Lecture 9: Uncertainty, Sensitivity, and Other Analysis Techniques

Uncertainty and sensitivity are central to sound engineering economic analysis. Future consequences—costs, benefits, useful life, and salvage value—are rarely known with precision. The goal is not just to select the best estimate, but to understand how uncertainty affects decisions.

I. Addressing Uncertainty in Analysis

Economic analysis requires evaluating future consequences, which are inherently difficult to describe accurately.

A. Range of Estimates (Scenarios)

Replace single values with a range of possible estimates for critical parameters:

  • Optimistic Estimate: Most favorable outcome
  • Most Likely Estimate: Best single projection
  • Pessimistic Estimate: Least favorable outcome

Analyzing these scenarios helps determine if the decision is sensitive to the estimated range of values.

B. Weighted Averages & Expected Value

  • Weighted Mean (Project Management): Assigns weights to optimistic, most likely, and pessimistic estimates (1:4:1)
  • Expected Value (EV): Probability-weighted average of all possible outcomes. Probabilities can be based on historical data or expert judgment.
  • Decision Rule: Maximize expected Present Worth (EV(PW)) or minimize expected Equivalent Uniform Annual Cost (EV(EUAC)).

C. Risk Measurement

  • Probability of Loss: Chance of negative present worth
  • Standard Deviation (\(\sigma\)): Measures dispersion of outcomes; higher \(\sigma\) means greater risk
  • Risk vs. Return: Compare expected return (IRR or PW) against risk (standard deviation) to identify efficient projects

II. Sensitivity & Breakeven Analysis

Sensitivity and breakeven analysis test the stability and criticality of estimates relative to the final decision.

A. Sensitivity Analysis

Measures how much an input estimate must change to reverse a decision.

  • Criticality: If a small variation in an estimate causes a large change in the final measure of merit, the decision is sensitive to that estimate.
  • Application: Focus effort on improving critical estimates.

B. Breakeven Analysis

Pinpoints the point of indifference—the exact condition under which two alternatives yield equivalent results.

  • Calculation: Determines the value of a variable (e.g., production volume, initial cost, years) at which measures of merit are equal.
  • Application Example: Find the economic life required for a project to break even (NPW = 0).

C. Graphical & What-If Analysis

  • Graphical Representation: Breakeven charts display the economic measure versus the variable being changed. Useful for visualizing sensitivity and changeover points.
  • What-If Analysis: Spreadsheet-based approach to change one or many input estimates and observe the effect on outcomes (e.g., Benefit-Cost Ratio, Present Worth Index).

III. Alternative Measures of Merit

Beyond Present Worth, Annual Worth, and Rate of Return, several techniques address specific decision contexts.

A. Future Worth (FW) Analysis

  • Concept: Calculates the equivalent total worth of cash flows at a future date.
  • Relationship: Related to PW and EUAW; positive PW implies positive FW.
  • Calculation: Move all cash flows to the future using compound amount factors.

B. Benefit-Cost (B/C) Ratio

  • Core Calculation: \(\text{B/C Ratio} = \frac{\text{Equivalent Worth of Benefits}}{\text{Equivalent Worth of Costs}}\)
  • Decision Rule: \(\text{B/C} \geq 1\) is justified.
  • Public Sector: Numerator = public benefits; Denominator = sponsor costs.
  • Private Sector (Present Worth Index): Numerator = all subsequent costs/benefits; Denominator = initial investment.
  • Incremental Analysis: Use incremental B/C ratio (\(\Delta B / \Delta C\)) for mutually exclusive alternatives.

C. Payback Period

  • Purpose: Time until accumulated net benefits equal initial investment.
  • Limitations: Ignores time value of money and consequences after payback; quick metric of liquidity.

IV. Expected Value Concepts

Expected value introduces probabilistic uncertainty into economic decision-making.

  • Concept: Probability-weighted average of all possible outcomes.
  • Application: Use expected monetary value (e.g., expected PW or annual cost) when probabilities can be assigned to estimates.

Summary

  • Use ranges and probabilities to address uncertainty
  • Apply sensitivity and breakeven analysis to test decision stability
  • Consider alternative measures of merit for different contexts
  • Use expected value for probabilistic analysis