40 years; see the Google Stock Ticker for the S&P 500 for the behavior of the stock market from 1975 through today. The opportunity for huge financial return compared to a savings account is why many people invest much of their money for retirement in the stock exchange; however, the huge volatility and varied rate of return can profoundly affect returns, and in the worst case scenario can completely wipe out peoples' investments and keep them from retiring.
To help deal with this risk to their retirement savings, investors often put some of their money stocks and the rest in investments such as bonds, which are safer than stocks - in other words, they have a much lower risk of going down in value than a stock would. However, they also have much lower rates of return, meaning that their value is much less likely to go up substantially compared to stocks. Depending on an individual investor's ability to tolerate uncertainty, they may choose different ratios of stocks to bonds, and stocks with varied amounts of risk - riskier stocks can both make more money and lose more money!
An additional concern when investing is inflation - the tendency in the price of goods and services to go up over time. This gradual increase in cost means that, over time, a dollar buys less and less. In the United States over the past 50 years, inflation has hovered around 2% per year, although with significant swings (see this chart for more information). As a result, investments with rates of return lower than inflation are typically not a smart choice if you're trying to increase the amount of money available to you for retirement!
In this section, we're going to make a simple model of the investment returns of an individual who is saving for retirement. This model assumes that they put money into their investment portfolio at a constant rate, and that their stock portfolio grows at a constant rate. Our assumptions are:
Using these assumptions, make a simple program to calculate (A) how much money will be in their retirement account every year from age 25 through 65 for all six possible options (low vs. high investment, and low/medium/high growth rate of the stock market); (B) how much inflation decreases the effective value of their investment each year; and, (C) how much money will be available to this person per year when they retire, in today's dollars (in other words, taking into account how inflation changes the purchasing power of their savings). Show all of this information in three ways:
Hint 1: You can create an empty list by typing my_list = [], and can append items to that list with my_list.append(val), where val is a number or string.
Hint 2: You can append an element to the end of a numpy array using the append method, though the syntax is different. For example: my_array = np.append(my_array,13.0) will append the number 13 to the end of the numpy array my_array.
Hint 3: You can create functions that take lists, arrays, and other types of variables as inputs, and also returns several variables (even of different types). For example:
def example_func(my_list, my_array):
my_list.append(2)
my_array ** 2
return my_list, my_arraywill take a list and an array as an input, and return a new list and array that are modified versions of the originals. For example:
new_list, new_array = example_func(my_old_list, my_old_array)
will return new_list and new_array, which are modified versions of my_old_list and my_old_array.
It is important to know that if you modify a list or an array within a function, it will stay modified after you call the function even if you don't return it explicitly! (So, in the example above, both my_old_list and my_old_array will be changed.)
If you want to actually copy the list or an array and only modify the copies, you have to use new_list = list(old_list) for lists, and new_array = np.copy(old_array) for numpy arrays.
In [1]:
# put your code for Section 1 here, adding additional cells right below this as necessary!
%matplotlib inline
import matplotlib.pyplot as plt
# function to make a simple calculation of the change in investment portfolio
# value over time
def investment_calculator_simple(salary, salary_increase, salary_contribution,
inflation, investment_return,
starting_age, retirement_age):
# empty lists for the age of the person and yearly investment portfolio value
age = []
investment_yearly = []
# total money in investment portfolio
investment_total = 0.0
# total amount of inflation (compared to present year)
inflation_total = 1.0
# loop over year, calculating change in investment portfolio every year
for year in range(starting_age, retirement_age+1):
# add some fraction of your salary
yearly_contribution = salary * salary_contribution
# increase investments by some contribution, and increase by some yield
investment_total = (investment_total + yearly_contribution)*investment_return
# salary and inflation both go up
salary *= salary_increase
inflation_total *= inflation
# print things out in a table, and use formatting to make it pretty
print('{0:d} {1:.0f} {2:.0f} {3:.0f} {4:.3f}'.format(year,salary,yearly_contribution,
investment_total,inflation_total))
# keep track of what the investment amount is in a given year
age.append(year)
investment_yearly.append(investment_total)
# return a ton of stuff so we can plot, etc.
return age, investment_yearly, investment_total, inflation_total
# for each of the models (low vs. high investment rate, low/med/high market return)
# run the model.
age, inv_low_low, inv_low_low_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.05,
1.02, 1.05, 25,65)
age, inv_high_low, inv_high_low_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.10,
1.02, 1.05, 25,65)
age, inv_low_med, inv_low_med_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.05,
1.02, 1.08, 25,65)
age, inv_high_med, inv_high_med_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.10,
1.02, 1.08, 25,65)
age, inv_low_high, inv_low_high_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.05,
1.02, 1.11, 25,65)
age, inv_high_high, inv_high_high_total, inflation_total = investment_calculator_simple(50000.0, 1.03, 0.10,
1.02, 1.11, 25,65)
In [2]:
plt.plot(age,inv_low_low, 'r-',
age,inv_high_low, 'r--',
age,inv_low_med, 'b-',
age,inv_high_med, 'b--',
age,inv_low_high, 'k-',
age,inv_high_high, 'k--',
linewidth=3)
plt.title('simple model')
plt.xlabel('age')
plt.ylabel('investment amount [$]')
Out[2]:
In [3]:
print("For low contribution/low return, available yearly amount is $",
"{:.0f}".format(inv_low_low_total*0.05))
print("For high contribution/low return, available yearly amount is $",
"{:.0f}".format(inv_high_low_total*0.05))
print("For low contribution/med return, available yearly amount is $",
"{:.0f}".format(inv_low_med_total*0.05))
print("For high contribution/med return, available yearly amount is $",
"{:.0f}".format(inv_high_med_total*0.05))
print("For low contribution/high return, available yearly amount is $",
"{:.0f}".format(inv_low_high_total*0.05))
print("For high contribution/high return, available yearly amount is $",
"{:.0f}".format(inv_high_high_total*0.05))
print("")
print("$1 in 2016 is equal to $","{:.3f}".format(1.0/inflation_total),"in 2056")
print("")
print("The effective value of each one (in 2016 dollars) is:")
print("")
print("low contribution/low return, $",
"{:.0f}".format(inv_low_low_total*0.05/inflation_total))
print("high contribution/low return, $",
"{:.0f}".format(inv_high_low_total*0.05/inflation_total))
print("low contribution/med return, $",
"{:.0f}".format(inv_low_med_total*0.05/inflation_total))
print("high contribution/med return, $",
"{:.0f}".format(inv_high_med_total*0.05/inflation_total))
print("low contribution/high return, $",
"{:.0f}".format(inv_low_high_total*0.05/inflation_total))
print("high contribution/high return, $",
"{:.0f}".format(inv_high_high_total*0.05/inflation_total))
Some questions about this model:
ANSWERS
| beginning age | yearly income at retirement (inflation-adjusted) |
|---|---|
| 20 | \$164,165 |
| 25 | \$105,949 |
| 30 | \$67,882 |
| 35 | \$43,013 |
| 40 | \$26,788 |
| 45 | \$16,223 |
As you saw in the links above, the return from the stock market and the inflation rate both change significantly from year to year, as do the raises that people tend to get. This can result in substantial differences in investment return compared to the simple model in the previous section.
You're now going to improve upon your previous model by introducing one element of randomness into the modeling process: namely, the rate of return of their investments. We'll keep all of the assumptions from the previous section except #5. In this version of the model, the investment portfolio now fluctuates in growth every year, and the amount depends on the risk the investor chooses to take. The low growth rate/low risk version fluctuates between 2%-8%; the medium growth rate/medium risk version fluctuates between 3% and 13% each year; and the high growth rate/high risk version fluctuates between 1% and 21% each year. (Assume that the return in any given year is totally random, and varies linearly between the maximum and minimum number give.) The investor's yearly raises and inflation rate will stay constant.
Given that there is quite a bit of randomness, it's important to run each of the six investment scenarios (low vs. high investment fraction, low/medium/high growth/risk rate) many times to get a sense of the range of possible outcomes. We're going to explore this in several ways:
In [4]:
# put your code for section 2 here, adding additional cells as necessary!
import random
def investment_calculator_2(salary, salary_increase, salary_contribution,
inflation, investment_return_low, investment_return_high,
starting_age, retirement_age):
age = []
investment_yearly = []
investment_total = 0.0
inflation_total = 1.0
for year in range(starting_age, retirement_age+1):
yearly_contribution = salary * salary_contribution
investment_total = (investment_total + yearly_contribution)*random.uniform(investment_return_low,
investment_return_high)
salary *= salary_increase
inflation_total *= inflation
#print('{0:d} {1:.0f} {2:.0f} {3:.0f} {4:.3f}'.format(year,salary,yearly_contribution,
# investment_total,inflation_total))
age.append(year)
investment_yearly.append(investment_total)
return age, investment_yearly, investment_total, inflation_total
age, inv_low_low, inv_low_low_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.05,
1.02, 1.02, 1.08, 25,65)
age, inv_high_low, inv_high_low_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.10,
1.02, 1.02, 1.08, 25,65)
age, inv_low_med, inv_low_med_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.05,
1.02, 1.03, 1.13, 25,65)
age, inv_high_med, inv_high_med_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.10,
1.02, 1.03, 1.13, 25,65)
age, inv_low_high, inv_low_high_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.05,
1.02, 1.01, 1.21, 25,65)
age, inv_high_high, inv_high_high_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.10,
1.02, 1.01, 1.21, 25,65)
In [5]:
plt.plot(age,inv_low_low, 'r-',
age,inv_high_low, 'r--',
age,inv_low_med, 'b-',
age,inv_high_med, 'b--',
age,inv_low_high, 'k-',
age,inv_high_high, 'k--',
linewidth=3)
plt.title("slightly more complex model")
plt.xlabel('age')
plt.ylabel('investment amount [$]')
Out[5]:
In [6]:
import numpy as np
number_of_trials = 10000
low_low_end_vals = []
high_high_end_vals = []
for i in range(number_of_trials):
age, inv_low_low, inv_low_low_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.05,
1.02, 1.02, 1.08, 25,65)
low_low_end_vals.append(inv_low_low_total*0.05)
age, inv_high_high, inv_high_high_total, inflation_total = investment_calculator_2(50000.0, 1.03, 0.10,
1.02, 1.01, 1.21, 25,65)
high_high_end_vals.append(inv_high_high_total*0.05)
In [7]:
plt.hist(low_low_end_vals,bins=20,normed=True)
plt.title("available salary: low contribution/low return scenario")
low_low_end_vals = np.array(low_low_end_vals)
meanval = low_low_end_vals.mean()
stddev = low_low_end_vals.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the low contribution/low return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
In [8]:
plt.hist(high_high_end_vals,bins=20,normed=True)
plt.title("available salary: high contribution/high return scenario")
high_high_end_vals = np.array(high_high_end_vals)
meanval = high_high_end_vals.mean()
stddev = high_high_end_vals.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the high contribution/high return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
Question: How the outcomes for this model different, both qualitatively and quantitatively, than the model in section 1?
The mean values for the two models I show the distribution for (the low investment/low yield model and high investment/high yield model) are pretty much the same as in the simple model; however, there is a pretty substantial variation in outcome now for each of the models, with the riskier model having a substantially larger range of possible outcomes. In other words, depending on market fluctuations, the final amount available to spend varies tremendously!
In this section, we're going to build upon what you did in Section 2 and introduce some additional elements of randomness into our model to make it more realistic and to see how that affects the outcomes. We will keep assumptions #1, 2, and 6 from the first section the same, but we'll change the others as follows:
Given that there is quite a bit of randomness, it's important to run each of the six investment scenarios (low vs. high investment fraction, low/medium/high growth/risk rate) many times to get a sense of the range of possible outcomes. We're going to explore this in several ways:
In [9]:
# put your code for section 3 here, adding additional cells as necessary!
def investment_calculator_3(salary, salary_increase_low, salary_increase_high, salary_contribution,
inflation_low, inflation_high,
investment_return_low, investment_return_high,
starting_age, retirement_age):
age = []
investment_yearly = []
investment_total = 0.0
inflation_total = 1.0
for year in range(starting_age, retirement_age+1):
yearly_contribution = salary * salary_contribution
investment_total = (investment_total + yearly_contribution)*random.uniform(investment_return_low,
investment_return_high)
salary *= random.uniform(salary_increase_low, salary_increase_high)
inflation_total *= random.uniform(inflation_low, inflation_high)
#print('{0:d} {1:.0f} {2:.0f} {3:.0f} {4:.3f}'.format(year,salary,yearly_contribution,
# investment_total,inflation_total))
age.append(year)
investment_yearly.append(investment_total)
return age, investment_yearly, investment_total, inflation_total
age, inv_low_low, inv_low_low_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.05,
1.00, 1.04, 1.02, 1.08, 25,65)
age, inv_high_low, inv_high_low_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.10,
1.00, 1.04, 1.02, 1.08, 25,65)
age, inv_low_med, inv_low_med_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.05,
1.00, 1.04, 1.03, 1.13, 25,65)
age, inv_high_med, inv_high_med_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.10,
1.00, 1.04, 1.03, 1.13, 25,65)
age, inv_low_high, inv_low_high_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.05,
1.00, 1.04, 1.01, 1.21, 25,65)
age, inv_high_high, inv_high_high_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.10,
1.00, 1.04, 1.01, 1.21, 25,65)
In [10]:
plt.plot(age,inv_low_low, 'r-',
age,inv_high_low, 'r--',
age,inv_low_med, 'b-',
age,inv_high_med, 'b--',
age,inv_low_high, 'k-',
age,inv_high_high, 'k--',
linewidth=3)
plt.title("most complex model")
plt.xlabel('age')
plt.ylabel('investment amount [$]')
Out[10]:
In [11]:
number_of_trials = 1000
low_low_end_vals = []
high_high_end_vals = []
low_low_end_vals_infl_adj = []
high_high_end_vals_infl_adj = []
for i in range(number_of_trials):
age, inv_low_low, inv_low_low_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.05,
1.00, 1.04, 1.02, 1.08, 25,65)
low_low_end_vals.append(inv_low_low_total*0.05)
low_low_end_vals_infl_adj.append(inv_low_low_total*0.05/inflation_total)
age, inv_high_high, inv_high_high_total, inflation_total = investment_calculator_3(50000.0, 1.01, 1.05, 0.10,
1.00, 1.04, 1.01, 1.21, 25,65)
high_high_end_vals.append(inv_high_high_total*0.05)
high_high_end_vals_infl_adj.append(inv_high_high_total*0.05/inflation_total)
In [12]:
# IGNORING INFLATION
plt.hist(low_low_end_vals,bins=20,normed=True)
plt.title("available salary: low contribution/low return scenario")
low_low_end_vals = np.array(low_low_end_vals)
meanval = low_low_end_vals.mean()
stddev = low_low_end_vals.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the low contribution/low return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
In [13]:
# TAKING INTO ACCOUNT INFLATION
plt.hist(low_low_end_vals_infl_adj,bins=20,normed=True)
plt.title("available salary: low contribution/low return scenario (inflation-adjusted)")
low_low_end_vals_infl_adj = np.array(low_low_end_vals_infl_adj)
meanval = low_low_end_vals_infl_adj.mean()
stddev = low_low_end_vals_infl_adj.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the low contribution/low return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
In [14]:
# IGNORING INFLATION
plt.hist(high_high_end_vals,bins=20,normed=True)
plt.title("available salary: high contribution/high return scenario")
high_high_end_vals = np.array(high_high_end_vals)
meanval = high_high_end_vals.mean()
stddev = high_high_end_vals.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the high contribution/high return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
In [15]:
# TAKING INTO ACCOUNT INFLATION
plt.hist(high_high_end_vals_infl_adj,bins=20,normed=True)
plt.title("available salary: high contribution/high return scenario (inflation-adjusted)")
high_high_end_vals_infl_adj = np.array(high_high_end_vals_infl_adj)
meanval = high_high_end_vals_infl_adj.mean()
stddev = high_high_end_vals_infl_adj.std()
print("the average return is $",'{:.2f}'.format(meanval),
"per year for the high contribution/high return scenario")
print("with a standard deviation of $",'{:.2f}'.format(stddev))
Question: Think about the information that you get from the model in this section compared to both the original model in Section 1 and the somewhat more advanced model in Section 2. What parts of the models agree and disagree? What additional information do you get from this model? How does it differ from Section 2's model?
This model gives a large spread of answers compared to the original model, and answers that are comparable to section 2 (though the latter is true only for the distribution at the end; taking into account variable contributions and variable inflation can mean that the amount that the investment portfolio can change each year is very large. We get a bit of additional information compared to section 2, in that it has a different spread of final values. It isn't too different from section 2's model, though. This might be because all of the randomness is uniform (i.e., values only change linearly and are random from year to year)
Question: Also, think about other information that you can get from this model. What can it tell you about how an investment portfolio might behave? And, if you were an investment advisor, how would you use these models to give your clients advice?
This model provides some sense of what the possible variation in investment yields are over time - in other words, how much randomness there is in a year-over-year sense, and how much the total variation in final outcome there might be. I would show the outcomes of this type of model to my clients to explain what the difference of possible outcomes is between the various investment strategies and investment contributions, and explain that the high risk/high reward option will yield more money on average but with a wider range of possible gains.
Question: Looking at the graphs of inflation and the stock market, and thinking about peoples' careers and savings habits, how might you improve your model to more accurately reflect the possible outcomes for an investor?
There are a variety of ways that this model could be improved:
Write your answer here
Write your answer here