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A Mathematical Approach for Cost and Schedule Risk Attribution Fred Y A Mathematical Approach for Cost and Schedule Risk Attribution Fred Y. Kuo NASA Johnson Space Center Abstract The level of sophistication in cost estimate has progressed substantially in the last decade. This is especially true at NASA where, spearheaded by the Cost Analysis Division (CAD) of NASA Headquarters, cost and schedule risks in terms of confidence level have been codified. Cost estimate community now has a very good understanding on how to calculate portfolio risks, identify risk drivers and quantify their impacts, aided by various off-the-shelf simulation engines. However, one thing lacking in all these tools is the attribution of individual risks to the overall S-Curve. In this paper, the author attempts to delineate the concept of risk attribution in a portfolio of cost risks, develop mathematical formulation for this attribution. The author also attempts to extend the same formulation to the quantification of schedule risks by creating a portfolio of risks and tasks that affects the critical path. This approach has great potential in delineating and quantifying risk impacts on cost and schedule, and provides a metrics for trade studies in risk mitigation. Background and Motivation There are many off-the-shelf cost and schedule risk analysis tools on the market that allow user to perform cos/schedule risk analysis and generate S-Curves so that different confidence level can be estimated. One important aspect of risk analysis is to quantify the risk impact of each risk on the project. Tools like Primavera Risk Analysis allows user to perform sensitivity analysis with the aforementioned intent. However, the result is more confusing than illuminating. Figure 1 and Figure 2 serve to illustrate this point. Figure 1 Cost Risk Sensitivity (Post-mitigated) Cost Sensitivity 63 - Firmware Effort Growth 16% 80 - Inadequate V&V Schedule for L5/6 D&T 6% 51 - Availability of Legacy Simulators -5% 75 - Inadequate V&V Schedule for L3/4 I&T 3% 62 - Software Size Growth (EI) 3% 52s - Availability of Main Mission Antennas 3% Figure 2 Schedule Risk Sensitivity (Pre-mitigated) Duration Sensitivity 80 - Inadequate V&V Schedule for L5/6 D&T 29% 75 - Inadequate V&V Schedule for L3/4 I&T 8% 41 - Microcontroller Development -6% 63 - Firmware Effort Growth 6% 76 - Early Ops Schedule 6% 49 - Interface Definitions/Documentation -5% The definitions of cost and duration sensitivities are as follows: 1. The cost sensitivity of a risk event is a measure of the correlation between the occurrence of any of its impacts and the cost of the project (or a key task). 2. The duration sensitivity of a risk event is a measure of the correlation between the occurrence of any of its impacts and the duration (or dates) of the project (or a key task). After working with this tool for many years and after a number of calls to the technical support, I am still befuddled by these two charts, especially the negative values. Somehow the risk of “availability of simulators and emulators” for testing is negatively correlated with project costs and schedule duration just does not sit well with common sense. The problem of using Correlation as a measure is that it is not a good risk metrics. There is little useful information that can be gleaned from these two charts. A more useful and intuitively appealing approach is to calculate the contribution of each risk to a predefined risk measure so that the results are unambiguous. The advantages of this approach are manifold, not least of which is to provide a clear metrics for trade studies for a cost/benefit analysis for risk mitigation. Some of the current tool allows user to perform this manually by turning on/off each risk, rerun the simulation and records the difference. This approach becomes untenable if a large set of discrete risks has to be entertained. Therefore, it will be of value if a mathematical formulation can be developed and be easily integrated with existing cost estimate tools so that more insightful information regarding risk attribution can be obtained. Defining Risk Measure The financial industry went through a similar struggle on Wall Street, namely, how to capture the potential loss of a portfolio of assets with a certain probability. The Risk Metrics group of JP Morgan Bank came up with the concept of Value-at-Risk (VaR), and the concomitant methodology and process in calculating this number in the 1990s. Despite many criticisms, VaR is without question the most widely used and recognized risk metrics in the financial industry for risk management. The notable innovation of VaR is that it recognizes the portfolio volatility as a quantitative risk measure from which a portfolio loss both in terms of magnitude and probability can now be calculated. This same idea can be extended to the cost and schedule risk analysis. A project, with its constituents of subprojects or systems, can be considered as a portfolio. Therefore, the cost of a project is the statistical sum of its constituent parts. From cost estimate viewpoint, each item of the portfolio can be represented by a cost curve or its parametric equivalent of expected value (mean) and standard deviation. Borrowing from financial industry, the proper measure for risk is the concept of “volatility”; the dispersion around the mean or standard deviation. This risk measure is intuitively appealing as even cost estimate practitioners seem to have the same notion that the “steepness” of the cost curve is a measure of the “risk” of the estimate. It also turns out that by choosing portfolio standard deviation as the risk measure, Euler’s theorem on Homogeneous Function provides a general method for additively decomposing this risk measure into its specific contributions. Appendix A describes the theorem. Mathematical Formulation The portfolio standard deviation can be represented as: ′ 휎푝 = √푤 Σ푤 Where 휎11 ⋯ 휎1푛 Σ = ( ⋮ ⋱ ⋮ ) is the covariance matrix 휎푛1 ⋯ 휎푛푛 푤 = [ 푤1, 2, … , 푤푛] is a vector of portfolio weights 푤′ is the transpose of 푤. The advantage of choosing 휎푝as the risk measure is that now we can use Euler’s theorem on Homogeneous Function of Degree One to decompose risks as: 휕휎푝 휕휎푝 휕휎푝 휎푝 = 푤1 + 푤2 + . + 푤푛 휕푤1 휕푤2 휕푤푛 Note that 휕휎푝 푀퐶푅1 = is defined as the marginal contribution to risk measure by risk #1 휕푤1 Then 퐶푅1 = 푤1 ∗ 푀퐶푅1 is the contribution to risk measure by risk #1, and the total risk is the summation of each of the risk contribution 퐶푅 휎푝 = 퐶푅1 + 퐶푅2+ . + 퐶푅푛 So the percent contribution from each risk is: 퐶푅푖 푃퐶푅= 휎푝 Derivation of Marginal Contribution to Risk The marginal contribution to risk (MCR) is just the partial derivative of 휎푝 with respect to weights. In matrix notation, the derivation is: 1 ′ −1 휕휎푝 휕(푤 Σ푤)2 Σ푤 Σ푤 = = (푤′Σ푤) 2 (Σ푤) = = 휕푤 휕푤 1 휎 (푤′Σ푤)2 푝 Where, 휕휎푝 푡ℎ Σ푤 is the 푖 row of 휕푤푖 휎푝 Example for a portfolio of 2 Risks ′ 휎푝 = √푤 Σ푤 2 2 휎1 휎12 푤1 푤1휎1 + 푤2휎12 Σ푤= ( 2 ) (푤 ) =( 2 ) 휎12 휎2 2 푤2휎2 + 푤1휎12 2 푤1휎1 +푤2휎12 Σ푤 휎푝 푀퐶푅1 = ( 2 ) =( ) 휎푝 푤2휎2 +푤1휎12 푀퐶푅2 휎푝 2 2 퐶푅1 푤1 휎1 +푤1푤2휎12 퐶푅1 = 푤1 푀퐶푅1 ; 푃퐶푅1 = = 휎푝 휎푝 2 2 퐶푅2 푤2 휎2 +푤1푤2휎12 퐶푅2 = 푤2 푀퐶푅2 ; 푃퐶푅2 = = 휎푝 휎푝 푛 It is obvious that ∑=1 푃퐶푅 = 1 One of the advantages of this formulation is that it is not just limited to risks. The same formulation can be extended to include opportunities by merely reversing the signs of weights in the equation. Numerical Examples The following numerical examples serve to illustrate the validity of this formulation. In this example, a portfolio of 2 cost risks is considered and each is a lognormal distribution. The properties of the distribution and the resulting calculations are shown in the following table. Mean SD Covariance w(i) MCR(i) CR(i) PCR(i) Risk 1 10 0.2 0.0162 0.3338 0.1663 0.0555 0.3759 Risk 2 20 0.15 0.0162 0.6662 0.1383 0.0921 0.6240 Portfolio 30 0.15 This result shows that Risk 1 contributes 37.6% of the overall variance, and Risk 2 contributes 62.4% of the overall variance. In the next example, a portfolio of 5 risks with different risk characteristics were considered. The following table shows the type of risks and their parametric values. Type Mean SD W(i) MCR(i) CR(i) PCR(i) Risk 1 Lognormal 9.981 2.004 0.118 0.146 0.017 0.123 Risk 2 Lognormal 19.957 3.013 0.235 0.117 0.028 0.196 Risk 3 Triangular 18.312 4.236 0.216 0.189 0.041 0.292 Risk 4 Triangular 11.658 3.046 0.137 0.200 0.028 0.197 Risk 5 Normal 24.962 2.981 0.294 0.092 0.027 0.193 Portfolio 84.411 11.881 1.000 0.140 1.000 From this table, one can see that Risk 3 has the largest contribution to the variance even though it is the third from the weight stand point. The plots of risk contribution to portfolio mean and standard deviation are shown in Figure 3 and 4. Figure 3 Figure 4 It should be emphasized that the sum of contribution from all risks is 100%. These results are more intuitively appealing because it is unambiguous and self-consistent. The other advantage of this approach is that we can mix risks with “opportunities” by reversing the sign of weights as long as the sum of the weights is 1. This allows the “opportunity” to reduce the variance of the portfolio.
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