Michelle purchases a retirement annuity that will pay her $2,500 at the end of every six months for the first nine years and $200 at the end of every month for the next three years. The annuity earns interest at a rate of 5.9% compounded quarterly. a. What was the purchase price of the annuity? Round to the nearest cent b. How much interest did Michelle receive from the annuity? Round to the nearest cent

To solve this problem, we need to calculate the present value of the annuity payments for both periods and then sum them up to find the purchase price. After that, we will determine the total interest received.

Step-by-Step Solution:

Part (a): Calculate the Purchase Price of the Annuity

1. First Period (First 9 Years):

   • Payment: $2,500 every six months
   • Duration: 9 years
   • Number of payments: $9 \text{ years} \times 2 \text{ payments/year} = 18 \text{ payments}$
   • Interest rate: 5.9% compounded quarterly

   Since the payments are semi-annual, we need to convert the quarterly interest rate to a semi-annual rate.

   Quarterly interest rate: $\frac{5.9\%}{4} = 1.475\%$

   Semi-annual interest rate: $(1 + 0.01475)^2 - 1 \approx 0.0297 \text{ or } 2.97\%$

   Present value of an ordinary annuity formula: $PV_1 = PMT \times \frac{1 - (1 + r)^{-n}}{r}$ Where:

   • $PMT = 2500$
   • $r = 0.0297$
   • $n = 18$

   $PV_1 = 2500 \times \frac{1 - (1 + 0.0297)^{-18}}{0.0297}$

   Calculate $PV_1$: $PV_1 = 2500 \times \frac{1 - (1.0297)^{-18}}{0.0297} \approx 2500 \times 14.486 \approx 36,215.00$

2. Second Period (Next 3 Years):

   • Payment: $200 every month
   • Duration: 3 years
   • Number of payments: $3 \text{ years} \times 12 \text{ payments/year} = 36 \text{ payments}$
   • Interest rate: 5.9% compounded quarterly

   Monthly interest rate: $(1 + 0.01475)^{1/3} - 1 \approx 0.0049 \text{ or } 0.49\%$

   Present value of an ordinary annuity formula: $PV_2 = PMT \times \frac{1 - (1 + r)^{-n}}{r}$ Where:

   • $PMT = 200$
   • $r = 0.0049$
   • $n = 36$

   $PV_2 = 200 \times \frac{1 - (1 + 0.0049)^{-36}}{0.0049}$

   Calculate $PV_2$: $PV_2 = 200 \times \frac{1 - (1.0049)^{-36}}{0.0049} \approx 200 \times 32.898 \approx 6,579.60$

3. Total Present Value (Purchase Price): Since the second period starts after the first period ends, we need to discount $PV_2$ back to the present value at the end of the first period.

   Discount factor for 9 years at 5.9% compounded quarterly: $(1 + 0.01475)^{36} \approx 1.6487$

   Present value of $PV_2$ at the end of the first period: $PV_2' = \frac{6579.60}{1.6487} \approx 3,990.00$

   Total present value (purchase price): $PV = PV_1 + PV_2' \approx 36,215.00 + 3,990.00 \approx 40,205.00$

   Therefore, the purchase price of the annuity is approximately $40,205.00.

Part (b): Calculate the Total Interest Received

1. Total Payments:

   • First period: $2500 \times 18 = 45,000$
   • Second period: $200 \times 36 = 7,200$
   • Total payments: $45,000 + 7,200 = 52,200$

2. Total Interest: $\text{Total Interest} = \text{Total Payments} - \text{Purchase Price} = 52,200 - 40,205 \approx 11,995.00$

   Therefore, Michelle received approximately $11,995.00 in interest from the annuity.