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) Show that if the annual rate of interest $r$ is compounded continuously, then the amount $A$ after $t$ years is $A=P e^{r t}$, where $P$ is the initial investment. (b) If an initial investment of $P=\$ 5000$ earns $2 \%$ interest compounded continuously, how much is the investment worth after 10 years? (c) How long does it take an investment of $\$ 10,000$ to double if it is invested at $2.4 \%$ compounded continuously? (d) Show that the rate of change of $A$ with respect to $t$ when the interest rate $r$ is compounded continuously is $\frac{d A}{d t}=r A$.

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) Show that if the annual rate of interest $r$ is compounded continuously, then the amount $A$ after $t$ years is $A=P e^{r t}$, where $P$ is the initial investment.
(b) If an initial investment of $P=\$ 5000$ earns $2 \%$ interest compounded continuously, how much is the investment worth after 10 years?
(c) How long does it take an investment of $\$ 10,000$ to double if it is invested at $2.4 \%$ compounded continuously?
(d) Show that the rate of change of $A$ with respect to $t$ when the interest rate $r$ is compounded continuously is $\frac{d A}{d t}=r A$.

Key Concepts

Compound Interest

Compound interest refers to the process where interest is added to the principal so that, from that moment on, the interest that has been added also earns interest. This leads to a faster growth of the investment compared to simple interest, where only the principal earns interest.

Continuous Compounding

Continuous compounding is the limit of the compound interest process as the number of compounding periods per year tends to infinity. In this case, the formula for the accumulated amount becomes A = P e^(rt), where e is the base of the natural logarithm. This model describes a scenario in which the growth of the investment is truly seamless and continuous.

Exponential Growth

Exponential growth describes processes where the rate of change of a quantity is directly proportional to the current amount. In the context of continuous compounding, the investment grows exponentially over time, which is mathematically expressed as A = P e^(rt).

Differential Equations in Financial Modeling

The relationship dA/dt = rA is a differential equation that represents the instantaneous rate of change of the investment value under continuous compounding. This equation formalizes the concept that the rate of increase of the investment is proportional to its current value, encapsulating the fundamental principle behind exponential growth in financial contexts.

Transcript

00:01 They want us to consider compounding interest.

00:04 And so basically they said that, you know, if n is the periods per, if this is the yearly rate of interest, and then is the number of periods per year that was compounded, then this is the over, you know, you basically this right here is the principle after you put in, well, they use p here, which is i don't like.

00:30 But anyway, a, i guess what they may be called a crude value.

00:35 So we have p times 1 plus r over n to the n times t power.

00:44 Now, so this would be in years, and this would be in periods, you know, compounds per year.

00:52 And this is the yearly compounding rate.

00:56 So, you know, we could figure out what that was.

00:59 You know, if we're compounding quarterly, then this is four.

01:02 You know over you know five years then this would be five so we could figure this out right for whatever n and t and r and p we have p is what the money we put in um obviously when t equals zero we just get p right just get back what we put in so now um what we can do they want us to figure out what this limit is so as n goes to infinity that means we're compounding continually so you know the compounding rate is compounding rate or frequency is infinite so the compounding period is zero basically continuously so we can do the same trick that we did for some of the previous problems where we can look at this as one over n over r and then write n times t as n as n times um n times r times t over r right so what we get this thing and this thing looking the same and then we can take the r t out of the of the limit because that doesn't depend on n and then we can redefine m as n over r and if n goes to infinity then so does m um i guess if r was zero that would be a little bit of a weird situation but if r were zero that would be a little bit of a weird situation but if r were zero then basically you have no interest.

02:36 And so you're just, no matter how long you wait, you just get that.

02:39 So that's like basically r is zero when you cram your money onto your mattress.

02:45 So now we recognize this thing as e.

02:49 And then we have this multiplied by p out here that i've got to carry along.

02:53 And so we just get this is p times e to the rt.

02:59 So we get exponentially growing.

03:02 So if we compound continually, we have.

03:04 Have exponential growth.

03:07 Now, let's see here.

03:09 Where are we going to go from there? so from there, they tell us we have $5 ,000.

03:17 The annual interest rate is 2%.

03:19 So we got to write that as 0 .02.