What initial investment would I have to make at 6% per year compounded semi-annually in order to end up with $1,000,000 in 29 years if I didn't make any monthly contributions.
a) estimate using the rule of 72. Explain how you came up with your estimate.
b) calculate exactly.
a) ... they still use that ??
b) x(1.03)^58 = 1000000
x = $ 180,069.84
a) To estimate the initial investment required to reach $1,000,000 in 29 years with a 6% annual interest rate compounded semi-annually, we will use the rule of 72. The rule of 72 is a simple approximation that helps estimate the time it takes for an investment to double or the rate of return needed to double an investment.
The formula for the rule of 72 is:
Time to double = 72 / Interest rate
In this case, we want to calculate the approximate time it takes for an investment to double from the initial amount to $1,000,000 with a 6% interest rate. Using the formula, we have:
Time to double = 72 / 6 = 12 years
Since our goal is to reach $1,000,000 in 29 years, we can estimate that the investment would double approximately twice in that time period. Therefore, we can estimate that the initial investment required would be around half of $1,000,000, which is $500,000.
b) To calculate the exact initial investment required to reach $1,000,000 in 29 years with a 6% annual interest rate compounded semi-annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Future value (in this case, $1,000,000)
P = Principal or initial investment (unknown)
r = Annual interest rate (6% or 0.06)
n = Number of times the interest is compounded per period (semi-annually, so 2)
t = Number of periods (29 years or 29 * 2 = 58 semi-annual periods)
We need to solve for P. Rearranging the formula, we have:
P = A / (1 + r/n)^(nt)
Plugging in the values, we get:
P = $1,000,000 / (1 + 0.06/2)^(2 * 29)
Calculating this equation will give us the exact initial investment required to reach $1,000,000 in 29 years with a 6% interest rate compounded semi-annually.