Meg's pension plan is an annuity with a guaranteed return of 9% interest per year (compounded semi-annually). She would like to retire with a pension of $70000 per semi-annum for 25 years. If she works 45 years before retiring, how much money must she and her employer deposit per semi-annum? (
Assume the 9% per annum interest stays fixed for the 70 years of Meg's life.
D=semi-annual deposit (in first 45 years)
=70000
m=number of periods while working = 2*45=90
W=semi-annual withdrawal (in last n=25 years)
n=number of periods while retired = 2*25=50
A=amount accumulated on Meg's retirement
R=semi-annual interest rate = 1.045
Capital required on Meg's retirement,
First calculate A,
A=W(R^n-1)/(R-1)=70000*(1.045^50-1)/(1.045-1)
=$12,495,211.98
To accumulate A over 45 years:
12495211.98=D(R^m-1)/(R-1)=D(1.045^90-1)/(1.045-1)
D=12495211.98(1.045-1)/(1.045^90-1)
=$10,910.286
Check me.
To determine how much money Meg and her employer need to deposit per semi-annum, we need to calculate the required annuity payment.
An annuity is a series of equal payments made at regular intervals over a specified period of time. In this case, Meg wants to receive $70,000 semi-annually for 25 years.
To calculate the required annuity payment, we can use the formula for the future value of an annuity:
FV = P * [(1 + r)^n - 1] / r,
where:
FV = future value of the annuity (the desired pension amount)
P = annuity payment (the amount to be deposited per semi-annum)
r = interest rate per semi-annum (9% per year compounded semi-annually, so r = 0.09 / 2)
n = number of periods (semi-annually for 25 years, so n = 25 * 2)
Substituting the given values into the formula, we have:
$70,000 = P * [(1 + 0.09 / 2)^(25 * 2) - 1] / (0.09 / 2).
Now we can solve for P to find the required annuity payment.
$70,000 = P * [(1.045)^50 - 1] / 0.045.
First, simplify the expression inside the brackets:
$70,000 = P * (1.045^50 - 1) / 0.045.
Next, multiply both sides of the equation by 0.045:
$70,000 * 0.045 = P * (1.045^50 - 1).
Substituting the values and solving for P:
$3,150 = P * (1.045^50 - 1).
Now, divide both sides of the equation by (1.045^50 - 1):
P = $3,150 / (1.045^50 - 1).
Calculating the expression (1.045^50 - 1) using a calculator:
(1.045^50 - 1) ≈ 49.6052.
Finally, substitute this value into the equation to find P:
P ≈ $3,150 / 49.6052.
Calculating this expression using a calculator:
P ≈ $63.39.
So, Meg and her employer must deposit approximately $63.39 per semi-annum into her pension plan.