Contingency is added to a project estimate to cover the inherent project risks. The estimated contingency should be neither too high nor too low. By adding the optimum contingency to the base estimate, the owner makes sure that there is no budget over run and the contractor ensures himself to quote the right price and win the bid. As part of the risk management process, risks are quantified by determining the risk contingency in a project. Instead of arbitrarily assigning a percentage for contingency, calculations can be performed using a threepoint estimate, with basic understanding of probability analysis and statistics. There are many commercial risk analysis software packages available, but the focus of this article is on how to calculate contingency using an Excel spreadsheet.
Contingency
Contingency is defined as "An amount added to an estimate to allow for items, conditions, or events for which the state, occurrence, or effect is uncertain and that experience shows will likely result, in aggregate, in additional costs" [1]. Contingency is described in many different ways across the industry but the common purpose of determining contingency is to account for the uncertainty or unexpected. Contingency has an inverse relationship with risk. A risk averse person may try to add more contingency and a risk tolerant person will opt for lowering it. The higher the risk, the lower the contingency will be and vice versa.
ThreePoint Estimates
In a threepoint estimate, the cost of an item is expected to be between a range of 'low' and 'high' values, with a 'most likely' value in between the two. Table1 shows threepoint estimates for five selected items in a project. For simplicity, we will assume that each of these values is independent of others. The estimator may have the estimated value as the 'most likely' value and the upper and lower values may be assigned based on experience or by parametric methods. If the estimated value is not available, the average of the three values may be used in the contingency analysis and calculations. The 'average', is calculated by dividing the sum by three (a + b + c)/3. Alternatively, this average can be arrived at by using ‘Monte Carlo simulation’ that is also possible with Excel spreadsheets, using the random value (RAND ()) function. We will use the 'most likely' values for our analysis and calculations in this article.
Table 1  Threepoint estimates
Triangular Distribution
Most of us may be able to recall terms like 'normal distribution', 'bell curve', etc., from basic statistics. Not all data may have the 'normal distribution' profile. For threepoint estimates, the approximated distribution profile is a triangular shape. Triangular distribution is used in our analysis and it is easy to visualize and understand.
Figure1 shows a triangular distribution for the three values of item 'A'. A random variable can range between $7,500 (a) to $12,000 (b) with a peak value of $8,250 (c). The area under the curve represents the probability and the total area of the triangle will be equal to one (1), for 100% probability of occurrence. Let us assume that A1 is the area of the triangle between 'a' and 'b' and A2 is the area of the triangle between 'b' and 'c'. The total area, i.e. A1 + A2 will be equal to one (1), representing 100% probability.
The height of the triangle, using the probability density function is 2/(ca):
A1 = 1/2 * (b  a) * 2/(c  a)  (1), A1 is the area of triangle on the left
or
A1 = (b  a)/(c  a)
A2 = 1/2 * (c  b) * 2/(c  a)  (2), A2 is the area of triangle on the right
or
A2 = (c  b)/(c  a)
Substituting values for a (7,500), b (8,250) and c (12,000) for item 'A', the cumulative probability at the most likely value, 8,250 will be (82507500) / (12,0007,500) = 0.17 or 17%. This means the likelihood of the expected value being less than or equal to $8,250 is 17%.
Figure 1  Triangular Distribution
The cumulative probability at the three values (a, b, and 'c') for item 'A' will be 0%, 17% and 100%.
Fig2 explains the probability of under run and over run for (b).
Figure 2  Probability of under run and over run
Analysis of Probability of Underrun
In our example (item 'A'), the probability of under run is 17% and over run 83% for the most likely value of 'A'. The contractor or the owner may not be comfortable with only 17% of under run. The chance of over run is more than the under run (4:1). Instead, the expected value for a higher probability of under run (80% or higher), may be acceptable. Below, Table2 indicates the 'actual' probability and the 'recommended' probability for items 'A' to 'E'. The probability of over run is reduced to 5% for items A, B and E. For item C, the probability of under run is reduced to 75% from 80%, as the owner's or contractor's risk analysis (RA) team feels confident to go with 75% instead of 80%. Similarly, for item D, the actual probability of the 'most likely' value is not changed, as the team feels confident with 90% under run probability.
Table 2  Actual probability vs. recommended probability
Calculating Expected Values
The next step is to find the expected value 'x' for the recommended probability of under run. The recommended probability is the area of the triangle up to the expected value. As we know the areas (recommended probabilities), substituting for P (x), we can find the value of the expected value (x). The equations shown in Table3 are statistical equations used for the cumulative density function (CDF). Using Excel spreadsheetd the expected value calculated for the suggested probability of under run for items 'A' to 'E' are tabulated in Table 4. The Table also indicates the formulas built into the Excel cells.
Diagrammatic representation for the recommended probability for item 'A' and the corresponding expected value is shown below (Fig3).
Figure 3  Recommended probability for item 'A'
Table 3  Statistical equations used for the cumulative density function (CDF)
Table 4  Suggested probability of under run
Analysis of the Estimate Totals
Initially the total of the 'low', 'most likely' and the 'high' values are 55,500, 74,050 and 93,100, respectively. The probability of the 'most likely' total value (74,050) is 0.49 or 49%. After incorporating the recommended probabilities, the expected value total is 86,091. If we calculate the probability of under run for this overall total, it is 0.93 or 93%. This is much better than the initial value of 49%. We can also perform the probability analysis just for the three totals but performing it for the individual items and then totaling all the expected values is more precise.
Contingency Calculation
The contingency factor is calculated by dividing the 'expected values' by the 'most likely' values (see Table 5). There are many methods for calculating contingency using triangular distribution and one of them has been explained in this article.

Calculate the probability of under run for the 'most likely' or 'average' values

Evaluate the probability calculated and assign the probability of under run that the estimator or the team feels confident.

For the assigned probability, calculate the expected value (using inbuilt formulas in Excel cells)

Calculate contingency by dividing total of expected value for the selected items by the total of or 'most likely' or 'average' values for these items
"The triangular distribution is a way of thinking about variation that corresponds rather well to whatif analysis in business. Its finite range and simple form are more understandable than a normal distribution. It is more versatile than a normal distribution because it can be skewed in either direction. Yet it has some of the nice properties of a normal distribution, such as a distinct mode." [2]
The method enables whatif analyses for various assumed probabilities of under run for a given item. The change in values are easily noticeable in an Excel spreadsheet with the inbuilt statistical formulas in Excel cells. The model is easy to explain and beneficial to both the owner and contractor in contingency analysis and risk quantification.
References:
[1] AACE International, Skills and Knowledge of Cost Engineering, 6th edition
[2] Online Learning Center  www.mhhe.com/doane4e  ‘Applied Statistics in Business and Economics’ – Chapter 7  Continuous Probability Distributions David P. Doane, Oakland University, Lori E. Seward, University of Colorado
Recommended Reading:
AACE International, Risk Analysis and Contingency Determination Using Range Estimating, RP41R08
Curran, Michael W., Range Estimating: Measuring Uncertainty and Reasoning with Risk, Cost
Engineering, Vol. 31, No. 03, AACE International, Morgantown, WV, USA, 1989
Dr. Amarjit Sing PE, Sharilyn Shiramizu and Kamal Gantam, Bid Risk and Contingency Analysis, Cost Engineering, Vol. 49/No. 12 December 2007
Dr. Osama Moselhi PE, Risk Assessment and Contingency Estimating, D&RM/A.06.5 1997 AACE International Transactions