Compound Interest
Compound Interest
Simple Interest
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
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Simple interest model: after t years, P increases by Prt
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
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Simple interest model: after t years, P increases by Prt
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Interest earned: Prt
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
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Simple interest model: after t years, P increases by Prt
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Interest earned: Prt
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Accumulated Amount:
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
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Simple interest model: after t years, P increases by Prt
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Interest earned: Prt
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Accumulated Amount: Principal + Interest earned after t
years
Compound Interest
Simple Interest
Determine how the value of an initial capital grows over time if
one assumes that it earns interest at a fixed rate
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P Principal (Initial capital)
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r interest rate (nominal rate, stated rate)
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Simple interest model: after t years, P increases by Prt
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Interest earned: Prt
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Accumulated Amount: Principal + Interest earned after t
years
P → P + Prt = P(1 + rt).
Compound Interest
Compound Interest: (Annual Interest)
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year:
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ).
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
P(1 + r )
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
P(1 + r ) →
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
P(1 + r ) → P(1 + r )(1 + r ) = P(1 + r )2
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
P(1 + r ) → P(1 + r )(1 + r ) = P(1 + r )2
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Accumulated amount by the end of year t
Compound Interest
Compound Interest: (Annual Interest)
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As before, we have a principal P and an interest rate
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Accumulated Amount after one year: P + Pr = P(1 + r )
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Annually compound interest model: The Accumulated
Amount after the first year is the principal of the investment
for the second year.
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Start year 2 with a principal equal to A(1 + r ). By the end of
year 2 we have
P(1 + r ) → P(1 + r )(1 + r ) = P(1 + r )2
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Accumulated amount by the end of year t
A = P(1 + r )t
Compound Interest
Example
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
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Find an expression for the Accumulated amount for the
annually compound interest model after t years
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
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Find an expression for the Accumulated amount for the
annually compound interest model after t years
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After how many years do we see an Accumulated Amount
equivalent to increasing the principal by 30%
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
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Find an expression for the Accumulated amount for the
annually compound interest model after t years
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After how many years do we see an Accumulated Amount
equivalent to increasing the principal by 30% (i.e. after how
many years do we have that A > P + 0.3P)
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
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Find an expression for the Accumulated amount for the
annually compound interest model after t years
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After how many years do we see an Accumulated Amount
equivalent to increasing the principal by 30% (i.e. after how
many years do we have that A > P + 0.3P)
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Ans: t > 3.409053
Compound Interest
Example
Given a principal P and a nominal rate of 8 percent
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Find an expression for the Accumulated amount for the
annually compound interest model after t years
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After how many years do we see an Accumulated Amount
equivalent to increasing the principal by 30% (i.e. after how
many years do we have that A > P + 0.3P)
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Ans: t > 3.409053
A
After 4 years
= 1.360489
P
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Compound Interest
Compound Interest (interest compounded periodically)
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
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After 1-period:
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
r
After 1-period: A → A 1 +
m
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Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
r
After 1-period: A → A 1 +
m
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After 1-year
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
r
After 1-period: A → A 1 +
m
r m
After 1-year A → A 1 +
m
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Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
r
After 1-period: A → A 1 +
m
r m
After 1-year A → A 1 +
m
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After t-years
Compound Interest
Compound Interest (interest compounded periodically)
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Start with P and an annual rate r as before.
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Divide the year into m periods of equal duration and assume
that the interest is compounded m times a year.
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Simple interest rate per conversion period equals r /m
r
After 1-period: A → A 1 +
m
r m
After 1-year A → A 1 +
m
r mt
After t-years A → A 1 +
m
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Compound Interest
Example
Compound Interest
Example
If r = 0.02, what results in a higher Accumulated Amount,
compounding bi-annually or quarterly?
Compound Interest
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
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
m
0.02
Graph of f (m) = 1 +
m
Compound Interest
m
0.02
Graph of f (m) = 1 +
m
Compound Interest
Example
Compound Interest
Example
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f (m) is increasing
Compound Interest
Example
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f (m) is increasing
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f (4) > f (2)
Compound Interest
Example
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f (m) is increasing
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f (4) > f (2)
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Quarterly is better than bi-annually
Compound Interest
Example
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f (m) is increasing
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f (4) > f (2)
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Quarterly is better than bi-annually
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In general one obtains a marginally better return if one
compounds interest more frequently.
Compound Interest
Present Value and Effective Interest
Compound Interest
Present Value and Effective Interest
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Present Value of an investment
Compound Interest
Present Value and Effective Interest
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Present Value of an investment
r −mt
P =A 1+
m
Compound Interest
Present Value and Effective Interest
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Present Value of an investment
r −mt
P =A 1+
m
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Effective Interest
Compound Interest
Present Value and Effective Interest
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Present Value of an investment
r −mt
P =A 1+
m
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Effective Interest
r m
reff = 1 +
−1
m
Compound Interest
Present Value and Effective Interest
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Present Value of an investment
r −mt
P =A 1+
m
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Effective Interest
r m
reff = 1 +
−1
m
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A = P(1 + reff )t
Compound Interest
Example
Compound Interest
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
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
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?
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5 years:
Compound Interest
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?
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5 years: 23400.26
Compound Interest
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?
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5 years: 23400.26
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10 years
Compound Interest
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?
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5 years: 23400.26
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10 years 1825.24.
Compound Interest
Interest compounded continuously
Compound Interest
Interest compounded continuously
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Compound “more and more often”.
Compound Interest
Interest compounded continuously
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Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
Compound Interest
Interest compounded continuously
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Compound “more and more often”.
r mt
From A = P 1 +
, let m → ∞.
m
r mt
= e rt .
lim 1 +
m→∞
m
Compound Interest
Interest compounded continuously
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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
Interest compounded continuously
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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