1. **Problem statement:** We compare two loan options: Lendo with 2.95% annual interest over 15 years (180 months), and Kliklån with 18.58% annual interest over 48 months, both with monthly repayments. --- ### Opgave 1: Lendo 2. **Given:** - Principal $A_0 = 25000$ - Annual nominal interest rate $r_{annual} = 2.95\% = 0.0295$ - Number of months $n = 180$ 3. **Find the monthly interest rate (terminsrente):** The monthly interest rate $r_{monthly}$ is the annual rate divided by 12: $$r_{monthly} = \frac{0.0295}{12} = 0.0024583$$ 4. **Calculate the effective annual interest rate:** The effective annual rate (EAR) accounts for compounding monthly: $$EAR = \left(1 + r_{monthly}\right)^{12} - 1 = \left(1 + 0.0024583\right)^{12} - 1$$ Calculate: $$EAR = 1.0024583^{12} - 1 = 1.0300 - 1 = 0.0300 = 3.00\%$$ 5. **Calculate the monthly payment $M$ using the annuity formula:** $$M = A_0 \times \frac{r_{monthly} (1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1}$$ Calculate numerator and denominator: $$\text{num} = 0.0024583 \times (1.0024583)^{180}$$ $$\text{den} = (1.0024583)^{180} - 1$$ Calculate powers: $$(1.0024583)^{180} = e^{180 \times \ln(1.0024583)} \approx e^{180 \times 0.002455} = e^{0.4419} \approx 1.555$$ So: $$\text{num} = 0.0024583 \times 1.555 = 0.003823$$ $$\text{den} = 1.555 - 1 = 0.555$$ Then: $$M = 25000 \times \frac{0.003823}{0.555} = 25000 \times 0.00689 = 172.25$$ 6. **Calculate total payments:** $$\text{Total} = M \times n = 172.25 \times 180 = 31005$$ --- ### Opgave 2: Kliklån 7. **Given:** - Principal $A_0 = 25000$ - Annual nominal interest rate $r_{annual} = 18.58\% = 0.1858$ - Number of months $n = 48$ 8. **Find the monthly interest rate (terminsrente):** $$r_{monthly} = \sqrt[12]{1 + r_{annual}} - 1 = \sqrt[12]{1.1858} - 1$$ Calculate: $$r_{monthly} = e^{\frac{\ln(1.1858)}{12}} - 1 = e^{\frac{0.1704}{12}} - 1 = e^{0.0142} - 1 = 1.0143 - 1 = 0.0143$$ 9. **Calculate the effective annual interest rate:** $$EAR = (1 + r_{monthly})^{12} - 1 = (1.0143)^{12} - 1 = 1.1858 - 1 = 0.1858 = 18.58\%$$ 10. **Calculate the monthly payment $M$:** $$M = A_0 \times \frac{r_{monthly} (1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1}$$ Calculate powers: $$(1.0143)^{48} = e^{48 \times \ln(1.0143)} = e^{48 \times 0.0142} = e^{0.6816} \approx 1.977$$ Calculate numerator and denominator: $$\text{num} = 0.0143 \times 1.977 = 0.0283$$ $$\text{den} = 1.977 - 1 = 0.977$$ Then: $$M = 25000 \times \frac{0.0283}{0.977} = 25000 \times 0.02895 = 723.75$$ 11. **Calculate total payments:** $$\text{Total} = M \times n = 723.75 \times 48 = 34740$$ --- **Final answers:** - Lendo monthly payment: **172.25** - Lendo total payment: **31005** - Kliklån monthly payment: **723.75** - Kliklån total payment: **34740**