interest

interest

This entry mostly concerns the mathematical aspects of

interest

. Also, we assume that the value of a unit amount of money does not change with time (so there is no consideration regarding inflation, etc…)

Let’s first look at two common examples in real life:

1.

An

individual

deposits a certain amount of money into an entity (a bank, for example) for a period of time, interest is accrued at the end of the time period payable to the individual by the entity.

2.

One borrows a certain amount of money from an entity (a loan company, for example) for a period of time, interest is accrued at the end of the time period payable to the entity by the borrower.

In both examples, two parties, or

entities

are involved. Let’s call one the lender

L

and the other the borrower

B

. Between

L

and

B

, there is an initial

transaction

where a certain amount of money

M

, called the

principal

, is transferred from one to another.

M

can be seen as a

function

of

L

and

B

:

M

=

M

⁢

(

L

,

B

)

and has the

property

that

M

⁢

(

L

,

B

)

+

M

⁢

(

B

,

L

)

=

0

. This property basically says that

B

borrowing

M

from

L

is

equivalent

to

L

borrowing

-

M

from

B

. Let us agree that

M

⁢

(

L

,

B

)

≥

0

.

At a later time

t

, an

interest

is accrued. Put it another way,

M

has “grown” to

M

⁢

(

t

)

. Set the initial transaction at time

0

, then

M

=

M

⁢

(

0

)

. From this point of view,

interest

i

, is the “additional” amount of money earned, or accrued, between

0

and

t

, or

i

=

M

⁢

(

t

)

-

M

⁢

(

0

)

. If we want to emphasize

L

and

B

as the additional

variables

, then the

definition

, fully stated, becomes

i

⁢

(

L

,

B

,

t

)

:=

M

⁢

(

L

,

B

,

t

)

-

M

⁢

(

L

,

B

,

0

)

.

We can calculate interest between any two time periods. For example, if we know

i

⁢

(

L

,

B

,

t

)

and

i

⁢

(

L

,

B

,

s

)

, then the interest earned from

t

to

s

is simply the

difference

of the two:

i

⁢

(

L

,

B

,

t

,

s

)

:=

i

⁢

(

L

,

B

,

s

)

-

i

⁢

(

L

,

B

,

t

)

=

M

⁢

(

L

,

B

,

s

)

-

M

⁢

(

L

,

B

,

t

)

.

Keep in mind is that in this definition, only one transaction has taken place: the initial one at

0

(we assume that at time

t

, an interest has been merely been accrued but no actual transfer of money is taking place). Interests resulting from additional transactions that happened between

0

and

t

need to be evaluated separately.

The last paragraph merely says that

i

is

additive

with respect to

transactions

: if

L

1

loans

M

1

to

B

1

at

t

1

, and

L

2

loans

M

2

to

B

2

at

t

2

, then the

total

interest earned by the lenders from the borrowers at time

t

is

i

⁢

(

L

,

B

,

t

)

=

i

1

⁢

(

L

1

,

B

1

,

t

)

+

i

2

⁢

(

L

2

,

B

2

,

t

)

,

where

L

is

L

1

and

L

2

considered as a single lending entity, and

B

is

B

1

and

B

2

considered as a single borrowing entity.

Remarks

.

•

Note that

t

does not always have to be

positive

. It is possible that we are interested in finding what

i

is at time

t

≤

0

. For example, one may want to know the amount of interest lost had he saved his money in a bank and earned interest 30 years earlier, well before his current retirement age.

•

Another point worth stressing is that the defintion only works if we stick to the same units. If

M

is expressed as U.S. Dollars, then that needs to be the unit of choice for money throughout. Any other denominations or currencies need to be converted. The same goes with time

t

.

References

1

S. G. Kellison,

Theory of Interest

, McGraw-Hill/Irwin, 2nd Edition, (1991).

Title

interest

Canonical

name

Interest

Date of creation

2013-03-22 16:39:51

Last modified on

2013-03-22 16:39:51

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

6

Author

CWoo (3771)

Entry type

Definition

Classification

msc 00A69

Classification

msc 00A06

Classification

msc 91B28

Generated on Tue Feb 6 22:21:11 2018 by

LaTeXML