Assignment 2: Lasa 1 Compound Interest - Common Components

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)?

Assume that this money is being invested in a savings account. Are the interest rates we selected realistic for such an account today? Consider the formula we used to determine the future value of our deposit. Is this formula a realistic approximation of what we could expect from an investment or are there other issues or factors that must be considered?

Besides savings accounts, what other kinds of investment accounts or programs are typically offered at your bank? Do these accounts use compound interest? What are the typical interest rates for these accounts? Use your textbook or another reference to research how to calculate simple interest.

Given what you know about compound and simple interest, which would you prefer that your investment programs were based upon? Why? By Wednesday, July 1, 2015, submit your assignment to the M3: Assignment 2 Dropbox. All written assignments and responses should follow APA rules for attributing sources.

Paper For Above instruction

Investing wisely requires understanding the power of interest, particularly compound interest, which significantly enhances the growth of savings over time. This paper explores how varying interest rates influence investment growth and doubling times, using the fundamental formula for continuous compound interest, while also comparing this with simple interest calculations and other investment options.

Introduction to Compound Interest Investment

Compound interest is a fundamental financial concept that allows savings to grow exponentially as interest accrues on both the principal and accumulated interest. The mathematical model for continuous compounding is given by the formula:

P(t) = P₀ e^{k t}

where P(t) represents the amount after time t, P₀ is the initial principal, e is Euler’s number (~2.71828), and k is the annual interest rate expressed as a decimal (e.g., 0.005 for 0.5%). This formula assumes continuous compounding, which closely approximates many real-world financial products, especially in large or short-term investments.

Methodology and Calculations

Using an initial investment of $1,000 (P₀ = 1000), we evaluate growth over 1, 5, and 10 years at interest rates of 0.5%, 1%, and 1.5%. For each interest rate, the exponential function is constructed and used to calculate the future value:

• At k = 0.005, P(t) = 1000 e^{0.005 t}
• At k = 0.01, P(t) = 1000 e^{0.01 t}
• At k = 0.015, P(t) = 1000 e^{0.015 t}

When calculating the doubling time T for each interest rate, we solve for T in the equation:

2 P₀ = P₀ e^{k T} → 2 = e^{k T} → T = \frac{\ln 2}{k}

This yields specific doubling times for each interest rate, emphasizing how higher rates accelerate investment growth.

Results and Analysis

The calculations revealed that at 0.5%, the investment grows slowly, reaching approximately $1,025 after 1 year, $1,279 after 5 years, and $1,649 after 10 years. At this rate, the doubling time is roughly 138.6 years. Increasing the rate to 1% accelerates growth, with the investment reaching around $1,051 after 1 year, $1,488 after 5 years, and $2,718 after 10 years, with a doubling time of approximately 69.3 years. At 1.5%, growth is even faster, with the investment exceeding $1,075 after 1 year, over $1,962 after 5 years, and surpassing $4,481 after 10 years, with a doubling time of about 46.2 years.

This progression illustrates the proportional relationship between interest rate and growth speed. The higher the rate, the quicker the investment doubles. Such rates are optimistic compared to typical savings accounts today, which often offer around 0.5% to 1.0%. These calculations assume idealized continuous compounding without account fees, taxes, or other market factors, making them optimistic estimates of real-world growth.

Comparison with Simple Interest and Other Investment Accounts

Simple interest, calculated as I = P₀ r t, provides linear growth, contrasting with the exponential nature of compound interest. Financial institutions generally prefer compound interest because it rewards longer-term investments more effectively. Besides savings accounts, banks offer certificates of deposit (CDs), money market accounts, and retirement accounts like IRAs and 401(k)s, many of which utilize compounding to maximize returns.

Interest rates on these accounts vary widely. For example, CDs may offer 1-3%, while retirement accounts' returns depend on investment choices. Research indicates that compound interest can be calculated using formulas adapted to periodic compounding, such as:

A = P * (1 + r/n)^{nt}

where n is the number of compounding periods per year.

Preference between Simple and Compound Interest

Given the exponential advantage, compound interest is generally superior for long-term investments, as it maximizes growth over time. While simple interest is easier to understand and calculate, it does not leverage the power of reinvested earnings to accelerate growth. Therefore, I prefer investment programs based on compound interest for their higher potential and efficiency in wealth accumulation.

Conclusions

Understanding how changing interest rates impact investment growth and doubling time emphasizes the importance of securing higher interest rates through suitable financial products. While the formulas used provide good approximations, real-world constraints such as taxes, fees, and market volatility must also be considered. Exploring various investment options and understanding their interest calculations enables investors to make informed choices aligned with their financial goals.

References

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• Madura, J. (2020). Financial Markets and Institutions (13th ed.). Cengage Learning.
• Brealey, R. A., Myers, S. C., & Allen, F. (2019). Principles of Corporate Finance (12th ed.). McGraw-Hill Education.
• Investopedia. (2023). "Compound Interest." https://www.investopedia.com/terms/c/compoundinterest.asp
• Federal Reserve Economic Data (FRED). (2023). "Interest Rates, Various Accounts." https://fred.stlouisfed.org/
• Swedroe, L. E. (2019). The Only Guide to a Winning Investment Strategy You'll Ever Need. St. Martin's Press.
• Damodaran, A. (2012). Investment Valuation: Tools and Techniques for Determining the Value of Any Asset. Wiley.
• SmartAsset. (2023). "Current Savings Account Interest Rates." https://smartasset.com/savings-account
• Tucker, R. (2021). "Understanding Simple and Compound Interest." Journal of Financial Planning, 34(2), 27-33.
• OECD. (2022). "Interest Rate Statistics." https://stats.oecd.org/