1. **Problem Statement:** Delos borrowed 80,000,000 euros two years ago on a 6-year amortizing loan at 8.626% annual interest. We need to find: a. The original annual principal and interest payments. b. The outstanding principal after two years of payments. c. The new annual payments if the loan is extended by two more years (total 6 years remaining) at the same interest rate. 2. **Formula and Explanation:** For an amortizing loan, the annual payment $A$ is calculated by the annuity formula: $$A = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $P$ = principal (80,000,000 euros) - $r$ = annual interest rate (8.626% = 0.08626) - $n$ = total number of payments (6 years) This payment includes both principal and interest. 3. **Calculate original annual payment:** $$r = 0.08626$$ $$n = 6$$ $$P = 80,000,000$$ Calculate numerator: $$r(1+r)^n = 0.08626 \times (1.08626)^6 = 0.08626 \times 1.6174 = 0.1395$$ Calculate denominator: $$(1+r)^n - 1 = 1.6174 - 1 = 0.6174$$ Annual payment: $$A = 80,000,000 \times \frac{0.1395}{0.6174} = 80,000,000 \times 0.2259 = 18,072,000$$ (Rounded to two decimals: 18,072,000.00 euros) 4. **Outstanding principal after 2 years:** Outstanding principal after $k$ payments is: $$P_k = P \times \frac{(1+r)^n - (1+r)^k}{(1+r)^n - 1}$$ For $k=2$: $$(1+r)^2 = (1.08626)^2 = 1.1809$$ Calculate numerator: $$1.6174 - 1.1809 = 0.4365$$ Denominator is 0.6174 (from above) $$P_2 = 80,000,000 \times \frac{0.4365}{0.6174} = 80,000,000 \times 0.7069 = 56,552,000$$ (Rounded: 56,552,000.00 euros) 5. **New annual payments if loan extended by 2 years (now 6 years remaining) at same rate:** New principal $P' = 56,552,000$ New term $n' = 6$ Calculate numerator: $$r(1+r)^{n'} = 0.08626 \times (1.08626)^6 = 0.1395$$ Denominator: $$(1+r)^{n'} - 1 = 0.6174$$ New annual payment: $$A' = 56,552,000 \times \frac{0.1395}{0.6174} = 56,552,000 \times 0.2259 = 12,778,000$$ (Rounded: 12,778,000.00 euros) 6. **Comparison:** Original payment: 18,072,000.00 euros New payment: 12,778,000.00 euros Reduction: $$18,072,000 - 12,778,000 = 5,294,000$$ This is a significant reduction in annual payments. **Final answers:** - a. Original annual payment: 18,072,000.00 euros - b. Outstanding principal after 2 years: 56,552,000.00 euros - c. New annual payment after restructuring: 12,778,000.00 euros (significant reduction)