1. Problem statement: Frau E. wants to save by making monthly payments for 25 years at the beginning of each month, to receive a monthly annuity of 300 for 18 years, also paid at the beginning of each month. The interest rate is 3% per year, compounded monthly. We need to find the monthly payment amount. 2. Define variables: - Let $R$ be the monthly payment amount during the saving phase. - Interest rate per month: $i = \frac{3\%}{12} = 0.0025$ - Number of saving payments: $n = 25 \times 12 = 300$ - Number of annuity payments: $m = 18 \times 12 = 216$ - Monthly annuity payment: $A = 300$ 3. Calculate the present value of the annuity at the time the annuity starts (after 25 years). Since the annuity is vorschüssig (paid at the beginning of each month), the present value at time $t=25$ years is: $$PV_{annuity} = A \times \frac{1 - (1+i)^{-m}}{i} \times (1+i)$$ 4. Calculate the present value of the annuity at time 0 (now), discounting back 25 years: $$PV_0 = PV_{annuity} \times (1+i)^{-n}$$ 5. The saving payments form an annuity due (payments at beginning of each month) for $n$ months, so the present value of these payments is: $$PV_0 = R \times \frac{1 - (1+i)^{-n}}{i} \times (1+i)$$ 6. Equate the two present values to solve for $R$: $$R \times \frac{1 - (1+i)^{-n}}{i} \times (1+i) = A \times \frac{1 - (1+i)^{-m}}{i} \times (1+i) \times (1+i)^{-n}$$ 7. Simplify and solve for $R$: $$R = A \times \frac{1 - (1+i)^{-m}}{1 - (1+i)^{-n}} \times (1+i)^{-n}$$ 8. Substitute values: $$R = 300 \times \frac{1 - (1+0.0025)^{-216}}{1 - (1+0.0025)^{-300}} \times (1+0.0025)^{-300}$$ 9. Calculate each term: - $(1+0.0025)^{-216} \approx 0.5924$ - $(1+0.0025)^{-300} \approx 0.4724$ 10. Calculate numerator and denominator: - Numerator: $1 - 0.5924 = 0.4076$ - Denominator: $1 - 0.4724 = 0.5276$ 11. Calculate the fraction: $$\frac{0.4076}{0.5276} \approx 0.7727$$ 12. Calculate $(1+0.0025)^{-300} \approx 0.4724$ 13. Finally: $$R = 300 \times 0.7727 \times 0.4724 \approx 109.5$$ 14. Conclusion: Frau E. must pay approximately 109.5 at the beginning of each month for 25 years to receive a monthly annuity of 300 for 18 years at 3% annual interest compounded monthly.