interest rate

interest rate

An

interest rate

, loosely speaking, is the rate in which

interest

accumulates over time. There are different ways of measuring this change. In other words, there are different types of interest rates. Suppose we are given the following setup:

1.

there is a borrower

B

and a lender

L

and that’s it

2.

at time

0

, a transaction takes place where

L

loans

M

to

B

3.

at times

t

1

and

t

2

, the interests accrued are

i

⁢

(

t

1

)

and

i

⁢

(

t

2

)

Interest rate

. The

interest rate

is defined as the value

r

⁢

(

t

1

,

t

2

)

:=

1

M

⁢

i

⁢

(

t

2

)

-

i

⁢

(

t

1

)

t

2

-

t

1

.

If we set

M

⁢

(

t

)

=

M

+

i

⁢

(

t

)

, then

M

⁢

(

0

)

=

M

and

r

⁢

(

t

1

,

t

2

)

=

1

M

⁢

(

0

)

⁢

M

⁢

(

t

2

)

-

M

⁢

(

t

1

)

t

2

-

t

1

.

M

⁢

(

t

1

)

can be interpreted as the “accumulated” principal at

t

1

, although the transaction of the interest

actually

being added to the principal is not assumed.

Effective interest rate

. Another way of measuring the rate in which interest change with respect to

t

is known as the

effective interest rate

. It is defined as:

eff

.

⁡

r

⁢

(

t

1

,

t

2

)

:=

1

M

⁢

(

t

1

)

⁢

i

⁢

(

t

2

)

-

i

⁢

(

t

1

)

t

2

-

t

1

=

1

M

⁢

(

t

1

)

⁢

M

⁢

(

t

2

)

-

M

⁢

(

t

1

)

t

2

-

t

1

.

Unlike the ordinary interest rate, effective interest rate measures the changes in interest relative to the principal at the beginning of the time period that is being measured, rather than the original principal.

Discount rate

. The

discount rate

is defined as:

d

⁢

(

t

1

,

t

2

)

:=

1

M

⁢

(

t

2

)

⁢

i

⁢

(

t

2

)

-

i

⁢

(

t

1

)

t

2

-

t

1

=

1

M

⁢

(

t

2

)

⁢

M

⁢

(

t

2

)

-

M

⁢

(

t

1

)

t

2

-

t

1

.

This is very

similar

to the

definition

of the effective interest rate. The

difference

here is the we are interested in looking at changes in interest relative to the end of the time period. The following relationship is useful:

1

d

⁢

(

t

1

,

t

2

)

+

1

eff

.

⁡

r

⁢

(

t

1

,

t

2

)

=

t

2

-

t

1

.

Discount rate is handy when one wants to know the current, or present value of some amount of money in the future, for example, looking at the present value of the total mortgage to be paid 30 years into the future.

Others

. If we include the effect of inflation, or any changes affecting the value of money not due to interest, we have

1.

real interest rate - the interest rate calculated based on the “real” value of money, adjusted for inflation, and

2.

nominal interest rate - the interest rate calculated based on the “face”, or “unadjusted” value of money.

Interest rate

continuous

with respect to time

. When

M

is a

differentiable function

with respect to

t

, we may define what is called the

instantaneous interest rate

:

r

⁢

(

t

)

=

1

M

⁢

d

⁢

M

⁢

(

t

)

d

⁢

t

,

and the corresponding

instantaneous effective interest rate

eff

.

⁡

r

⁢

(

t

)

=

1

M

⁢

(

t

)

⁢

d

⁢

M

⁢

(

t

)

d

⁢

t

.

References

1

S. G. Kellison,

Theory of Interest

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

Title

interest rate

Canonical

name

InterestRate

Date of creation

2013-03-22 16:39:54

Last modified on

2013-03-22 16:39:54

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 91B28

Classification

msc 00A69

Classification

msc 00A06

Related topic

SimpleInterest

Related topic

CompoundInterest

Defines

effective interest rate

Defines

instantaneous interest rate

Defines

instantaneous effective interest rate

Defines

nominal interest rate

Defines

real interest rate

Defines

discount rate

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