Chapter 21. Savings Model
Arithmetic Growth and Simple
Interest
What’s important
• Bring your calculator. Practice the correct way
to use the calculator.
• Remember the formula and terminology.
• Write your reasoning( formula you use) in
details when you take any test. Answer
without any reasoning will get zero point.
• I won’t allow you to share calculator with
others.
• You cannot use your cell phone as a calculator.
Terminology
Principal – The initial
balance of the savings
account.
Example:
You are opening a savings
account today. You
deposited $50. Then, $50
is principal.
Interest – Money earned
on a savings account or a
loan.
Example:
If at the end of the year,
your savings account has
$51, then the interest is
$1.
Simple Interest
The method of paying interest only on the
initial balance in an account, not on any
accrued interest.
Simple Interest
Example
You deposited $1000. An annual simple interest
rate is 1%. Then,
Your Account Balance:
After a year: 0.01 X 1000 + 1000 = 1010.
After two years: 2X(0.01X1000)+1000=1020
After n years: n X (0.01X1000)+1000
Simple Interest
Example
You deposited $1000. An annual simple interest
rate is 1%.
That is, at the end of each year, you get $10 as
an interest no matter how many years you
deposit.
Simple Interest
For a principal P and an annual rate of interest r,
after t years,
Interest 𝐼 = 𝑃 × 𝑟 × 𝑡
Total amount
𝐴= 𝑃+𝐼 = 𝑃+ 𝑃 × 𝑟 × 𝑡
= 𝑃 × 1 + 𝑟 × 𝑡 = 𝑃(1 + 𝑟𝑡)
Use of Simple Interest
• Private loans between individuals (easy to
calculate)
• Commercial loans for less than a year (Doesn’t
make much differences from the compound
interest)
Example
Let’s suppose that you have exhausted the amount
that you can borrow under federal loan programs
and need a private direct student loan for $10,000.
National City Corporation quoted a rate in May
2008 of 5.7% for the 2007-08 school year. It offers
an interest-only repayment option, under which
you make monthly interest payments while you are
in school and pay on the principal only after
graduation. Under this plan, National City earns
simple interest from you while you are in school.
How much monthly interest would you pay for such
a $10,000 loan?
Answer
• Student loan= $10,000.
• Annual rate= Simple interest 5.7%
• Monthly interest= 𝑃𝑟𝑡 = $10,000 ×
1
0.057 ×
= $47.50
12
Arithmetic Growth(Linear Growth)
Growth by a constant amount in each time
period
Example: simple interest
Geometric Growth
and Compound Interest
Compound Interest
Interest that is paid on both the
original principal and accumulated
interest
Compound Interest
Suppose that you deposited $10,000 into the
savings account with annual compound interest
rate of 10%.
After
….year
later
1
2
Interest
Savings total
10% × 10,000
= 0.1 × 10,000
= 1000
10,000 + 1,000
= 11,000
10% × 11,000
= 0.1 × 11,000
= 1,100
11,000 + 1,100
= 12,100
After
….year
later
Interest
Savings total
Compound Interest
1
10% × 10,000
= 0.1 × 10,000
= 1000
10,000 + 1,000
= 11,000
2
10% × 11,000
= 0.1 × 11,000
= 1,100
11,000 + 1,100
= 12,100
3
10% × 12,100
= 0.1 × 12,100
= 1,210
12,100 + 1,210
= 13,310
Nominal Rate
Any state rate of interest for a specified length
of time. Nominal rate have broad( vague )
meaning. It doesn’t indicate or take into account
whether or how often interest is compounded.
Annual Interest Rate of 10%
(Nominal Rate)
That might mean
• Annual interest rate of 10% compounded
quarterly.
( every three months, 4 times a year )
• Annual interest rate of 10% with simple
interest……
Annual interest rate of 10% compounded
quarterly with $1,000 Principal
It is compounded every three months, that is, 4
times a year.
Annual interest rate of 10% in this case means
2.5% quarterly.
Annual interest rate of 10% compounded
quarterly with $1,000 Principal
After
0
month
3
months
6
months
9
months
12
months
Balance
1,000
0.025 × 1,000 + 1,000 = 1,025
0.025 × 1,025 + 1,025 = 1,050.63
0.025 × 1,050.63 + 1,050.63 = 1,076.90
0.025 × 1,076.90 + 1,076.90 = 1,103.82
Annual interest rate of 10%
with simple interest
After a year later,
0.1 × 1,000 + 1,000 = 1,100
Annual interest rate of 10%
with simple interest vs. compound interest
with $1,000 a year later
Simple
interest
Quarterly
Compound
interest
A year later
Interest amount
Effective interest
compared to the
principal
$1,100
100
10%
$1,103.82 103.82
10.382%
Effective Rate
and Annual Percentage Rate
If we know the effective rate and annual percentage rate for
the compound interest, then it is easier to know how much
you earn interest.
Effective rate: the rate of simple interest that would realize
exactly as much interest over the same length of time.
Annual percentage yield(APY): Effective rate for a year
the money after a year − the principal
× 100 %
Principal
Annual interest amount
=
× 100(%)
Principal
Rate per Compounding Period
Terminology
Meaning
Number of
compounding per
year
Annually
Bi-annually
Every year
1
Twice a year 2
Quarterly
Monthly
Four times a 4
year
Every month 12
Daily
Every day
365
Rate when nominal
annual rate is r.
𝑟
𝑟
2
𝑟
4
𝑟
12
𝑟
365
Rate per Compounding Period
Terminology
Annually
Bi-annually
Quarterly
Monthly
Daily
Periodic rate when nominal annual rate
is 12%
12%
12
= 6%
2
12
= 3%
4
12
= 1%
12
12
= 0.03%
365
Nominal annual interest rate of 10%
with Principal $1000
After…… years
Balance in dollars
0
1000
1
1000 + 1000 × 0.1 = 1000(1 + 0.1)
2
1000 1 + 0.1 + 1000 × 1 + 0.1 × 0.1
= 1000 1 + 0.1 1 + 0.1 = 1000(1 + 0.1)2
3
n
1000(1 + 0.1)2 +1000(1 + 0.1)2 × 0.1 =
1000(1 + 0.1)2 (1+0.1)=1000(1 + 0.1)3
1000(1 + 0.1)𝑛
Nominal annual interest rate of 10%
with Principal $1000
1000(1 + 0.1)𝑛
Number of years
Principal
Interest rate per year
Number of
compounding
Generalization
Interest rate per
compounding
period
Compound Interest Formula
An initial principal 𝑃 in an account that pays interest
at a periodic interest rate 𝑖 per compounding period
grows after 𝑛 compounding periods to
𝒏
𝑨 = 𝑷(𝟏 + 𝒊) = 𝑷(𝟏 +
𝒓 𝒏
)
𝒎
Remark:
𝑟
,
𝑚
Periodic interest rate 𝑖 = where r is an annual
rate of interest and m is the number of
compounding per year.
Compound Interest Formula for an
annual nominal rate r, t years later
𝒏
𝑨 = 𝑷(𝟏 + 𝒊) = 𝑷(𝟏 +
𝒓 𝒏
𝒓 𝒎𝒕
) = 𝑷(𝟏 + )
𝒎
𝒎
𝑡: Number of years
m: Number of compounding per year
r: nominal annual interest rate
n: Number of compounding
Geometric Growth
( Exponential Growth )
Growth proportional to the amount
present( Not to the principal )
Example
Suppose that you have a principal of P=$1000
invested at 10% nominal interest per year with
annual compounding. How much balance will
you have 10 years later?
Answer
Suppose that you have a principal of P=$1000 invested
at 10% nominal interest per year with annual
compounding. How much balance will you have 10
years later?
$1000(1 + 0.10)1×10 = $1000 ∙ 1.1010 = $2593.74
Use of Calculator( Scientific )
(1) Calculate what are in
the parenthesis.
(2) Calculate the exponent
and then raise to the
power.
(3) Multiply by the
principal.
However, there are many
other correct ways to use
the calculator.
Example
Suppose that you have a principal of P=$1000
invested at 10% nominal interest per year with
quarterly compounding. How much balance will
you have 10 years later?
Answer
Suppose that you have a principal of P=$1000 invested
at 10% nominal interest per year with quarterly
compounding. How much balance will you have 10
years later?
$1000(1 +
0.10 4×10
)
=$1000(1.025)40 =
4
$2685.06
Example
Suppose that you have a principal of P=$1000
invested at 10% nominal interest per year with
monthly compounding. How much balance will
you have 10 years later?
Answer
Suppose that you have a principal of P=$1000 invested at 10%
nominal interest per year with monthly compounding. How
much balance will you have 10 years later?
0.10 12×10
$1000(1 +
)
= $1000(1.0083)120 = $2707.04
12
Basic Useful Formula
1
𝑛
1
𝑛
• If 𝑎 = 𝑏, then 𝑎 = 𝑏 .
• (𝑎𝑚 )𝑛 = 𝑎𝑚𝑛
• (𝑎 + 𝑏)𝑛 ≠ 𝑎𝑛 + 𝑏 𝑛
Example
A man borrowed $29,000 for two years under
simple interest. At the end of the two years his
balance due was $31,900. What annual simple
interest rate did he pay?
Answer
A man borrowed $29,000 for two years under simple interest. At
the end of the two years his balance due was $31,900. What
annual simple interest rate did he pay?
5%
Example
Suppose you invest $6000 and would like your
investment to grow to $8000 in five years. What
interest rate, compounded monthly, would you
have to earn in order for this to happen?
Answer
Suppose you invest $6000 and would like your investment to
grow to $8000 in five years. What interest rate, compounded
monthly, would you have to earn in order for this to happen?
5.77%
Group Activity
You have $3500 that you invest at 7% simple
interest. How long will it take for your balance to
reach $4235?
Answer
You have $3500 that you invest at 7% simple interest. How long will it take for
your balance to reach $4235?
Three years
Group Activity
Merrie borrowed $1000 from her parents,
agreeing to pay them back when she graduated
from college in five years. If she paid interest
compounded quarterly at 5%, how much would
she owe at the end of the five years?
Answer
Merrie borrowed $1000 from her parents, agreeing to pay them back when she
graduated from college in five years. If she paid interest compounded quarterly at 5%,
how much would she owe at the end of the five years?
$1282
Group Activity
Merrie borrowed $500 from her parents,
agreeing to pay them back when she graduated
from college in four years. If she paid interest
compounded daily at 16%, how much would she
owe at the end of the four years?
Answer
Merrie borrowed $500 from her parents, agreeing to pay them back when she
graduated from college in four years. If she paid interest compounded daily at 16%,
how much would she owe at the end of the four years?
$948
Effective Rate
What percent of interest do you earn after n
compounding period compared to the principal?
Effective rate:
(𝟏 + 𝒊)𝒏 −𝟏
the money after − the principal
=
𝑷𝒓𝒊𝒏𝒄𝒊𝒑𝒂𝒍
𝑛: number of compounding
𝑟
i: interest rate per compounding period ( )
𝑚
r: nominal interest rate
Annual Percentage Yield (APY)
What percent of interest do you get after a year
compared to the principal?(i.e. as an annual
simple interest rate)
Effective rate = (𝟏 +
𝒓 𝒎
) −𝟏
𝒎
m: number of compounding per year
APY for 4% of nominal interest
with $10,000 deposit( Principal)
Account type
Number of
Balance after a year
compounding
Interest
amount
% of increase
after a year
(APY)
Simple interest
1
$10400
400
4%
Compounding
bi-annually
2
$10404=
404
4.04%(=
404
)
0.04 2
10000(1 +
)
2
10000
Compounding
quarterly
4
$10406.04 from
10406.0401
406.04 4.0604%
Compounding
monthly
12
$10407.42 from
10407.41543
407.42 4.0742%
Example
With a nominal annual rate of 6% compounded
monthly, what is the APY?
Answer
With a nominal annual rate of 6% compounded monthly, what is
the APY?
0.06 12
(1 +
) −1 = 0.0617 = 6.17%
12
Example
Suppose that the monthly statement from the
fund reports a beginning balance(P) of $7373.93
and a closing balance(A) of $7382.59 for 28
days(n). What is the effective daily rate?
Example
Suppose that the monthly statement from the fund reports a beginning
balance(P) of $7373.93 and a closing balance(A) of $7382.59 for 28 days(n).
What is the effective daily rate?
Effective rate= (𝟏 + 𝒊)𝒏 −𝟏
𝑛: number of compounding, i: interest rate per
𝑟
compounding period ( )
𝑚
Difference of money after 28 days
= Principal × 𝟏 + 𝒊 𝒏 − 𝟏
What is i?
Example
Beginning balance(P): $7373.93, Closing
balance(A): $7382.59 for 28 days(n).
Difference of money after 28 days = Principal × 𝟏 + 𝒊 𝒏 − 𝟏
𝟕𝟑𝟖𝟐. 𝟓𝟗 − 𝟕𝟑𝟕𝟑. 𝟗𝟑 = 𝟕𝟑𝟕𝟑. 𝟗𝟑 × [( 𝟏 + 𝒊 𝟐𝟖 − 𝟏]
𝟕𝟑𝟖𝟐. 𝟓𝟗 = 𝟕𝟑𝟕𝟑. 𝟗𝟑 × (𝟏 + 𝒊)𝟐𝟖
𝟕𝟑𝟖𝟐. 𝟓𝟗
= (𝟏 + 𝒊)𝟐𝟖
𝟕𝟑𝟕𝟑. 𝟗𝟑
(1 + 𝑖)28 = 1.0011744077
1
28
1 + 𝑖 = (1.001174408) =1.000041919
𝑖 = 0.000041919
So, the daily effective rate is 0.004919%.
Example
Angela invests in a savings account that pays 4%
interest compounded monthly. What is the APY
for this account?
Answer
Angela invests in a savings account that pays 4% interest
compounded monthly. What is the APY for this account?
4.07%
Example
You have $4300 that you invest at 5% simple
interest. How long will it take for your balance to
reach $7525?
Example
You have $4300 that you invest at 5% simple interest. How long
will it take for your balance to reach $7525?
15 years
Example
Suppose you invest in an account that pays 5%
interest, compounded quarterly. You would like
your investment to grow to $5000 in 16 years.
How much would you have to invest in order for
this to happen?
Answer
Suppose you invest in an account that pays 5% interest,
compounded quarterly. You would like your investment to grow
to $5000 in 16 years. How much would you have to invest in
order for this to happen?
$2258
Simple Interest Versus Compound Interest
Continuous Compounding
𝐴 = 𝑃𝑒 𝑟𝑡
About Number e
• 𝑒 ≈ 2.71828
1 𝑚
+ )
𝑚
• The number (1
approaches when m
gets larger and larger.
Continuous Compounding
Continuous compounding is the method of
calculating interest in which the amount of
interest is what compound interest tends
toward with more and more frequent
compounding.
Continuous Interest Formula
𝐴 = 𝑃𝑒 𝑟𝑡
A: Balance after t years when a principal P is
continuously compounded
r: nominal annual rate
Example
What is the balance after 1 year if the principal
is $1000 and 10% continuous compounding?
Example
What is the balance after 1 year if the principal is $1000 and 10% continuous
compounding?
1000 ∙ 𝑒 0.10 = $1105.17
Example
What is the balance after 5 years if the principal
is $1000 and 10% continuous compounding?
Example
What is the balance after 5 years if the principal is $1000 and 10% continuous
compounding?
1000 ∙ 𝑒 (5)∙(0.10) = 1648.72