To solve these questions, we'll follow step-by-step financial calculations.
### a. The value of the annuity
First, use the future value of an annuity formula compounded annually:
[tex]\[ FV = P \times \left( \frac{(1 + r)^t - 1}{r} \right) \][/tex]
where:
- [tex]\(FV\)[/tex] is the future value of the annuity
- [tex]\(P\)[/tex] is the periodic deposit (also called the payment per period)
- [tex]\(r\)[/tex] is the annual interest rate (as a decimal)
- [tex]\(t\)[/tex] is the time in years
Given data:
- Periodic deposit ([tex]\(P\)[/tex]): [tex]$51,000
- Annual interest rate (\(r\)): 0.07 (7%)
- Time (\(t\)): 30 years
Plugging in the values into the formula:
\[ FV = 51,000 \times \left( \frac{(1 + 0.07)^{30} - 1}{0.07} \right) \]
Calculate the future value:
\[ FV = 51,000 \times \left( \frac{(1.07)^{30} - 1}{0.07} \right) \]
Using these specific parameters, the future value of the annuity, after making all the calculations, would be:
\[ FV = \$[/tex]4,817,500 \]
Thus, the value of the annuity is \[tex]$4,817,500.
### b. The interest
To find the interest earned, we need to calculate the total amount deposited and then subtract that from the future value.
The total amount deposited over 30 years is:
\[ \text{Total deposit} = \text{Periodic deposit} \times \text{Time} \]
Given:
- Periodic deposit: $[/tex]51,000
- Time: 30 years
[tex]\[ \text{Total deposit} = 51,000 \times 30 = 1,530,000 \][/tex]
Next, we need to subtract the total deposit from the future value to find the interest earned:
[tex]\[ \text{Interest earned} = FV - \text{Total deposit} \][/tex]
Given the future value ([tex]\(FV\)[/tex]) as:
[tex]\[ FV = 4,817,500 \][/tex]
[tex]\[
\text{Interest earned} = 4,817,500 - 1,530,000 = 3,287,500
\][/tex]
Thus, the interest earned is \[tex]$3,287,500.
### Summary
a. The value of the annuity is \$[/tex]4,817,500.
b. The interest is \$3,287,500.
