The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?

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BlackTom AI · Accepted Answer

{"reasoning": "To determine the maximum amount you are willing to pay today for the annuity, we need to calculate the present value of the annuity. The annuity pays $770 per quarter for 25 years, and you want to earn a minimum rate of return of 6.2 percent per annum. Since the payments are quarterly, we must adjust the annual interest rate to a quarterly rate and the number of periods to quarters.\n\n1. **Quarterly Interest Rate**: The annual interest rate of 6.2% must be converted to a quarterly rate. This is done by dividing the annual rate by the number of quarters in a year:\n   \n   $$ r_{quarterly} = \frac{6.2\%}{4} = 1.55\% = 0.0155 $$\n\n2. **Number of Periods**: Since the annuity pays quarterly for 25 years, the total number of periods is:\n   \n   $$ n = 25 \times 4 = 100 \text{ quarters} $$\n\n3. **Present Value of Annuity Formula**: The present value $ PV $ of an annuity can be calculated using the formula:\n   \n   $$ PV = C \times \frac{1 - (1 + r)^{-n}}{r} $$\n   \n   where $ C $ is the cash flow per period ($770), $ r $ is the quarterly interest rate (0.0155), and $ n $ is the total number of periods (100).\n\n4. **Calculation**:\n   \n   $$ PV = 770 \times \frac{1 - (1 + 0.0155)^{-100}}{0.0155} $$\n   \n   First, calculate $(1 + 0.0155)^{-100}$:\n   \n   $$ (1 + 0.0155)^{-100} \approx 0.2115 $$\n   \n   Then, calculate the present value:\n   \n   \[ PV = 770 \times \frac{1 - 0.2115}{0.0155} \approx 770 \times \frac{0.7885}{0.0155}

The Good Life Insurance Co. wants to sell you an annuity which will pay you $770 per quarter for 25 years. You want to earn a minimum rate of return of 6.2 percent. What is the most you are willing to pay as a lumpsum today to buy this annuity?单项选择题

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