To find the present value of an annuity, we can use the formula for the present value of an ordinary annuity:
\[ PV = P \times \frac{1 - (1 + r)^{-nt}}{r} \]
Where:
- \( PV \) = present value of the annuity
- \( P \) = payment per period
- \( r \) = interest rate per period
- \( n \) = number of periods per year
- \( t \) = number of years
In this case:
- \( P = 700 \) (monthly deposit)
- \( r = \frac{0.05}{12} = \frac{0.05}{12} \approx 0.00416667 \) (monthly interest rate since it is 5% per year)
- \( n = 12 \) (number of deposits per year)
- \( t = 7 \) (years)
Now, calculate the total number of periods:
\[ nt = 12 \times 7 = 84 \text{ months} \]
Substituting the values into the formula:
\[ PV = 700 \times \frac{1 - (1 + 0.00416667)^{-84}}{0.00416667} \]
Now calculate \( (1 + 0.00416667)^{-84} \):
\[ (1 + 0.00416667)^{-84} \approx (1.00416667)^{-84} \approx 0.6885 \]
Then compute \( 1 - (1 + r)^{-nt} \):
\[ 1 - 0.6885 \approx 0.3115 \]
Now plug it into the present value formula:
\[ PV = 700 \times \frac{0.3115}{0.00416667} \]
Calculating \( \frac{0.3115}{0.00416667} \):
\[ \frac{0.3115}{0.00416667} \approx 74.76 \]
Now multiply by \( P \):
\[ PV \approx 700 \times 74.76 \approx 52332 \]
Therefore, the present value of the annuity is:
\[ PV \approx 52332.00 \]
So, the present value of the annuity is $52,332.00.