Question

In December 2017, a DUI accident resulted in Vinnie liable for substantial payment to the victims. Although he denies being intoxicated, he agreed to make payments starting in January 2018 for the next 14 years. He will be required to pay $6,000 every 6 months (January 1 and July 1). Vinnie has decided to purchase an annuity contract that promises to make the required payments for him. The interest rate on the contract is 6% per year, compounded semiannually. How much must vinnie invest in the contract if he purchases it on January 1, 2018?

Answer 1

To determine how much Vinnie needs to invest in an annuity contract to cover semiannual payments of $6,000 for 14 years at a 6% annual interest rate, compounded semiannually, one must use the present value of an annuity formula. By inserting the given variables into the formula, Vinnie can calculate the present value that he must invest today.

Step-by-step explanation:

The question at hand involves calculating the present value of an annuity to determine how much Vinnie must invest in a contract. Given the conditions that the interest rate is 6% per year, compounded semiannually, and considering the annuity payments of $6,000 every six months for 14 years. First, we must recognize this as a present value of an annuity problem, where the annuity formula is applied:

PV = Pmt [(1 - (1 + r)^-n) / r]

Where PV is the present value of the annuity, Pmt is the payment per period, r is the interest rate per period, and n is the total number of payments.

Since the interest rate is compounded semiannually, we have r = (6% / 2) = 0.03 or 3% per period. The total number of payments, given semiannual payments over 14 years, is n = 14 * 2 = 28. Thus:

PV = $6,000 [1 - (1 + 0.03)^-28] / 0.03

Calculating this, Vinnie needs to invest a certain amount today, which can be determined by solving the above equation.