Final answer:
To determine the most one should pay for an annuity offering $10,000 semi-annual payments for 30 years at a 7.2% annual interest rate, the present value of the annuity formula is applied. By inserting the known values into the formula, we obtain the maximum amount that would make the investment worth it, emphasizing the importance of early saving and the effects of compound interest over time.
Step-by-step explanation:
The question at hand involves calculating the present value of an annuity. To decide the most you should pay for the annuity that makes $10,000 semi-annual payments for 30 years with a 7.2% annual interest rate, we utilize the present value formula for an annuity. The formula for the present value of an annuity is PVA = PMT × ((1 - (1 + r)^-n) / r), where PVA is the present value of the annuity, PMT is the payment amount per period, r is the interest rate per period, and n is the total number of payments.
Here, PMT is $10,000, r is 7.2% per year or 0.036 (3.6%) per semi-annual period, and n is 60 periods (30 years × 2 payments per year). Plugging these values into the formula gives us the maximum amount you should be willing to pay for this annuity.
It's important to start saving money early to benefit from the power of compound interest, as shown by the example of investing $3,000 at a 7% real annual rate of return. In the example provided, after 40 years, the investment grows nearly fifteen times, demonstrating the significant impact that time and interest rates have on investments.