Questions: Calculate the amount of money that an individual must have invested today to begin receiving payments of 44,000 annually, paid at the beginning of the payment period, if the investment earns 11.25% interest compounded monthly with a remaining balance of 125,000 at the end of the 30 year term.

Solution Steps

To solve this problem, we need to determine the present value of an annuity due, which is a series of payments made at the beginning of each period. The investment earns interest compounded monthly, and we also need to account for a remaining balance at the end of the term. We will use the formula for the present value of an annuity due and adjust it to include the future value of the remaining balance.

The annual interest rate is given as \(11.25\%\). To find the monthly interest rate, we divide the annual rate by the number of compounding periods per year (12 months):

\[ \text{Monthly Interest Rate} = \frac{0.1125}{12} = 0.009375 \]

The total number of payments over 30 years, with monthly compounding, is:

\[ \text{Total Payments} = 30 \times 12 = 360 \]

The present value of an annuity due, where payments are made at the beginning of each period, is calculated using the formula:

\[ PV_{\text{annuity due}} = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \times (1 + r) \]

where \(P = 44000\), \(r = 0.009375\), and \(n = 360\). Substituting these values, we get:

\[ PV_{\text{annuity due}} = 44000 \times \left( \frac{1 - (1 + 0.009375)^{-360}}{0.009375} \right) \times (1 + 0.009375) \approx 4572661.9628 \]

The present value of the future value of the remaining balance is calculated using the formula:

\[ PV_{\text{future value}} = \frac{FV}{(1 + r)^n} \]

where \(FV = 125000\), \(r = 0.009375\), and \(n = 360\). Substituting these values, we get:

\[ PV_{\text{future value}} = \frac{125000}{(1 + 0.009375)^{360}} \approx 4345.0439 \]

The total present value required is the sum of the present value of the annuity due and the present value of the future value:

\[ \text{Total Present Value} = PV_{\text{annuity due}} + PV_{\text{future value}} \approx 4572661.9628 + 4345.0439 = 4577007.0067 \]

Final Answer