Continuously Compounded Interest Use the following...

Continuously Compounded Interest Use the following discussion: Suppose an initial investment, called the principal $P,$ earns an annual rate of interest $r,$ which is compounded $n$ times per year. The interest earned on the principal $P$ in the first compounding period is $P\left(\frac{r}{n}\right)$, and the resulting amount A of the investment after one compounding period is $A=P+P\left(\frac{r}{n}\right)=P\left(1+\frac{r}{n}\right) .$ After $k$ compounding periods, the amount $A$ of the investment is $A=P\left(1+\frac{r}{n}\right)^{k}$. Since in $t$ years there are nt compounding periods, the amount A after t years is $$ A=P\left(1+\frac{r}{n}\right)^{n t} $$ When interest is compounded so that after $t$ years the accumulated amount is $A=\lim _{n \rightarrow \infty} P\left(1+\frac{r}{n}\right)^{n t},$ the interest is said to be compounded continuously. A bank offers a certificate of deposit (CD) that matures in 10 years with a rate of interest of $3 \%$ compounded continuously. (See Problem 89.) Suppose you purchase such a CD for $\$ 2000$ in your IRA. (a) Write an equation that gives the amount $A$ in the $\mathrm{CD}$ as a function of time $t$ in years. (b) How much is the CD worth at maturity? (c) What is the rate of change of the amount $A$ at $t=3 ?$ At $t=5$ ? At $t=8$ ? (d) Explain the results found in (c).

Continuously Compounded Interest Use the following discussion:

Suppose an initial investment, called the principal $P,$ earns an annual rate of interest $r,$ which is compounded $n$ times per year. The interest earned on the principal $P$ in the first compounding period is $P\left(\frac{r}{n}\right)$, and the resulting amount A of the investment after one compounding period is $A=P+P\left(\frac{r}{n}\right)=P\left(1+\frac{r}{n}\right) .$ After $k$ compounding periods, the amount $A$ of the investment is $A=P\left(1+\frac{r}{n}\right)^{k}$. Since in
$t$ years there are nt compounding periods, the amount A after t years is
$$
A=P\left(1+\frac{r}{n}\right)^{n t}
$$
When interest is compounded so that after $t$ years the accumulated amount is $A=\lim _{n \rightarrow \infty} P\left(1+\frac{r}{n}\right)^{n t},$ the interest is said to be compounded continuously.
A bank offers a certificate of deposit (CD) that matures in 10 years with a rate of interest of $3 \%$ compounded continuously. (See Problem 89.) Suppose you purchase such a CD for $\$ 2000$ in your IRA.
(a) Write an equation that gives the amount $A$ in the $\mathrm{CD}$ as a function of time $t$ in years.
(b) How much is the CD worth at maturity?
(c) What is the rate of change of the amount $A$ at $t=3 ?$ At $t=5$ ? At $t=8$ ?
(d) Explain the results found in (c).

Key Concepts

Compound Interest

Compound interest is the process of earning interest not only on the initial amount of money (the principal) but also on the accumulated interest from previous periods. This recursive accumulation makes compound interest a powerful tool for growing investments over time and is fundamental to understanding various financial calculations and investment strategies.

Continuous Compounding

Continuous compounding is a concept where the frequency of compounding interest is increased to its theoretical limit, occurring an infinite number of times per period. This leads to the formula A = P · e^(rt), representing a model for how investments grow under constant, uninterrupted accumulation of interest, and introduces the importance of the exponential function in finance.

Exponential Growth in Finance

Exponential growth in finance refers to the type of growth characterized by a constant relative rate of change, as seen in continuously compounded interest. When an investment's value increases by a fixed percentage over time, the resulting growth can be modeled using exponential functions, providing a clear framework for predicting future values of investments.

Instantaneous Rate of Change

The instantaneous rate of change of an investment’s value is found by differentiating the growth function with respect to time. This derivative measures how quickly the investment is growing at any specific moment, offering valuable insights into short-term performance and the sensitivity of the investment's value to time.

Limit Process in Calculus

The limit process in calculus is used to define concepts like continuous compounding by taking the compounding frequency to infinity. This process transitions the discrete model of compound interest to a continuous model, highlighting the bridge between algebraic formulas and calculus in financial mathematics.

Transcript

00:01 We're told we have a cd, a certificate of deposit, that is compounding continuously at a rate of 3%.

00:12 And they ask us, if we put in $2 ,000, we want to get an equation for the amount.

00:21 Write an equation gives the amount a of the investment after a certain period of years.

00:31 And so we just get 2 ,000 e to the 0 .03.

00:34 This is the interest rate 3 % times t.

00:37 This was our initial principle.

00:40 So after 10 years, they ask us to find what is the value after 10 years.

00:47 And so this winds up being 0 .3.

00:51 And so 2 ,000 times e to the 0 .3 is 2 ,700, very close to 2 ,700.

00:56 So we made about over 10 years, we made about $700.

01:01 So not, again, three percent is pretty poor.

01:13 So, i mean, think about it as, you know, basically inflation, something after 10 years, something that cost this much if we had 3 % inflation that was, you know, compounding continuously, it would cost this much after, in 10 years from now.

01:30 So then they say, okay, well, we know, we know this relationship.

01:33 Ship before, that the a, the t equals r times a.

01:38 We showed that in the previous problem.

01:40 And so they ask us to figure out what this is at three years...

Please give Ace some feedback