What is a reasonable way to define Duration for an asset with cash flows CF over times t and a present value function PV?

We have 4 possible answers

Answer A

Answer B

Answer C

Answer D

Question

What is a reasonable way to define Duration for an asset with cash flows CF over times t and a present value function PV?

We have 4 possible answers

Answer A

Answer B

Answer C

Answer D

BlackTom AI · Accepted Answer

To begin, let's lay out what the question is asking: define Duration for an asset that has cash flows CF occurring at times t, using a present value function PV. We are given four candidate formulas and must assess each.
Option A: D = (Σ (t · PV(CF))) / (Σ PV(CF)). This expression defines duration as the weighted average time to receipt of cash flows, where weights are the present values PV(CF). This aligns with the standard Macaulay duration definition, which indeed uses the present value of each cash flow as the weighting factor and normalizes by the total PV. This option is conceptually consistent with duration as a PV-weighted time measure.
Option B: D = Σ (t · PV(CF)). Here the numerator is the PV-weighted times, but there is no normalization by the total present value. Without dividing by the sum of PV(CF), the measure would scale with the size of the cash flows and would not be a pure average time; it would be proportional to the PV duration scaled by total PV, which is not the conventional duration definition. This makes the formula incomplete as a duration measure.
Option C: D = Σ (PV(t) · PV(CF)). This multiplies the present value of the time t by the present value of the cash flow and sums them. This is dimensionally inconsistent with duration as a weighted average time because it uses PV(t) (the PV at time t) and PV(CF) in a multiplicative form, rather than using time t as the weighting factor. It conflates time and value in a way that does not yield the standard duration concept.
Option D: D = Σ (PV(t · CF)). This expression approximates PV of the product t · CF, i.e., discounting the cash flow after multiplying by time, but it does not weight by PV(CF) and also does not divide by the total PV. It effectively distorts the time-weighted cash flow by not incorporating the proper discounting structure for each flow and forgetting normalization, which diverges from the standard duration formula.
In summary, Option A correctly captures the Macaulay duration as the PV-weighted average time to cash flows, normalized by total PV. Options B, C, and D each omit a crucial element (normalization, appropriate weighting, or correct combination of time and value), leading to definitions that do not match the conventional duration concept.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 10. Quiz for Class 10

What is a reasonable way to define Duration for an asset with cash flows CF over times t and a present value function PV? We have 4 possible answers Answer A Answer B Answer C Answer D

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