To compare the three types of investments over time, we can use the formulas for each interest calculation method.
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Simple Interest (SI): The formula for calculating the future value with simple interest is: \[ A = P(1 + rt) \] where:
- \(A\) is the amount of money accumulated after n years, including interest.
- \(P\) is the principal amount (the initial sum of money).
- \(r\) is the annual interest rate (in decimal).
- \(t\) is the time the money is invested for in years.
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Compound Interest (CI) - Compounded Semi-Annually: The formula for compound interest when compounded semi-annually is: \[ A = P\left(1 + \frac{r}{n}\right)^{nt} \] where:
- \(n\) is the number of times that interest is compounded per year (for semi-annual compounding, \(n = 2\)).
- The other variables are the same as in the simple interest formula.
In this case, the formula becomes: \[ A = P\left(1 + \frac{r}{2}\right)^{2t} \]
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Continuous Compounding: The formula for continuously compounded interest is: \[ A = Pe^{rt} \] where:
- \(e\) is the base of the natural logarithm (approximately equal to 2.71828).
Comparison Over Time
To analyze the growth of each investment over time, let's consider how each account performs as \(t\) increases, assuming the same principal \(P\) and annual interest rate \(r\).
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Simple Interest generally grows linearly, as the interest earned each year is the same percentage of the initial principal.
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Compounded Interest (Semi-Annual) grows at a faster rate than simple interest because interest is earned on both the initial principal and the interest that has already been added to the account twice a year.
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Continuous Compounding produces the highest return out of all three methods. It consistently earns interest on the interest for an infinite number of compounding periods, resulting in exponential growth.
Conclusion
Over time:
- Continuous Compounding yields the highest returns, followed by Compounded Semi-Annually, and then Simple Interest which grows the slowest.
- The differences will become more pronounced over longer periods due to the effect of compounding interest. As time increases, the gap in final amounts among the three methods widens, favoring continuous compounding the most.
In summary:
- \( A_{SI} = P(1 + rt) \)
- \( A_{CI(Semi-Annually)} = P\left(1 + \frac{r}{2}\right)^{2t} \)
- \( A_{CI(Continuously)} = Pe^{rt} \)
For practical purposes, using a specific numerical example with actual values can show clearer distinctions over set periods.