1.3: A Trillion Dollars of Student Loans

Last updated
Save as PDF

Page ID: 147896

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$ \newcommand{\dsum}{\displaystyle\sum\limits} $

$ \newcommand{\dint}{\displaystyle\int\limits} $

$ \newcommand{\dlim}{\displaystyle\lim\limits} $

$ \newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\id}{\mathrm{id}}$

$ \newcommand{\Span}{\mathrm{span}}$

$ \newcommand{\kernel}{\mathrm{null}\,}$

$ \newcommand{\range}{\mathrm{range}\,}$

$ \newcommand{\RealPart}{\mathrm{Re}}$

$ \newcommand{\ImaginaryPart}{\mathrm{Im}}$

$ \newcommand{\Argument}{\mathrm{Arg}}$

$ \newcommand{\norm}[1]{\| #1 \|}$

$ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\AA}{\unicode[.8,0]{x212B}}$

$ \newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$ \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$ \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$ \newcommand{\vectorC}[1]{\textbf{#1}} $

$ \newcommand{\vectorD}[1]{\overrightarrow{#1}} $

$ \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} $

$ \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} $

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $

$\newcommand{\longvect}{\overrightarrow}$

$ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $

$\newcommand{\avec}{\mathbf a}$ $\newcommand{\bvec}{\mathbf b}$ $\newcommand{\cvec}{\mathbf c}$ $\newcommand{\dvec}{\mathbf d}$ $\newcommand{\dtil}{\widetilde{\mathbf d}}$ $\newcommand{\evec}{\mathbf e}$ $\newcommand{\fvec}{\mathbf f}$ $\newcommand{\nvec}{\mathbf n}$ $\newcommand{\pvec}{\mathbf p}$ $\newcommand{\qvec}{\mathbf q}$ $\newcommand{\svec}{\mathbf s}$ $\newcommand{\tvec}{\mathbf t}$ $\newcommand{\uvec}{\mathbf u}$ $\newcommand{\vvec}{\mathbf v}$ $\newcommand{\wvec}{\mathbf w}$ $\newcommand{\xvec}{\mathbf x}$ $\newcommand{\yvec}{\mathbf y}$ $\newcommand{\zvec}{\mathbf z}$ $\newcommand{\rvec}{\mathbf r}$ $\newcommand{\mvec}{\mathbf m}$ $\newcommand{\zerovec}{\mathbf 0}$ $\newcommand{\onevec}{\mathbf 1}$ $\newcommand{\real}{\mathbb R}$ $\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$ $\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$ $\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$ $\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$ $\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$ $\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$ $\newcommand{\laspan}[1]{\text{Span}\{#1\}}$ $\newcommand{\bcal}{\cal B}$ $\newcommand{\ccal}{\cal C}$ $\newcommand{\scal}{\cal S}$ $\newcommand{\wcal}{\cal W}$ $\newcommand{\ecal}{\cal E}$ $\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$ $\newcommand{\gray}[1]{\color{gray}{#1}}$ $\newcommand{\lgray}[1]{\color{lightgray}{#1}}$ $\newcommand{\rank}{\operatorname{rank}}$ $\newcommand{\row}{\text{Row}}$ $\newcommand{\col}{\text{Col}}$ $\renewcommand{\row}{\text{Row}}$ $\newcommand{\nul}{\text{Nul}}$ $\newcommand{\var}{\text{Var}}$ $\newcommand{\corr}{\text{corr}}$ $\newcommand{\len}[1]{\left|#1\right|}$ $\newcommand{\bbar}{\overline{\bvec}}$ $\newcommand{\bhat}{\widehat{\bvec}}$ $\newcommand{\bperp}{\bvec^\perp}$ $\newcommand{\xhat}{\widehat{\xvec}}$ $\newcommand{\vhat}{\widehat{\vvec}}$ $\newcommand{\uhat}{\widehat{\uvec}}$ $\newcommand{\what}{\widehat{\wvec}}$ $\newcommand{\Sighat}{\widehat{\Sigma}}$ $\newcommand{\lt}{<}$ $\newcommand{\gt}{>}$ $\newcommand{\amp}{&}$ $\definecolor{fillinmathshade}{gray}{0.9}$

INTRODUCTION

Taking out a loan is usually a necessary step to prepare for a large purchase, such as a car, a house, or a college education. While there are online tools and specific equations used to calculate the amount of a monthly payment for these loans, you can usually use a much simpler set of steps to find an accurate estimate for a monthly payment. Other variables that contribute to the affordability of a loan may be harder (or impossible) to calculate. These might be things like estimating your monthly paycheck or the average starting salary for a person with a college degree in your field.

What variables contribute to determining the amount of a monthly payment on a loan?
What variables outside of the loan contribute to determining whether or not one can afford a loan?

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

estimation is useful in many situations and can be done with simple as well as more complex calculations.
the interest rate of a loan and the length of a loan repayment have a significant impact on the size of each loan payment.
the interest rate and the length of a loan repayment impact the total amount repaid.

By the end of this collaboration, you should be able to

compute estimations related to personal budget/finance issues.
estimate the total interest charged on a loan.
estimate monthly payments for a variety of loans.
determine if an estimate is an underestimate or overestimate.

PROBLEM SITUATION 1: ESTIMATING COSTS

Melinda is purchasing a car and needs to borrow $12,000 to make the purchase. She needs to decide between two loans. Both loans have an Annual Percentage Rate (APR) of 5%, but one has a term of 3 years while the other has a term of 6 years.

(1) Compare the two loans by estimating the monthly payment for each loan. To figure this out, work through the following questions in your group.

(a) Estimate how much the monthly payments would be if there were no interest?

3-year loan =

6-year loan =

(b) How much would the payments be if the interest was charged on the entire original principal for the entire term of the loan? Round to the nearest dollar.

3-year loan =

6-year loan =

3-year loan =

6-year loan =

(d) Use your estimates in Question 1(c) to estimate the total amount Melinda would pay for each loan.

3-year loan =

6-year loan =

(e) Use your estimates in Questions 1(c) and 1(d) to estimate the amount of interest Melinda would pay for each loan.

3-year loan =

6-year loan =

(f) Write a few sentences explaining to Melinda the advantages and disadvantages of each loan. One thing you might consider is the value of the car over time.

PROBLEM SITUATION 2: PAYING FOR COLLEGE

Sandra is eligible for a subsidized Stafford loan to help her pay for college. Subsidized Stafford loans are loans available to low- and medium-income families to help pay for college. While a student is in college, the U.S. government pays the interest on the loan. So, if a student borrows $5,000, then he/she still owes only $5,000 when he/she graduates from college (or leaves early).

Note that loan repayment for a Stafford loan begins six months after graduation (or after the student has left college). That is the point when interest begins to accrue (be charged). Once repayment begins, payments are typically made monthly.

(2) Sandra is not sure whether she should take the subsidized loan. She is concerned about being able to pay it back (her other option is to work part-time and attend school part-time). Suppose her financial aid award letter says she is qualified to borrow up to $4,000 per year.

How much (in total) will Sandra owe if she borrows $4,000 per year and it takes her five years to graduate from college? Note: This total is called her “loan balance.”

(3) Typically, individuals are given a 10-year repayment period. Payments are made monthly.

(a) How many payments (total) will Sandra make during the repayment period?

(b) Pretend that no interest is charged on Sandra’s loan. If she only has to repay the principal, she takes 10 years to repay, and she repays the same amount each month, what would be the amount of each monthly payment? Round to the nearest cent.

(c) Any estimate will not be exact, but it is important to have an understanding as to why the “actual” answer is more or less than an estimate and why. Is the estimate from Question 3(b) more or less than the “actual” answer?

(4) (a) If Sandra is charged an interest rate of 6.8% per year and she takes the full 10 years to repay, how much would her monthly payments be to pay off the loan? To figure this out, use an estimation strategy. You don’t need exact calculations or formulas.

(b) How much total interest will Sandra pay over the 10 years?

(c) Do you think your group’s estimate in Question 4(b) is an overestimate or an underestimate of the actual interest Sandra will pay? Why?

(5) Use the Average Balance Method estimate from Question 4(a). Assume that Sandra has the following minimum living expenses:

Rent and utilities cost $1000/month
Food and eating out cost $300/month
Car payment and insurance cost $300/month
Entertainment, gas, and cell phone cost $200/month
Clothing and miscellaneous household expenses cost $100/month
Health Insurance costs $100/month

You should also expect that about 30% of Sandra’s “gross income” (her actual salary) will be deducted from her paycheck to cover various taxes, Medicare, and Social Security. Estimate how much Sandra would have to earn annually to be able to afford the loan payment.

(6) Do you think this is a reasonable salary for Sandra to expect to earn if she has a bachelor’s degree? Why or why not?

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

Search

Text Color

Text Size

Margin Size

Font Type