A.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?
B.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?
C.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?
A. To find out how much the court should invest today, we can use the present value formula for an annuity:
PV = PMT * [1 - (1 / (1 + r)^n)] / r
Where:
PV = Present Value (amount to be invested today)
PMT = Payment (amount to be received each year)
r = Interest rate
n = Number of years
Given:
PMT = $11,000
r = 5% (or 0.05)
n = 10 years
Substituting the given values into the formula:
PV = $11,000 * [1 - (1 / (1 + 0.05)^10)] / 0.05
Calculating the equation:
PV = $11,000 * [1 - (1 / 1.62889462677)] / 0.05
PV = $11,000 * (1 - 0.61391386754) / 0.05
PV = $11,000 * 0.38608613246 / 0.05
PV = $84,305.29
Therefore, the court should invest approximately $84,305.29 today.
B. To find out how much Mary will have in the account at the end of the seventh year, we can use the future value formula for compound interest:
FV = PV * (1 + r)^n
Where:
FV = Future Value (amount in the account at the end of the seventh year)
PV = Present Value (initial deposit)
r = Interest rate
n = Number of years
Given:
PV = $33,000
r = 7% (or 0.07)
n = 7 years
Substituting the given values into the formula:
FV = $33,000 * (1 + 0.07)^7
Calculating the equation:
FV = $33,000 * (1.07)^7
FV = $33,000 * 1.54881529577
FV = $51,033.09
Therefore, Mary will have approximately $51,033.09 in the account at the end of the seventh year.
C. To find out how much the year-end payments should be, we can use the future value formula for an annuity:
FV = PMT * [(1 + r)^n - 1] / r
Where:
FV = Future Value (target savings for the furniture)
PMT = Payment (year-end deposits to the savings account)
r = Interest rate
n = Number of years
Given:
FV = $10,000
r = 6% (or 0.06)
n = 3 years
Substituting the given values into the formula:
$10,000 = PMT * [(1 + 0.06)^3 - 1] / 0.06
Calculating the equation:
$10,000 = PMT * (1.191016 - 1) / 0.06
$10,000 = PMT * 0.191016 / 0.06
$10,000 = PMT * 3.1836
Dividing both sides by 3.1836:
$10,000 / 3.1836 = PMT
PMT = $3,141.29
Therefore, the year-end payments should be approximately $3,141.29.
A. To calculate the amount that should be invested today, we can use the present value formula for an ordinary annuity. The formula is:
PV = PMT * [(1 - (1 + r)^-n) / r]
Where:
PV = Present value (amount to be invested today)
PMT = Payment per period ($11,000 per year)
r = Interest rate per period (5% or 0.05)
n = Number of periods (10 years)
Plugging in the values, we can calculate:
PV = $11,000 * [(1 - (1 + 0.05)^-10) / 0.05]
= $11,000 * [(1 - (1.05)^-10) / 0.05]
= $11,000 * [(1 - 0.6139) / 0.05]
= $11,000 * (0.3861 / 0.05)
≈ $11,000 * 7.722
≈ $85,342.86
Therefore, the court should invest approximately $85,342.86 today to ensure that there is nothing left in the account after the final payment is made.
B. 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 = Future amount (amount Mary will have in the account at the end of the seventh year)
P = Principal amount (initial deposit of $33,000)
r = Interest rate per period (7% or 0.07)
n = Number of periods (7 years)
Plugging in the values, we can calculate:
A = $33,000 * (1 + 0.07)^7
= $33,000 * (1.07)^7
≈ $33,000 * 1.71815
≈ $56,471.95
Therefore, Mary will have approximately $56,471.95 in the account at the end of the seventh year.
C. To calculate the year-end payments for Mary and Joe, we can use the future value formula for an ordinary annuity. The formula is:
FV = PMT * [((1 + r)^n - 1) / r]
Where:
FV = Future value ($10,000)
PMT = Payment per period (year-end deposits)
r = Interest rate per period (6% or 0.06)
n = Number of periods (3 years)
We need to find the value of PMT. Rearranging the formula:
PMT = FV * (r / ((1 + r)^n - 1))
Plugging in the values, we can calculate:
PMT = $10,000 * (0.06 / ((1 + 0.06)^3 - 1))
= $10,000 * (0.06 / (1.191016 - 1))
= $10,000 * (0.06 / 0.191016)
≈ $10,000 * 0.313933
≈ $3,139.33
Therefore, Mary and Joe should make year-end payments of approximately $3,139.33 to have $10,000 in the savings account by the end of three years.