Questions: Mr. Quijano decided to sell their farm and to deposit the fund in a bank. After computing the interest, they learned that they may withdraw Php480,000.00 yearly for 8 years starting at the end of 6 years when it is time for him to retire. How much is the fund deposited if the interest rate is 5% converted annually?

Solution Steps

To solve this problem, we need to determine the present value of an annuity. Mr. Quijano wants to withdraw Php480,000.00 yearly for 8 years, starting at the end of 6 years. We will use the formula for the present value of an annuity to calculate how much needs to be deposited initially. The interest rate is 5% per annum.

1. Calculate the present value of the annuity at the start of the withdrawal period (end of year 5).
2. Discount this present value back to the present time (year 0) to find the initial deposit amount.

To find the present value of the annuity that Mr. Quijano will withdraw, we use the formula for the present value of an annuity:

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

where:

• \( P = 480,000 \) (annual withdrawal),
• \( r = 0.05 \) (interest rate),
• \( n = 8 \) (number of withdrawals).

Substituting the values, we get:

\[ PV = 480,000 \times \left( \frac{1 - (1 + 0.05)^{-8}}{0.05} \right) \approx 3,102,342.1245 \]

Next, we need to discount the present value of the annuity back to the present time (year 0) using the formula for the present value of a lump sum:

\[ PV_0 = \frac{FV}{(1 + r)^t} \]

where:

• \( FV = 3,102,342.1245 \) (present value at year 5),
• \( r = 0.05 \) (interest rate),
• \( t = 5 \) (years until withdrawal starts).

Substituting the values, we find:

\[ PV_0 = \frac{3,102,342.1245}{(1 + 0.05)^5} \approx 2,430,766.2319 \]

Final Answer