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.
To find the value of M, we need to use the formula for the future value of periodic deposits, which is given by:
FV = P * [(1 + r)^n - 1) / r]
Where:
FV = Future Value
P = Deposit amount
r = Interest rate per period
n = Number of periods
In this case, Felicity intends to deposit $M into the fund at the beginning of each month for the first 25 years (300 months), and $2M for the remaining 15 years (180 months).
Let's calculate the future value for both periods and then set up an equation to find M.
1. First 25 years:
For this period, P = M, r = 6% per annum compounded monthly (or 0.06/12 = 0.005), and n = 300 months.
FV1 = M * [(1 + 0.005)^300 - 1] / 0.005
2. Remaining 15 years:
For this period, P = 2M, r = 6% per annum compounded monthly (same as before), and n = 180 months.
FV2 = 2M * [(1 + 0.005)^180 - 1] / 0.005
3. Total future value:
Since Felicity wants to have $1,000,000 in her fund when she retires, we can set up the equation:
FV1 + FV2 = $1,000,000
Substituting the values of FV1 and FV2:
M * [(1 + 0.005)^300 - 1] / 0.005 + 2M * [(1 + 0.005)^180 - 1] / 0.005 = $1,000,000
Now, we can solve this equation to find the value of M.
To find the value of M, we can use the formula for the future value of an ordinary annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value
P = periodic payment
r = interest rate per period
n = number of periods
For the first 25 years (300 months), Felicity will be depositing M into the fund at the beginning of each month. The interest rate per period is 6% per annum compounded monthly, which can be converted to a monthly interest rate by dividing it by 12, and then converting it to a decimal:
r = 6% / 12 / 100 = 0.005
n = 300 (as there are 300 months in 25 years)
Now, we can substitute these values into the formula and solve for the future value (FV) needed:
FV = M * [(1 + 0.005)^300 - 1] / 0.005
For the remaining 15 years (180 months), Felicity will be depositing $2M into the fund at the beginning of each month. Using the same formula:
FV = 2M * [(1 + 0.005)^180 - 1] / 0.005
Now, we can set the sum of these two future values equal to the desired final amount of $1,000,000:
(M * [(1 + 0.005)^300 - 1] / 0.005) + (2M * [(1 + 0.005)^180 - 1] / 0.005) = $1,000,000
Simplifying this equation, we can solve for M:
[(1 + 0.005)^300 - 1] + 2 * [(1 + 0.005)^180 - 1] = (1,000,000 * 0.005) / M
Calculating the left-hand side of the equation, we get:
[1.98272679354] + 2 * [1.13878374896] ≈ 5000 / M
Simplifying the equation further:
4.26029429145 ≈ 5000 / M
Now, we can solve for M:
M = 5000 / 4.26029429145
M ≈ $1,173.41
Therefore, the value of M is approximately $1,173.41.