1. **State the problem:** Ashley buys an annuity paying $2500 every six months for 11 years, then $100 every month for 4 years. Interest rate is 2.1% compounded quarterly. We want to find: a) The purchase price (present value) of the annuity. b) The total interest received. 2. **Formulas and rules:** - Present value of an annuity formula: $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ where $P$ is payment per period, $i$ is interest rate per period, $n$ is number of periods. - Interest rate per period must match payment frequency. - For different payment frequencies, calculate present value of each part separately and sum. 3. **Calculate interest rate per period:** - Annual nominal rate $r = 0.021$ compounded quarterly means quarterly rate $i_q = \frac{0.021}{4} = 0.00525$. - To find effective rates for 6-month and monthly periods: - 6 months = 2 quarters, so effective 6-month rate: $$i_{6m} = (1 + i_q)^2 - 1 = (1 + 0.00525)^2 - 1 = 0.0105$$ - Monthly rate from quarterly rate: $$i_{m} = (1 + i_q)^{\frac{1}{3}} - 1 = (1.00525)^{\frac{1}{3}} - 1 \approx 0.001747$$ 4. **Calculate number of periods:** - For first 11 years, payments every 6 months: $n_1 = 11 \times 2 = 22$ - For next 4 years, payments every month: $n_2 = 4 \times 12 = 48$ 5. **Calculate present value of first annuity part:** $$PV_1 = 2500 \times \frac{1 - (1 + 0.0105)^{-22}}{0.0105}$$ Calculate: $$ (1 + 0.0105)^{-22} = (1.0105)^{-22} \approx 0.7893$$ So: $$PV_1 = 2500 \times \frac{1 - 0.7893}{0.0105} = 2500 \times \frac{0.2107}{0.0105} = 2500 \times 20.0667 = 50166.75$$ 6. **Calculate present value of second annuity part:** Payments start after 11 years, so discount back 11 years first. - Present value of second annuity at time 11 years: $$PV_{2,11} = 100 \times \frac{1 - (1 + 0.001747)^{-48}}{0.001747}$$ Calculate: $$ (1 + 0.001747)^{-48} = (1.001747)^{-48} \approx 0.9201$$ So: $$PV_{2,11} = 100 \times \frac{1 - 0.9201}{0.001747} = 100 \times \frac{0.0799}{0.001747} = 100 \times 45.74 = 4574.00$$ - Discount $PV_{2,11}$ back 11 years to present using 6-month periods: $$PV_2 = PV_{2,11} \times (1 + 0.0105)^{-22} = 4574.00 \times 0.7893 = 3609.15$$ 7. **Total purchase price:** $$PV = PV_1 + PV_2 = 50166.75 + 3609.15 = 53775.90$$ 8. **Calculate total amount received:** - First part total payments: $$2500 \times 22 = 55000$$ - Second part total payments: $$100 \times 48 = 4800$$ - Total payments: $$55000 + 4800 = 59800$$ 9. **Calculate interest received:** $$\text{Interest} = \text{Total payments} - \text{Purchase price} = 59800 - 53775.90 = 6024.10$$ **Final answers:** a) Purchase price = $53775.90$ b) Interest received = $6024.10$