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by Mai
Compound interest “ is a terminology where interest on a loan or deposit is calculated based on both the initial principal and the accumulated interest from previous periods.” (JASON FERNANDO) Compound interest allows a principal amount to grow over time. It is also commonly used in our world, especially in banks or loans, where we put money in for interest rates to gain more savings for purposes such as creating investment accounts, as well as student loans, or when you are simply borrowing money. (Kate Ashford, and Benjamin Curry) Compound interest is 1 of the foundations for our economy as it is used mostly in our world, as many people in our society make and develop small businesses from compound interest as it also allows them to save money for further investments and much more. The pattern of compound interest can be described with exponential growth as the money gained or loss is proportional to the compound rate.
The concept of money and interest rate can be seen as having an exponential growth as every year, it is multiplied with a certain amount of money in the bank. The more money you have in the bank, the more it’ll increase when multiplied with the same rate over time, therefore, leading to a bigger amount of money when we add the money and interest together. Time also plays an important role in helping interest rates grow considering how time keeps on moving and how the money from the interest rate would grow more and more depending on those factors. but the important fact is the principal amount, as the principal amount is bigger then it would also link with the increase in the added tax amount.
Figure 1: The comparison between the amount of money saved at age 25 and 35 according to compound interest (Andy Kiersz)
To find the compound interest you have to multiply original money with interest rate + 1 and raise the power of the compound period. (JASON FERNANDO) As you can see here 25, years old started first so when 35 years old started slower, this means when 35 years old started then 25 years old was a large quantity of money and as the money grows exponentially, it would lead to a larger sum of money. So, they would then yield more money, and as compounded interest grows exponentially, the 25-year-old person will be able to yield more money as the interest has accumulated throughout the years and increase the amount of money owned rapidly. In contrast, at the same point in time, the 35-year-old person started with only the initial amount of money, while the 25-year-old person already accumulated throughout the years and increased the quantity of money (according to exponential growth). Through this example, it lets you see that money grows exponentially and it adds/doubles up by time. This also affects time because time can help change the amount of money you have, the earlier you start the more money you would grow and the interest rate would be higher. The dependent variable is considered the subject that is growing exponentially and here we can see its money, as for our independent variable that is considered the thing that is changing, then we can see its time in years as time changes which then changes the dependent variable. Independent variable change, which then leads to dependent variable growing, and that loop continues to repeat and repeat until you end the cycle of the dependent and independent variable. In mathematics, there's a way to calculate your compound interest, this formula is :
In this formula:
A equals to the final amount of money
P equals to initial principal balance ( starting amount )
r equals to the interest rate ( the amount of money it add after a period of time )
n equals to the number of times you compound within 1 year ( usually 1 time if the bank gives you interest 1 time a year, but it can also be 2 considering banks can give you the interest half year but this is used for situations where you save money, but if you are in a loan then n is replaced by the amount of times you pay. )
t equals to the number of years you have to pay
To find A which is the final amount of money, you will start with the initial amount (p). The initial amount is multiplied with 1 plus the interest rate (r). The rate is divided by the amount of time compounded within a year. Then this rate is raised to the power of the number of compound periods multiplied by the number of years compound.
From my research , the average interest rate from saving is 0.06% , making r : 0.06 (Matthew Goldberg) while the compound interest for saving accounts are daily, monthly , quarterly ( every 3 months ) and yearly (Motley Fool Staff.). In this example we are using yearly for the compound interest as taking interest from each year would be more convenient rather than going every month or every 3 months, average banks also take interest yearly. Here we can use 100$ as the starting amount (P) , r would be 0.06%, n is 1 for a year as the bank compounds 1 time a year. and t would be x, t is x because we should keep it like this so it's easier to monitor, this makes the formula become the one on the right:
Figure 2: correlation between the interest rate in 10 years with a starting amount of 100 dollars
Figure 3 : detailed, zoomed out version of figure 2
As you can see from the figures above, those are the graphs I made using Desmos by plotting the variables in the formula, as you can see in my graph the interest rate of 0.06% and the dots move up very fast due to exponential growth, and also due to exponential growth if the money value increase, we would be able to observe a larger amount of money. The starting value of 100 was a bit too big for the graph so the image could not be detailed as other graphs. The highest point in the graph is 179 dollars which are in 10 years, so as you can see each dot represents each year and how it increases the value, the first dot ( first year ) value was 106, then for the 2 years ( the next dot ) it was 112 and then it went to 119, then 126, 134, then on the 6 year its 142, then 150, 159, 169 then after that in the last year, which is the final year it went up till 179. So we can see in the first year it was 106 but in the period of 10 years it increased to 179, that shows that it went up by 73 which is nearly the amount of money we originally have and it would grow even larger in the following years .
Figure 4 : A graph describing compound interest and simple interest from elioministeri.com
The graph that I model myself is very similar to the graphs that I could find on the internet, as you see the graph shows a positive slope as the money increase, the linear is very steep because the dots represents the year and its amount, so the years can be close to each other but as the years increase it starts to spread out more, this graph can continue for long but it's not infinite as your age would one day turn old, or one day you would use the money for something, not everyone saves their money forever as money is also a finite resource in the human world money also needs to be conserved ( SHANE EDE), but yet the interest would grow all the time.
The exponential growth was modeled quite accurately, as exponential growth is when “a pattern of data shows an increase over time, creating a curve of exponential function" (JAMES CHEN) so I find that the exponential growth was modeled accurately as in my graph. The amount of money increased over time in the data, it increased by 1.06 and kept increasing. If you look at my graph it shows an example of a curve of an exponential function.
The exponential graph model money really good as you can see it show how money increased over time and let us observe how it grew, so it model money perfectly as it doesn't include many numbers but models with an image and a slope which then helps us observe to see how the change is, as images are easier to learn than words. This knowledge then can be applied in real-life situations when we want to save money, make an investment, loan, or when we want to increase our wealth with a small amount of money to start with. Through this model, it helps us with the concept of saving money smartly so that we can use it in future use, as compound interest can be seen as the foundation of our economy. Besides that, overall, this exponential growth model has given us the general mathematics concepts in order to understand more about where we are placing our money, what happens behind that, and with that knowledge, you can be more conscious and can carry it out in life for your job and much more.
Work Cited:
Andy Kiersz. The same basic math concept behind your retirement account's growth explains why it feels like the robots are taking over. INSIDER, 2 Feb. 2019, www.businessinsider.com/exponential-growth-moores-law-compound-interest-2019-2.
JAMES CHEN. Exponential Growth. Investopedia , 18 Oct. 2021, www.investopedia.com/terms/e/exponential-growth.asp.Accessed 29 Dec. 2021.
JASON FERNANDO . Compound Interest. Investopedia , 16 Feb. 2021, www.investopedia.com/terms/c/compoundinterest.asp. Accessed 29 Dec. 2021.
Kate Ashford, and Benjamin Curry. The Life-Changing Magic Of Compound Interest. forbes , 21 Nov. 2020, www.forbes.com/advisor/investing/compound-interest/. Accessed 29 Dec. 2021
Matthew Goldberg. What is the average interest rate for savings accounts? Bankrate , 30 Nov. 2021, www.bankrate.com/banking/savings/average-savings-interest-rates/#:~:text=The%20national%20average%20interest%20rate,ll%20earn%20on%20your%20saving. Accessed 29 Dec. 2021
Motley Fool Staff. How Often Is Interest Accrued on a Savings Account? The Motel Fool, 24 Aug. 2017, www.fool.com/saving/how-often-is-interest-accrued-on-a-savings-account.aspx. Accessed 29 Dec. 2021
SHANE EDE. Money is a Finite Resource. beatingbroke , 6 Apr. 2012, www.beatingbroke.com/money-is-a-finite-resource/. Accessed 29 Dec. 2021
WHY IS THE GAP BETWEEN RICH AND POOR GROWING? elioministeri.com , 24 Aug. 2017, https://elioministeri.com/why-is-the-gap-between-rich-and-poor-growing/ Accessed 29 Dec. 2021
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