Question

The amount of $2,800 is deposited into a sinking fund at the end of every semi-annual period for 5 years. Assume the interest rate is 7.5%compounded semi-annually.

What is the account balance after 1 year?

Answer 1

The account balance of the sinking fund after 1 year, with a 7.5% interest rate compounded semi-annually and semi-annual deposits of $2,800, will be $5,805. This includes one period of interest on the first payment.

Step-by-step explanation:

The question relates to the calculation of the future value of a series of semi-annual payments into a sinking fund with an interest rate that is compounded semi-annually. Since the deposits are made at the end of each semi-annual period, we are dealing with an ordinary annuity situation. After 1 year, two payments of $2,800 will have been made into this fund.

To calculate the balance after 1 year, we can use the future value of a single sum formula for the first deposit after it has been in the account for one period (6 months), and simply add the amount of the second deposit, since it does not earn interest:

Future Value = $2,800 (1 + 0.075/2)^1 + $2,800

The calculation will be:

=$2,800 (1 + 0.0375)^1 + $2,800

=$2,800 (1.0375) + $2,800

=$2,800 × 1.0375 + $2,800

=$2,805 + $2,800

=$5,805

The account balance after 1 year would be $5,805. This takes into account one period of interest on the first payment, as the second payment would not have had time to accrue interest by the end of the year.

Answer 2

The account balance after 1 year, with semi-annual deposits of $2,800 and an interest rate of 7.5% compounded semi-annually, would be $5,705.

Step-by-step explanation:

The subject of this question is mathematics, and it involves calculating the balance of a sinking fund with regular deposits and compound interest. We want to find out the account balance after 1 year when an amount of $2,800 is deposited at the end of every semi-annual period for 5 years with an interest rate of 7.5% compounded semi-annually. To compute this, we can use the future value of an annuity formula, which is P × [(1 + r)^n - 1]/r, where P is the regular deposit amount, r is the interest rate per period, and n is the total number of periods.

After 1 year, there will be 2 periods because deposits are made semi-annually. The first deposit would have compounded for one period, and the second deposit would not have compounded yet since it's made at the end of the period. The calculation is as follows:

• First deposit future value: $2,800 × (1 + 0.075/2)
• Second deposit future value: $2,800 (since it does not compound)

Adding both deposits together gives us the total balance after 1 year.

The future value of the first deposit is $2,800 × (1 + 0.075/2)^1 = $2,800 × 1.0375 = $2,905

So, the total account balance after 1 year is $2,905 (first deposit with interest) + $2,800 (second deposit without interest) = $5,705.