Your parents will retire in 27 years. They currently have $360,000 saved, and they think they will need $2,400,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds? Round your answer to two decimal places.

To solve for the annual interest rate needed for your parents to reach their retirement goal of $2,400,000 in 27 years with a current savings of $360,000, we can use the formula for the future value of a single lump sum, which is derived from the compound interest formula:

Future Value = Present Value x (1 + Interest Rate)^Number of Periods

Here, the Present Value (PV) is $360,000, the Future Value (FV) is $2,400,000, and the Number of Periods (n) is 27 years. We need to solve for the Interest Rate (r).

Let's rearrange the formula to solve for the interest rate:

Interest Rate = (Future Value / Present Value)^(1/Number of Periods) - 1

Plugging in the numbers:

Interest Rate = ($2,400,000 / $360,000)^(1/27) - 1

Interest Rate = (6.6667)^(1/27) - 1

Now we calculate this using a calculator:

Interest Rate ≈ (1.082857) - 1

Interest Rate ≈ 0.082857 or 8.29% when rounded to two decimal places.

So, your parents must earn an annual interest rate of approximately 8.29% to reach their goal of $2,400,000 at retirement in 27 years.

Extra: Understanding the concept of compound interest is essential here. Compound interest is interest calculated on the initial principal, which also includes all the accumulated interest from previous periods on a deposit or loan. Over time, compound interest will grow at a faster rate than simple interest, because it includes interest on the interest that has already accumulated.

In this calculation, the formula for the future value of a lump sum investment takes the present value and applies compound interest over a set number of periods (years, in this case). It's a powerful tool for predicting how an investment will grow over time.

Furthermore, the Annual Interest Rate is fundamentally the percentage that the investment will grow each year. When we solve for the interest rate in the formula, we're looking for the consistent annual growth percentage that would result in the original sum of money reaching the desired future value at the end of the investment period (27 years for your parents' retirement).

Finally, it's important to note that the 8.29% interest rate needed is an idealized figure. In reality, investment returns can fluctuate, and your parents might need to adjust their savings or expectations based on actual return rates and other factors such as inflation, additional contributions, withdrawals, fees, and taxes. Financial planning for retirement generally involves careful consideration of these elements and often the assistance of a financial advisor.