To determine the required semiannual deposit, we can use the formula for the future value of an annuity:
FV = P * [(1 + r/n)^(nt) - 1] / (r/n)
Where:
FV = future value
P = periodic deposit
r = interest rate per period
n = number of compounding periods per year
t = number of years
Given that the engineer wishes to have a fund of P400,000 at the end of each year for the last five years (periods 10 to 14), we can calculate the future value of this annuity and then solve for the periodic deposit.
First, let's calculate the number of compounding periods per year:
n = 12 (compounded monthly)
Since the engineer wants to make semiannual deposits, we need to adjust the number of years and the interest rate accordingly. The total number of periods (t) will be double the number of years (14).
t = 28
Next, we need to convert the interest rate from an annual rate (5%) to a monthly rate by dividing it by the number of compounding periods per year:
r = 5% / 12
r = 0.05 / 12
r = 0.0041667
Now, substitute the given values into the future value formula and solve for P:
400,000 = P * [(1 + 0.0041667/12)^(12*28) - 1] / (0.0041667/12)
Simplifying the equation:
400,000 = P * [(1.0041667)^(336) - 1] / 0.0041667
Now, solve for P:
P = 400,000 * (0.0041667) / [(1.0041667)^(336) - 1]
Using a calculator, we find:
P ≈ 4,967.28
Therefore, the engineer needs to make a semiannual deposit of approximately P4,967.28 to achieve the desired fund amount of P400,000 at the end of each year for the last five years of the 14-year period.
