Finding the Monthly Compounding Interest Rate

Introduction

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When dealing with financial investments, understanding the concept of compound interest is crucial. This article will explore the step-by-step process to determine the monthly compounding interest rate necessary for an investment of $360,000 to yield $9,000 over an 8-month period. We will utilize the compound interest formula and present a thorough explanation of the calculations involved.

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Understanding Compound Interest

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Compound Interest Formula

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Compound interest is calculated using the formula:

r r [ A P left(1 frac{r}{n}right)^{nt} ]r r

Where:

r r - A is the amount of money accumulated after n months, including interest.r - P is the principal amount (initial investment amount).r - r is the annual interest rate (decimal).r - n is the number of times interest is compounded per year.r - t is the time the money is invested for, in years.

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Initial Conditions and Variables

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In this specific scenario:

r r - A 360,000 9,000 369,000 (the final amount after 8 months).r - P 360,000 (the initial investment).r - n 12 (since interest is compounded monthly).r - t frac{8}{12} frac{2}{3} (the time the money is invested, in years).

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Solving for the Monthly Interest Rate

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We can rearrange the compound interest formula to solve for the monthly interest rate ( r ).

r r [ 369,000 360,000 left(1 frac{r}{12}right)^{12 times frac{2}{3}} ]

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This simplifies to:

r r [ 369,000 360,000 left(1 frac{r}{12}right)^8 ]

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Dividing both sides by 360,000:

r r [ frac{369,000}{360,000} left(1 frac{r}{12}right)^8 ]

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Calculating the left side:

r r [ 1.025 left(1 frac{r}{12}right)^8 ]

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Next, take the eighth root of both sides:

r r [ 1 frac{r}{12} approx 1.025^{frac{1}{8}} ]

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Calculating ( 1.025^{frac{1}{8}} ):

r r [ 1 frac{r}{12} approx 1.003066 ]

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Now, subtract 1 from both sides:

r r [ frac{r}{12} approx 0.003066 ]

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Finally, multiply by 12 to find ( r ):

r r [ r approx 0.03679 ] r r

Converting to a percentage:

r r [ r approx 3.68% ] r r

Alternative Method: Using Algebra

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Alternatively, we can use the algebraic approach:

r r [ 9,000 360,000 left(1 frac{r}{12}right)^8 - 360,000 ]

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Let ( 1 frac{r}{12} x ). Then:

r r [ x^8 - 1 frac{9,000}{360,000} frac{1}{40} ]

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This simplifies to:

r r [ x^8 frac{41}{40} 1.025 ]

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Since ( x 1.0031 ), we have:

r r [ 1 frac{r}{12} 1.0031 ]

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Therefore:

r r [ frac{r}{12} approx 0.0031 ] r r

Converting to a percentage:

r r [ r approx 3.72% ] r r

Comparison with Simple Interest

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To compare, we can also calculate the simple interest:

r r [ Y P cdot r cdot t ]r r

Where:

r r - ( Y 9,000 ) (final additional amount).r - ( P 360,000 ) (initial investment).r - ( n 12 ) (times compounded per year).r - ( t frac{8}{12} frac{2}{3} ) (time in years).

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The simple rate is:

r r [ r frac{12 cdot Y}{P cdot t} frac{12 cdot 9,000}{360,000 cdot frac{2}{3}} 0.0375 ] r r

Thus, the simple rate is 3.75%, while the compound rate is approximately 3.71%.

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Conclusion

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In conclusion, the monthly compounding interest rate required for an initial investment of $360,000 to yield $9,000 over an 8-month period is approximately 3.71%. This calculation demonstrates the power of compound interest compared to simple interest, highlighting the importance of understanding financial concepts for investment planning and analysis.