Assignment 2 Lasa 1 Compound Interest: A Common Component

Assignment 2 Lasa 1 Compound Interesta Common Component Of Investing

Assignment 2: LASA 1: Compound Interest A common component of investing money is to take advantage of a financial institution’s willingness to pay compound interest. Compound interest is basically interest paid on a deposit that continually accumulates interest. In general, the formula for compound interest can be represented by the following exponential function: In this formula, P(t) represents the total money in the account after t years given the interest rate k which is compounded continuously. In this assignment, you will use this formula to explore the affect that compound interest can have over a period of time and at different interest rates.

Directions: Select an amount of money that you would like to invest (for example $1000.00). This will be your P₀ value. Let your interest rate be k = 0.5%. Write out the exponential function using the P₀ and k values you have. Determine the value of your investment after 1, 5, and 10 years. Now, find the doubling time T for your investment.

In other words, at what time would your initial deposit double in value? Repeat steps 3 through 5 for k = 1%. Repeat steps 3 through 5 for k = 1.5%. In a Microsoft Word document, prepare a report that includes answers to the following:

• Report the results of the calculations you performed above.
• What effect did changing the interest rate have on the rate at which your investment grew?
• What effect did changing the interest rate have on the doubling time (time until your initial deposit doubled in size)?
• Assuming the money is invested in a savings account, are the interest rates used realistic today?
• Consider the formula used; is it a realistic approximation of expected returns or are other factors involved?
• Explore other investment accounts offered at banks—do they use compound interest, and what are typical rates?
• Compare and contrast simple and compound interest—their calculation methods and advantages.
• State which method you prefer for your investments and justify your choice.

By Wednesday, August 6, 2014, submit your assignment to the M3: Assignment 2 Dropbox. All responses should adhere to APA attribution rules.

Paper For Above instruction

Investment growth through compound interest reflects the power of earning interest on both the principal and accumulated interest from previous periods. Understanding how various interest rates influence investment growth and doubling time is crucial for making informed financial decisions. This paper explores the mathematical modeling of continuous compound interest at different rates, analyzes the effects on investment value over time, and evaluates the realism of these models in the context of modern banking and investment options.

To start, I selected an initial investment amount of $1,000. The exponential formula for continuous compounding is given by P(t) = P₀ e^{kt}, where P₀ is the principal amount, k is the interest rate expressed as a decimal, and t is time in years. For the base case, I used an interest rate of 0.5%, or 0.005 in decimal form. The function becomes P(t) = 1000 e^{0.005t}.

Calculating the investment value after 1, 5, and 10 years yielded insights into the growth pattern. For k = 0.5%:

• After 1 year: P(1) ≈ 1000 e^{0.0051} ≈ 1000 * 1.0050125 ≈ $1,005.01
• After 5 years: P(5) ≈ 1000 e^{0.0055} ≈ 1000 * 1.025315 ≈ $1,025.32
• After 10 years: P(10) ≈ 1000 e^{0.00510} ≈ 1000 * 1.051271 ≈ $1,051.27

Next, I calculated the doubling time T, which occurs when P(T) = 2000. Setting up the equation:

2000 = 1000 * e^{0.005T} → 2 = e^{0.005T} → ln(2) = 0.005T → T = ln(2)/0.005 ≈ 0.6931/0.005 ≈ 138.62 years.

This indicates that at 0.5%, it would take approximately 138.62 years for the initial investment to double, signifying a slow growth rate.

Repeating this process for higher rates: 1% and 1.5%, reveals how interest rates influence growth

• At 1% (k=0.01): T ≈ 69.31 years.
• At 1.5% (k=0.015): T ≈ 46.21 years.

These results confirm that increasing the interest rate significantly accelerates both the growth of the investment and the time to double its size. For example, doubling time at 1.5% roughly halves compared to 0.5%, illustrating an inverse relationship between interest rate and doubling period.

Regarding the realism of these rates today, they are somewhat optimistic considering current savings account rates in many banks typically hover below 1%, often around 0.05% to 0.5%, with higher rates offered by investments like certificates of deposit or government bonds but still below 2%. As such, the model simplifies the dynamics, assuming continuous compounding without account fees or taxes, which might not be reflective of actual returns.

The formula used assumes perpetual, continuous compounding, which is a mathematical idealization. Real-world factors such as fluctuating interest rates, account fees, inflation, and taxes affect actual returns and make the model a simplified approximation. Over longer periods, these factors can significantly reduce effective gains.

Besides savings accounts, several other investment instruments utilize compound interest, including mutual funds, retirement accounts (such as IRAs and 401(k)s), and certain types of bonds. These often compound interest periodically (annually, semi-annually, quarterly), with rates varying widely depending on the product, risk level, and market conditions.

Conversely, simple interest is calculated solely on the principal, using the formula I = P r t, which results in linear growth over time. While easier to compute, it generally yields lower returns compared to compound interest for the same rates and periods. Given the advantages of compound interest—particularly its exponential growth potential—most financial institutions favor products that pay compound interest.

In my opinion, investment programs based on compound interest are preferable due to their greater long-term growth potential. The exponential nature of compounding allows individuals to maximize returns by reinvesting earned interest, whereas simple interest offers more predictable but lower growth. For long-term savings goals, compound interest provides a more effective strategy, aligning with the principles of maximizing wealth accumulation. However, investors must also consider risk, liquidity, and other factors alongside the compounding effects.

References

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