PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (230)
Orphanage
Unclass'd
Unproven (519)
Corrections (17)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] simple interest (Example)

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,2t,\ldots$, 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$
$ 2t$ $ M_0$ $ i$ $ 2i$
$ 3t$ $ M_0$ $ i$ $ 3i$
$ \vdots$ $ \vdots$ $ \vdots$ $ \vdots$
$ nt$ $ M_0$ $ i$ $ ni$

The “total” interest $ i(nt)$ earned (accrued) at the end of time $ nt$ is $ ni$. If the account is closed and the money withdrawn at the end of $ nt$, and the total amount of money received is

$\displaystyle M(nt)=M_0+ni.$

The interest rate associated with the simple interest as presented above between two time periods, say $ at$ and $ bt$, is given by

$\displaystyle r(at,bt)=\frac{1}{M_0}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_0t},$
which does not depend on the choice of $ a$ and $ b$. In other words, the original principal $ M_0$, the amount of interest $ i$, and the length of the initial time interval $ t$ are enough to determine the interest rate.

Remark.

  • The expression for the effective interest rate for simple interest is a bit more complicated:
    $\displaystyle \operatorname{eff.}r(at,bt)=\frac{1}{M(at)}\frac{i(bt)-i(at)}{bt-at}= \frac{1}{M_0+ai}\frac{i}{t},$
    which decreases with increasing $ a$. Imagine as $ a$ becomes very large, the increase in interest has practically no impact on the “accumulated” principal $ M(at)$.
  • More generally, we say that an interest is simple if its interest rate $ r$ is constant with respect to time $ t$. Solving
    $\displaystyle r=\frac{1}{M_0}\frac{i(t)-i(0)}{t-0}$
    for $ i(t)$, we get $ i(t)=M_0rt$, or that the accrued interest is a linear function of $ t$. It grows directly proportionally with respect to time.



"simple interest" is owned by CWoo.
(view preamble)

View style:

See Also: compound interest, interest rate


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: function, simple, increasing, effective interest rate, expression, interest rate, closed, structure, interval, length, periods, characteristics, interest
There is 1 reference to this entry.

This is version 4 of simple interest, born on 2007-02-05, modified 2007-12-18.
Object id is 8875, canonical name is SimpleInterest.
Accessed 993 times total.

Classification:
AMS MSC: 00A06 (General :: General and miscellaneous specific topics :: Mathematics for nonmathematicians )
 00A69 (General :: General and miscellaneous specific topics :: General applied mathematics)
 91B28 (Game theory, economics, social and behavioral sciences :: Mathematical economics :: Finance, portfolios, investment)
Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add example | add (any)