Interest and TVM Dewey Sykes plans to open a business in four years when he retires. How much must he invest today to have \( \$ 8,000 \) when he retires if the bank pays \( 10 \% \) annually, compounded quarterly?

To determine how much Dewey must invest today, we can use the present value formula for compound interest: \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] Where: - \(PV\) = present value (the amount to invest today) - \(FV\) = future value (\$8,000) - \(r\) = annual interest rate (0.10) - \(n\) = number of times interest is compounded per year (4 for quarterly) - \(t\) = number of years until retirement (4) First, plug the numbers into the formula: \[ PV = \frac{8000}{(1 + \frac{0.10}{4})^{4 \times 4}} = \frac{8000}{(1 + 0.025)^{16}} = \frac{8000}{(1.025)^{16}} = \frac{8000}{1.439} \approx 5557.24 \] So, Dewey must invest approximately \$5,557.24 today to have \$8,000 upon retirement. --- Understanding the concept of time value of money (TVM) is key for anyone entering the world of finance. It reflects how the value of money changes over time due to factors like interest rates and inflation. Hence, money today is worth more than the same amount in the future because of its potential earning capacity—this is the cornerstone of investing! If you want to get an in-depth understanding of TVM, consider checking out "The Future of Finance" where you’ll find fascinating insights into not just interest, but the evolution of how we value money over time and its implications for personal finance and investing!