simple interest

simple interest

Suppose a bank account is opened at time

0

and

M

0

is deposited into the account. A

simple interest

is

interest

with the following characteristics:

1.

it is earned at subsequent time periods

t

,

2

⁢

t

,

…

, where

t

is the length of the initial time interval (1 for 1 month, 12 for 1 year, etc…)

2.

the interest earned at the end of each time period is the same regardless of the time period

The following table illustrates the structure of the simple interest.

time period at

principal

interest

interest accrued

0

M

0

0

0

t

M

0

i

i

2

⁢

t

M

0

i

2

⁢

i

3

⁢

t

M

0

i

3

⁢

i

⋮

⋮

⋮

⋮

n

⁢

t

M

0

i

n

⁢

i

T

⁢

h

⁢

e

⁢

`

⁢

`

⁢

t

⁢

o

⁢

t

⁢

a

⁢

l

′′

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

i(nt)

e

⁢

a

⁢

r

⁢

n

⁢

e

⁢

d

⁢

(

a

⁢

c

⁢

c

⁢

r

⁢

u

⁢

e

⁢

d

)

⁢

a

⁢

t

⁢

t

⁢

h

⁢

e

⁢

e

⁢

n

⁢

d

⁢

o

⁢

f

⁢

t

⁢

i

⁢

m

⁢

e

nt

i

⁢

s

ni

.

I

f

t

h

e

a

c

c

o

u

n

t

i

s

c

l

o

s

e

d

a

n

d

t

h

e

m

o

n

e

y

w

i

t

h

d

r

a

w

n

a

t

t

h

e

e

n

d

o

f

nt

,

a

n

d

t

h

e

t

o

t

a

l

a

m

o

u

n

t

o

f

m

o

n

e

y

r

e

c

e

i

v

e

d

i

s

M

⁢

(

n

⁢

t

)

=

M

0

+

n

⁢

i

.

T

⁢

h

⁢

e

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

r

⁢

a

⁢

t

⁢

e

⁢

a

⁢

s

⁢

s

⁢

o

⁢

c

⁢

i

⁢

a

⁢

t

⁢

e

⁢

d

⁢

w

⁢

i

⁢

t

⁢

h

⁢

t

⁢

h

⁢

e

⁢

s

⁢

i

⁢

m

⁢

p

⁢

l

⁢

e

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

a

⁢

s

⁢

p

⁢

r

⁢

e

⁢

s

⁢

e

⁢

n

⁢

t

⁢

e

⁢

d

⁢

a

⁢

b

⁢

o

⁢

v

⁢

e

⁢

b

⁢

e

⁢

t

⁢

w

⁢

e

⁢

e

⁢

n

⁢

t

⁢

w

⁢

o

⁢

t

⁢

i

⁢

m

⁢

e

⁢

p

⁢

e

⁢

r

⁢

i

⁢

o

⁢

d

⁢

s

,

s

⁢

a

⁢

y

at

a

⁢

n

⁢

d

bt

,

i

s

g

i

v

e

n

b

y

r

⁢

(

a

⁢

t

,

b

⁢

t

)

=

1

M

0

⁢

i

⁢

(

b

⁢

t

)

-

i

⁢

(

a

⁢

t

)

b

⁢

t

-

a

⁢

t

=

i

M

0

⁢

t

,

w

⁢

h

⁢

i

⁢

c

⁢

h

⁢

d

⁢

o

⁢

e

⁢

s

⁢

n

⁢

o

⁢

t

⁢

d

⁢

e

⁢

p

⁢

e

⁢

n

⁢

d

⁢

o

⁢

n

⁢

t

⁢

h

⁢

e

⁢

c

⁢

h

⁢

o

⁢

i

⁢

c

⁢

e

⁢

o

⁢

f

a

a

⁢

n

⁢

d

b

.

I

n

o

t

h

e

r

w

o

r

d

s

,

t

h

e

o

r

i

g

i

n

a

l

p

r

i

n

c

i

p

a

l

M_0

,

t

h

e

a

m

o

u

n

t

o

f

i

n

t

e

r

e

s

t

i

,

a

n

d

t

h

e

l

e

n

g

t

h

o

f

t

h

e

i

n

i

t

i

a

l

t

i

m

e

i

n

t

e

r

v

a

l

t

a

⁢

r

⁢

e

⁢

e

⁢

n

⁢

o

⁢

u

⁢

g

⁢

h

⁢

t

⁢

o

⁢

d

⁢

e

⁢

t

⁢

e

⁢

r

⁢

m

⁢

i

⁢

n

⁢

e

⁢

t

⁢

h

⁢

e

⁢

i

⁢

n

⁢

t

⁢

e

⁢

r

⁢

e

⁢

s

⁢

t

⁢

r

⁢

a

⁢

t

⁢

e

.

𝐑𝐞𝐦𝐚𝐫𝐤

.

•

The expression for the effective interest rate for simple interest is a bit more complicated:
eff.r(at,bt)=

1

M(at)

i(bt)-i(at)

bt-at

=

1

M

0

+ai

i

t

,
which decreases with increasing

a

. Imagine as

a

becomes very large, the increase in interest has practically no impact on the “accumulated” principal

⁢
M

(

⁢
a
t

)

.

•

More generally, we say that an interest is simple if its interest rate

r

is constant with respect to time

t

. Solving
r=

1

M

0

i(t)-i(0)

t-0

for

⁢
i

(
t
)

, we get

=

⁢
i

(
t
)

⁢

M
0

r
t

, or that the accrued interest is a linear function of

t

. It grows directly proportionally with respect to time.

⁢

Title

simple interest

Canonical name

SimpleInterest

Date of creation

2013-03-22 16:40:06

Last modified on

2013-03-22 16:40:06

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

7

Author

CWoo (3771)

Entry type

Example

Classification

msc 00A06

Classification

msc 00A69

Classification

msc 91B28

Related topic

CompoundInterest

Related topic

InterestRate

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