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'interest rate'
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| Title of object: |
interest rate |
| Canonical Name: |
InterestRate |
| Type: |
Definition |
| Created on: |
2007-02-04 22:23:42 |
| Modified on: |
2007-02-05 12:54:05 |
| Classification: |
msc:00A06, msc:00A69, msc:91B28 |
| Defines: |
effective interest rate, simple interest, compound interest, compounded continuously |
| Synonyms: |
interest rate=continuously compounded
interest rate=continuously-compounded |
Preamble:
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Content:
This following assumes the value of a unit amount of money is constant with respect to time.
An \emph{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:
\begin{enumerate}
\item there is a borrower $B$ and a lender $L$ and that's it
\item at time $0$, a transaction takes place where $L$ loans $M$ to $B$
\item at times $t_1$ and $t_2$, the interests accrued are $i(t_1)$ and $i(t_2)$
\end{enumerate}
\textbf{Definition 1}. The \emph{interest rate} is defined as the value $$r(t_1,t_2):=\frac{1}{M}\frac{i(t_2)-i(t_1)}{t_2-t_1}.$$
The simplest kind of interest is known as the \emph{simple interest}. It is defined as an interest where the interest rate is a constant $r$. Let's see what this means. Set $t_1=t$ and $t_2=t+n$. Solving the above equation to get $$i(t+n)=i(t)+Mrn.$$
Setting $t=0$ above, we see the interest accrued is a linear function of time. If interest is accrued at a fixed interval $n=1$ (monthly, semiannually, etc...), the interest collected at the end of each interval is the same: $i(t+1)-i(t)=Mr$ is a fixed amount.
\textbf{Definition 2}. Another way of measuring the rate in which interest change with respect to $t$ is known as the \emph{effective interest rate}. It is defined as:
$$\operatorname{eff.}r(t_1,t_2):=\frac{1}{M(t_1)} \frac{i(t_2)-i(t_1)}{t_2-t_1},$$
where $M(t_1)=M+i(t_1)$, and can be interpreted as the ``accumulated'' principal at $t_1$ (the transaction that the interest at $t_1$ is \emph{actually} added to the principal is not assumed). The above definition of $r$ can be restated as
$$\operatorname{eff.}r(t_1,t_2):=\frac{1}{M(t_1)} \frac{M(t_2)-M(t_1)}{t_2-t_1}.$$
The next simplest kind of interest is known as the \emph{compound interest}. It is defined as the interest where the effective interest rate $r(t_1,t_2)$ only depends on $t_2-t_1$, or $r(t_1,t_2)=c(t_2-t_1)$ for some function $c$. Now, suppose $M$ is deposited at $t_1=0$ and interest is collected at a fixed unit time interval $t_2-t_1=1$. After the first interval, $cM(0)=M(1)-M(0)$, or $M(1)=(c+1)M$. After another interval, $M(2)=(c+1)M(1)=(c+1)^2M$. $n$ intervals later, we see that the interest collected is $$i(n)=M(n)-M=(c+1)^nM-M=\big( (c+1)^n-1\big) M.$$ Compound interest process can be interpreted as follows: at the end of each fixed time interval, interest is calculated based on the principal at the beginning of the time interval, then a transaction takes place where the interest is added to the principal. The principal ``grows'' or ``compounded'' exponentially as time increases.
Looking back at the simple interest, we see that the effective interest rate is
$$\operatorname{eff.}r=\frac{1}{M(t_1)}\frac{i(t_2)-i(t_1)}{t_2-t_1}= \frac{1}{M+Mrt_1} \frac{Mrt_2-Mrt_1}{t_2-t_1} =\frac{r}{1+rt_1}$$
decreases as $t_1$ increases. It is not as ``effective'' compared to the compound interest because the principal stays constant and does not ``grow''.
\textbf{Remark}. When $M$ is a differentiable function with respect to $t$, we may define what is called the \emph{instantaneous interest rate}: $$r(t)=\frac{1}{M}\frac{dM(t)}{dt},$$ and the corresponding \emph{instantaneous effective interest rate} $$\operatorname{eff.}r(t)=\frac{1}{M(t)}\frac{dM(t)}{dt}.$$
An interest is said to be \emph{compounded continuously} if it is differentiable with respect to time $t$ and its instantaneous effective interest rate is a constant $r$. If we solve the corresponding differential equation, we see that for a continously compounded interest, $$M(t)=Me^{rt}.$$
The effective interest rate of a continuously compounded interest is $$\frac{1}{M(t_1)}\frac{M(t_2)-M(t_1)}{t_2-t_1} =\frac{1}{Me^{rt_1}}\frac{Me^{rt_2}-Me^{rt_1}}{r_2-r_1} =\frac{e^{r(t_2-t_1)}-1}{r_2-r_1}.$$
Since it is a function of $r_2-r_1$, interest compounded continuously is a compounded interest.
\begin{thebibliography}{8}
\bibitem{sk} S. G. Kellison, {\em Theory of Interest}, McGraw-Hill/Irwin, 2nd Edition, (1991).
\end{thebibliography} |
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