9. Nicki deposits \( € 200 \) at the end of each quarter in her savings account. The money earns \( 5.5 \% \) (AER). How much will the investment be worth at the end of four years? State clearly any assumptions that you make.

To calculate the future value of Nicki's investment, we can use the formula for the future value of an annuity. Since Nicki is making regular deposits at the end of each quarter, we will treat this as an ordinary annuity. ### Assumptions: 1. The interest rate is compounded quarterly. 2. The annual effective rate (AER) of \( 5.5\% \) is converted to a quarterly interest rate. 3. The deposits are made at the end of each quarter. ### Step 1: Convert the annual interest rate to a quarterly interest rate The quarterly interest rate can be calculated as follows: \[ \text{Quarterly interest rate} = \frac{5.5\%}{4} = \frac{0.055}{4} = 0.01375 \] ### Step 2: Determine the number of deposits Nicki is making deposits at the end of each quarter for 4 years. Since there are 4 quarters in a year, the total number of deposits is: \[ \text{Total number of deposits} = 4 \text{ years} \times 4 \text{ quarters/year} = 16 \text{ deposits} \] ### Step 3: Use the future value of an annuity formula The future value \( FV \) of an ordinary annuity can be calculated using the formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( P \) = amount of each deposit (\( € 200 \)) - \( r \) = quarterly interest rate (\( 0.01375 \)) - \( n \) = total number of deposits (16) ### Step 4: Plug in the values Now we can calculate the future value: \[ FV = 200 \times \frac{(1 + 0.01375)^{16} - 1}{0.01375} \] Calculating \( (1 + 0.01375)^{16} \): \[ (1 + 0.01375)^{16} \approx 1.233 \] Now substituting this back into the formula: \[ FV = 200 \times \frac{1.233 - 1}{0.01375} \] \[ FV = 200 \times \frac{0.233}{0.01375} \approx 200 \times 16.96 \approx 3392 \] ### Conclusion The investment will be worth approximately \( € 3392 \) at the end of four years.