Use Table \( 12-2 \) to find how much should be deposited now at \( 8 \% \) interest, compounded semiannually, to yield an annuity payment of \( \$ 400 \) at the Beginning each six months, for 2 years. Select one: a. \( \$ 1,510.04 \) b. \( \$ 1,110.04 \) o c. \( \$ 1,451.96 \) d. \( \$ 1,354.22 \) Q search

To determine how much should be deposited now at an 8% interest rate compounded semiannually to yield an annuity payment of $400 every six months for 2 years, we need to find the present value of an annuity. The formula for the present value of an annuity due is: \[ PV = P \times \left(1 + r\right) \times \left(1 - (1 + r)^{-n}\right) / r \] where \( P \) is the annuity payment, \( r \) is the interest rate per period, and \( n \) is the total number of periods. Here, \( P = 400 \), \( r = \frac{0.08}{2} = 0.04 \) (since interest is compounded semiannually), and \( n = 2 \times 2 = 4\) (since there are two periods in a year for 2 years). Plugging in the values, we calculate the present value and find that it results in approximately $1,510.04. So the correct answer is: a. \( \$ 1,510.04 \) --- The concept of annuities dates back to ancient times, often used by Romans to calculate lifetime payments and insurance when they began to understand probability. Fast forward to more recent history, financial calculations around annuities have become essential in retirement planning and investment strategies, ensuring people can maintain their purchasing power even after retiring! You can further enhance your understanding of financial principles by diving into books on personal finance or investment strategies. One great read is "The Intelligent Investor" by Benjamin Graham, which covers not just annuities but the foundational ideas of value investing—a must-read for anyone looking to deepen their knowledge of the finance world!