1. **Problem Statement:** Delos borrowed 80,000,000 euros two years ago with 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 Rules:** For an amortizing loan, the annual payment $A$ is calculated by the formula: $$A = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$$ where: - $P$ = principal (initial loan amount) - $r$ = annual interest rate (decimal) - $n$ = total number of payments (years) The loan balance after $k$ payments is: $$B_k = P(1+r)^k - A \times \frac{(1+r)^k - 1}{r}$$ 3. **Calculations:** **a. Original annual payment:** - $P = 80,000,000$ - $r = 0.08626$ - $n = 6$ Calculate $A$: $$A = 80,000,000 \times \frac{0.08626(1+0.08626)^6}{(1+0.08626)^6 - 1}$$ Calculate $(1+0.08626)^6$: $$1.08626^6 \approx 1.628894$$ Then: $$A = 80,000,000 \times \frac{0.08626 \times 1.628894}{1.628894 - 1} = 80,000,000 \times \frac{0.1405}{0.628894} \approx 80,000,000 \times 0.2234 = 17,872,000$$ Rounded to two decimals: 17,872,000.00 euros **b. Outstanding principal after 2 years:** Calculate balance after 2 payments: $$B_2 = 80,000,000 \times 1.08626^2 - 17,872,000 \times \frac{1.08626^2 - 1}{0.08626}$$ Calculate $1.08626^2 \approx 1.180\,$ Calculate numerator for payment part: $$1.180 - 1 = 0.180$$ Calculate denominator: $$0.08626$$ So: $$B_2 = 80,000,000 \times 1.180 - 17,872,000 \times \frac{0.180}{0.08626} = 94,400,000 - 17,872,000 \times 2.086 = 94,400,000 - 37,280,000 = 57,120,000$$ Rounded: 57,120,000.00 euros **c. New annual payments if loan extended by 2 years (now 4 years remaining) on balance 57,120,000:** - New principal $P' = 57,120,000$ - $r = 0.08626$ - $n' = 4$ Calculate new payment $A'$: $$A' = 57,120,000 \times \frac{0.08626(1+0.08626)^4}{(1+0.08626)^4 - 1}$$ Calculate $(1.08626)^4 \approx 1.3895$ Then: $$A' = 57,120,000 \times \frac{0.08626 \times 1.3895}{1.3895 - 1} = 57,120,000 \times \frac{0.1198}{0.3895} = 57,120,000 \times 0.3075 = 17,560,000$$ Rounded: 17,560,000.00 euros **Comparison:** Original payment: 17,872,000.00 euros New payment: 17,560,000.00 euros The reduction is: $$17,872,000 - 17,560,000 = 312,000$$ This is a modest reduction, not very significant. --- **Final answers:** - a. Original annual payment: 17,872,000.00 euros - b. Outstanding principal after 2 years: 57,120,000.00 euros - c. New annual payment if extended 2 more years: 17,560,000.00 euros (modest reduction)