You are comparing two annuities that offer regular payments of $2,500 for five years and pay .75 percent interest per month. You will purchase one of these today with a single lump sum payment. Annuity A will pay you monthly, starting today, while Annuity B will pay monthly, starting one month from today. Which one of the following statements is correct concerning these two annuities?

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BlackTom AI · Accepted Answer

First, restate the setup: there are two 60-payment annuities of 2,500 each, with a monthly interest rate of 0.75% (i = 0.0075). Annuity A starts payments immediately (an annuity due), while Annuity B starts payments one month from today (an ordinary annuity). We will compare present values (PV) and future values (FV) to see how they differ between the two.

Option 1: Annuity B has a smaller present value than Annuity A. This aligns with the standard relationship PV_due = PV_ordinary × (1 + i). Since Annuity A is due, its present value is larger than Annuity B’s for the same payment amount, number of payments, and interest rate. The math is PV_due = 2,500 × [1 + i] × [1 − (1 + i)^−n]/i, while PV_ordinary = 2,500 × [1 − (1 + i)^−n]/i. Because (1 + i) > 1, PV_due > PV_ordinary, meaning Annuity B’s PV is indeed smaller.

Option 2: Annuity B is an annuity due. This is incorrect because Annuity B begins payments one month from today, which characterizes an ordinary annuity, not a due. Annuity due would have payments starting today.

Option 3: These two annuities have both equal present and equal future values. This is false given the timing difference. The initial payment timing creates a PV and FV advantage for the due version (Annuity A) over the ordinary version (Annuity B). Therefore neither PV nor FV are equal between the two.

Option 4: Annuity A has a smaller future value than Annuity B. This cannot be right because the future value of a due annuity is larger than that of an ordinary annuity when payments and terms are the same, since each payment is invested an extra period. Hence FV_A > FV_B, not smaller.

Option 5: These annuities have equal present values but unequal future values. This is contradicted by the standard relationship between PVs for due and ordinary annuities with identical terms and payments; the present values are not equal (PV_due > PV_ordinary), and the future values are also not equal (FV_due > FV_ordinary).

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