To build up $30,000 in seven years at 4.0% interest with semi-annual payments, how much is needed?

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To find how much needs to be contributed regularly to accumulate $30,000 in seven years at a 4.0% annual interest rate with semi-annual compounding, we use the future value of an annuity formula. This formula allows us to calculate the payment needed to reach a certain future sum when interest is compounded over time.

Given:

• Future value (FV) = $30,000

• Interest rate (I) = 4.0% per annum, compounded semi-annually, which translates to 2% per period (0.04/2).

• Number of total periods (n) = 7 years * 2 = 14 periods.

The formula for the future value of an annuity is:

[ FV = P \cdot \frac{(1 + r)^n - 1}{r} ]

where:

• ( FV ) is the future value,

• ( P ) is the payment per period,

• ( r ) is the interest rate per period, and

• ( n ) is the total number of periods.

Rearranging the formula to solve for ( P ) gives us:

[ P = \frac{FV \cdot r}{(