# 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. 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. 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 $0$ $t$ principal interest interest accrued $M_{0}$ $0$ $0$ $M_{0}$ $i$ $i$ $M_{0}$ $i$ $2i$ $M_{0}$ $i$ $3i$ $\vdots$ $\vdots$ $\vdots$ $M_{0}$ $i$ $ni$

${}\end{center}\inner@par Thetotal^{\prime\prime}interest$i(nt)$earned(accrued)attheendoftime$nt$is$ni$.Iftheaccountisclosedandthemoneywithdrawnattheendof$nt$,andthetotalamountofmoneyreceivedis$$M(nt)=M_{0}+ni.$$\inner@par Theinterestrateassociatedwiththesimpleinterestaspresentedabovebetweentwotimeperiods% ,say$at$and$bt$,isgivenby$$r(at,bt)=\frac{1}{M_{0}}\frac{i(bt)-i(at)}{bt-at}=\frac{i}{M_{0}t},$$whichdoesnotdependonthechoiceof$a$and$b$.Inotherwords,theoriginalprincipal$M_0$,theamountofinterest$i$,andthelengthoftheinitialtimeinterval$t${{{areenoughtodeterminetheinterestrate.\inner@par\textbf{Remark}.\begin{itemize} \itemize@item The expression for the effective interest rate for simple % interest is a bit more complicated: \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). \itemize@item More generally, we say that an interest is \emph{simple} if its % interest rate r is constant with respect to time t. Solving r=\frac{1}{M_{0}}\frac{i(t)-i(0)}{t-0} for i(t), we get i(t)=M_{0}rt, or that the accrued interest is a linear % function of t. It grows directly proportionally with respect to time. \end{itemize}\begin{flushright}\begin{tabular}[]{|ll|}\hline 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\\ \hline}\end{tabular}}}\end{flushright}\end{document}$