MATHEMATICAL PROBLEMS AND SOLUTUIONS
A more conservative investment account pays 3% interest. If you deposit $5 a day into this account,
how much will you have after 10 years? How much is from interest?
Answer
d = $5 the daily deposit
r = 0.03 3% annual rate
n = 365 since we’re doing daily deposits, we’ll compound daily
t = 10 we want the amount after 10 years
0.03 365×10
5[(1+ 365 ) − 1]
A = 0.03 = $21, 282.07
365
We would have deposited a total of , so $3,032.07 is from interest
$5 ⋅ 365 ⋅ 10 = $18, 250
This page titled 2.6: Annuities is shared under a CC BY-SA license and was authored, remixed, and/or
curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
9.4: Annuities by David Lippman is licensed CC BY-SA 3.0. Original source:
Section 2.5.1 Exercises
You wish to have $3000 in 2 years to buy a fancy new stereo system. How much should you
deposit each quarter into an account paying 8% compounded quarterly?
You deposit $200 each month into an account earning 3% interest compounded monthly.
How much will you have in the account in 30 years?
How much total money will you put into the account?
How much total interest will you earn?
You deposit $1000 each year into an account earning 8% compounded annually.
How much will you have in the account in 10 years?
How much total money will you put into the account?
How much total interest will you earn?
Raymond has determined he needs to have $800,000 for retirement in 30 years. His account
earns 6% interest.
How much would he need to deposit in the account each month?
How much total money will he put into the account?
How much total interest will he earn?
This page titled 2.6.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.6 Learning Objectives
Use the payout annuity formula to determine the regular withdrawal amount
Determine the amount needed in an account to allow for a given payout amount
Determine when to use the annuity formula and when to use the payout formula
In the last section you learned about annuities. In an annuity, you start with nothing, put money into
an account on a regular basis, and end up with money in your account.
In this section, we will learn about 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. The formula is derived in a similar way as we did for
savings annuities. The details are omitted here.
Payout Annuity Formula
−tn
d [1 − (1 + ) ]
0 =
( )
is the balance in the account at the beginning (starting amount, or principal).
0
is the regular withdrawal (the amount you take out each year, each month, etc.)
d
is the annual interest rate (in decimal form. Example: )
is the number of compounding periods in one year. 5% = 0.05
n
is the number of years we plan to take withdrawals
Like with annuities, the compounding frequency is not always explicitly given, but is determined by
how often you take the withdrawals.
When do you use this
Payout annuities assume that you take money from the account on a regular schedule (every month,
year, quarter, etc.) and let the rest sit there earning interest.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Example 1
After retiring, you want to be able to take $1000 every month for a total of 20 years from your
retirement account. The account earns 6% interest. How much will you need in your account when
you retire?
Solution
In this example,
d = $1000
the monthly withdrawal
r = 0.06 6% annual rate
since we’re doing monthly withdrawals, we’ll compound monthly
t = 20 since we're taking withdrawals for 20 years
We’re looking for ; how much money needs to be in the account at the beginning.
0
Putting this into the equation:
0.06 −20(12)
1000 [1 −(1 + ) ]
0 =
0.06
( )
1000 × (1 − (1.005)−240)
0 = (0.005)
1000 × (1 − 0.302)
P0 = (0.005) = $139, 600
You will need to have $139,600 in your account when you retire.
Notice that you withdrew a total of $240,000 ($1000 a month for 240 months). The difference
between what you pulled out
and what you started with is the interest earned. In this case it isin interest. $240, 000 − $139, 600 =
$100, 400
Evaluating negative exponents on your calculator
With these problems, you need to raise numbers to negative powers. Most calculators have a
separate button for negating a number that is different than the subtraction button. Some calculators
label this [(-)], some with [+/-] . The button is often near the = key or the decimal point.
If your calculator displays operations on it (typically a calculator with multiline display), to calculate
something like:
1.005[∧][(−)]240
−240 you'd type
If your calculator only shows one value at a time, then usually you hit the (-) key after a number to
negate it, so you'd hit:
1.005[yx]240[(−)] =
Give it a try - you should get −240
Example 2
You know you will have $500,000 in your account when you retire. You want to be able to take
monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8%
interest. How much will you be able to withdraw each month?
Solution
In this example,
We’re looking for d.
8% annual rate
n = 12 since we’re doing monthly withdrawals
since were taking withdrawals for 30 years
0 = $500, 000 we are beginning with $500, 000
In this case, we’re going to have to set up the equation, and solve for .
−30(12)
d [1 − (1 + ) ]
500, 000 =
)
(1 − (1.00667)−360)
500, 000 = (0.00667)
500, 000 = d(136.232)
500, 000
d = = $3670.21
You would be able to withdraw $3,670.21 each month for 30 years.
Try it Now 1
A donor gives $100,000 to a university, and specifies that it is to be used to give annual scholarships
for the next 20 years. If the university can earn 4% interest, how much can they give in scholarships
each year?
Answer
d = unknown
r = 0.04 4% annual rate
n = 1 since we’re doing annual scholarships
t = 20 since were taking withdrawals for 20 years
P0 = $100, 000 we are starting with $100, 000
0.04 −20×1
d [1 − (1 + 1 ) ]
100, 000 = 0.04
Solving for gives $7,358.18 each year that they can give in scholarships.
d
It is worth noting that usually donors instead specify that only interest is to be used for scholarship,
which makes the
original donation last indefinitely. If this donor had specified that, a year would have been
$100, 000(0.04) = $4, 000
available.
This page titled 2.7: Payout Annuities is shared under a CC BY-SA license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years
We’re looking for , the starting amount of the loan.
0
−5(2)
200 [1 −(1 + ) ]
0 =
( )
P0 = 200 (1 − (1.0025)−60)
(0.0025)
0 = 200(1 − 0.861) = $11, 120
(0.0025)
You can afford a loan.
$11, 120
You will pay a total of $12,000 ($200 per month for 60 months) to the loan company. The difference
between the amount you
pay and the amount of the loan is the interest paid. In this case, you’re paying interest total.
$12, 000 − $11, 120 = $880
2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years2.7.1: Exercises
Section 2.6.1 Exercises
You want to be able to withdraw $30,000 each year for 25 years. Your account earns 8%
interest compounded annually.
How much do you need in your account at the beginning?
How much total money will you pull out of the account?
How much of that money is interest?
How much money will I need to have at retirement so I can withdraw $60,000 a year for 20
years from an account earning 8% compounded annually?
How much do you need in your account at the beginning
How much total money will you pull out of the account?
How much of that money is interest?
You have $500,000 saved for retirement. Your account earns 6% interest compounded
monthly. How much will you be able to pull out each month, if you want to be able to take
withdrawals for 20 years?
Loren already knows that he will have $500,000 when he retires. If he sets up a payout
annuity for 30 years in an account paying 10% interest compounded monthly, how much could the
annuity provide each month?
This page titled 2.7.1: Exercises is shared under a not declared license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.7 Learning Objectives
Determine the payment on an installment loan
Determine amount that can be borrowed on an installment loan given the amount of the loan
payment
In the last section, you learned about payout annuities.
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.
One great thing about loans is that they use exactly the same formula as a payout annuity. 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.
Loans Formula
r −tn
0 = d [1 − (1 + n ) ]
r
n )
is the balance in the account at the beginning (the principal, or amount of the loan).
0
is your loan payment (your monthly payment, annual payment, etc)
is the annual interest rate in decimal form.
r
is the number of compounding periods in one year.
is the length of the loan, in years
t
Like before, the compounding frequency is not always explicitly given, but is determined by how
often you make payments.
When do you use this
The loan formula assumes that you make loan payments on a regular schedule (every month, year,
quarter, etc.) and are paying interest on the loan.
Compound interest: One deposit
Annuity: Many deposits.
Payout Annuity: Many withdrawals
Loans: Many payments
Example 1
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 loan can you
pay off with $200 per month?
Solution
In this example,
d = $200
the monthly loan payment
r = 0.03 3% annual rate
since we’re doing monthly payments, we’ll compound monthly
t = 5 since we’re making monthly payments for 5 years
Example 2
You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the
loan is for 30 years. How much will your monthly payments be?
Solution
In this example,
We’re looking for .
r = 0.06 6% annual rate
since we’re doing monthly payments, we’ll compound monthly
since we’re making monthly payments for 30 years
P0 = $140, 000 the starting loan amount
In this case, we’re going to have to set up the equation, and solve for .
d
−30(12)
d [1 − (1 + ) ]
140, 000 =
)
(1 − (1.005)−360)
140, 000 = (0.005)
140, 000 = d(166.792)
140, 000
d = = $839.37
You will make payments of $839.37 per month for 30 years.
You're paying a total of to the loan company: per month for 360 months. You are paying a total of
$302, 173.20 $839.37
in interest over the life of the loan.
$302, 173.20 − $140, 000 = $162, 173.20
Try it Now 1
Janine bought $3,000 of new furniture on credit. Because her credit score isn’t very good, the store is
charging her a fairly high interest rate on the loan: 16%. If she agreed to pay off the furniture over 2
years, how much will she have to pay each month?
Answer
d = unknown
r = 0.16 16% annual rate
n = 12 since we’re doing monthly payments, we’ll compound monthly
t = 2 2 year to repay
0 =3,000 the starting loan amount $3,000 loan
0.16 −2×12
d [1 − (1 + 12 ) ]
3, 000 = 0.16
Solving for gives 12 as monthly payments.
d $146.89
In total, she will pay to the store, meaning she will pay
in interest over the two years.
$3, 525.36 $525.36
This page titled 2.8: Installment Loans is shared under a CC BY-SA license and was authored, remixed,
and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.8.1: Exercises
Section 2.7.1 Exercises
You can afford a $700 per month mortgage payment. You’ve found a 30 year loan at 5%
interest.
How big of a loan can you afford?
What would be the total payments made on the loan?
How much of that money is interest?
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?
What would be the total payments made on the loan?
How much of that money 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?
You decide finance a $12,000 car at 3% compounded monthly for 4 years. What will your
monthly payments be? How much interest will you pay over the life of the loan?
You want to buy a $200,000 home. You plan to pay 10% as a down payment, and take out a
30 year loan for the rest.
How much is the loan amount going to be?
What will your monthly payments be if the interest rate is 5%?
What will your monthly payments be if the interest rate is 6%?
Lynn bought a $300,000 house, paying 10% down, and financing the rest at 6% interest for
30 years.
Find her monthly payments.
How much interest will she pay over the life of the loan?
Emile bought a car for $24,000 three years ago. The loan had a 5 year term at 3% interest
rate, making monthly payments. How much does he still owe on the car?
A friend bought a house 15 years ago, taking out a $120,000 mortgage at 6% for 30 years,
making monthly payments. How much does she still owe on the mortgage?
