Like annuities, perpetuities are characterized by equal dollar value payments that are regular and periodic. Unlike annuities, however, perpetuities have a shorter duration.

Question

BlackTom AI · Accepted Answer

Perpetuities provide equal periodic payments forever; annuities provide equal periodic payments for a fixed, finite term. This distinction about duration is the key to judging the statement.

1. Relevant knowledge points and concepts
- An annuity: a series of equal payments made at regular intervals for a specified number of periods (finite n).
- A perpetuity: a series of equal payments made at regular intervals that continue indefinitely (n → ∞).
- Present value formulas: PV(annuity) = PMT × [1 − (1 + r)^−n] / r. PV(perpetuity) = PMT / r (assuming r > 0).

2. Step-by-step reasoning
- The statement claims perpetuities have a shorter duration than annuities. Duration here refers to the length of time payments are made.
- By definition, a perpetuity’s payments never stop, so its duration is infinite.
- An annuity’s payments stop after n periods, so its duration is finite.
- Infinite duration is clearly not “shorter” than a finite duration.

3. Calculations and intermediate steps (illustrative)
- Take PMT = $100 and r = 5% for comparison.
- PV(perpetuity) = 100 / 0.05 = $2,000.
- For an annuity of n = 20: (1 + r)^−n = 1.05^−20 ≈ 0.3769.
- PV(20-year annuity) = 100 × [1 − 0.3769] / 0.05 = 100 × 0.6231 / 0.05 = 100 × 12.462 = $1,246.20.
- The perpetuity lasts forever (infinite n); the annuity stops after 20 periods. The perpetuity’s payments continue indefinitely, hence longer duration.

4. Why the correct answer is correct
- Because perpetuities never end, they have infinite duration. The statement claiming perpetuities have a shorter duration contradicts the definitions and is therefore false.

5. Why the other option (True) is incorrect
- As noted, True would assert perpetuities are shorter, which contradicts the proper definitions: annuities are finite, perpetuities infinite. Thus True is incorrect.

6. Conclusion
- The correct response is False, which agrees with the provided answer.

Like annuities, perpetuities are characterized by equal dollar value payments that are regular and periodic. Unlike annuities, however, perpetuities have a shorter duration.True/False

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