After Placing $8,000 In A Savings Account Paying Annual Comp

After Placing 8000 In A Savings Account Paying Annual Compound In

Calculate the future value of an $8,000 deposit in a savings account with 8% annual compound interest over 8 years. Determine the most beneficial choice among receiving $2,000 today, $10,000 in 10 years, or $31,000 in 29 years, assuming a 12% annual return. For a loan of $180,000 with a 10% monthly compounded interest rate, find the repayment periods for monthly payments of $3,500 and $4,000. Compute the present value of a perpetual, growing cash flow of $80,000 at the end of one year, growing at 5%, discounted at 10%. Lastly, determine the lump sum needed today to fund withdrawals of $9,000 annually for 8 years starting 11 years from now, plus an $18,000 final year amount, with a 6% interest rate.

Paper For Above instruction

Financial mathematics provides essential tools for making informed decisions about savings, investments, loans, and other financial instruments. The problems presented involve fundamental concepts such as compound interest, present and future value calculations, amortization of loans, and valuation of perpetuities and delayed annuities. This paper explores these concepts in detail, illustrating their applications through the specific examples provided.

Future Value of a Lump Sum Investment

The initial problem involves calculating the future value of an $8,000 deposit in a savings account offering 8% annual compound interest over 8 years. The compound interest formula is given by:

FV = PV × (1 + r)^n

where PV is the present value ($8,000), r is the annual interest rate (8% or 0.08), and n is the number of years (8). Substituting these values:

FV = 8000 × (1 + 0.08)^8 ≈ 8000 × 1.85093 ≈ $14,807.44

Hence, after 8 years, the deposit will grow to approximately $14,807.44. This calculation demonstrates how compound interest accumulates over time, emphasizing the importance of the time horizon and the interest rate in investment growth.

Optimal Choice of Investment Timing Based on Discounted Values

The second scenario involves choosing between receiving $2,000 today, $10,000 in 10 years, or $31,000 in 29 years, with an assumed return rate of 12%. To compare these options, we calculate the present value (PV) of each using:

PV = Future Value / (1 + r)^n

For $2,000 today, PV is simply $2,000. For $10,000 in 10 years:

PV = 10,000 / (1 + 0.12)^10 ≈ 10,000 / 3.10585 ≈ $3,220.77

For $31,000 in 29 years:

PV = 31,000 / (1 + 0.12)^29 ≈ 31,000 / 23.958 ≈ $1,292.65

Comparing these present values, accepting $2,000 now yields the highest immediate value. Alternatively, if available for investment at 12%, receiving $2,000 today is optimal. The future sums, discounted to their present equivalent, show that immediate receipt provides the most value, underscoring the significance of the time value of money.

Loan Amortization with Monthly Compounding

Loan repayment calculations involve understanding the amortization schedule, especially with monthly compounding interest. The loan of $180,000 at 10% annual interest compounded monthly has a monthly interest rate:

i = 0.10 / 12 ≈ 0.00833333

The monthly payment p is given by the amortization formula:

p = PV × [i / (1 - (1 + i)^-n)]

where n is the number of payments. To find the number of months (n) for payments of $3,500:

n = -ln(1 - (PV × i) / p) / ln(1 + i)

Calculating:

n = -ln(1 - (180,000 × 0.00833333) / 3,500) / ln(1 + 0.00833333)


= -ln(1 - 1,500 / 3,500) / ln(1.00833333)


= -ln(1 - 0.42857)/ 0.0083219


= -ln(0.57143) / 0.0083219 ≈ 0.5607 / 0.0083219 ≈ 67.38 months

which is approximately 5.62 years. Similarly, for $4,000 monthly payments:

n = -ln(1 - (180,000 × 0.00833333) / 4,000) / ln(1.00833333)


= -ln(1 - 1,500 / 4,000) / 0.0083219


= -ln(1 - 0.375) / 0.0083219


= -ln(0.625) / 0.0083219 ≈ 0.470 / 0.0083219 ≈ 56.56 months

This corresponds to approximately 4.71 years. These calculations highlight how increasing monthly payments decreases the loan payoff period significantly, which is crucial for debt management.

Present Value of a Growing Perpetuity

The problem involves calculating the present value of a perpetuity that pays $80,000 at the end of one year, with payments growing at 5% indefinitely, discounted at 10%. The formula for a growing perpetuity is:

PV = C / (r - g)

where C is the initial cash flow ($80,000), r is the discount rate (10% or 0.10), and g is the growth rate (5% or 0.05).

PV = 80,000 / (0.10 - 0.05) = 80,000 / 0.05 = $1,600,000

This illustrates that the present value of a perpetuity with growth exceeding the discount rate would be infinite or undefined, but since growth is less than the discount rate here, the formula provides a finite value, emphasizing the potential for perpetual income streams in financial planning.

Valuing a Deferred Annuity with Final Lump Sum

Finally, to determine today's lump sum such that starting 11 years from now, one can withdraw $9,000 annually for 8 years and an additional $18,000 in the 8th year, with a 6% interest rate, involves two parts: the present value at year 11 of the withdrawals, and then discounting that amount back to present.

First, calculate the present value at year 11 of the 8-year annuity of $9,000:

PV at year 11 = 9000 × [1 - (1 + i)^-8] / i


= 9000 × [1 - (1 + 0.06)^-8] / 0.06


= 9000 × [1 - 1/1.59385] / 0.06


= 9000 × [1 - 0.6274] / 0.06


= 9000 × 0.3726 / 0.06 ≈ 9000 × 6.210 ≈ $55,890

Adding the final $18,000 in year 18, discounted back 11 years at 6% per year:

PV of lump sum = 18000 / (1 + 0.06)^{7} ≈ 18000 / 1.5036 ≈ $11,974.56

The total present value at year 11 is approximately:

$55,890 + $11,974.56 ≈ $67,864.56

Finally, discount this combined value back to today over 11 years:

PV today = 67,864.56 / (1 + 0.06)^{11} ≈ 67,864.56 / 1.8983 ≈ $35,763.24

Therefore, an initial deposit of approximately $35,763.24 today would suffice to meet these future withdrawal needs, illustrating how the time value of money and present value calculations can aid in planning long-term financial goals.

Conclusion

The applications examined demonstrate the foundational importance of compound interest, the time value of money, and financial valuation techniques in personal and corporate finance. Whether investing savings, comparing cash flow options, paying off loans, valuing perpetual streams, or planning future withdrawals, these concepts provide critical tools to make informed financial decisions. The calculations underscore that understanding the effects of interest rates, compounding frequency, and timing significantly impacts the magnitude of financial outcomes, guiding individuals and businesses alike toward more optimal financial strategies.

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