4. Pat plans to retire at 57 , and will receive a cash payment of \( \$ 50000 \) from a profit sharing plan. The total sum received will be used to set up an annuity with an insurance company at \( 8 \% / \) a compounded semi-annually until age 65 . How large is each payment, if Pat is to receive two equal payments per year? ( 4 marks)

To determine how large each payment is, we need to first calculate the future value of the cash payment of \( \$50,000 \) when it is invested in an annuity at \( 8\% \) compounded semi-annually for 8 years (from age 57 to 65). 1. The interest rate per period (semi-annual) is \( \frac{8\%}{2} = 4\% \) or \( 0.04 \). 2. The number of periods (payments) is \( 8 \times 2 = 16 \). 3. We use the future value formula for an annuity: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where \( P \) is the annuity payment, \( r \) is the interest rate per period, and \( n \) is the number of periods. We don't have \( FV \) immediately, as we first need to calculate the future value of the lump sum. \[ FV = PV \times (1 + r)^n \] In this case, \( PV = 50000 \), \( r = 0.04 \), and \( n = 16 \): \[ FV = 50000 \times (1 + 0.04)^{16} \] Calculating it: \[ FV \approx 50000 \times (1.04)^{16} \approx 50000 \times 1.8009 \approx 90045 \] Now we set up the equation for an annuity to find the payment amount. Using the future value of the annuity formula: \[ 90045 = P \times \frac{(1 + 0.04)^{16} - 1}{0.04} \] Solving for \( P \): \[ 90045 = P \times \frac{1.8009 - 1}{0.04} \] \[ 90045 = P \times 20.0225 \] \[ P \approx \frac{90045}{20.0225} \approx 4498.85 \] Therefore, each payment that Pat will receive is approximately **$4498.85**.