Questions: Calculate the future value. a. Use the compound interest formula, FV = PV(1+i)^N, to find the future value of the current amount in the investment. After 10 years, the 33,000.00 will grow to b. Determine the annuity type. - Ordinary Simple Annuity - Ordinary General Annuity - Simple Annuity Due - General Annuity Due c. Identify the following pieces of information to be used to calculate the future value of the annuity. Periodic Payment: PMT = 1300 Number of Payments per Year: PY = 12 Total Number of Payments: N = 120 Annual Interest Rate: r = 0.062 Number of Compoundings per Year: CY = 2 d. Determine the total future value of the investment (incorporating your answer from part a above).

Solution Steps

To solve the given problem, we need to break it down into parts:

1. Calculate the future value of the present amount using the compound interest formula.
2. Determine the type of annuity.
3. Identify the necessary pieces of information for the annuity calculation.
4. Calculate the total future value incorporating both the present value and the annuity payments.

• Use the compound interest formula \( FV = PV(1 + i)^N \) to find the future value of the current amount in the investment.
• Here, \( PV = 33000 \), \( i = \frac{0.062}{2} \) (since the interest is compounded semi-annually), and \( N = 10 \times 2 \) (since the interest is compounded semi-annually for 10 years).

• Determine the annuity type based on the given information. Since payments are made at the beginning of each period, it is a "General Annuity Due".

• Identify the periodic payment, number of payments per year, total number of payments, annual interest rate, and number of compoundings per year.

• Calculate the future value of the annuity using the formula for the future value of an annuity due.
• Add the future value of the present amount (from part a) to the future value of the annuity to get the total future value.

Using the compound interest formula:

\[ FV = PV(1 + i)^N \]

where:

• \( PV = 33000 \)
• \( i = \frac{0.062}{2} = 0.031 \)
• \( N = 10 \times 2 = 20 \)

Calculating the future value:

\[ FV_{\text{present}} = 33000(1 + 0.031)^{20} \approx 60769.72 \]

Since payments of \( PMT = 1300 \) are made at the beginning of each period, the annuity type is classified as:

\[ \text{Annuity Type} = \text{General Annuity Due} \]

The following pieces of information are identified for the annuity calculation:

• Periodic Payment: \( PMT = 1300 \)
• Number of Payments per Year: \( PY = 12 \)
• Total Number of Payments: \( N = 120 \)
• Annual Interest Rate: \( r = 0.062 \)
• Number of Compoundings per Year: \( CY = 2 \)

Using the formula for the future value of an annuity due:

\[ FV_{\text{annuity due}} = PMT \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) (1 + r/n) \]

where:

• \( r/n = \frac{0.062}{12} \approx 0.00516667 \)
• \( n = 12 \)
• \( t = 10 \)

Calculating the future value of the annuity due:

\[ FV_{\text{annuity due}} \approx 1300 \left( \frac{(1 + 0.00516667)^{120} - 1}{0.00516667} \right) (1 + 0.00516667) \approx 216484.15 \]

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

\[ \text{Total } FV = FV_{\text{present}} + FV_{\text{annuity due}} \approx 60769.72 + 216484.15 \approx 277253.87 \]

Final Answer