Please answer these questions:
1. After a protracted legal case, Joe won a settlement that will pay him $11,000 each year at the end of the year for the next ten years. If the market interest rates are currently 5%, exactly how much should the court invest today, assuming end of year payments, so there will be nothing left in the account after the final payment is made?
2. Mary just deposited $33,000 in an account paying 7% interest. She plans to leave the money in this account for eight years. How much will she have in the account at the end of the seventh year?
3. Mary and Joe would like to save up $10,000 by the end of three years from now to buy new furniture for their home. They currently have $1500 in a savings account set aside for the furniture. They would like to make three equal year end deposits to this savings account to pay for the furniture when they purchase it three years from now. Assuming that this account pays 6% interest, how much should the year end payments be?
Show all work for each assignment and explain each step carefully.
Sure! Let's solve each question step by step and explain the process.
1. To find out how much the court should invest today, we can use the concept of present value. The present value of an annuity formula can be used to calculate the initial investment. The formula is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value or initial investment
PMT = Annual payment
r = Interest rate
n = Number of years
In this case, Joe will receive $11,000 each year for ten years, and the interest rate is 5%. We need to calculate the present value (PV) or initial investment that will be required.
Using the formula:
PV = $11,000 * (1 - (1 + 0.05)^(-10)) / 0.05
PV = $11,000 * (1 - (1.05)^(-10)) / 0.05
By evaluating this equation, you will find that the court should invest approximately $85,765.69 today to ensure there is nothing left in the account after the final payment is made.
2. To calculate the amount Mary will have in the account at the end of the seventh year, we can use the compound interest formula. The formula is:
A = P * (1 + r)^n
Where:
A = Final amount
P = Initial deposit
r = Interest rate
n = Number of years
In this case, Mary deposited $33,000 in an account with a 7% interest rate. She plans to keep the money in the account for eight years, but we need to find the amount at the end of the seventh year.
Using the formula:
A = $33,000 * (1 + 0.07)^7
By evaluating this equation, you will find that Mary will have approximately $50,965.76 in the account at the end of the seventh year.
3. To determine the year-end payment required for Mary and Joe to save up $10,000 in three years, we can use the future value of an annuity formula. The formula is:
FV = PMT * ((1 + r)^n - 1) / r
Where:
FV = Future value or target amount
PMT = Year-end payment
r = Interest rate
n = Number of years
In this case, they currently have $1500 in a savings account, and they want to save up $10,000 in three years with a 6% interest rate.
Using the formula:
$10,000 = PMT * ((1 + 0.06)^3 - 1) / 0.06
To solve for PMT, rearrange the formula:
PMT = ($10,000 * 0.06) / ((1 + 0.06)^3 - 1)
By evaluating this equation, you will find that the year-end payments should be approximately $2,950.33 in order to save up $10,000 in three years.
Remember to plug in the values accurately and use the appropriate formulas for each question to get the correct answers.