Compound Interest and derivatives of exponential
functions
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
I
After 1-period:
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
r
After 1-period: P 1 +
m
I
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
r
After 1-period: P 1 +
m
I
I
After 1-year
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
r
After 1-period: P 1 +
m
r m
After 1-year P 1 +
m
I
I
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
r
After 1-period: P 1 +
m
r m
After 1-year P 1 +
m
I
I
I
After t-years
Compound Interest and derivatives of exponential functions
Compound Interest (interest compounded periodically)
I
Start with P and an annual rate r as before.
I
Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
I
Apply the Simple Interest model to each conversion period
using the rate r /m.
I
Accumulated Amount after k-periods = Principal at the
beginning of the (k + 1)-st period.
r
After 1-period: P 1 +
m
r m
After 1-year P 1 +
m
r mt
After t-years P 1 +
m
I
I
I
Compound Interest and derivatives of exponential functions
Example
Compound Interest and derivatives of exponential functions
Example
If r = 0.02, what results in a higher Accumulated Amount,
compounding bi-annually or quarterly?
Compound Interest and derivatives of exponential functions
Example
If r = 0.02, what results in a higher Accumulated Amount,
compounding bi-annually or quarterly?
0.02 mt
P →P 1+
m
Compound Interest and derivatives of exponential functions
Example
If r = 0.02, what results in a higher Accumulated Amount,
compounding bi-annually or quarterly?
0.02 mt
P →P 1+
m
0.02 m
Graph of f (m) = 1 +
m
Compound Interest and derivatives of exponential functions
m
0.02
Graph of f (m) = 1 +
m
Compound Interest and derivatives of exponential functions
m
0.02
Graph of f (m) = 1 +
m
Compound Interest and derivatives of exponential functions
Example
Compound Interest and derivatives of exponential functions
Example
I
f (m) is increasing
Compound Interest and derivatives of exponential functions
Example
I
f (m) is increasing
I
f (4) > f (2)
Compound Interest and derivatives of exponential functions
Example
I
f (m) is increasing
I
f (4) > f (2)
I
Quarterly is better than bi-annually
Compound Interest and derivatives of exponential functions
Example
I
f (m) is increasing
I
f (4) > f (2)
I
Quarterly is better than bi-annually
I
In general one obtains a marginally better return if one
compounds interest more frequently.
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
I
Present Value of an investment
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
I
Present Value of an investment
r −mt
P =A 1+
m
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
I
Present Value of an investment
r −mt
P =A 1+
m
I
Effective Interest
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
I
Present Value of an investment
r −mt
P =A 1+
m
I
Effective Interest
r m
reff = 1 +
−1
m
Compound Interest and derivatives of exponential functions
Present Value and Effective Interest
I
Present Value of an investment
r −mt
P =A 1+
m
I
Effective Interest
r m
reff = 1 +
−1
m
I
A = P(1 + reff )t
Compound Interest and derivatives of exponential functions
Example
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years?
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years? What about 10 years?
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years? What about 10 years?
I
5 years:
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years? What about 10 years?
I
5 years: 2340.26
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years? What about 10 years?
I
5 years: 2340.26
I
10 years
Compound Interest and derivatives of exponential functions
Example
If interest is compounded quarterly at at rate of 5%, what is the
principal needed in order for the Accumulated Amount to be $3000
over a period of 5 years? What about 10 years?
I
5 years: 2340.26
I
10 years 1825.24.
Compound Interest and derivatives of exponential functions
Interest compounded continuously
Compound Interest and derivatives of exponential functions
Interest compounded continuously
I
Compound “more and more often”.
Compound Interest and derivatives of exponential functions
Interest compounded continuously
I
I
Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
Compound Interest and derivatives of exponential functions
Interest compounded continuously
I
I
I
Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
r mt
= e rt .
lim 1 +
m→∞
m
Compound Interest and derivatives of exponential functions
Interest compounded continuously
I
I
I
I
Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
r mt
= e rt .
lim 1 +
m→∞
m
Model of interest compounded continuously
Compound Interest and derivatives of exponential functions
Interest compounded continuously
I
I
I
I
Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
r mt
= e rt .
lim 1 +
m→∞
m
Model of interest compounded continuously
A = Pe rt
Compound Interest and derivatives of exponential functions
Example
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years?
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years:
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compare to the quarterly case
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compare to the quarterly case
I
5 years:
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compare to the quarterly case
I
5 years: 2340.26
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compare to the quarterly case
I
5 years: 2340.26
I
10 years
Compound Interest and derivatives of exponential functions
Example
If interest is compounded continuously at at rate of 5%, what is
the principal needed in order for the Accumulated Amount to be
$3000 over a period of 5 years? What about 10 years?
I
5 years: 2336.402
I
10 years 1819.592.
Compare to the quarterly case
I
5 years: 2340.26
I
10 years 1825.24.
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
I
What is lim
h→0
eh − 1
?
h
Compound Interest and derivatives of exponential functions
Using R
Compound Interest and derivatives of exponential functions
Using R
Compound Interest and derivatives of exponential functions
Using R
Compound Interest and derivatives of exponential functions
Using R
Compound Interest and derivatives of exponential functions
Graph of x(e 1/x − 1)
Compound Interest and derivatives of exponential functions
1.0
1.1
1.2
1.3
p
1.4
1.5
1.6
1.7
Graph of x(e 1/x − 1)
0
200
400
600
800
1000
x
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
I
What is lim
h→0
eh − 1
?
h
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
I
What is lim
h→0
I
eh − 1
?
h
Ans: The limit equals 1
Compound Interest and derivatives of exponential functions
Derivative of Exponential Functions
d x
e = lim
h→0
dx
e x+h − e x
h
eh − 1
h
d x
e = e x lim
h→0
dx
I
What is lim
h→0
I
eh − 1
?
h
Ans: The limit equals 1 =⇒
d x
e = ex .
dx
Compound Interest and derivatives of exponential functions