Sadeeq wants to save for a down payment on a house and plans to deposit $700 every month into an annuity for the next 7 years. If the annuity interest rate is 5 percent per year, what is the present value of the annuity? Round your answer to the nearest hundredth

1 answer

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.