If Naomi invests in a stock portfolio, her returns for 10 or more years will average 10%-12%. Naomi realizes that the stock market has higher returns because it is a more risky investment than a savings account or a CD. She wants her calculations to be conservative so she decides to use 8% to calculate possible stock market earnings. How much will she need to invest annually to accumulate a million dollars in the stock market?
1000,000 = P (1 + r/n) ^ n-1 / r / n
1000,000 = P (1+ 0.08/30/0.08/30)^30 -1
1000,000= PV (1.083171739 -1/0.00266666666) ^ 30 -1
1000,000= PV (1.083171739 – 1/ 0.00266666666)
1000,000= PV (0.083171739/0.0026666666)
PV = $32,062.17
To accumulate a million dollars, Naomi will need to invest $32,062.17 annually in the stock market.
To calculate how much Naomi needs to invest annually to accumulate a million dollars in the stock market, we can use the concept of compound interest.
First, let's calculate the number of years, 'n', required to reach a million dollars. We'll assume an 8% annual return on her investments.
The formula for calculating future value using compound interest is:
FV = PV × (1 + r)^n
Where:
FV = Future Value (in this case, one million dollars)
PV = Present Value (the initial investment)
r = Annual interest rate (8% or 0.08 as a decimal)
n = Number of years
Substituting the values into the formula:
1,000,000 = PV × (1 + 0.08)^n
Next, we need to determine the value of 'n'. Taking the logarithm of both sides of the equation, we can solve for 'n':
log(1,000,000) = log(PV × (1 + 0.08)^n)
Using log properties, we can rewrite the equation as:
log(1,000,000) = log(PV) + n × log(1 + 0.08)
Simplifying further:
6 = log(PV) + n × log(1.08)
Let's assume that Naomi plans to invest the same amount annually, denoted as 'A'. If her investments grow at an average rate of 8%, we can rewrite the equation using this annual investment amount:
6 = log(A) + n × log(1.08)
To isolate 'A', we subtract log(A) from both sides of the equation:
6 - log(A) = n × log(1.08)
Now, we substitute the desired rate of return, 8%, into the equation:
6 - log(A) = n × log(1.08)
To solve for the value of 'A', we need to determine the value of 'n'. Based on the given information, Naomi wants to accumulate a million dollars in the stock market over 10 or more years, assuming an 8% return.
Using trial and error or an iterative approach, we can calculate the approximate value of 'n' that achieves this goal. Here, we will start with 10 years.
6 - log(A) = 10 × log(1.08)
Let's solve this equation iteratively and find the value of 'A' that satisfies it.