Study guide
14.2 Monte Carlo Simulation & Probability Distributions in Cost/Schedule
AACE Certified Cost Professional CCP
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1 Chapter 1: Introduction & Total Cost Management (TCM) Framework
1.1 AACE CCP Exam Structure, Eligibility & the In-Exam Memo 1.2 The Total Cost Management (TCM) Framework & PDCA Cycle 1.3 Strategic Asset Management vs. Project Control Processes
2 Chapter 2: Cost Estimating Fundamentals & Classification
2.1 AACE Recommended Practice 18R-97 Estimate Classification System 2.2 Direct vs. Indirect Costs and Cost Element Breakdown 2.3 Quantity Takeoff, Standard Cost Databases & Pricing Sources
3 Chapter 3: Conceptual & Parametric Estimating Techniques
3.1 Capacity Factoring & the Six-Tenths Rule 3.2 Parametric Estimating, Lang Factors & Hand Factors 3.3 Cost Indices, Location Adjustments & Escalation Modeling
4 Chapter 4: Cost Engineering Economics & Time Value of Money
4.1 Simple & Compound Interest, Equivalence & Cash Flow Diagrams 4.2 Discrete Compounding Formulas & Uniform Series Factors 4.3 Nominal vs. Effective Interest Rates & Continuous Compounding
5 Chapter 5: Project Profitability & Capital Investment Decision-Making
5.1 Net Present Value (NPV) & Present Worth Analysis 5.2 Internal Rate of Return (IRR) & Incremental IRR Analysis 5.3 Payback Period, Benefit-Cost Ratio (BCR) & Breakeven Analysis
6 Chapter 6: Depreciation Methods, Taxes & Asset Valuation
6.1 Classical Depreciation Methods (Straight-Line, DDB, Sum-of-Years-Digits) 6.2 Modified Accelerated Cost Recovery System (MACRS) & Tax Shield 6.3 After-Tax Cash Flow Analysis & Asset Replacement Economics
7 Chapter 7: Project Scope, Work Breakdown Structure (WBS) & Baselines
7.1 Work Breakdown Structure (WBS) Principles & the 100% Rule 7.2 Organizational Breakdown Structure (OBS) & Control Accounts 7.3 Establishing the Performance Measurement Baseline (PMB)
8 Chapter 8: Critical Path Method (CPM) Planning & Scheduling Logic
8.1 Precedence Diagramming Method (PDM) & Dependency Relationships 8.2 Forward Pass, Backward Pass & Float Calculations (Total vs. Free Float) 8.3 Schedule Baseline Development, Calendar Constraints & Milestones
9 Chapter 9: Schedule Optimization, Compression & Probabilistic PERT
9.1 Schedule Crashing & Least-Cost Slope Optimization 9.2 Fast-Tracking Risks, Overlap Analysis & Execution Strategies 9.3 PERT Three-Point Estimating & Probabilistic Duration Variance
10 Chapter 10: Resource Management, Productivity & Learning Curves
10.1 Resource Loading, Leveling vs. Resource Smoothing 10.2 Labor Productivity Factors, Work Sampling & Efficiency Loss 10.3 Learning Curve Theory (Wright Cumulative vs. Crawford Unit Models)
11 Chapter 11: Earned Value Management (EVM) Core Principles & Metrics
11.1 ANSI/EIA-748 EVM Framework & Core Metrics (PV, EV, AC) 11.2 EVM Variances & Performance Indices (CV, SV, CPI, SPI) 11.3 Work Progress Measurement Techniques (0/100, 50/50, Milestone Weighting)
12 Chapter 12: EVM Forecasting, Indices & Baseline Change Control
12.1 Estimate at Completion (EAC) & Estimate to Complete (ETC) Formulas 12.2 To-Complete Performance Index (TCPI) & Variance at Completion (VAC) 12.3 Integrated Change Control, Baseline Revisions & Trend Management
13 Chapter 13: Risk Management Planning & Qualitative Risk Analysis
13.1 Risk Management Framework & Risk Breakdown Structure (RBS) 13.2 Qualitative Risk Assessment & Probability-Impact (P-I) Matrix 13.3 Risk Response Strategies & Threat/Opportunity Management
14 Chapter 14: Quantitative Risk Analysis, Monte Carlo & Contingency Estimating
14.1 Expected Monetary Value (EMV) & Decision Tree Analysis 14.2 Monte Carlo Simulation & Probability Distributions in Cost/Schedule 14.3 Contingency Estimating & AACE Recommended Practice 40R-08
15 Chapter 15: Contract Types, Procurement Strategy & Commercial Terms
15.1 Contract Types & Risk Allocation (Firm Fixed Price, Cost-Plus, T&M) 15.2 Project Delivery Systems & Procurement Lifecycle (DBB, DB, EPC, CMAR) 15.3 Commercial Terms, Bonds, Retainage & Claims Management
16 Chapter 16: Quality Management, Value Engineering & Technical Communication
16.1 Quality Management & Cost of Quality (CoQ) 16.2 Value Engineering, Function Analysis System Technique (FAST) & Life Cycle Costing 16.3 The CCP Memo, Cost Report Writing, Data Presentation & AACE Ethics 16.4 AACE CCP Capstone Review & High-Yield Exam Strategies
17 Chapter 17: Cost Accounting, Budget Administration & Cost System Governance
17.1 Historical Cost Databases, Data Organization & Benchmarking 17.2 Committed Costs, Accruals, Work-in-Place & Invoice Validation 17.3 Budget Creation, Operating Budgets & Funding Requirement Forecasting 17.4 Cost System Governance, the Code of Accounts & Preserving Project Controls Data
18 Chapter 18: Estimate Preparation, Review & Cost Driver Analysis
18.1 ROM, Summary-Level and Detailed Estimates: Preparing and Reviewing 18.2 Verifying the Basis of Estimate & Analyzing Estimates 18.3 Cost Drivers, Change Order Impact & Resource Utilization Analysis
19 Chapter 19: Interfacing with Other Disciplines & Stakeholders
19.1 Coordinating with Internal Disciplines & Supporting Project Leadership 19.2 Communicating Cost Estimates and Cost/Schedule Key Performance Indicators 19.3 Supporting Tenders, Procurement, Audits, Field Verification & Recovery
20 Chapter 20: Cost Reporting, Dashboards & Profitability Analysis
20.1 Cost Reports, Performance Dashboards & the Reporting Cycle 20.2 Variance Reports & Explaining Project Financial Health 20.3 Profit Margin Analysis & Profitability Enhancement
21 Chapter 21: Cost Support to Scheduling & the CCP Exam Memo
21.1 Cost and Resource Loading Summaries from the Project Schedule 21.2 Translating Schedule Change into Cost Impact 21.3 Writing the CCP Exam Memo: A Worked Domain 7 Walkthrough
14.2 Monte Carlo Simulation & Probability Distributions in Cost/Schedule
Key Takeaways Explain This Topic AI
Monte Carlo Simulation (MCS) replaces static single-point deterministic estimates with probabilistic range forecasts by iteratively sampling input probability distributions across thousands of iterations to generate a Cumulative Distribution Function (CDF / S-curve).
Latin Hypercube Sampling (LHS) stratifies the input cumulative distribution into equal-probability non-overlapping intervals, ensuring complete tail coverage and achieving statistical convergence in significantly fewer iterations (typically 1,000 to 5,000) than Simple Random Sampling.
Probability distribution selection must reflect underlying physical and market behavior: Triangular for empirical 3-point estimates; Beta/PERT for smooth weighted central tendencies; Lognormal for positively skewed items with unbounded right tails (severe cost/schedule overruns); and Normal for symmetrical, independent natural variations.
Simulation outputs define risk-adjusted confidence levels: P50 (median) represents the 50% probability cost, while capital project sanction budgets are typically set at P70 to P90 to establish adequate contingency reserves.
In Quantitative Schedule Risk Analysis (QSRA), 'Merge Bias' causes deterministic CPM schedules to systematically underestimate project completion dates because parallel paths converge at merge nodes, where the maximum duration of all converging paths governs the successor start date.
Last updated: August 2026
14.2 Monte Carlo Simulation & Probability Distributions in Cost/Schedule
This section covers 6.D review/interpret risk analysis and impact on a specific project . The blueprint verb is interpret , not run . CCP items on Monte Carlo typically hand you an output — an S-curve, a P-value, a tornado diagram — and ask what it means for a specific project decision, rather than asking you to configure the simulation.
Deterministic cost and schedule estimating relies on single-point values (e.g., 'Labor will cost $ 4 , 500 , 000 \text{\textdollar}4,500,000 $ 4 , 500 , 000 ' or 'Turbine installation will take 45 days' ). However, complex industrial engineering projects rarely behave deterministically. As observed by mathematician Sam L. Savage, relying on single-point averages leads to the 'Flaw of Averages' : plans based on average assumptions will, on average, fail .
Under AACE International Total Cost Management (TCM) standards, Monte Carlo Simulation (MCS) is the recognized standard for quantitative risk analysis (QRA). By replacing single-point inputs with probability distribution functions (PDFs) and recalculating the project model across thousands of randomized iterations, MCS generates a complete Cumulative Distribution Function (CDF) —commonly known as the S-Curve —allowing project stakeholders to establish risk-adjusted budgets and schedules at defined confidence levels (e.g., P 50 , P 80 , P 90 P_{50}, P_{80}, P_{90} P 50 , P 80 , P 90 ).
1. Monte Carlo Simulation Mechanics & Sampling Computational Algorithms +-----------------------------------------------------------------------------+
- | MONTE CARLO SIMULATION COMPUTATIONAL FLOW |
- | |
- | 1. DEFINE DETERMINISTIC BASE MODEL: |
- | Cost Model: Total Cost = \sum (Quantity_i \times Unit_Rate_i) |
- | Schedule Network: CPM Critical Path Logic (Precedence Diagramming) |
- | |
- | 2. ASSIGN PROBABILITY DISTRIBUTIONS TO UNCERTAIN VARIABLES: |
- | Assign appropriate PDFs (Triangular, Beta/PERT, Lognormal) to items |
- | |
- | 3. DEFINE CORRELATION COEFFICIENTS (r_ij): |
- | Model dependencies between interrelated items (e.g., labor rates) |
- | |
- | 4. ITERATIVE STOCHASTIC SAMPLING (5,000 - 10,000 RUNS): |
- | Draw random values from each input PDF; compute total project outcome |
- | |
- | 5. GENERATE STATISTICAL OUTPUT DISTRIBUTIONS: |
- | Construct Histogram (PDF) and Cumulative Probability S-Curve (CDF) |
- +-----------------------------------------------------------------------------+
- Simple Random Sampling vs. Latin Hypercube Sampling (LHS)
A critical distinction in simulation mechanics is the method used to sample input distributions: +-----------------------------------------------------------------------------+
- | SAMPLING METHOD COMPARISON |
- | |
- | A. SIMPLE RANDOM SAMPLING (Standard Monte Carlo): |
- | - Draws pseudo-random numbers purely by chance. |
- | - Risk of "clustering" in central regions and under-sampling tails. |
- | - Requires very large sample sizes (e.g., 20,000+ iterations) to reach |
- | stable convergence in extreme percentiles (P10, P90). |
- | |
- | B. LATIN HYPERCUBE SAMPLING (LHS - Stratified Sampling): |
- | - Divides the cumulative probability range [0, 1] into N equal, non- |
- | overlapping intervals (where N = number of iterations). |
- | - Samples exactly once from each interval without replacement. |
- | - Guarantees uniform coverage across the entire distribution, including|
- | extreme low-probability / high-impact tails. |
- | - Reaches statistical convergence with 70% to 80% fewer iterations |
- | (typically 1,000 to 5,000 runs). |
- +-----------------------------------------------------------------------------+
- 2. Common Probability Distribution Functions in Cost Engineering
Selecting the correct probability distribution is vital. Fitting the wrong distribution to a cost or schedule element introduces model error that invalidates simulation outputs. +-----------------------------------------------------------------------------+
- | COMMON COST ENGINEERING DISTRIBUTIONS |
- | |
- | 1. TRIANGULAR: Tri(min, mode, max) -> Tri(a, m, b) |
- | - Mean: \mu = (a + m + b) / 3 |
- | - Variance: \sigma^2 = (a^2 + m^2 + b^2 - am - ab - bm) / 18 |
- | |
- | 2. BETA / PERT: PERT(min, mode, max) -> PERT(a, m, b) |
- | - Mean: \mu = (a + 4m + b) / 6 |
- | - Standard Deviation: \sigma = (b - a) / 6 |
- | - Variance: \sigma^2 = (b - a)^2 / 36 |
- | |
- | 3. LOGNORMAL: Lognormal(\mu_L, \sigma_L) |
- | - Strictly positive (X > 0); highly positively skewed (long right tail)|
- | - Models unbounded upside cost growth / severe schedule delays |
- | |
- | 4. NORMAL (GAUSSIAN): N(\mu, \sigma^2) |
- | - Symmetrical bell curve (Mean = Median = Mode) |
- | - Governed by Central Limit Theorem for aggregated independent items |
- | |
- | 5. UNIFORM: U(min, max) -> U(a, b) |
- | - Constant probability density: f(x) = 1 / (b - a) |
- | - Mean: \mu = (a + b) / 2; Models total uncertainty / maximum ignorance|
- +-----------------------------------------------------------------------------+
- Comparative Distribution Selection Matrix
Distribution Parameters Symmetry / Skewness Primary Cost Engineering Use Cases Key Pitfalls / Cautions
Triangular a a a (Min), m m m (Mode), b b b (Max) Asymmetrical or Symmetrical Standard 3-point expert elicitation when only rough boundaries and most likely values are known. Straight linear slopes overemphasize extreme corner bounds; sharp apex is unnatural.
Beta / PERT a a a (Min), m m m (Mode), b b b (Max) Asymmetrical or Symmetrical Detailed engineering estimates, labor productivity rates, task durations. Weights the mode heavily ( 4 × 4\times 4 × ); smooth polynomial tails taper gracefully to zero.
Lognormal μ L , σ L \mu_L, \sigma_L μ L , σ L (Log Mean & Std) Positively Skewed (Long right tail) Commodity price spikes, severe labor disruption, major design growth, catastrophic claims. Bounded by zero on left, but right tail is unbounded; can generate extreme outlier values if unchecked.
Normal μ \mu μ (Mean), σ \sigma σ (Std Dev) Symmetrical (Zero skewness) Standardized, highly repeatable fabrication processes with historical quality control data. Unbounded on both tails; can theoretically generate impossible negative cost or duration values.
Uniform a a a (Min), b b b (Max) Symmetrical (Flat) Early conceptual screening when estimators have zero basis to identify a most likely mode. Assumes equal likelihood across entire range; unrealistic for mature engineering items.
3. Interpreting Simulation Outputs: S-Curves and Percentiles
The primary output of a Monte Carlo cost simulation is the Cumulative Distribution Function (CDF) , commonly plotted as an S-Curve . +-----------------------------------------------------------------------------+
- | CONFIDENCE PERCENTILE METRICS |
- | |
- | - P10 (Optimistic): 10% probability that final cost will be <= P10 |
- | (90% chance of cost overrun). |
- | |
- | - P50 (Median): 50% probability that final cost will be <= P50 |
- | (Equal chance of overrun or underrun). |
- | |
- | - Mean (Expected Value): Statistical arithmetic average of all runs. |
- | In positively skewed models: Mean > P50. |
- | |
- | - P80 (Sanction Budget): 80% probability that final cost will be <= P80 |
- | (Standard corporate investment benchmark). |
- | |
- | - P90 (Conservative): 90% probability of non-exceedance; used for |
- | firm fixed-price bidding and high-risk scopes. |
- +-----------------------------------------------------------------------------+
- Mathematical Derivation of Contingency from CDF
Under AACE standards, the quantitative contingency requirement is derived directly from the CDF percentile corresponding to the organization's risk tolerance: Contingency Amount ( P 80 ) = Simulated Cost at P 80 − Deterministic Base Estimate \text{Contingency Amount } (P_{80}) = \text{Simulated Cost at } P_{80} - \text{Deterministic Base Estimate} Contingency Amount ( P 80 ) = Simulated Cost at P 80 − Deterministic Base Estimate Contingency Percentage ( P 80 ) = ( Simulated Cost at P 80 − Base Estimate Base Estimate ) × 100 \text{Contingency Percentage } (P_{80}) = \left( \frac{\text{Simulated Cost at } P_{80} - \text{Base Estimate}}{\text{Base Estimate}} \right) \times 100 Contingency Percentage ( P 80 ) = ( Base Estimate Simulated Cost at P 80 − Base Estimate ) × 100
Sensitivity Analysis: Tornado Diagrams
During simulation, software tracks the correlation between each individual input variable and the total project outcome. This is displayed as a Tornado Diagram :
Ranks risk drivers in descending order of their impact on total cost/schedule variance.
Uses Spearman Rank Correlation Coefficients ( r s r_s r s ) or Pearson Correlation ( r r r ) .
Pinpoints the 'Vital Few' risk drivers (e.g., piping craft labor productivity, steel market escalation) that contribute 80 % 80\% 80% of total bottom-line uncertainty.
4. Correlation and Dependencies Between Cost Elements
In complex capital projects, cost elements rarely move independently. For example, if severe local wage inflation impacts structural steel labor, it will almost certainly impact piping, electrical, and mechanical labor simultaneously. +-----------------------------------------------------------------------------+
- | THE CRITICAL IMPACT OF CORRELATION (r_ij) |
- | |
- | 1. ZERO CORRELATION (r = 0) ASSUMPTION (The Independence Fallacy): |
- | - When items are modeled as independent, positive variances in one item|
- | cancel out negative variances in another (Diversification Effect). |
- | - Result: ARTIFICIALLY NARROWS THE TOTAL S-CURVE; severely understates |
- | P80/P90 contingency requirements. |
- | |
- | 2. POSITIVE CORRELATION (0 < r <= 1.0): |
- | - Items vary in tandem. When labor rates rise, all labor accounts rise.|
- | - Result: WIDENS THE TOTAL S-CURVE SPREAD; increases standard deviation|
- | and establishes realistic, robust contingency amounts. |
- | |
- | 3. CORRELATION MATRIX (Pearson / Spearman): |
- | - Cost engineers construct an n x n matrix defining correlation |
- | coefficients between related WBS accounts. |
- +-----------------------------------------------------------------------------+
- 5. Quantitative Schedule Risk Analysis (QSRA) & Merge Bias
Applying Monte Carlo simulation to project schedules is known as Quantitative Schedule Risk Analysis (QSRA) . QSRA replaces deterministic activity durations with probability distributions and runs Critical Path Method (CPM) forward and backward passes on every iteration.
The Merge Bias Phenomenon (Parallel Path Convergence)
In deterministic CPM scheduling, the total project duration is governed strictly by the single longest path. However, when two or more parallel activity paths converge at a common milestone (a merge node ), deterministic schedules systematically underestimate project completion dates. +-----------------------------------------------------------------------------+
- | THE MERGE BIAS MATHEMATICAL PROOF |
- | |
- | Suppose three independent parallel paths converge at Milestone M: |
- | - Path 1 has an 80% probability of finishing by Day 100: P(T_1 <= 100) = 0.80
- | - Path 2 has an 80% probability of finishing by Day 100: P(T_2 <= 100) = 0.80
- | - Path 3 has an 80% probability of finishing by Day 100: P(T_3 <= 100) = 0.80
- | |
- | DETERMINISTIC FALLACY: |
- | Project manager assumes Milestone M has an ~80% chance of on-time finish. |
- | |
- | PROBABILISTIC TRUTH (Joint Probability): |
- | Because Milestone M cannot start until ALL THREE paths are complete: |
- | P(Milestone M <= 100) = P(T_1 <= 100) * P(T_2 <= 100) * P(T_3 <= 100) |
- | = 0.80 * 0.80 * 0.80 = 0.512 (51.2%) |
- | |
- | CONCLUSION: |
- | Merge bias drops the actual milestone confidence from 80% down to 51.2%! |
- +-----------------------------------------------------------------------------+
- Schedule Criticality Index ( C I CI C I )
Unlike deterministic CPM where an activity is either 100 % 100\% 100% on the critical path or not on it ( 0 % 0\% 0% ), QSRA calculates the Criticality Index ( C I CI C I ) : C I = ( Number of Iterations Activity is on the Critical Path Total Simulation Iterations ) × 100 CI = \left( \frac{\text{Number of Iterations Activity is on the Critical Path}}{\text{Total Simulation Iterations}} \right) \times 100 C I = ( Total Simulation Iterations Number of Iterations Activity is on the Critical Path ) × 100
An activity with high float in the deterministic schedule may have a C I = 45 % CI = 45\% C I = 45% , meaning that in nearly half of all simulated scenarios, duration growth consumes its float and pushes it onto the driving critical path.
6. Comprehensive Worked Step-by-Step Engineering Case Study
Subsea Pipeline Installation Cost Risk Model: A cost engineer builds a risk model for three major work packages with the following three-point estimates:
Package 1: Subsea Trenching & Dredging (Beta/PERT Distribution):
- Optimistic ( a 1 a_1 a 1 ): $ 12 , 000 , 000 \text{\textdollar}12,000,000 $ 12 , 000 , 000
- Most Likely ( m 1 m_1 m 1 ): $ 18 , 000 , 000 \text{\textdollar}18,000,000 $ 18 , 000 , 000
- Pessimistic ( b 1 b_1 b 1 ): $ 36 , 000 , 000 \text{\textdollar}36,000,000 $ 36 , 000 , 000
Package 2: Pipe Lay Vessel Charter (Triangular Distribution):
- Minimum ( a 2 a_2 a 2 ): $ 20 , 000 , 000 \text{\textdollar}20,000,000 $ 20 , 000 , 000
- Mode ( m 2 m_2 m 2 ): $ 25 , 000 , 000 \text{\textdollar}25,000,000 $ 25 , 000 , 000
- Maximum ( b 2 b_2 b 2 ): $ 42 , 000 , 000 \text{\textdollar}42,000,000 $ 42 , 000 , 000
Package 3: Hyperbaric Tie-in & Hydrotesting (Normal Distribution):
- Mean ( μ 3 \mu_3 μ 3 ): $ 10 , 000 , 000 \text{\textdollar}10,000,000 $ 10 , 000 , 000
- Standard Deviation ( σ 3 \sigma_3 σ 3 ): $ 2 , 000 , 000 \text{\textdollar}2,000,000 $ 2 , 000 , 000
Deterministic Base Estimate (Sum of Most Likely / Mode values): Base Cost = $ 18 , 000 , 000 + $ 25 , 000 , 000 + $ 10 , 000 , 000 = $ 53 , 000 , 000 \text{Base Cost} = \$18,000,000 + \$25,000,000 + \$10,000,000 = \mathbf{\$53,000,000} Base Cost = $18 , 000 , 000 + $25 , 000 , 000 + $10 , 000 , 000 = $53 , 000 , 000
Step 1: Calculate Mean and Variance for Each Package
Package 1 (Beta/PERT): μ 1 = a 1 + 4 m 1 + b 1 6 = $ 12 M + 4 ( $ 18 M ) + $ 36 M 6 = $ 12 M + $ 72 M + $ 36 M 6 = $ 120 M 6 = $ 20 , 000 , 000 \mu_1 = \frac{a_1 + 4m_1 + b_1}{6} = \frac{\$12M + 4(\$18M) + \$36M}{6} = \frac{\$12M + \$72M + \$36M}{6} = \frac{\$120M}{6} = \mathbf{\$20,000,000} μ 1 = 6 a 1 + 4 m 1 + b 1 = 6 $12 M + 4 ( $18 M ) + $36 M = 6 $12 M + $72 M + $36 M = 6 $120 M = $20 , 000 , 000 σ 1 = b 1 − a 1 6 = $ 36 M − $ 12 M 6 = $ 24 M 6 = $ 4 , 000 , 000 \sigma_1 = \frac{b_1 - a_1}{6} = \frac{\$36M - \$12M}{6} = \frac{\$24M}{6} = \mathbf{\$4,000,000} σ 1 = 6 b 1 − a 1 = 6 $36 M − $12 M = 6 $24 M = $4 , 000 , 000 σ 1 2 = ( $ 4 , 000 , 000 ) 2 = 16.0 × 10 12 ( $ 2 ) \sigma_1^2 = (\$4,000,000)^2 = 16.0 \times 10^{12} \; \text{($\$^{2}$)} σ 1 2 = ( $4 , 000 , 000 ) 2 = 16.0 × 1 0 12 ( $ 2 )
Package 2 (Triangular): μ 2 = a 2 + m 2 + b 2 3 = $ 20 M + $ 25 M + $ 42 M 3 = $ 87 M 3 = $ 29 , 000 , 000 \mu_2 = \frac{a_2 + m_2 + b_2}{3} = \frac{\$20M + \$25M + \$42M}{3} = \frac{\$87M}{3} = \mathbf{\$29,000,000} μ 2 = 3 a 2 + m 2 + b 2 = 3 $20 M + $25 M + $42 M = 3 $87 M = $29 , 000 , 000 σ 2 2 = a 2 2 + m 2 2 + b 2 2 − a 2 m 2 − a 2 b 2 − m 2 b 2 18 \sigma_2^2 = \frac{a_2^2 + m_2^2 + b_2^2 - a_2 m_2 - a_2 b_2 - m_2 b_2}{18} σ 2 2 = 18 a 2 2 + m 2 2 + b 2 2 − a 2 m 2 − a 2 b 2 − m 2 b 2 σ 2 2 = 20 2 + 25 2 + 42 2 − ( 20 ) ( 25 ) − ( 20 ) ( 42 ) − ( 25 ) ( 42 ) 18 = 400 + 625 + 1764 − 500 − 840 − 1050 18 = 399 18 = 22.167 × 10 12 \sigma_2^2 = \frac{20^2 + 25^2 + 42^2 - (20)(25) - (20)(42) - (25)(42)}{18} = \frac{400 + 625 + 1764 - 500 - 840 - 1050}{18} = \frac{399}{18} = \mathbf{22.167} \times 10^{12} σ 2 2 = 18 2 0 2 + 2 5 2 + 4 2 2 − ( 20 ) ( 25 ) − ( 20 ) ( 42 ) − ( 25 ) ( 42 ) = 18 400 + 625 + 1764 − 500 − 840 − 1050 = 18 399 = 22.167 × 1 0 12 σ 2 = 22.167 ≈ $ 4 , 708 , 150 \sigma_2 = \sqrt{22.167} \approx \mathbf{\$4,708,150} σ 2 = 22.167 ≈ $4 , 708 , 150
Package 3 (Normal): μ 3 = $ 10 , 000 , 000 \mu_3 = \mathbf{\$10,000,000} μ 3 = $10 , 000 , 000 σ 3 = $ 2 , 000 , 000 \sigma_3 = \mathbf{\$2,000,000} σ 3 = $2 , 000 , 000 σ 3 2 = ( $ 2 , 000 , 000 ) 2 = 4.0 × 10 12 ( $ 2 ) \sigma_3^2 = (\$2,000,000)^2 = 4.0 \times 10^{12} \; \text{($\$^{2}$)} σ 3 2 = ( $2 , 000 , 000 ) 2 = 4.0 × 1 0 12 ( $ 2 )
Step 2: Combine Statistical Moments (Assuming Independence)
Total Expected Cost (Mean μ total \mu_{\text{total}} μ total ): μ total = μ 1 + μ 2 + μ 3 = $ 20 , 000 , 000 + $ 29 , 000 , 000 + $ 10 , 000 , 000 = $ 59 , 000 , 000 \mu_{\text{total}} = \mu_1 + \mu_2 + \mu_3 = \$20,000,000 + \$29,000,000 + \$10,000,000 = \mathbf{\$59,000,000} μ total = μ 1 + μ 2 + μ 3 = $20 , 000 , 000 + $29 , 000 , 000 + $10 , 000 , 000 = $59 , 000 , 000
(Notice that the Expected Mean of $ 59.0 M \text{\textdollar}59.0M $ 59.0 M is already $ 6.0 M \text{\textdollar}6.0M $ 6.0 M above the deterministic base estimate of $ 53.0 M \text{\textdollar}53.0M $ 53.0 M due to positive skewness!).
Total Variance ( σ total 2 \sigma_{\text{total}}^2 σ total 2 ) and Standard Deviation ( σ total \sigma_{\text{total}} σ total ): σ total 2 = σ 1 2 + σ 2 2 + σ 3 2 = 16.0 + 22.167 + 4.0 = 42.167 × 10 12 \sigma_{\text{total}}^2 = \sigma_1^2 + \sigma_2^2 + \sigma_3^2 = 16.0 + 22.167 + 4.0 = \mathbf{42.167} \times 10^{12} σ total 2 = σ 1 2 + σ 2 2 + σ 3 2 = 16.0 + 22.167 + 4.0 = 42.167 × 1 0 12 σ total = 42.167 × 10 12 = $ 6 , 493 , 612 \sigma_{\text{total}} = \sqrt{42.167 \times 10^{12}} = \mathbf{\$6,493,612} σ total = 42.167 × 1 0 12 = $6 , 493 , 612
Step 3: Compute Risk-Adjusted Percentile Budgets and Required Contingency
By the Central Limit Theorem, the sum of multiple distributions approximates a Normal distribution with μ = $ 59 , 000 , 000 \mu = \text{\textdollar}59,000,000 μ = $ 59 , 000 , 000 and σ = $ 6 , 493 , 612 \sigma = \text{\textdollar}6,493,612 σ = $ 6 , 493 , 612 :
P 50 P_{50} P 50 (Median = Mean for Normal): $ 59 , 000 , 000 \mathbf{\text{\textdollar}59,000,000} $ 59 , 000 , 000
P 80 P_{80} P 80 Confidence Level ( Z 0.80 = + 0.8416 Z_{0.80} = +0.8416 Z 0.80 = + 0.8416 ): P 80 = μ total + ( 0.8416 × σ total ) = $ 59 , 000 , 000 + ( 0.8416 × $ 6 , 493 , 612 ) = $ 59 , 000 , 000 + $ 5 , 465 , 024 = $ 64 , 465 , 024 P_{80} = \mu_{\text{total}} + (0.8416 \times \sigma_{\text{total}}) = \$59,000,000 + (0.8416 \times \$6,493,612) = \$59,000,000 + \$5,465,024 = \mathbf{\$64,465,024} P 80 = μ total + ( 0.8416 × σ total ) = $59 , 000 , 000 + ( 0.8416 × $6 , 493 , 612 ) = $59 , 000 , 000 + $5 , 465 , 024 = $64 , 465 , 024
P 90 P_{90} P 90 Confidence Level ( Z 0.90 = + 1.2816 Z_{0.90} = +1.2816 Z 0.90 = + 1.2816 ): P 90 = $ 59 , 000 , 000 + ( 1.2816 × $ 6 , 493 , 612 ) = $ 59 , 000 , 000 + $ 8 , 322 , 213 = $ 67 , 322 , 213 P_{90} = \$59,000,000 + (1.2816 \times \$6,493,612) = \$59,000,000 + \$8,322,213 = \mathbf{\$67,322,213} P 90 = $59 , 000 , 000 + ( 1.2816 × $6 , 493 , 612 ) = $59 , 000 , 000 + $8 , 322 , 213 = $67 , 322 , 213
Step 4: Contingency Allocation Table +-----------------------------------------------------------------------------+
- | CONTINGENCY DETERMINATION TABLE |
- | |
- | Metric Cost ($) Contingency ($) Contingency (%)|
- | -------------------- --------------- --------------- ---------------|
- | Deterministic Base $53,000,000 $0 0.0% |
- | P50 (Median) $59,000,000 +$6,000,000 +11.3% |
- | P80 (Sanction Budget) $64,465,024 +$11,465,024 +21.6% |
- | P90 (High Confidence) $67,322,213 +$14,322,213 +27.0% |
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Important
Exam Takeaway: Setting a budget at the deterministic base estimate ($53M) provides less than a 17 % 17\% 17% probability of success ( Z = ( 53 − 59 ) / 6.494 = − 0.924 → P = 17.8 % Z = (53-59)/6.494 = -0.924 \rightarrow P = 17.8\% Z = ( 53 − 59 ) /6.494 = − 0.924 → P = 17.8% ). To achieve corporate governance compliance at P 80 P_{80} P 80 , management must authorize $ 11 , 465 , 024 \text{\textdollar}11,465,024 $ 11 , 465 , 024 ( 21.6 % 21.6\% 21.6% ) in contingency .