1. **State the problem:** Ashley buys an annuity paying $2,500 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:** The present value (PV) of an annuity is calculated by: $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ where $P$ is the payment per period, $i$ is the interest rate per period, and $n$ is the number of periods. Since the interest is compounded quarterly (4 times a year), the quarterly interest rate is: $$i_q = \frac{2.1}{100} \div 4 = 0.00525$$ We must adjust the interest rate per payment period for each annuity phase. 3. **Calculate present value of first annuity (11 years, payments every 6 months):** - Number of payments: $11 \times 2 = 22$ - Interest rate per 6 months (2 quarters): $$i_1 = (1 + i_q)^2 - 1 = (1 + 0.00525)^2 - 1 = 0.0105$$ - Payment per period: $2500$ Calculate PV: $$PV_1 = 2500 \times \frac{1 - (1 + 0.0105)^{-22}}{0.0105}$$ Calculate $(1 + 0.0105)^{-22}$: $$= (1.0105)^{-22} = \frac{1}{(1.0105)^{22}} \approx \frac{1}{1.254} = 0.797$$ So: $$PV_1 = 2500 \times \frac{1 - 0.797}{0.0105} = 2500 \times \frac{0.203}{0.0105} = 2500 \times 19.33 = 48325$$ 4. **Calculate present value of second annuity (4 years, payments every month):** - Number of payments: $4 \times 12 = 48$ - Interest rate per month (quarterly rate converted to monthly): $$i_2 = (1 + i_q)^{\frac{1}{3}} - 1 = (1.00525)^{\frac{1}{3}} - 1 \approx 0.00175$$ - Payment per period: $100$ Calculate PV at the time the second annuity starts (after 11 years): $$PV_2 = 100 \times \frac{1 - (1 + 0.00175)^{-48}}{0.00175}$$ Calculate $(1 + 0.00175)^{-48}$: $$= (1.00175)^{-48} = \frac{1}{(1.00175)^{48}} \approx \frac{1}{1.088} = 0.919$$ So: $$PV_2 = 100 \times \frac{1 - 0.919}{0.00175} = 100 \times \frac{0.081}{0.00175} = 100 \times 46.29 = 4629$$ 5. **Discount $PV_2$ back 11 years to present value:** - Number of quarters in 11 years: $11 \times 4 = 44$ - Discount factor: $$DF = (1 + i_q)^{-44} = (1.00525)^{-44} = \frac{1}{(1.00525)^{44}} \approx \frac{1}{1.254^2} = \frac{1}{1.572} = 0.636$$ So present value of second annuity: $$PV_2^{present} = 4629 \times 0.636 = 2945$$ 6. **Total purchase price:** $$PV = PV_1 + PV_2^{present} = 48325 + 2945 = 51270$$ 7. **Calculate total amount received:** - First annuity total payments: $$2500 \times 22 = 55000$$ - Second annuity total payments: $$100 \times 48 = 4800$$ - Total payments: $$55000 + 4800 = 59800$$ 8. **Calculate interest received:** $$\text{Interest} = \text{Total payments} - \text{Purchase price} = 59800 - 51270 = 8530$$ **Final answers:** a) Purchase price of the annuity is approximately $51270.00. b) Interest received from the annuity is approximately $8530.00.