Given An Exponential Function For Compounding Interest, A(X) = P(1.01)x, What Is The Rate Of Change?

Developing Financial Intuition

Rarely is it the case these days that you invest $100 of your money at, say, v% per year and become
$5 every year (known as simple interest). Why is this non the case? Considering involvement is ofttimes compounded, which means that the 5% interest is paid on the full current balance. Permit's illustrate this in a comparison of tables:

Unproblematic Interest
Yr	Balance	Interest	Year-End Balance
0	$100.00	.05(100) = 5	$105.00
1	$105.00	.05(100) = v	$110.00
two	$110.00	.05(100) = 5	$115.00
3	$115.00	.05(100) = five	$120.00

Interest paid on original remainder only: constant rate of growth

Chemical compound Interest
Year	Balance	Involvement	Year-Terminate Balance
0	$100.00	.05(100) = 5	$105.00
1	$105.00	.05(105) = 5.25	$110.25
2	$110.25	.05(110.25) = 5.51	$115.76
3	$115.76	.05(115.7625) = 5.79	$121.55

Involvement paid on overall balance: abiding per centum growth

Although non a huge difference, discover that the balances go on to grow slightly further apart as the years go by. This difference is easier to see in a graph comparing the balances:

If were to await at the balances 20 years downwardly the line, we would run into a more substantial departure:

After twenty years, chemical compound interest brings in $73.60 more profit than uncomplicated interest. Y'all might be saying, "this difference is insignificant over a twenty yr flow," and by that yous have a valid point. Keep in mind that this is based on a one-time investment of $100. Over a twenty-yr menstruation, you will have earned:

[latex]\displaystyle\frac{{\${265.33}}}{{\${100}}}={2.65}[/latex]

[latex]\displaystyle{2.65}-{1}={1.65}={165}%[/latex] gain

This represents near tripling the original amount (2.65 times the original, to be more exact). With simple interest, this gain would simply be:

[latex]\displaystyle\frac{{\${200}}}{{100}}-{1}={1.00}={100}%[/latex] gain

The simple interest amount is double the original balance.

At this point, you might exist wondering how it is that we obtained the 20-year balances. Certainly, we tin can approximate these balances based on the graph given, only even then we need a style to generate the graph.

For unproblematic involvement, this is quite simple. Suppose the periodic interest charge per unit, that is, the involvement paid per catamenia (i.e. per year, per month, per day, etc.), is represented as a decimal and assigned to the variable
i. And so, commencement calculate the regular involvement amount by multiplying the charge per unit by the initial deposit, or the principle, P.

regular interest paid = P × i

This amount volition be paid over
fourth dimension periods, so the total amount of involvement is

total interest over
t periods = N × P × i

For example, if the principle is
P = $500 and the interest charge per unit is i = x% per year for N = 8 years, then the regular involvement paid is P × i = $500 × .10 = $50 per year. Paid over 8 years, nosotros get:

full interest over 10 years = viii × $500 × .x = $400

To go our total residual, we must add this amount back to the original principle to get:

P + N × P × i = $500 + $400 = $900

We often telephone call this the accumulated amount, or
A. More than by and large,

A = P + Northward × P ×

=
P(i + Ni)

Simple Involvement Residual Formula

If involvement is paid according to a elementary involvement schedule and we ascertain

A = accumulated balance or future value

P = principal invested

N = number of periods

i = periodic interest rate

Then
A = P(i + Ni)

Example 1

Verify that the 20-yr residuum for a $100 investment at 5% yearly interest is $200 by using the simple interest balance formula.

Solution

We have that
P = 100, N = xx, i = .05 so

A = 100(1 + 20 × .05)

= 100(two)

= $200

Edifice a Chemical compound Interest Formula

For chemical compound interest the idea is fairly uncomplicated. Recall that growth by a percent is called
exponential growth. To calculate a new amount, we must account for 100% of the original amount, plus the periodic growth rate, say , written as a decimal, Then, there volition be a total of of the original amount after one period.

For example, suppose that a population grows by 3% every twelvemonth. Next year there will be a total of 103% of the corporeality this twelvemonth. We write this as i + .03 = 1.03 to represent a decimal. This is called the
growth factor and is what we multiply past to obtain the new amount. The 3% represents the growth charge per unit and is usually the value reported by banks, the media, etc. when describing growth.

Suppose the population is 1,000. Next yr the population is expected to be m(i.03) = one,030.

What will this amount be in 2 years?

Bold the same growth rate of iii%, nosotros simply apply the growth factor to the 1-year corporeality:

ane,030(1.03) ≈ 1,061

Or, alternatively nosotros can write

[1000(one.03)]1.03 = k(one.03)
²

Do you see the pattern? The exponent simply represents the number of time periods that nosotros require to pass. If we wanted to know the population afterwards 10 years, we would multiply 1000 by ane.03 a total of 10 times, or

1000(1.03)
¹⁰ ≈ 1,344

This same idea applies to compound involvement!

Chemical compound Interest Balance Formula

If interest is paid according to a compound involvement schedule, where interest is paid on the
current balance and we define

A = accumulated balance or futurity value

P = principal invested

N = number of periods

i = periodic interest rate

And then A = P(ane + i) ^North

Example 2

Confirm that if you invest $100 for 20 years at an annual interest rate of v% compounded annually, that you volition have a rest of $253.33.

Solution

We have
P = 100, i = .05, Northward = 20, so

A = 100(1 + .05)^twenty

=100(one.05)²⁰

≈ 200(2.6533)

= $265.33

Discover in Case 2 the wording "compounded annually." This merely specifies how often the interest is paid. The values of
and should reflect the compounding menstruum specified.

Historically, banks have decided that offer a
nominal annual rate, or Annual Per centum Rate (APR). These are identical terms. This is simply a name for the charge per unit, considering information technology is rarely paid once each year. Instead, a depository financial institution will place how often interest is compounded. Some of the common ones are listed beneath:

Compounding Period	Number of Almanac Compoundings
Annually	one
Semi-annually	two
Quarterly	4
Monthly	12
Weekly	52
Daily	365

Note: Weeks and days vary depending on year. For ease of use, we ignore this particular.

Do you think that a monthly compounding schedule means a very generous bank? Not in the manner you lot might expect. Suppose a bank offers you a nominal annual rate of 12% compounded monthly. They exercise
not really pay you 12% each month. Instead you receive a pro-rated percentage every month, which is an equal fraction of the 12% per flow. Since there are 12 periods per year, you would receive 12%/12 months = 1%/month.

Example 3

A depository financial institution offers you a nominal annual charge per unit of five% compounded monthly. You invest $100 and plan on keeping information technology invested for 20 years. Calculate your balance after 20 years. And so, compare this to the value found in case 2 based on annual compounding and comment on the effect of compounding periods.

Solution

Nosotros have that
P = 100. Since the compounding menstruum is i calendar month, we must express i and Due north in terms of months. Since there are 12 months per year, in that location are N = 12 × 20 = 240 periods in the investment. Farther, the periodic charge per unit is [latex]\displaystyle{i}=\frac{{.05}}{{{12}\ {m}{o}{n}{t}{h}{s}}}\approx{.00417}{\quad\text{or}\quad}{.417}%[/latex] per month. We calculate

A = 100(one + .00417)²⁴⁰

≈ $271.48
We found that if involvement is paid once a twelvemonth, then the 20-yr accumulated balance is $265.33, which is $6.fifteen less than when interest is compounded monthly. Thus, increasing compounding frequency increases total balance. However, this deviation is not very much.

Effect of Compounding Frequency on Accumulated Balance (Time to come Value),

As the frequency of compounding interest increases, so does the accumulated balance.

To see this more clearly, consider the diverse compounding periods below, and the balance of $100 afterwards twenty years at 5%:

Compounding Period	Rest	Differences
Annually	$265.33
Semi-annually	$268.51	$three.18
Quarterly	$270.15	$1.64
Monthly	$271.26	$1.12
Weekly	$271.70	$0.43
Daily	$271.81	$0.11

We tin run into that, while the residue is slightly larger than that of the previous compounding period, the differences become quite small as the frequency increases more and more.

Case 4

Is 12% given annually the same thing as i% given monthly? Why or why not?

Solution

Suppose a person deposits
P = $100. Then, at the end of one yr the residuum will exist one.12(100) = $112, if interest is paid one time. But, the interest under monthly compounding (one% per month) will exist:

100(1.01)¹² ≈ $112.68

This deviation occurs due to the fact that monthly compounding pays 1% of the
current balance. Later on the first calendar month, there is a residuum of 100(1.01) = 101, but one month later the balance is 101(1.01) = 102.01, which is more than a $one increase. A rate of 12% annually is the same as $ane per month, an amount less than would be received equally of the 2d calendar month and across compared to monthly compounding.

Annual Percentage Yield

So, if 12% once is not the same as 1% 12 times, what pct
is the per centum paid over a year for 1% paid 12 times? To find the per centum that $112.68 is of the original amount, nosotros carve up:

[latex]\displaystyle\frac{{112.68}}{{100}}={1.1268}[/latex]

This means that the overall growth was 12.68%, a percentage larger than 12. Remember that the rate of 12% is called the nominal annual rate. The rate that you
really become later on compounding is taken into business relationship is called the annual percentage yield (APY).

We present a formal style to calculate this:

Since the APY is over a yr (
annual percentage yield), we take the chemical compound interest formula over the course of 1-twelvemonth but and only business ourselves with a $1 investment (since 1 = 100%). Decrease one from the effect, and so that nosotros only account for the growth, not the original 100%.

[latex]\displaystyle{A}{P}{Y}={1}{\left({1}+\frac{{r}}{{north}}\right)}^{{{n}\times{1}}}-{ane}={\left({1}+\frac{{r}}{{due north}}\right)}^{{{northward}}}-{one}[/latex]

Thus,
[latex]\displaystyle{A}{P}{Y}={\left({ane}+\frac{{r}}{{n}}\correct)}^{{{n}}}-{1}[/latex]

Alternatives to the formula?

Admittedly! If the amount invested is different than $1, calculate what it volition become in one year. Take the year-end corporeality, split up it by the original, and subtract 1.

Instance five

Let'due south say you invest $325 at ten% compounded semi-annually (twice a year) for five years. What is the APY?

Solution

Since we want the
annual percentage yield, we don't need to worry nearly the duration of the investment. We will compute the reply using the formula, and the intuitive way:

APY Formula	Intuitively
[latex]\displaystyle{\left({1}+\frac{{.1}}{{2}}\correct)}^{{ii}}-{1}={ane.1025}-{one}[/latex]	Using TVM Solver, $325 will be $358.3125 in one year. Discover the ratio of new to former.
[latex]\displaystyle={.1025}[/latex]	[latex]\displaystyle\frac{{\ne{west}}}{{{o}{fifty}{d}}}=\frac{{358.3125}}{{325}}={1.1025}[/latex]
[latex]\displaystyle={10.25}%[/latex]	This means that the growth is 10.25%. The ones place tells us that the new is 100% of the old, and then some.

In my opinion, information technology is much easier to empathize and recall the intuitive arroyo on the right. Needless to say, yous'll become the same reply.

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Milos Podmanik, Past the Numbers, "Chemical compound Interest and Exponential Growth," licensed under a CC BY-NC-SA three.0 license.

Source: https://courses.lumenlearning.com/finitemath1/chapter/reading-compound-interest-and-exponential-growth/

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