Final answer:
To calculate Jeff and Sarah's required lump-sum investment, one needs to calculate the present value of the annuity payments as of the payment start date using a 4% quarterly compounded rate, and then discount it back to the investment start date using a 6.6% monthly compounded rate. This two-step process involves the present value of an ordinary annuity formula followed by a present value of a lump sum calculation.
Step-by-step explanation:
Jeff and Sarah are looking to invest a lump-sum amount that will allow them to provide their daughter with a monthly payment of $1,000 for three years and nine months starting from August 1, 2024. This payment plan is referred to as an ordinary annuity. They plan to make the investment on August 1, 2014. The investments will earn a 6.6% compounded monthly rate during the accumulation stage and later, 4% compounded quarterly during the payment stage.
The future value of an annuity can be calculated using the formula for the future value of an ordinary annuity. However, since the annuity payments are to be made in the future, we must also calculate the present value of those payments at the start date of the annuity payments (August 1, 2024) using the 4% quarterly rate, and then determine the lump-sum investment required on August 1, 2014, utilizing the 6.6% monthly compounded rate.
The process requires calculating the present value of the annuity payments as of August 1, 2024 (just before the payments start), and then discounting that present value back to August 1, 2014. To do this, we will use the formulas for both the present value of an annuity and the present value of a lump sum (using the respective interest rates and compounding frequencies).