In this unit, we study three kinds of growth models–linear, exponential, and logistic–that describe physical phenomena all around us, then apply the same theory to study applications in finance.

1 Growth Models

1.1 Linear Growth

Linear growth occurs when the same number is added (or subtracted) each time period.

Example: Suppose I have $20 in my piggy bank, and I add $5 each week.

Question:

How much money do I have in 4 weeks?

Let’s model this. Let $n$ be the number of weeks, and $P_n$ be the amount of money in my piggy bank after $n$ weeks. Then:

$n=0$ corresponds to now, so $P_0=\$20$;
$n=1$ corresponds to one week from now, so $P_1=\$20+\$5 = \$25$;
$n=2$ corresponds to two weeks from now, so $P_2=\$25+\$5 = \$30$; etc.

Since we are adding $5 each time period, we could say the amount at time $n$ is equal to the amount at the previous time $n-1$ plus $5, or

\[P_n = P_{n-1} + 5, \text{ } P_0=20.\]

This is an example of a recursive formula, a formula for finding a result. (finding $P_n$) that uses a previous result ($P_{n-1}$).

How much do you have after a year?

You could use this information alone to find the amount of money you have at the end of the year (find $P_1$, $P_2$, $P_3$, etc. up to $P_52$). But this would take a long time! More simply you could simply say, “I put in $5 each week for 52 weeks, that’s $\$5*52=\$260$, so $\$20+\$260=\$280$ total.”

This reasoning can be applied to make an explicit formula, a formula that allows you to calculate any $P_n$ directly. If I put in $5 each week for $n$ weeks, that’s $5n$ dollars added for $n$ weeks. Thus, $P_n=20+5n$.

In general, linear growth can be modeled as follows.

The value $d$ is also called a constant rate of change. This is the same thing as slope ($m$ in the formula $y=mx+b$).

The population of elk in a national forest was measured to be 12,000 in 2003 and was measured again to be 15,000 in 2007.

Assuming linear growth, what is the constant rate of change, i.e. how much does the elk population increase in 1 year?

(15000-12000)/(2007-2003)

## [1] 750

Find a recursive formula for elk population (n is years after 2003):
1. $P_n = 750 + 12,000n$
2. $P_n = 12,000+ 750n$
3. $P_n = 750 + P_{n-1}, P_o=12,000$
4. $P_n = 12,000 + P_{n-1}, P_o=15,000$
Find an explicit formula for the elk population (n is years after 2003).
1. $P_n = 750 + 12,000n$
2. $P_n = 12,000+ 750n$
3. $P_n = 750 + P_{n-1}, P_o=12,000$
4. $P_n = 12,000 + P_{n-1}, P_o=15,000$
Use #3 to find what will the elk population be in 2014 (assuming linear growth).

The cost, in dollars, of a gym membership for n months can be described by the explicit equation $P_n = 70 + 30n$. What does this equation tell us?

Written Homework:

The number of stay-at-home fathers in Canada has been growing steadily. While the trend is not perfectly linear, it is fairly linear. Use the data from 1976 and 2010 to find the rate of change between 1976 and 2010. (Round your answer to the nearest whole number.)

Year	StayAtHomeFathers
1976	20610
1984	28725
1991	43530
2000	47665
2010	53555

Find an explicit formula for the number of stay-at-home fathers.
Use this formula to predict the number of stay-at-home fathers in 2025. (Round your answer to the nearest whole number.)

1.2 Exponential Growth

Suppose that every year, only 10% of the fish in a lake have surviving offspring. If there were 100 fish in the lake last year, there would now be 110 fish. If there were 1000 fish in the lake last year, there would now be 1100 fish. Absent any inhibiting factors, populations of people and animals tend to grow by a percent of the existing population each year.

Suppose we start with 1000 fish in a lake ($P_0=1000$), and the population increases by 10% by the next year.

Question:

What is the new population after the first year ($P_1$)?

We take the old population $P_0$ and add 10% of that population:

\[P_1 = P_0 + 0.10*P_0\]

We can simplify this as follows.

\[P_1 = P_0 + 0.10*P_0 = (1+0.10)P_0 = 1.10P_0 = 1.1*1000=1100\]

Note that while 10% is the growth rate, the growth multiplier (or growth factor) is 1.10.

Question:

What is the new population after the second year ($P_2$)?

After the next period, the population grows by 10% again, so:

\[P_2 = P_1 + 0.10*P_1 = (1+0.10)P_1 = 1.10P_1 = 1.1*1100=1210\] \[P_3 = P_2 + 0.10*P_2 = (1+0.10)P_2 = 1.10P_2 = 1.1*1210=1331\]

Notice this is not the same thing as if the population were to grow by 100 fish per year. Rather than growing by a constant number (adding each time period), it is growing by a constant percentage (multiplying each time period). This results in repeated multiplication.

$P_1 = 1.10 P_0$
$P_2 = 1.10 P_1 = 1.10 (1.10 P_0) = (1.10)^2 P_0$
$P_3 = 1.10 P_2 = 1.10 (1.10 P_1) = 1.10 (1.10 (1.10 P_0) ) = (1.10)^3 P_0$

By this pattern, we see $P_n = (1.10)^n 1000$ or $P_n = 1000(1.10)^n$. This allows us to quickly calculate the population in 20 years:

$P_{20} = (1.10)^{20}(1000) = 6727.$

These ideas are summarized below.

Tacoma’s population in 2000 was about 200 thousand and had been growing by about 9% each year.
1. Write a recursive formula for the population of Tacoma
2. Write an explicit formula for the population of Tacoma
If this trend continues, what will Tacoma’s population be in 2016?
When does this model predict Tacoma’s population to exceed 400 thousand?

Hint: Once you set this up, you may use Wolfram Alpha to solve this for n.

Written Homework:

In 1987, the world population reached 5 billion people, at which point, the population was growing 2% per year. Assuming that the world population follows an exponential growth model, find the projected world population in 2024. (Give your answer in billions, rounded to the nearest tenth.)
Look up the current population. What does this say about the actual growth rate of the world population?

1.3 Logistic Growth

To model a situation using exponential growth assumes growth while continuing at the same rate. This is not the case in many natural phenomena. Often the growth rate decreases as a population reaches a maximum sustainable population, called a carrying capacity. If the growth rate decreases linearly until the growth rate is zero at the carrying capacity, K, we can model the actual growth rate at time $n$ as $\big(1-\frac{P_{n-1}}{K}\big)$ times the unconstrained growth rate. This gives the following recursive formula.

This kind of logistic growth does not have an easy explicit formula. However, when we model growth as a continuous phenomenon, we can derive an explicit formula for the population at time $t$:

\[P(t) = \frac{K}{1+Ae^{-rt}}, \text{ where }A=\frac{K-P_0}{P_0}.\]

Below is a comparison of exponential and logistic growth. The asymptote represents the carrying capacity.

Written Homework:

The wolf population in Yellowstone National Park in January 2020 was 94 and is projected to grow 4%/year under a carrying capacity of 300.

What is the projected wolf population for 2021? (Round your answer to the nearest whole number.)
What is the projected wolf population for 2022? (Use your answer from part A and round your answer to the nearest whole number.)

Extra Practice:

A national park has a current population of 2000 bison. They reproduce annually at a rate of 5%. Experts say the park can sustain a population of 10,000 bison.

What is the next year’s population?
What is the current population 2 years from now?
What is the current population 5 years from now? (Hint: Use a spreadsheet.)
How long until the population reaches 5000 bison? (Hint: Use a spreadsheet.)

2 Finance

2.1 Simple Interest

A friend asks to borrow $300 and agrees to repay it in a month with 3% interest.

Question:

How much interest will you earn in one month?

You earn $300*0.03 = $9 interest.

In other words, he owes you $300+$9=$309.

This situation is called simple one-time interest. If your interest rate is $r$, your initial amount is $P_0$, than your interest is $P_0*r$, and the amount you owe (or gain) is \[P_0+P_0*r=P_0(1+r)\].

Now suppose he doesn’t pay you after a month but says he will after another month, with another 3% interest on the original $300 he owes you.

Question:

How much will he owe you after the second month?

How much will he owe you after a year (12 months)?

This situation is called simple interest over time:

Question:

What kind of formula is simple interest over time?

Linear
Exponential
Quadratic
Logistic

Question:

A friend lends you $200 for a week, which you agree to repay with 5% one-time interest. How much will you have to repay?

2.2 Compound Interest

Back to your friend who, after a month owes you $309 and can’t pay. You say, “Alright, you owe me $309, so next month, you gotta pay me 3% interest on what you owe me now.” He says, “OK.” How much does he owe you after the second month?

He owes you $309 plus 3% interest on $309, so $318.27.

309+309*0.03

## [1] 318.27

This situation is called compound interest:

What kind of formula is compound interest?

Linear
Exponential
Quadratic
Logistic

You deposit $300 in a savings account earning 2% interest compounded annually. How much will you have in the account in 10 years?

Homework:

You deposit $2000 in an account earning 3% interest compounded monthly. How much will you have in the account in 20 years? (Round your answer to the nearest cent.)
How much interest will you earn in 20 years?
Calculate the doubling time for this investment. (Round your answer to the nearest tenth of a year.)

2.3 Annuity

The next day, your friend comes back and says “I need another $300 this month, and next month, and next month, and you can add 3% to what I owe you each month. I’ll pay you back at the end of the year.” What does he owe you at the end of the year?

We can model this recursively as follows. Each month $n$, the amount he owes you, $P_n$ is equal to what he owed you from the previous month, $P_{n-1}$, plus 3% interest on $P_{n-1}$, plus $300: \[P_n = P_{n-1}+P_{n-1}*0.03 + 300, P_0=300.\] It’s not easy to solve this formula explicitly, but with a little trick (from Calculus II), we could derive the explicit formula: \[P_n = \frac{300((1+0.03)^n-1)}{0.03}.\]

So this is what your friend will owe you after a year. $4257.61!

300*((1+0.03)^12-1)/0.03

## [1] 4257.609

This situation is called annuity. An annuity is a fixed sum of money invested or paid to someone each time period (usually monthly or annually), typically for a long period. Below is the general annuity formula:

In the example with your friend, we could have used this formula, whereby the number of compounding periods per year is $k=12$, the annual interest rate is $3\%*12=36\%$, and $\frac{r}{k}=\frac{36\%}{12}=3\%$ is the monthly interest rate.

Most retirement plans are examples of annuities. Below are the three main types of retirement annuities:

A 401K is a tax-deferred plan offered by employers, meaning you put money in without paying taxes on it. This reduces your taxable income, BUT, you then pay taxes later on what you withdraw from your account. Many companies have it set so that they will match the money you put in up to a certain percentage of your salary.
Traditional IRA (Individual Retirement Account) - a tax-deferred plan like a 401K, but set up for individuals, and has annual contribution limits
Roth IRA - a plan whereby you pay taxes on the money you put in, but don’t pay taxes on what you take out (so interest is not taxed!). Like Traditional IRAs, Roth IRAs are subject to contribution limits.

Question:

If you deposit $100 each month into a traditional IRA earning 5% annual interest compounded monthly, how much will you have in the account after 20 years?

Homework:

If you deposit $150 each month into a Roth IRA earning 6% annual interest compounded monthly, how much will you have in the account after 30 years? (Round your answer to the nearest cent.)

You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 6% annual interest compounded monthly. How much do you need to deposit each month to meet your retirement goal?

2.4 Loans

Now, back to your friend who was borrowing money and now owes you $4257.61 after recklessly borrowing your hard-earned money. Fortunately for you, he just got a somewhat lucrative job and wants to start paying you back. He says he wants to pay the same amount back to you each month for 12 months. How much should he pay you each month?

You decide this is a good plan IF he keeps paying 3% interest on whatever he owes you.

“Fair enough,” he agrees.

How do we calculate this? First, find a recursive formula for this situation. Let’s call the amount he owes you each month $d$.

\[P_n = P_{n-1}+P_{n-1}*0.03 - d, P_0=\$4257.61.\]

Like the annuity formula, it’s not obvious to solve this formula explicitly, but with the same trick from (Calculus II), we could derive the explicit formula: \[P_0 = \frac{d\Big((1-(1+0.03)^{-n}\Big)}{0.03}=\$4257.61.\]

There are $n=12$ months he will make payments, so solving this for $d$ gives:

4257.61*0.03/(1-(1+0.03)^(-12))

## [1] 427.7284

Below is the general formula for calculating loan payments, following the principles outlined in this example.

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive of a car can you afford? In other words, what amount of loan can you pay off with $200 per month?

200*(1-(1+0.03/12)^(-12*5))/(0.03/12)

## [1] 11130.47

Question:

You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 4%, and the loan is for 30 years.

How much will your monthly payments be?

RHS = (1-(1+0.04/12)^(-30*12))/(0.04/12)
140000/RHS

## [1] 668.3814

How much interest are you paying over the life of the loan?

668.38*12*30 - 140000

## [1] 100616.8

Homework:

Marie can afford a $250 per month car payment. She’s found a 5-year loan at 7% interest. How expensive of a car can she afford? (Round your answer to the nearest cent.)
How much total money will Marie pay the loan company?
How much of the money Marie paid is interest?
You want to buy a $25,000 car. The company is offering a 2% interest rate for 48 months (4 years). What will your monthly payments be? (Round your answer to the nearest cent.)

3 Group Project Problems

In your group, select one of the following problems you want to work on together. Work together to find a solution (or solutions) to the given problem(s) and/or related questions you find interesting. Then, as a group, write a 1-2 page write-up presenting:

the problem or question you worked on,
a solution and the tools and reasoning you used to arrive at a solution, and
the significance of the result and how it can contribute toward better decision-making.

Make sure you answer all the questions in your problem and edit your write-up to make sure it is readable with no grammar or spelling errors.

Each group will also make a short video presentation of their work, so keep in mind, that your work will be made public for other students to view and study.

3.1 Fish Population Management

Recursive equations can be very handy for modeling complicated situations for which explicit equations would be hard to interpret. As an example, consider a lake in which 2000 fish currently reside. The fish population seems to be growing by 10% each year, but every year about 100 fish are harvested from the lake by people fishing.

Write a recursive equation for the number of fish in the lake after n years.
Calculate the population after the first 10 years. What does the population appear to be doing in the long run?
Do you think this model will be accurate in the long run?
What is the number of fish that could be harvested each year without causing the fish population to change in the long run?
Officials determine that 4000 fish is the ideal population for this lake in terms of the overall health of the ecosystem surrounding it. What recommendations would you have for harvesting fish from this lake?

3.2 Forensic Science

A hotter object in a colder room will decrease in temperature exponentially, approaching the room temperature according to the formula \[T_n = a(1-r)^n+T_r,\] where:

$T_n$ is the temperature of the object after $n$ minutes,
$r$ is the rate at which temperature is changing,
$a$ is a constant, and
$T_r$ is the temperature of the room.

Forensic investigators use this formula to predict the time of death of a homicide victim. Suppose that when a victim was discovered ($n = 0$), the body was 85 degrees F. After 20 minutes, the temperature of the body was measured again to be 80 degrees. The temperature of the room was 70 degrees F.

Use the given information above to find a formula for the temperature of the body. Hint: First, use the information $T_n=85$ at $n=0$ to find $a$. (Remember anything to the zero power is 1). Then, use the information $T_n = 80$ at $n = 20$ to find $r$, possibly using the Desmos graphing calculator or an online equation solver (here is one). After solving for $a$ and $r$, you have a formula for $T_n$ in terms of n!
If the body started at 98.6 degrees F, when did the victim die?

3.3 Logistic Growth

Read the section on Logistic Growth and/or find some videos that explain logistic growth (e.g. here is one example). Use this information to explain what logistic growth is when it occurs, and what formula(s) are used to model it. Then use this understanding to answer the following questions.

One hundred trout are seeded into a lake. Absent any constraints, the trout population will grow by 70% a year. The lake can sustain a maximum of 2000 trout. Using the logistic growth model,
1. Write a recursive formula for the number of trout
2. Use your recursive formula to calculate the number of trout after 1, 2, 3, 4, and 5 years.
Now model the same fish population growing 70% per year without constraint, and calculate the number of trout after 1, 2, 3, 4, and 5 years.
Compare your answers to 1. and 2. above using a table or graph. Discuss the reason for any noted differences.

3.4 Should you Refinance?

Your original mortgage

Suppose that 10 years ago you bought a home for $130,000, paying 10% as a down payment, and financing the rest at 6% interest for 30 years.

1a. On your existing mortgage (the one you got 10 years ago), how much money did you pay as your down payment?
1b How much money was your existing mortgage (loan) for?
1c. What is your current monthly payment on your existing mortgage?
1d. How much will you pay in monthly payments over the life of this existing loan?
1e. How much total interest will you pay over the life of this existing loan?

Where are you now?

Now, 10 years after you first took out the loan, you check your loan balance. Only part of your payments have been going to pay down the loan; the rest has been going toward interest. You see that you still have $102,637 left to pay on your loan. Your house is now valued at $180,000.

At this point:

2a. How much of the original loan have you paid off?
(i.e., how much have you reduced the loan balance by? Keep in mind that interest is charged each month - it’s not part of the loan balance.)
2b. How much money have you paid to the loan company so far (over the last 10 years)? 2c. How much interest have you paid so far (over the last 10 years)? 2d. How much equity do you have in your home (equity is value minus remaining debt)?

Refinancing

Since interest rates have dropped, you consider refinancing your mortgage at a lower 4.5% rate. To refinance, you would pay a refinancing fee of 2% of the total mortgage, which is added to your new mortgage amount. You also notice that if you refinance, you are going to be making payments on your home for another 30 years. Note that these 30 years of payments are in addition to the 10 years you’ve already been paying, making 40 years of total payments.

If you took out the new 30-year mortgage at 4.5%:

3a. How much would your new mortgage be?
3b. What would your new monthly payments be?
3c. How much will you pay in monthly payments over the life of the new loan?
3d. How much interest would you pay over the life of the new loan?
3d. How much less will you pay each month because of the lower monthly payment?
3e. How much total interest will you be paying on house payments? Consider the interest you paid over the first 10 years of your original loan, as well as interest on your refinanced loan.

Conclusion

Discussion of your answers to the above questions. Conclude with a discussion of the question: Should you refinance? (There isn’t a correct answer to this. Give your opinion and justification.) Some questions to consider would be:

how much total money are you paying in each case?
how much interest are you paying in each case?

3.5 Maxing out your Credit

Suppose that as a Freshman you apply for the Chase Unlimited Freedom credit card and receive your credit card with a $2000 credit limit. This credit card has a 0% annual percentage rate (APR) for the first 15 months, meaning you don’t pay ANY interest during this time. After that, if you do not pay your bill, you owe 30% APR, which is accrued monthly.

After these 15 months, you have maxed out your credit and have a $2000 bill due the next month. You think perhaps you should wait to pay your credit card off until you graduate and get a job to pay for it. (In this case, where no payments are made, there are often additional fees owed; however, we will not consider this here in this example.)

Suppose after 15 months, you decide not to make your minimum payment on your credit card:
1. In this case, you will owe a 30% annual interest rate. How much of the 30% annual interest is owed after one month?
2. How much total interest do you owe after the first month?
3. How much is your credit card balance after the first month?
Suppose you cannot pay your credit card after the first two months.
1. How much interest do you owe after the second month?
2. How much do you owe on your credit card after the second month?
Suppose you cannot pay your credit card after the first three months.
1. How much interest do you owe after the third month?
2. How much do you owe on your credit card after the third month?
Use Google Sheets or Microsoft Excel to repeat the calculations from steps 2-3, for up to 36 months!
1. Follow the steps described in this video or together in class.
2. Based on your calculations, how much do you owe after 36 months?
3. How much interest have you accrued?
In this step, we’ll use the compound interest formula to calculate how much you would owe after 36 months.
1. If you have not yet, familiarize yourself with the compound interest formula. Identify what each variable means.
2. Use the compound interest formula to calculate how much you owe after 36 months.
3. Compare with your answer in 4b.
Suppose after 15 months, you decide to get a part-time job to pay $100/month on your credit card. Let’s do an experiment and compute how long it would take to pay off your credit card that way.
1. Modify the formula in cell B3 to subtract 100 from each month’s balance.
2. Then copy the formula in cell B3 down as you did in step 4b(iii).
3. How long does it take to pay off your bill at this rate?
4. After the last month in which you would pay your credit card off, how much total money would you have paid? (Make sure you include the total of all the payments you have made plus the last month’s interest.)
5. At this point, how much total interest have you paid? Compare with your answer from 4c.
(Optional) Suppose that, from the very beginning of college, you decided to get a part-time job and paid $100/month on your credit card. Let’s find out how much you would pay that way.
1. Suppose again, you spend $2000 on your credit card during the first 15 months, but pay $100/month b. to pay it off. How much do you owe on your credit card after the first 15 months?
2. If you continue to pay $100/month on your credit card after the first 15 months, how long would it take to pay it off?
3. After the last month in which you would pay your credit card off, how much total money would you have paid? (Again, make sure you include the total of all the payments you have made plus the last month’s interest.) By the time you pay your credit card off, how much interest have you paid?

Based on this, discuss with your group/class how this informs your approach to credit cards and loans. Write a 2-paragraph summarizing: The tools used for finding your results. Results and their significance. How can these results contribute to better decision-making?

3.6 Car Loans

You are shopping for a car for $20,000 and have enough money in your account to pay upfront, but then your salesman says he can get you a low-interest loan, 4% for 3 years. He argues that by investing your $20,000 at a higher interest rate, you will come out ahead. Is he right?

You are smart and know to expect hidden fees. You ask the salesman, “What are the fees for the loan?”

After rummaging around in the back room, the salesman comes back and says, “You’d have a 3% fee,” which would be added to the total loan amount.

So, you have two options:

Option 1: Take the loan and invest the $20,000 elsewhere.
Option 2: Do not take the loan, but invest what you would be paying every month.

Question 0: Based on your understanding of the problem, in which option do you think you would end up with more money? Why?

Now, to determine in which case you will come out ahead, you decide to compute your 3-year gain in each case.

Option 1 - Take the Loan

If you take the loan:
Question 1a) How much is the total loan with the 3% lender fees included?
Question 1b) What would be your monthly payment for this loan?
Question 1c) How much would you pay throughout this loan?

If you take the loan and follow the salesman’s advice, you can invest the $20,000 from your bank account straight into index funds.

Question 1d) Suppose you invest the $20,000 and earn 5% annual interest, compounded monthly. How much would this investment be worth after 3 years?

Option 2 - Do not take the loan

If you do not take the loan, you would have no monthly payment, so you could invest the same amount of money as your monthly payment (from 1b) each month.

Question 2 If you invest the money you would be paying against your car payment (from 1b) into index funds each month (earning 5% annual interest, compounded monthly), how much would this fund be worth after 3 years?

Conclusion

Question 3: Was the salesman right? Which option is better? Would you come out ahead? If so, by how much Which option would you choose and why?

3.7 Payday loans

Payday loans are short-term loans that you take out against future paychecks: The company advances money against a future paycheck. First, read more here to see how payday loans work.

Check ’n Go is one example of a payday loan loan company. Here are example loan rates for getting a Check ’n Go payday loan online. Use this rate (360% APR) with biweekly payment schedule for this problem.

Notice that if you are paying your loan after 14 days (2 weeks), this is the period. Since there are 52 weeks in a year, for this loan, there are 52/2 = 26 periods per year.

Suppose you decide to borrow $500 and that we will pay back the loan in 14 days. Determine the total amount that you would need to pay back and the effective loan rate.
If you cannot pay back the loan after 14 days, you will need to get an extension for another 14 days, paying an additional 14 days of interest on the original amount you owe in (a). Determine the total amount you will be paying for the now 28-day loan.
Suppose you wait a year to pay off this loan. How much are you paying?

In your write-up, include a discussion of the above question. Make sure you explain the general way in which payday loans work and your conclusions about whether or not (or when and when not) payday loans are a smart thing to do.

Unit 2 Notes - Growth Models and Finance

Paul Regier

1 Growth Models

1.1 Linear Growth

1.2 Exponential Growth

1.3 Logistic Growth

2 Finance

2.1 Simple Interest

2.2 Compound Interest

2.3 Annuity

2.4 Loans

3 Group Project Problems

3.1 Fish Population Management

3.2 Forensic Science

3.3 Logistic Growth

3.4 Should you Refinance?

3.5 Maxing out your Credit

3.6 Car Loans

3.7 Payday loans