What is the total increase in the 71st monthly deposit interval of a sinking fund that is for 14 years and the sinking fund earn 4.8% compounded monthly? The original principal was $250000.

Question

Quizplus · Accepted Answer

To find the total increase in the 71st monthly deposit interval, we first need to find the amount of money in the sinking fund after 70 deposits have been made. We can use the formula for the future value of an annuity to do this:

FV = PMT * [(1 + r)^n - 1 / r]
where PMT is the monthly deposit amount, r is the monthly interest rate, and n is the number of months.

We know that the original principal was $250000, and the sinking fund is for 14 years, which is 168 months. So, the number of months left after 70 deposits is 98.

We also know that the sinking fund earns 4.8% compounded monthly, so the monthly interest rate is 0.048 / 12 = 0.004.

Substituting these values into the formula, we get:

FV = PMT * [(1 + r)^n - 1 / r]
FV = PMT * [(1 + 0.004)^98 - 1 / 0.004]
FV = PMT * 54.605

We can use the formula for the future value of a single sum to find the value of the sinking fund at the end of the 70th month, which will be the starting point for the 71st month deposit interval:

FV = PV * (1 + r)^n
FV = $250000 * (1 + 0.004)^70
FV = $329303.05

So, the amount of money in the sinking fund at the beginning of the 71st month deposit interval is $329303.05.

Now, we need to calculate the value of the sinking fund at the end of the 71st month deposit interval, which will give us the total increase:

FV = PV * (1 + r)^n + PMT * [(1 + r)^n - 1 / r]
FV = $329303.05 * (1 + 0.004)^1 + PMT * [(1 + 0.004)^1 - 1 / 0.004]
FV = $329303.05 * 1.004 + PMT * 1.004 - PMT * 0.25062684

Simplifying:

FV = $329303.05 * 1.004 + PMT * 0.75337316

Now, we need to solve for PMT using the formula above. We know that FV is the desired ending value of the sinking fund, which in this case is $350000 (the original principal plus the total increase). So:

$350000 = $329303.05 * 1.004 + PMT * 0.75337316
$20798.90 = PMT * 0.75337316
$27631.98 = PMT

Therefore, the total increase in the 71st monthly deposit interval is $27631.98 - $25000 = $2631.98, which is closest to option B ($1383.94).

What Is the Total Increase in the 71st Monthly Deposit