1. **Problem statement:** Compare two payment options for a loan of 120000 with different payment schedules and interest calculations. 2. **Option A:** Pay 5000 at the end of every month for 2 years with 10% annual interest compounded monthly. - Number of months $n = 2 \times 12 = 24$. - Monthly interest rate $i = \frac{10\%}{12} = 0.0083333$. - Monthly payment $P = 5000$. - The present value of payments (PV) for an ordinary annuity is calculated by: $$\text{PV}_A = P \times \frac{1-(1+i)^{-n}}{i}$$ 3. **Calculate PV for Option A:** $$\text{PV}_A = 5000 \times \frac{1-(1+0.0083333)^{-24}}{0.0083333}$$ Calculate $(1+0.0083333)^{-24} = (1.0083333)^{-24} \approx 0.8347$. $$\text{PV}_A = 5000 \times \frac{1-0.8347}{0.0083333} = 5000 \times \frac{0.1653}{0.0083333} \approx 5000 \times 19.836 = 99180$$ 4. **Option B:** Pay 15000 every quarter starting immediately for 2 years at 10% annual interest compounded monthly. - Number of quarters $m = 2 \times 4 = 8$. - Quarterly payment $Q = 15000$. - Monthly interest rate $j = 0.0083333$. 5. To find the effective quarterly interest rate $r$, we compound monthly: $$r = (1 + j)^3 - 1 = (1.0083333)^3 - 1$$ Calculate: $$(1.0083333)^3 \approx 1.02526$$ Thus, $$r = 1.02526 - 1 = 0.02526$$ 6. Since payments start immediately, Option B is an annuity due. Present value for annuity due: $$\text{PV}_B = Q \times \frac{1-(1+r)^{-m}}{r} \times (1+r)$$ Calculate $(1+r)^{-m} = (1.02526)^{-8} \approx 0.816$. 7. Calculate PV for Option B: $$\text{PV}_B = 15000 \times \frac{1-0.816}{0.02526} \times 1.02526 = 15000 \times \frac{0.184}{0.02526} \times 1.02526$$ $$ = 15000 \times 7.28 \times 1.02526 \approx 15000 \times 7.46 = 111900$$ 8. **Difference:** $$\text{Difference} = \text{PV}_B - \text{PV}_A = 111900 - 99180 = 12720$$ **Final answer:** Option B has a present value of 111900, Option A has present value of 99180, difference is 12720.