Which of the following is true about perpetuities?

Question

BlackTom AI · Accepted Answer

Consider the question about perpetuities and examine each statement in turn to see what holds true under the standard definition.
Option a: 'If two perpetuities have the same present value and the same interest rate, they must have the same cash flows.' In the classic perpetuity model with a fixed annual cash flow C and discount rate r, the present value is PV = C / r. If PV is the same for both and r is the same, then C must equal PV × r, which uniquely determines the cash flow. Therefore, under the standard fixed-perpetuity assumption, this statement is true.
Option b: 'All of the above are true statements.' To evaluate this, we must check the other statements. Since option a is true as just explained, we need to assess options c and d as well to determine whether all are true.
Option c: 'All else equal, the value of a perpetuity is higher when the periodic cash flow is higher.' Given PV = C / r, a higher C directly increases PV when r stays fixed. Hence this statement is true.
Option d: 'All else equal, the value of a perpetuity is higher when the interest rate is lower.' With PV = C / r, reducing r increases PV, assuming C remains the same. Therefore this statement is also true.
Having found each of a, c, and d to be true under the standard perpetuity framework, option b, which asserts that all of the above are true, is consistent with the analysis of the other options.
In summary, the reasoning shows that each individual claim aligns with the fundamental formula PV = C / r for a fixed perpetuity, and consequently all statements are correct. The logic hinges on the direct relationships: C determines PV at a given r, and PV rises as C increases or as r decreases. The structure of these relationships explains why the set of statements is collectively true.

MCD2170 - T1 - 2025 Key Concept 4 Video quiz

Which of the following is true about perpetuities?

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