MATHEMATICS AND STATISTICS
You know that you will need $40,000 for your child’s education in 18 years. If your account earns 4%
compounded quarterly, how much would you need to deposit now to reach your goal?
Solution
We’re looking for .
P0
r = 0.04 4%
4 quarters in 1 year
t = 18 Since we know the balance in 18 years
A = $40, 000 The amount we have in 18 years
In this case, we’re going to have to set up the equation, and solve for .
0
0.04 18×4
40000 = 0 (1 + )
40000 = 0 (2.0471)
0 = 2.0471 = $19539.84
So you would need to deposit $19,539.84 now to have $40,000 in 18 years.
Rounding
It is important to be very careful about rounding when calculating things with exponents. In general,
you want to keep as many decimals during calculations as you can. Be sure to keep at least 3
significant digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually
give you a “close enough” answer, but keeping more digits is always better.
Example 3
To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest
compounded monthly for 30 years.
Solution
0 = $1000 the initial deposit
5%
n = 12 12 months in 1 year
t = 30 since we’re looking for the amount after 30 years
If we first compute , we find
0.00416666666667
Here is the effect of rounding this to different values:
r/n rounded to: Gives 30 to be:
$4208.59 $259.15
$4521.45 $53.71
0.00417$4473.09 $5.35
$4468.28 $0.54
0.0041667 $4467.80 $0.06
no rounding $4467.74
If you’re working in a bank, of course you wouldn’t round at all. For our purposes, the answer we got
by rounding to 0.00417, three significant digits, is close enough - $5 off of $4500 isn’t too bad.
Certainly keeping that fourth decimal place wouldn’t have hurt.
Using your calculator
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
When do you use this
Annuities assume that you put money in the account on a regular schedule (every month, year,
quarter, etc.) and let it sit there earning interest.
Compound interest assumes that you put money in the account once and let it sit there earning
interest.
Compound interest: One deposit
Annuity: Many deposits.
Example 1
A traditional individual retirement account (IRA) is a special type of retirement account in which the
money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month
into an IRA earning 6% interest, how much will you have in the account after 20 years?
Solution
In this example,
d = $100 the monthly deposit
r = 0.06 6% annual rate
since we’re doing monthly deposits, we’ll compound monthly
we want the amount after 20 years
Putting this into the equation:
0.06 20(12)
100 [(1 + ) − 1]
=
)
100 ((1.005)240 −1)
A = (0.005)
A = 100(3.310 −1)
(0.005)
A = 100(2.310) = $46200
(0.005)
The account will grow to $46,200 after 20 years.
Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months). The
difference between what you
end up with and how much you put in is the interest earned. In this case it is . $46, 200
−$24, 000 = $22, 200
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
In many cases, you can avoid rounding completely by how you enter things in your calculator. For
example, in the example above, we needed to calculate
12×30
A = 1000(1 + )
We can quickly calculate , giving 0.05 360.
12 ×30 = 360 A = 1000(1 + 12 )
Now we can use the calculator.
Type this
0.05[÷]12[=] 0.00416666666667
[+]11[=]
[yx ]360[=]
[×]1000[=]
4467.74431400613
Using your calculator continued
The previous steps were assuming you have a “one operation at a time” calculator; a more advanced
calculator will often allow you to type in the entire expression to be evaluated. If you have a
calculator like this, you will probably just need to enter:
1000
( 1
[×] [+]
0.05 12 )
[÷] [y
x]
360 .
[=]
This page titled 2.5: Compound Interest is shared under a CC BY-SA license and was authored,
remixed, and/or curated by Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia.
2.5.1: Exercises
Section 2.4.1 Exercises
You deposit $300 in an account earning 5% interest compounded annually. How much will
you have in the account in 10 years?
How much will $1000 deposited in an account earning 7% interest compounded annually be
worth in 20 years?
You deposit $2000 in an account earning 3% interest compounded monthly.
How much will you have in the account in 20 years?
How much interest will you earn?
You deposit $10,000 in an account earning 4% interest compounded monthly.
How much will you have in the account in 25 years?
How much interest will you earn?
How much would you need to deposit in an account now in order to have $6,000 in the
account in 8 years? Assume the account earns 6% interest compounded monthly.
How much would you need to deposit in an account now in order to have $20,000 in the
account in 4 years? Assume the account earns 5% interest compounded weekly.
This page titled 2.5.1: 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: Annuities
2.5 Learning Objectives
Use the annuity formula to determine the balance in an account
Use the annuity formula to determine what regular deposits are needed to achieve a goal
Determine when to use the compound interest formula and when to use the annuity formula
For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the
future by depositing a smaller amount of money from each paycheck into the bank. This idea is called
a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings
annuities.
An annuity can be described recursively in a fairly simple way. Recall that basic compound interest
follows from the relationship
m = (1 + ) m−1
For a savings annuity, we simply need to add a deposit, , to the account with each compounding
period:
d
m = (1 + ) m−1 + d
Taking this equation from recursive form to explicit form is a bit trickier than with compound interest.
It will be easiest to see by working with an example rather than working in general.
Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the
account is compounded with the same frequency as we make deposits unless stated otherwise. In this
example:
(6%)
r = 0.06
(12 compounds/deposits per year)
n = 12
(our deposit per month)
= $100
Writing out the recursive equation gives
m = (1 + ) m−1 +100 = (1.005) m−1 +100
Assuming we start with an empty account, we can begin using this relationship:
0 = 0
1 = (1.005) 0 +100 = 100
2 = (1.005) 1 +100 = (1.005)(100) +100 = 100(1.005) +100
= (1.005) 2 +100 = (1.005)(100(1.005) +100) +100 = 100(1.005)2 +100(1.005) +100
Continuing this pattern, after deposits, we’d have saved:
m
= 100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100
In other words, after months, the first deposit will have earned compound interest for months. The
second deposit will
m − 1
have earned interest for months. Last months deposit would have earned only one month worth of
interest. The most recent
− 2
deposit will have earned no interest yet.
This equation leaves a lot to be desired, though – it doesn’t make calculating the ending balance any
easier! To simplify things, multiply both sides of the equation by 1.005:
1.005 m = 1.005 (100(1.005)m−1 +100(1.005)m−2 + ⋯ +100(1.005) +100)
Distributing on the right side of the equation gives
1.005 m = 100(1.005)m +100(1.005)m−1 + ⋯ +100(1.005)2 +100(1.005) Now we’ll line this up with
like terms from our original equation, and subtract each side
1.005Pm = 100(1.005)m +100(1.005)m−1 + ⋯ +
100(1.005)
m = 100(1.005)m−1 + ⋯ + 100(1.005) +100
Almost all the terms cancel on the right hand side when we subtract, leaving
1.005Pm − Pm = 100(1.005)m − 100
Solving for m
0.005 m = 100 ((1.005)m − 1)
100 ((1.005)m − 1)
m 0.005
=
Replacing months with , where is measured in years, gives
100 ((1.005)12t − 1)
t 0.005
=
Recall 0.005 was and 100 was the deposit . 12 was , the number
of deposit each year. Generalizing this result, we get the
k
saving annuity formula.
Annuity Formula
tn
d [(1 + )−1]
A =
( )
is the balance in the account after t years.
is the regular deposit (the amount you deposit each year, each month, etc.)
is the annual interest rate in decimal form.
is the number of compounding periods in one year.
n
If the compounding frequency is not explicitly stated, assume there are the same number of
compounds in a year as there are deposits made in a year.
For example, if the compounding frequency isn’t stated:
If you make your deposits every month, use monthly compounding, .
If you make your deposits every year, use yearly compounding, .
If you make your deposits every quarter, use quarterly compounding, .
Etc. n = 4
Example 2
You want to have $200,000 in your account when you retire in 30 years. Your retirement account
earns 8% interest. How much do you need to deposit each month to meet your retirement goal?
Solution
In this example,
We’re looking for .
d
r = 0.08 8% annual rate
since we’re doing monthly deposits, we’ll compound monthly
30 years
A = $200, 000 The amount we want to have in 30 years
In this case, we’re going to have to set up the equation, and solve for .
30(12)
d [(1 + ) − 1]
200, 000 =
0.08
)
((1.00667)360 −1)
200, 000 = (0.00667)
200, 000 = d(1491.57)
200, 000
d = 1491.57 = $134.09
So you would need to deposit each month to have in 30 years if your account earns 8% interest
$134.09$200, 000
