CorBloc Ltd. is setting up a fund to pay back a $22,000,000 debt in 18 years. To accumulate this money they will make equal annual deposits at the beginning of each year into a fund earning 9% compounded quarterly. What will be the size of the annual deposits?

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Quizplus · Accepted Answer

Since the deposits are made at the beginning of each year, this is an annuity due. The formula to calculate the annual deposit (PMT) for an annuity due is: $PMT = \frac{FV}{\frac{((1 + r)^n - 1)}{r} \times (1 + r)}$, where FV is the future value ($22,000,000), r is the quarterly interest rate (9% annual rate compounded quarterly, so $r = \frac{9\%}{4} = 2.25\%\ or\ 0.0225$ per quarter), and n is the total number of deposits (since the fund earns interest compounded quarterly but deposits are made annually, we need to adjust n to reflect the total number of compounding periods, which is $18 \times 4 = 72$ quarters). However, since the deposits are made annually, we need to adjust the formula to reflect the annual deposit and annual compounding for the purpose of this calculation. The effective annual rate (EAR) can be used instead of the quarterly rate for simplification, calculated as $EAR = (1 + \frac{0.09}{4})^4 - 1$. The formula simplifies the calculation by using the annual rate and the number of years directly, avoiding the need to adjust for quarterly compounding in the deposit calculation. The correct calculation involves using the future value of an annuity due formula, taking into account the timing of deposits (beginning of the period) and the effective interest rate over the period. The given options suggest a misunderstanding in the calculation process, possibly not adjusting for the annuity due nature or misapplying the interest rate or compounding frequency. The correct approach involves calculating the present value of the annuity due, considering the effective annual rate and the timing of payments. The discrepancy in the provided answer indicates a need to revisit the calculation, ensuring the correct application of the annuity due formula, the effective interest rate, and the compounding effect over the specified period.