Felicity has just commenced work and is investigating superannuation funds.
She calculates that she will need $1 000 000 in her fund when she retires in 40
years. She finds a fund guaranteeing to pay interest on each deposit at a rate of
6% per annum compounded monthly.
She intends to deposit $M into the fund at the beginning of each month for the
first 25 years (300 months). For the remaining 15 years (180 months), she will
deposit $2M into the fund.
Find the value of M.
I will use the fact that she deposits $M for the entire period plus and addition $M for the last 15 years, (making it 2M for the last 15 years)
i = .06/12 = .005
M(1.005)^480 - 1)/.005 + M(1.005)^180 - 1)/.005 = 1000000
M( 1.005^480 -1 + 1.005^180 - 1)/.005 = 1000000
I get M = 438.15
check my calculations
or
amount of the first 300 payments of M at the end of 40 years
= M(1.005^300 - 1)/.005 (1.005)^180
amount of the last 180 payments of 2M
= 2M(1.005^180 - 1)/.005
M(1.005^300 - 1)/.005 (1.005)^180 + 2M(1.005^180 - 1)/.005 = 1000000
M(1700.672022 + 2(290.8187124) = 1000000
M(2282.309447) = 1000000
M = 438.15
clearly my first method is the easier of the two.
To find the value of M, we need to calculate the monthly deposit amount that Felicity needs to make in order to reach her retirement goal of $1,000,000.
Let's break down the problem into two parts: the first 25 years (300 months) and the remaining 15 years (180 months).
For the first 25 years, Felicity intends to make monthly deposits of $M. We can use the future value of an ordinary annuity formula to calculate the accumulated value of these deposits. The formula is:
A = P * (1 + r/n)^(nt) - 1 / (r/n)
Where:
A = accumulated value
P = monthly deposit
r = annual interest rate (as a decimal)
n = number of compounding periods per year
t = number of years
In this case, Felicity will deposit $M each month, the interest rate is 6% per annum compounded monthly (r = 0.06/12 = 0.005), the number of compounding periods per year is monthly (n = 12), and the number of years is 25 (t = 25).
So the accumulated value for the first 25 years is:
A1 = M * (1 + 0.005)^((12 * 25)) - 1 / (0.005)
For the remaining 15 years, Felicity intends to make monthly deposits of $2M. Again, we can use the same formula to calculate the accumulated value with the updated monthly deposit amount:
In this case, Felicity will deposit $2M each month, the interest rate, compounding periods, and the number of years remain the same.
So the accumulated value for the remaining 15 years is:
A2 = 2M * (1 + 0.005)^((12 * 15)) - 1 / (0.005)
Since the total accumulated value at retirement is $1,000,000, we can set up the equation:
A1 + A2 = $1,000,000
(M * (1 + 0.005)^((12 * 25)) - 1 / (0.005)) + (2M * (1 + 0.005)^((12 * 15)) - 1 / (0.005)) = $1,000,000
Now, we can solve this equation to find the value of M by using algebraic methods. Note that you might need a calculator or a tool like Excel for precise calculations.
Simplifying and solving the equation will give us the value of M, which represents the monthly deposit amount that Felicity needs to make to reach her retirement goal of $1,000,000.