1. **Problem statement:** Peter saves 550 at the start of each month for 25 years with an AER of 5.2%. We need to find: (i) The equivalent monthly compound interest rate. (ii) The lump sum he will receive at retirement. 2. **Formula for equivalent monthly rate:** The AER (Annual Equivalent Rate) relates to the monthly rate $r$ by: $$ (1 + r)^{12} = 1 + \text{AER} $$ where AER = 0.052. 3. **Calculate monthly rate:** $$ (1 + r)^{12} = 1.052 $$ Taking 12th root: $$ 1 + r = (1.052)^{\frac{1}{12}} $$ $$ r = (1.052)^{\frac{1}{12}} - 1 $$ Calculate: $$ r = e^{\frac{\ln(1.052)}{12}} - 1 $$ $$ r \approx e^{0.004238} - 1 \approx 1.004247 - 1 = 0.004247 $$ 4. **Answer for (i):** The monthly interest rate is approximately $0.004247$ or 0.4247% (6 significant figures). 5. **Formula for future value of annuity due:** Since Peter deposits at the start of each month, the future value $FV$ after $n$ months is: $$ FV = P \times \frac{(1 + r)^n - 1}{r} \times (1 + r) $$ where - $P = 550$ (monthly payment), - $r = 0.004247$ (monthly rate), - $n = 25 \times 12 = 300$ months. 6. **Calculate lump sum:** Calculate $(1 + r)^n$: $$ (1.004247)^{300} = e^{300 \times \ln(1.004247)} $$ $$ \approx e^{300 \times 0.004238} = e^{1.2714} \approx 3.566 $$ Then, $$ FV = 550 \times \frac{3.566 - 1}{0.004247} \times 1.004247 $$ $$ = 550 \times \frac{2.566}{0.004247} \times 1.004247 $$ $$ = 550 \times 604.23 \times 1.004247 $$ $$ = 550 \times 606.79 = 333,735. $$ 7. **Answer for (ii):** Peter will receive approximately 333735 at retirement (nearest euro).