3. What is the total amount that should be deposited today at \( 6.5 \% \) compounded quarterly order to withdraw \( P 200,000 \) and leave nothing in the fund at the end of 5 years? Solution:

To determine the total amount that should be deposited today, we can use the formula for present value in compound interest. The formula is: \[ PV = \frac{FV}{(1 + r/n)^{nt}} \] Where: - \( FV \) is the future value (or the amount to be withdrawn, \( P 200,000 \)) - \( r \) is the annual interest rate (6.5% or 0.065) - \( n \) is the number of times the interest is compounded per year (quarterly means \( n = 4 \)) - \( t \) is the number of years (5 years) Plugging in the numbers: \[ PV = \frac{200,000}{(1 + 0.065/4)^{4 \cdot 5}} \] \[ PV = \frac{200,000}{(1 + 0.01625)^{20}} \] \[ PV = \frac{200,000}{(1.01625)^{20}} \] \[ PV = \frac{200,000}{1.395489} \approx 143,000.17 \] Therefore, you would need to deposit approximately \( P 143,000.17 \) today to be able to withdraw \( P 200,000 \) after 5 years at a \( 6.5\% \) interest rate compounded quarterly. --- Did you know that compound interest can really be a game changer over time? It's often why starting to save early is so beneficial! Instead of just earning interest on your initial investment, you're also earning "interest on your interest" which can lead to exponential growth. Another fun fact: many people mistakenly think they need an immense amount of money to start investing. In reality, starting small and consistently contributing to a savings or investment account can accumulate significant rewards over time, thanks to the power of compound interest!