Final answer:
To find the annual rate of return needed for Peter to retire with $1 million, the future value of an annuity formula is used. The specific rate can be calculated with a financial calculator or an Excel function, as it involves an iterative process that is not easily solved by hand.
Step-by-step explanation:
To determine the annual rate of return Peter must earn to have $1 million in his retirement investment account, we can use the future value of an annuity formula:
The formula for the future value of an annuity is FV = Pmt * [((1 + r)^n - 1) / r], where:
- FV is the future value of the annuity (in this case, $1 million).
- Pmt is the annual payment (or contribution, which is $6,000).
- r is the annual interest rate (which we want to find).
- n is the number of periods (here it's 30 years).
In this situation, we have:
- FV = $1,000,000
- Pmt = $6,000
- n = 30
To solve for the rate r, we rearrange the formula and solve it, typically using a financial calculator or a software tool like Excel, as the equation cannot be algebraically rearranged to solve for r in a simple manner. The process entails plugging in the known values and iteratively computing or finding the rate that sets the equation equal to $1,000,000.
Although the manual calculation for r might not be straightforward, financial calculators and spreadsheets have built-in functions to solve for this interest rate. For example, in Excel, the RATE function can be used.