Meg's pension plan is an annuity with a guaranteed return of 7% interest per year (compounded annually). She would like to retire with a pension of $30000 per annum for 15 years. If she works 31 years before retiring, how much money must she and her employer deposit per annum? (Round your answer to the nearest cent.)
Assumption: the rate stays at 7% throughout her 31 years of deposits and the 15 years of withdrawals.
(What happens after the 15 years of retirement and her money is used up ? )
let her payment + employers share be $x per year
x( 1.07^31 - 1)/.07 = 30000(1 - 1.07^-15)/.07
you do the button pushing,
I got x = 2676.88
which has to be split between her and her employer. (1338.44 for Meg)
To calculate the amount of money Meg and her employer need to deposit per annum, we need to use the formula for the future value of an annuity.
The future value of an annuity formula is given by:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value of the annuity
P = deposit per annum
r = interest rate per compounding period
n = number of compounding periods
In this case, Meg wants to receive a pension of $30,000 per annum for 15 years after retiring. The interest rate is 7% per annum, compounded annually. The total number of compounding periods (years) is 15.
Plugging the values into the formula:
$30,000 = P * [(1 + 0.07)^15 - 1] / 0.07
Simplifying the equation:
$30,000 * 0.07 = P * [(1.07)^15 - 1]
$2,100 = P * [(1.07)^15 - 1]
Next, we calculate the future value of Meg's pension plan while she is still working. She works for 31 years, and we want to accumulate enough money so that she can receive $30,000 per annum for 15 years during retirement.
Using the same formula:
FV = P * [(1 + r)^n - 1] / r
We now solve for the future value:
FV = P * [(1 + 0.07)^31 - 1] / 0.07
Substituting the calculated future value of the pension plan ($2,100):
$2,100 = P * [(1.07)^31 - 1] / 0.07
We can now solve for P:
P = $2,100 * 0.07 / [(1.07)^31 - 1]
P ≈ $1,857.66762
Therefore, Meg and her employer must deposit approximately $1,857.67 per annum to accumulate enough money for Meg to receive a pension of $30,000 per annum for 15 years after retiring.