2.4: Installments
- Page ID
- 56770
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In this section, you will learn to:
- Use technology to
- Determine the payment schedule of a payout annuity (loan);
- Identify the amount of interest paid on each payment;
- Identify the amount paid on the balance of the payout annuity (loan);
- Identify the remaining balance of the payout annuity (loan);
- Identify the loan amount that is affordable based on a fixed budget amount;
So that they can analyze financial scenarios to make informed decisions.
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a Savings Annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.
There is also a variation called a Payout Annuity. With a payout annuity, you start with money in the account, and pull money out of the account on a regular basis. Any remaining money in the account earns interest. After a fixed amount of time, the account will end up empty. Payout annuities are typically used after retirement. Perhaps you have saved $500,000 for retirement, and want to take money out of the account each month to live on. You want the money to last you 20 years. This is a payout annuity.
Annuities assume that money is transferred on a regular schedule (every month, year, quarter, etc.) and let the amount sit there earning interest.
Compound interest assumes that you put money in the account once and let it sit there earning interest.
Compound interest: One deposit
Savings Annuity: Many deposits.
Payout Annuity: Many withdrawals
In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.
Loans are just payout annuities! To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests $10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.
With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.
To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will not have paid off $12,000 of the loan balance.
Amortization Calculators use the interested rates, compounding periods, time, and principal amount to determine the amount of interest and principal paid, as well as the remaining balance.
For example, suppose you had a $12,000 loan with 2.9% interest (compounded monthly) and you paid $1000 each month, you would owe an additional $191.87 after one year. See the payment schedule below:
| Month | Starting Balance | You Paid | Interest | Principal | Ending Balance | Total Interest |
|---|---|---|---|---|---|---|
| 1 | $12,000.00 | $1,000.00 | $29.00 | $971.00 | $11,029.00 | $29.00 |
| 2 | $11,029.00 | $1,000.00 | $26.65 | $973.35 | $10,055.65 | $55.65 |
| 3 | $10,055.65 | $1,000.00 | $24.30 | $975.70 | $9,079.95 | $79.95 |
| 4 | $9,079.95 | $1,000.00 | $21.94 | $978.06 | $8,101.90 | $101.90 |
| 5 | $8,101.90 | $1,000.00 | $19.58 | $980.42 | $7,121.48 | $121.48 |
| 6 | $7,121.48 | $1,000.00 | $17.21 | $982.79 | $6,138.69 | $138.69 |
| 7 | $6,138.69 | $1,000.00 | $14.84 | $985.16 | $5,153.52 | $153.52 |
| 8 | $5,153.52 | $1,000.00 | $12.45 | $987.55 | $4,165.98 | $165.98 |
| 9 | $4,165.98 | $1,000.00 | $10.07 | $989.93 | $3,176.04 | $176.04 |
| 10 | $3,176.04 | $1,000.00 | $7.68 | $992.32 | $2,183.72 | $183.72 |
| 11 | $2,183.72 | $1,000.00 | $5.28 | $994.72 | $1,189.00 | $189.00 |
| 12 | $1,189.00 | $1,000.00 | $2.87 | $997.13 | $191.87 | $191.87 |
Use this Amortization Calculator (Known Loan Amount) to find the minimum monthly payment on a $20,000 car loan with 2.9% interest over 6 years.
Solution
After entering a loan amount as 20000, the interest rate as .029, and the term in years as 6, you should receive a monthly payment of $302.98
Use the calculator linked above to solve the following problem:
This information can be truly eye-opening! The amount of interest paid over the life of these loans increases the price of the object you're buying by much more than expected. Knowing how to calculate the payment on a loan is important, but many times the question is reversed; that is, knowing how much your monthly budget will allow, what loan can you afford? Can you use the Amortization Calculator above to determine the loan amount you can afford with $300 a month payments for three years? What about 6 years? Although it works, guessing and checking the principal loan amount is inefficient.
Use this Amortization Calculator (Known Monthly Payment) to find the loan amount with a $500 monthly payment on a loan with 2.9% interest over 6 years.
Solution
After entering a payment amount as 500, the interest rate as .029, and the term in years as 6, you should receive a maximum loan amount of $33005.52
Use the calculator linked above to solve the following problem:
As you consider your financial future (i.e student loans, car loans, or mortgages) be sure to use these tools to help you make informed decisions about what you can afford!