You Wish To Deposit 500 Per Month Into An Account For 36 Mon

You Wish To Deposit 500 Per Month Into An Account For 36 Months A

You wish to deposit $500 per month into an account for 36 months. Assume your interest rate is equal to the prime interest rate.

a) How much do you have (total) in the account after 36 months?

b) How much of that total is interest?

Paper For Above instruction

Introduction

Saving money through regular deposits in a financial account is a common strategy to build wealth over time. Understanding how regular contributions, combined with interest accumulation, influence the total savings is essential for making informed financial decisions. This paper explores the calculation of the total amount accumulated after depositing $500 monthly over 36 months, assuming the prime interest rate, and determines the portion of that total attributable to interest earnings.

Calculating the Total Accumulation

The scenario involves making consistent monthly deposits of $500 into an interest-bearing account over a period of 36 months. The problem assumes that the interest rate matches the prime rate, which fluctuates over time but is often considered the baseline for short-term variable interest rates. For simplicity, and due to the lack of a specific prime rate value for the period, we will denote the interest rate as an annual nominal rate compounded monthly, represented by r.

The total accumulated amount can be modeled using the future value of an ordinary annuity formula, as deposits are made at the end of each month:

FV = P * \(\frac{(1 + i)^n - 1}{i}\)

Where:

  • P = monthly deposit = $500
  • i = monthly interest rate = annual interest rate / 12
  • n = total number of deposits = 36

Assuming the prime rate is, for example, 4% annually (as a typical historical average), then:

  • i = 0.04 / 12 ≈ 0.003333

Calculating the future value:

FV = 500 * \(\frac{(1 + 0.003333)^{36} - 1}{0.003333}\)

Computing the power term:

(1 + 0.003333)^{36} ≈ e^{36 \ln(1.003333)} ≈ e^{36 0.003327} ≈ e^{0.1198} ≈ 1.1277

Therefore:

FV ≈ 500 \(\frac{1.1277 - 1}{0.003333}\) ≈ 500 \(\frac{0.1277}{0.003333}\) ≈ 500 * 38.31 ≈ $19,155

Thus, at the end of 36 months, the total amount in the account would be approximately $19,155, given a 4% prime interest rate.

Calculating the Interest Earned

To find how much of the total is interest, we first determine the total amount deposited:

Deposits made:

  • $500/month * 36 months = $18,000

Interest earned is then: total accumulated - total deposits

Interest ≈ $19,155 - $18,000 = $1,155

Conclusion

Investing $500 monthly into an account with an assumed prime interest rate of 4% over 36 months results in an approximate total of $19,155. Of this, around $1,155 is attributable to interest earnings, demonstrating how regular deposits can grow through compounded interest over time. These calculations highlight the importance of understanding both deposit behavior and interest compounding in effective personal financial planning.

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