Questions: Patricia Richardson invested 4000 four times a year in an annuity due at Southern Trust Company for a period of 2 years at an interest rate of 12% compounded quarterly. Using the ordinary annuity table, calculate the total value of the annuity due at the end of the 2-year period.

Solution Steps

To solve this problem, we need to calculate the future value of an annuity due. An annuity due means payments are made at the beginning of each period. The formula for the future value of an annuity due is:

\[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \]

where:

• \( P \) is the payment amount per period,
• \( r \) is the interest rate per period,
• \( n \) is the total number of periods.

Given:

• \( P = 4000 \),
• Annual interest rate = 12%, so quarterly interest rate \( r = 0.12 / 4 = 0.03 \),
• Total number of periods \( n = 2 \times 4 = 8 \).

We have the following values for the annuity due calculation:

• Payment per period, \( P = 4000 \)
• Annual interest rate, \( r_{annual} = 0.12 \)
• Compounding periods per year, \( m = 4 \)
• Total number of years, \( t = 2 \)

The interest rate per period is calculated as: \[ r = \frac{r_{annual}}{m} = \frac{0.12}{4} = 0.03 \]

The total number of periods is given by: \[ n = t \times m = 2 \times 4 = 8 \]

Using the formula for the future value of an annuity due: \[ FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \times (1 + r) \] Substituting the known values: \[ FV = 4000 \times \left( \frac{(1 + 0.03)^8 - 1}{0.03} \right) \times (1 + 0.03) \] Calculating this gives: \[ FV \approx 36636.4245 \]

Final Answer