Understanding cash-flow gradients: in a time-phased cash-flow series, what statements correctly describe a uniform gradient and its sign?

Difficulty: Easy

A uniform gradient signifies that the cash flow increases or decreases by the same fixed amount each period.
Any period-by-period change (increase or decrease) is referred to as a gradient.
A gradient in cash flow can be positive (upward) or negative (downward).
All of these
None of these

Correct Answer: All of these

Explanation:

Introduction / Context:
Many engineering and business cash flows do not remain constant; they change by a fixed amount every period—think of scheduled maintenance ramp-ups or planned annual reductions. Such patterns are modeled using arithmetic (uniform) gradients, which are indispensable in present-worth and annual-worth analyses.

Given Data / Assumptions:

Cash-flow series indexed by periods t = 1, 2, …, n.
A constant increment (or decrement) G applied each period relative to the prior one.
Standard gradient factors are available for discounting/compounding.

Concept / Approach:
An arithmetic gradient adds a constant amount G each period to the base annuity; geometric gradients instead grow by a constant percentage. The sign of G establishes whether the series is rising (positive gradient) or falling (negative gradient). Correct identification of the gradient type is crucial to applying the right factor formulas.

Step-by-Step Solution:

Define gradient: difference between successive cash-flow amounts.Uniform (arithmetic) gradient: difference equals constant G each period.Assign sign: G > 0 → positive gradient; G < 0 → negative gradient.

Verification / Alternative check:

Model an example: A base annuity A with gradient G for n periods; convert series to present worth using gradient factor P/G at interest rate i.

Why Other Options Are Wrong:

Each of a–c is correct; “All of these” is the comprehensive answer.“None of these” conflicts with standard gradient definitions.

Common Pitfalls:

Confusing arithmetic gradients with geometric escalation; use the correct factor (P/G versus P/A with growth).Forgetting that gradients typically start in period 2 by convention in factor notation.

Final Answer:

All of these

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Understanding cash-flow gradients: in a time-phased cash-flow series, what statements correctly describe a uniform gradient and its sign?

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