1. Problem 6(a): Calculate the monthly repayment amount for a €5,000 loan with 1% monthly interest compounded monthly over 6 months. 2. The formula for the monthly repayment $R$ on a loan with principal $P$, monthly interest rate $i$, and number of months $n$ is: $$R = P \times \frac{i(1+i)^n}{(1+i)^n - 1}$$ where $i = 0.01$ (1% monthly), $P = 5000$, and $n = 6$. 3. Calculate $(1+i)^n = (1.01)^6$: $$ (1.01)^6 = 1.06152015 $$ 4. Substitute values into the formula: $$R = 5000 \times \frac{0.01 \times 1.06152015}{1.06152015 - 1} = 5000 \times \frac{0.0106152015}{0.06152015}$$ 5. Simplify the fraction: $$ \frac{0.0106152015}{0.06152015} \approx 0.1725 $$ 6. Calculate repayment: $$R = 5000 \times 0.1725 = 862.5$$ So, the monthly repayment is €862.50. 7. Problem 8: Calculate the fair price of a 10-year bond paying €30 every six months with AER 6% compounded bi-annually. 8. The bond pays €30 every 6 months for 10 years, so total payments $n = 20$ (2 per year × 10 years). 9. The periodic interest rate $i$ is 6% per year compounded bi-annually, so: $$i = \frac{6\%}{2} = 3\% = 0.03$$ 10. The present value (PV) of an annuity paying $C=30$ per period for $n=20$ periods at rate $i=0.03$ is: $$PV = C \times \frac{1 - (1+i)^{-n}}{i}$$ 11. Calculate $(1+i)^{-n} = (1.03)^{-20}$: $$ (1.03)^{20} = 1.8061 \Rightarrow (1.03)^{-20} = \frac{1}{1.8061} = 0.5537 $$ 12. Substitute values: $$PV = 30 \times \frac{1 - 0.5537}{0.03} = 30 \times \frac{0.4463}{0.03} = 30 \times 14.8767 = 446.3$$ 13. The bond also repays €2,000 at maturity (after 20 periods). The present value of this lump sum is: $$PV_{lump} = 2000 \times (1+i)^{-n} = 2000 \times 0.5537 = 1107.4$$ 14. Total fair price of the bond is: $$PV_{total} = PV + PV_{lump} = 446.3 + 1107.4 = 1553.7$$ So, the fair price to pay for this bond is approximately €1553.70.