After Placing 8000 In A Savings Account Paying Annual Compou

1after Placing 8000 In A Savings Account Paying Annual Compound Int

1. After placing $8,000 in a savings account paying annual compound interest of 7%, calculate the amount that will accumulate if it is left for 10 years?

2. After placing $13,000 in a savings account paying annual compound interest of 4%, Leona will accumulate what amount if she leaves the money in the bank for 3 years?

3. What is the present value of a $650 perpetuity discounted back to the present at 12%? What is the present value of the perpetuity?

4. What is the present value of a perpetual stream of cash flows that pays $80,000 at the end of one year and grows at a rate of 7% indefinitely? The rate of interest used to discount the cash flows is 9%.

5. How much do you have to deposit today so that beginning 11 years from now, you can withdraw $9,000 a year for the next eight years (periods 11 through 18) plus an additional amount of $18,000 in the last year (period 18)? Assume an interest rate of 8%. What is the amount of money you would need to deposit today?

Paper For Above instruction

This paper provides comprehensive solutions to various financial calculations involving compound interest, perpetuities, and present value valuations. The objective is to demonstrate a clear understanding of time value of money concepts, including future value (FV), present value (PV), and perpetuity valuation, applying relevant mathematical formulas and financial principles.

1. Future Value of a Lump Sum Investment

To calculate the future value of an initial lump sum invested at compound interest, we use the formula:

FV = PV × (1 + r)^n

where PV is the present value or initial investment, r is the annual interest rate, and n is the number of years. For the first scenario, PV = $8,000, r = 7% or 0.07, n = 10 years:

FV = 8000 × (1 + 0.07)^10 = 8000 × (1.07)^10 ≈ 8000 × 1.967151 ≈ $15,737.21

Therefore, after 10 years, the accumulated amount will be approximately $15,737.21.

2. Future Value of Another Investment

Similarly, for Leona’s investment, PV = $13,000, r = 4% or 0.04, n = 3 years:

FV = 13000 × (1 + 0.04)^3 = 13000 × 1.124864 ≈ $14,623.13

Leona will have approximately $14,623.13 after three years.

3. Present Value of a Perpetuity

The present value (PV) of a perpetuity, which provides a fixed payment indefinitely, is calculated as:

PV = C / r

where C is the annual payment, and r is the discount rate. Given C = $650 and r = 12% or 0.12:

PV = 650 / 0.12 ≈ $5,416.67

The present value of the perpetuity is approximately $5,416.67.

4. Present Value of a Growing Perpetuity

The present value of a growing perpetuity, which increases at a constant rate g, is given by:

PV = C / (r - g)

where C is the cash flow at the end of the first period ($80,000), r is the discount rate (9%, 0.09), and g is the growth rate (7%, 0.07):

PV = 80,000 / (0.09 - 0.07) = 80,000 / 0.02 = $4,000,000

Thus, the present value of the growing perpetuity is approximately $4,000,000.

5. Present Value of a Deferred Annuity with Final Lumpsum

This problem involves determining the initial deposit needed to fund a series of future withdrawals, starting 11 years from now, including an additional lump sum at year 18. The approach involves two main steps:

  1. Calculating the present value at year 10 of the series of withdrawals, using the discounted cash flow method.
  2. Discounting that amount back to present value today, at the given interest rate of 8%.

First, compute the present value at year 10 of the annuity payments from years 11 through 18:

PV at year 10 = \[ \sum_{k=11}^{18} \frac{Payment}{(1 + r)^{k-10}} \]

Since payments are annual and the rate is 8%, the present value of the 8-year annuity at year 10:

PV_{at year 10} = 9,000 × \frac{1 - (1 + 0.08)^{-8}}{0.08} ≈ 9,000 × 5.747

≈ $51,723.00

Additionally, the lump sum of $18,000 at year 18 needs to be discounted back 8 years to year 10:

PV of lump sum at year 10 = 18,000 / (1 + 0.08)^8 ≈ 18,000 / 1.8509 ≈ $9,717.43

The total present value at year 10:

Total PV at year 10 = 51,723 + 9,717.43 ≈ $61,440.43

Now, discount this amount back to the present (year 0):

PV today = 61,440.43 / (1 + 0.08)^10 ≈ 61,440.43 / 2.1589 ≈ $28,491.09

Thus, the required initial deposit today is approximately $28,491.09 to fund the future withdrawals and lump sum.

Conclusion

This analysis demonstrates the application of core financial formulas to determine future values, present values, and the valuation of perpetuities and deferred annuities. Mastery of these concepts facilitates informed financial decision-making and effective planning for both individual and institutional investments.

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