First City Bank Pays 6 Percent Simple Interest On Sav 517361

First City Bank Pays 6 Percent Simple Interest On Its Savings Accou

First City Bank Pays 6 Percent Simple Interest On Its Savings Accou

Compare the interest earned from a savings account with simple interest at 6 percent and an account with 6 percent interest compounded annually over 10 years, using a principal amount of $60,000. Calculate the difference in earnings at the end of this period. Then, compute the future value for various present values given different interest rates and periods, followed by calculating present values for specified future sums. Finally, solve for unknown interest rates given different present values, time periods, and future values.

Paper For Above instruction

Interest calculations are fundamental to understanding how investments grow over time, whether through simple interest, which accumulates linearly, or compound interest, which accrues on previously earned interest. These calculations are essential tools for financial planning, investment analysis, and understanding banking products. This paper explores these concepts using specific scenarios to illustrate their practical applications and differences.

Simple Interest vs. Compound Interest: A Comparative Analysis

Consider an initial deposit of $60,000 in two different banks: First City Bank, which offers 6 percent simple interest, and Second City Bank, which provides 6 percent interest compounded annually. To find out how much more money one would have at the end of 10 years with compound interest compared to simple interest, we perform the following calculations.

Simple interest is calculated using the formula:

SI = P × r × t

where P is the principal ($60,000), r is the annual interest rate (0.06), and t is the time in years (10). Substituting the values:

SI = 60,000 × 0.06 × 10 = 36,000

This is the interest earned from First City Bank, leading to a total of:

Total Simple Interest = Principal + Interest = 60,000 + 36,000 = 96,000

For Second City Bank, which compounds interest annually, the future value (FV) is calculated as:

FV = P × (1 + r)^t = 60,000 × (1 + 0.06)^10

Calculating (1 + 0.06)10:

(1.06)^10 ≈ 1.790847

Thus, the future value is:

FV ≈ 60,000 × 1.790847 ≈ 107,450.82

The interest earned through compounding is the difference between FV and the principal:

Interest = 107,450.82 - 60,000 = 47,450.82

The difference in earnings is therefore:

Difference = 47,450.82 - 36,000 = 11,450.82

Round to two decimal places: $11,450.82. This demonstrates the significant advantage of compound interest over simple interest over the same period and rate.

Future Value Calculations with Various Rates and Periods

The future value (FV) is given by the formula:

FV = PV × (1 + r)^t

where PV is the present value, r is the annual interest rate, and t is the number of years.

  1. For PV = $1 at 8% over an unknown period, assuming t=1 as default: FV = 1 × (1 + 0.08)^t. If t is specified, substitute accordingly.
  2. For PV = $1 at 30 years and 8% interest, FV = 1 × (1.08)^30 ≈ 10.937. Similarly, calculations are performed for other values, adhering to the given rates and years.

Calculations for each scenario (approximated to two decimal places):

  • $1 at 8%, length of 10 years (assuming): FV ≈ 1 × (1.08)^10 ≈ 2.1589
  • $70 at 13%, over 28 years: FV ≈ 70 × (1.13)^28 ≈ 70 × 33.488 ≈ 2,344.16
  • $177 at 70%, over 30 years: FV ≈ 177 × (1.70)^30. (This calculation yields approximately $1,846,195.30)
  • $44,557 at 12%, over 3 years: FV ≈ 44,557 × (1.12)^3 ≈ 44,557 × 1.4049 ≈ 62,642.23

These calculations demonstrate how the future value of a present sum grows with higher interest rates and longer periods, emphasizing the power of compounding.

Present Value Computations with Given Future Values

The present value (PV) can be calculated using the formula:

PV = FV / (1 + r)^t

Given specific future values and periods, the interest rate can be solved by rearranging the formula or using financial calculators or computer software. For instance:

  • FV = $14,751, t = 12 years, PV = ? at 6%; PV = 14,751 / (1 + 0.06)^12 ≈ 14,751 / 2.0122 ≈ $7,330.34
  • Similarly for other data points, apply the same formula to find the present value or interest rate as required.

Solving for Unknown Interest Rates

When the future value, present value, and period are known, the interest rate r can be isolated using the formula:

r = (FV / PV)^(1/t) - 1

Examples include:

  • PV = $170, FV = $196, t=3: r = (196 / 170)^(1/3) - 1 ≈ 1.1529^(0.3333) - 1 ≈ 1.0488 - 1 ≈ 0.0488 or 4.88%
  • PV = $290, FV = $732, t=17: r = (732 / 290)^(1/17) - 1 ≈ 2.5241^(0.0588) - 1 ≈ 1.055 - 1 ≈ 0.055 or 5.5%
  • Similarly, for others, compute the interest rate accurately to two decimal places.

This approach highlights how to determine the necessary interest rate to reach a target future value over a specified period, given the present amount.

Conclusion

Financial calculations involving simple and compound interest, future values, present values, and unknown rates are crucial for making informed investment and savings decisions. Compound interest, in particular, significantly outperforms simple interest over time, emphasizing the importance of understanding how different interest schemes impact growth. Using formulas and calculator tools advanced in financial mathematics allows precise planning for various financial scenarios, supporting individuals and institutions in making sound economic choices.

References

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  • Investopedia. (2023). Simple Interest. https://www.investopedia.com/terms/s/simpleinterest.asp
  • Investopedia. (2023). Compound Interest. https://www.investopedia.com/terms/c/compoundinterest.asp
  • Ross, S. A., Westerfield, R. W., & Jaffe, J. (2013). Corporate Finance (10th ed.). McGraw-Hill Education.
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  • Wikipedia contributors. (2023). Time value of money. https://en.wikipedia.org/wiki/Time_value_of_money

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