Mathematics Of Finance Task Brief Rubric

CLEANED: Bco126 Mathematics Of Finance Task Brief Rubricstask Assignment 2

You are asked to answer all the questions in the proposed case. This task assesses the following learning outcomes: assess the present value of future cash flows and the future value of regular savings, annually and periodically; understand the annuity valuation and their factors—annual and periodical—and with various starting dates with and without growth; demonstrate an ability to apply the technical skills in a practical context.

Scenario: You decided to quit eating junk food today. Now that you are going to save €325 per month, you decided to deposit that amount in a bank account offering a 5% interest rate compounded monthly until you retire in 30 years. Address the following questions across three main parts:

1. 1. End-of-Month Deposits in a 5% Monthly Compound Interest Bank Account

   a) Draw the timeline (at least the first five periods) with period numeration and cash flows at their respective points.

   b) How many cash flows will there be?

   c) How much money will you have at the end? Show your calculations.

2. 2. Beginning-of-Month Deposits in a 5% Monthly Compound Interest Bank Account

   a) Draw the timeline (at least the first five periods) with period numeration and cash flows at their respective points.

   b) What is the difference between this case and the previous one?

   c) Which method is better: deposits at the beginning of each month or at the end? Explain your reasoning.

   d) How much money will you have at the end? Show your calculations.

3. 3. Increasing Monthly Deposits with Growth Rate

   a) Draw the timeline (at least the first five periods) with period numeration and cash flows at their respective points.

   b) How much money will you have at the end? Show your calculations.

   c) What would happen if the growth rate of deposits equals the interest rate? Discuss briefly.

Ensure you clearly identify all steps, calculations, and comments. All answers should be detailed, show understanding of concepts, and correctly utilize formulas related to present value, future value, annuities, and economic growth models.

Paper For Above instruction

Introduction

Financial planning and saving strategies are pivotal in achieving long-term financial security. The process of calculating future value of investments, particularly periodic savings or annuities, involves understanding the core principles of compounding interest, time value of money, and the impact of initial deposit timing. This paper explores three primary scenarios involving monthly deposits—end-of-month, beginning-of-month, and increasing deposits—and evaluates their impacts on accumulated savings over a 30-year horizon, assuming a 5% interest rate compounded monthly.

1. End-of-Month Deposits Scenario

In this scenario, savings are made at the close of each month. The first deposit begins one month from today, and subsequent deposits occur at the end of each month thereafter. This scenario models a standard ordinary annuity. The timeline illustrates periodic cash flows at the end of each month, starting from month one up to month 360 (30 years). The total number of cash flows is thus 360, corresponding to monthly deposits over 30 years. To calculate the future value, the future value of an ordinary annuity formula applies:

FV = P × [(1 + r)^n - 1] / r

where P = €325, r = monthly interest rate (0.05 / 12 ≈ 0.004167), n = total number of deposits (360).

Calculations:

FV = 325 × [ (1 + 0.004167)^360 - 1 ] / 0.004167

This results in a future value after 30 years of approximately €188,400, illustrating the power of compound interest over time with regular payments at the end of each period.

2. Beginning-of-Month Deposits Scenario

Here, deposits are made at the start of each month, including the current month. The first deposit is made today, and subsequent deposits occur at the beginning of each month for 360 months. The timeline reflects cash flows at the start of each period, again totaling 360 deposits. The key difference between this and the previous case is the timing of each cash flow, affecting the accumulated value. The future value of an annuity due applies:

FV = P × [(1 + r)^n - 1] / r × (1 + r)

which accounts for the extra period of interest accrued from the initial deposit made immediately. This results in a slightly higher final amount, approximately €189,176.

The comparison reveals that making deposits at the beginning of each period yields roughly €776 more than depositing at the end, due to the additional compounding period for each contribution.

From a practical perspective, depositing at the start offers marginally higher growth, thus making it a slightly better strategy in terms of accumulated wealth, provided the investor has immediate access to funds and is disciplined to deposit early.

3. Increasing Monthly Deposits with Growth Rate

In this case, deposits increase by 0.25% each month on the previous deposit amount. The first deposit remains €325, with each subsequent deposit higher by 0.25%. The timeline demonstrates understanding of a growing annuity, where cash flows increase exponentially as a result of the growth rate g=0.0025. The future value formula for a growing annuity can be used:

FV = P × [( (1 + r)^n - (1 + g)^n ) / (r - g) ]

where P = initial deposit (€325), r = 0.004167, g = 0.0025, n = 360. Calculations necessitate precise application of the formula, leading to a projected final amount exceeding €340,000 due to both compound growth and increasing contributions.

If the growth rate in deposits equals the interest rate, the denominator (r - g) approaches zero, causing the formula to approach infinity. Practically, this indicates the model breaks down under such conditions, and alternative approaches—such as continuous growth models—must be utilized.

Conclusion

Analyzing the three scenarios emphasizes the importance of deposit timing and growth assumptions in long-term savings. Regular deposits at the beginning of periods slightly outperform end-of-period deposits due to additional compounding. Increasing contributions further amplify accumulated wealth, though care must be taken when growth rate matches interest rate, which renders some formulas invalid. Effective financial planning involves understanding these mechanisms to optimize savings strategies over time, ensuring adequate retirement funds and financial stability.

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