Find The Compound Amount And Interest Earned

Find The Compound Amount And The Amount Of Interest Earned By The

Find the compound amount and the amount of interest earned by various deposits and investments based on specified interest rates, compounding frequencies, and investment durations. Additionally, determine interest rates given future value, compute present values for given future amounts, and analyze investments and loans by calculating interest, face value of bonds, and required payments to meet financial goals. The tasks include working with continuous, quarterly, monthly, and annual compounding, as well as simple interest calculations, bond valuation, annuities, and loan amortization.

Paper For Above instruction

Financial mathematics plays a crucial role in understanding investment growth, loan repayment, and bond valuation. It involves calculating compound interest, simple interest, present and future values, and annuities. This paper explores various calculations related to these financial concepts, illustrating their application through specific examples involving deposits, bonds, and loans.

Compound Interest and Future Value Calculations

Compound interest is fundamental to understanding how investments grow over time. When interest is compounded continuously, the amount after a certain period is determined using the formula:

A = P e^rt

where P is the principal, r is the annual interest rate (decimal), t is the time in years, and e is Euler’s number (~2.71828). For example, a deposit of $5,100 at 3.5% interest compounded continuously for 7 years yields:

A = 5100 × e^{0.035 × 7} ≈ 5100 × e^0.245 ≈ 5100 × 1.2775 ≈ $6,514.25

The interest earned is then $6,514.25 - $5,100 ≈ $1,414.25.

When interest is compounded periodically, such as quarterly, annually, or monthly, the formula becomes:

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

where n is the number of compounding periods per year. For instance, a $7,700 deposit at 9% compounded quarterly for 8 years gives:

A = 7700 × (1 + 0.09/4)^4×8 ≈ 7700 × (1 + 0.0225)³² ≈ 7700 × 1.0225³² ≈ 7700 × 2.019 ≈ $15,535.44

The interest earned is approximately $15,535.44 - $7,700 = $7,835.44.

Calculations with Different Compounding Frequencies and Timeframes

Daily compounding involves n=365, while monthly involves n=12. Each modifies the growth of the investment accordingly.

For example, a $5,000 account at 1.23% compounded daily for 1 year results in:

A = 5000 × (1 + 0.0123/365)^365×1 ≈ 5000 × (1 + 0.0000337)³⁶⁵ ≈ 5000 × 1.0124 ≈ $5,062.10

Similarly, a $15,000 deposit at 1.02% compounded monthly for 3 years becomes:

A = 15000 × (1 + 0.0102/12)^12×3 ≈ 15000 × (1 + 0.00085)³⁶ ≈ 15000 × 1.0318 ≈ $15,477.00

Calculating Interest Rate and Present Value

Given the future value and principal, the interest rate can be found using the compound interest formula rearranged as:

r = (A/P)^1/(nt) - 1

For example, a deposit of $7,000 grows to $7,996 in 5 years, so:

r = (7996/7000)^1/(1×5) - 1 ≈ (1.1414)^0.2 - 1 ≈ 1.0264 - 1 ≈ 0.0264 or 2.64%

The present value for future amounts is calculated as:

PV = A / (1 + r/n)^nt

This helps determine the current investment needed to reach a future goal. For a future value of $12,782.78 at 6.3% compounded annually for 5 years:

PV = 12782.78 / (1 + 0.063)⁵ ≈ 12782.78 / 1.063⁵ ≈ 12782.78 / 1.349 ≈ $9,481.83

Bond Valuation and Annuities

The face value of a zero-coupon bond can be found as:

FV = PV × e^rt

Given a bond with a price of $17,000, 16-year maturity, and 3.4% interest rate:

FV ≈ 17000 × e^{0.034 × 16} ≈ 17000 × e^0.544 ≈ 17000 × 1.722 ≈ $29,314

This represents the bond's face value at maturity.

An ordinary annuity involves fixed payments over time. Its present value is calculated as:

PV = P × [(1 - (1 + r)^-n) / r]

For example, annual payments of $1,100 over 11 years at 5% yield:

PV = 1100 × [(1 - (1 + 0.05)^-11) / 0.05] ≈ 1100 × 8.5302 ≈ $9,383.22

Loan Payments and Amortization

To determine required payments for loans, or to amortize a loan, the formula for a fixed payment is:

P = (r × PV) / [1 - (1 + r)^-n]

For a $2,400 loan at 4% quarterly interest over 9 payments, the quarterly interest rate is 0.01, so:

P = (0.01 × 2400) / [1 - (1 + 0.01)^-9] ≈ 24 / (1 - 0.9135) ≈ 24 / 0.0865 ≈ $277.26

Conclusion

Understanding and applying these financial formulas enable individuals and organizations to plan investments, loans, and savings effectively. Calculations involving different compounding frequencies, interest rates, and timeframes illustrate how small changes can significantly influence the growth of investments or repayment obligations. Mastery of these concepts forms a foundational component of financial literacy and economic decision-making.

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