Questions: How much money would Sue have to invest today to have an annuity with a future value of 40,000 and future payments of 3000 per month for 3 years and 1000 per month for the next 5 years? Do your calculations using an interest rate that remains at 6% compounded monthly.

Solution Steps

Let \( FV = 40000 \) be the future value of the annuity. The annual interest rate is \( r = 0.06 \), which gives a monthly interest rate of \( i = \frac{r}{12} = 0.005 \). The monthly payments for the first phase are \( PMT_1 = 3000 \) for \( n_1 = 3 \times 12 = 36 \) months, and for the second phase, \( PMT_2 = 1000 \) for \( n_2 = 5 \times 12 = 60 \) months.

The present value of the first phase of the annuity can be calculated using the formula: \[ PV_1 = PMT_1 \times \left( \frac{1 - (1 + i)^{-n_1}}{i} \right) \] Substituting the values, we find: \[ PV_1 = 3000 \times \left( \frac{1 - (1 + 0.005)^{-36}}{0.005} \right) \]

Similarly, the present value of the second phase of the annuity is given by: \[ PV_2 = PMT_2 \times \left( \frac{1 - (1 + i)^{-n_2}}{i} \right) \] Substituting the values, we find: \[ PV_2 = 1000 \times \left( \frac{1 - (1 + 0.005)^{-60}}{0.005} \right) \]

The total present value of the annuity is the sum of the present values from both phases: \[ PV_{total} = PV_1 + PV_2 \] Substituting the calculated values: \[ PV_{total} = 98613.04871779581 + 51725.56075113097 \]

Final Answer