# The Mathematics of Project Contracts

## The Mathematics of Project Contracts

Mathematical formulas enter the picture when the project manager seeks to dollar-quantify the risk transferred or retained by a contracting activity. Each contract type, whether FP or CP, has a set of mathematical parameters. Table 9-1 provides a summary of the major contract types and the principal financial parameters for each. In the following paragraphs, we will present examples of how these parameters are applied to various contracts. Let's begin with FFP.

Table 9-1: Financial Parameters in Contracts

Contract Type

Contract Parameters

Firm fixed price (FFP)

Total price

Fixed price incentive fee (FPIF)

Target cost, target fee, target price, price ceiling, share ratio

Fixed price award fee (FPAF)

Total price, award fee, fee criteria

Cost plus fixed fee (CPFF)

Estimated target cost, fixed fee

Cost plus incentive fee (CPIF)

Target cost, target fee, target price, share ratio

Cost plus award fee (CPAF)

Target cost, target fee, target price, award fee, fee criteria

Cost plus percentage of costs (CPPC)

Actual cost, % uplift on cost

Time and materials (T&M)

Labor rates by labor categories, uplift fee on materials if any

### Fixed Price Contract Math

#### Firm Fixed Price Example

Scenario: A supplier is awarded an FFP contract for a special facility with certain specified features and functions. The contracted price is \$100,000 and the schedule is 120 days. The facility is ready in time with all required features and functions. The supplier announces that the actual cost was only \$80,000.

 Q: How much does the project owe the supplier on obtaining a certificate of occupancy?

 A: \$100,000. The price is firm and fixed regardless of whether the supplier overran or underran the estimates.

Let us move to another form of FP contract, one with an incentive fee, called FPI (fixed price incentive). In an incentive fee arrangement, some of the cost risk is retained by the project, but the supplier is given the opportunity to earn more or less fee commensurate with performance. There are several financial parameters in the FPI calculation.

#### Fixed Price Incentive Example

Scenario: A supplier is awarded an FPI contract to build a special facility with certain specified features and functions. The contract cost (risk) sharing parameters are negotiated between the project and the contractor. In this example the parameters are as follows: the contracted target price (TP) is \$100,000, the target cost (TC) is \$85,000, the target fee (TF) is \$15,000, the target return on cost (ROC) is equal to 15/85 = 17.6%, the ceiling price (CPr) is \$110,000, and the sharing ratio (SR) is 80%/20% applied to cost. The schedule is 120 days.

In this example, the facility is ready in time with all required features and functions. Actual cost is denoted AC, the contractor's share is denoted SRc, and in this example SRc is equal to 20%.

The FPI formula is:

Contractor payable = (TC - AC) * SRc + AC + TF CPr

Case 1. The supplier announces that the actual cost was only \$80,000, \$5,000 below target cost; the supplier is paid according to the formula

Contractor payable (\$000) = (\$85 - \$80) * 0.2 + \$80 + \$15 = \$96 \$110

ROC = (96 - 80)/80 = 20%

Case 2. The supplier announces that the AC = \$95,000, \$10,000 above TC:

Contractor payable (\$000) = (\$85 - \$95) * 0.2 + \$95 + \$15 = \$108 \$110

ROC = (108 - 95)/80 = 16.25%

Case 3. The supplier announces that the AC = \$105,000, \$20,000 above TC:

Contractor payable (\$000) = (\$85 - \$105) * 0.2 + \$105 + \$15 = \$116 \$110

In this case, the ceiling price is exceeded. The project's cost risk is capped at \$110,000, so the contractor is paid only \$110,000.

ROC = (110 - 105)/105 = 4.8%

It is instructive to understand what supplier cost, adjusted for incentive sharing, exactly equals the risk cap of the project. For this case, the contract situation has reached the point of total assumption (PTA). Costs above the cost at PTA are borne exclusively by the supplier, as shown in Figure 9-1. Below the PTA, costs are shared by the project according to the sharing ratio. The cost at PTA is defined by the formula

Figure 9-1: FPIF Risk.

Cost at PTA: (TC - AC) * SRc + AC + TF = CPr

Solving for AC: AC at PTA = (CPr - TF - TC * SRc)/(1 - SRc)

In the example that we have been discussing, the AC at PTA is (\$000):

AC at PTA = (\$110 - \$15 - \$85 * 0.2)/(1 - 0.2) = \$97.5

### Cost Plus Contract Math

Let us look at the CP contract vehicle and see how the project manager's cost risk is handled. Recall that the CP contract is primarily applied in situations where scope is indefinite and a cost estimate cannot be made to the precision necessary for an FP contract. In all CP contracts, the cost risk (as different from the fee) is largely retained by the project. On the other hand, the fee can either be fixed (no risk to the project) or made an incentive fee, usually on cost. With an incentive fee, the project assumes some of the risk of the total fee paid.

The simplest example of CP contracts is the CPFF (cost plus fixed fee), wherein the contractor is reimbursed cost [2] and receives a fixed fee that is negotiated and set before work begins. [3]

#### Cost Plus Fixed Fee Example

Scenario: A software program is to be developed for which the estimated cost is \$100,000. A fixed fee of \$8,000 is negotiated before the work starts. The estimated ROC is 8%. What is the contractor paid under the following cases?

Case 1: AC = \$90,000.

Contractor payment = \$90,000 + \$8,000 = \$98,000

ROC = 8/90 = 8.9%

Case 2: AC = \$110,000.

Contractor payment = \$110,000 + \$8,000 = \$118,000

ROC = 8/110 = 7.3%

To reduce the risk to the project, the project manager decides to create an incentive on cost such that on a 70%/30% share ratio, the contractor participates in any cost savings. Let's see how this might work in the example just discussed.

#### Cost Plus Incentive Fee Example

Scenario: A software program is to be developed for which the estimated target cost is \$100,000. An incentive fee (IF) of \$8,000 is negotiated before the work starts; the target price is \$108,000; there is no ceiling price. The project's cost liability is unlimited, as in all CP contracts, regardless of incentives. The sharing ratio is 70%/30%. SRc = 30%. The estimated target ROC is 8% = 8,000/100,000. What is the contractor paid under the following cases (\$000)?

Case 1: AC = \$90, \$10 under the estimated cost.

Contractor paid = AC + (TC - AC) * SRc + IF

Contractor paid (\$000) = \$90 + (\$100 - \$90) * 0.3 + \$8 = \$101.3

ROC = 11.3/90 = 12.6%

Case 2: AC = \$110, \$10 over the estimated cost.

Contractor paid (\$000) = \$110 + (\$100 - \$110) * 0.3 + \$8 = \$113.7

ROC = 3.7/110 = 3.4%

Now compare the two CP examples to see how the risk to the project manager changed with the introduction of the incentive fee on cost (\$000). The cases are illustrated in Figure 9-2:

• Case 1: AC = \$90. With the CPFF, the project paid \$98, whereas on the CPIF the project paid \$101.3.

• Case 2: AC = \$110. With the CPFF, the project paid \$118, whereas on the CPIF the project paid \$113.7.

• The risk range for the CPFF is from \$98 to \$118, a range of \$20; on the CPIF, the risk range is from \$101.3 to \$113.7, a range of a lesser figure of \$12.4. The risk range has been reduced by employing incentives:

• Risk range improvement = (20 - 12.4)/20 = 38%

Figure 9-2: CPIF Risk.

### Time and Materials Contract Math

A commonly employed contract vehicle for obtaining the services of temporary staff or to engage in highly speculative R&D is the time and materials (T&M) contract. The usual form of this contract is that the time charges are at a standard and fixed rate for a labor category, but there may be many different labor categories, each with a different labor rate, that are chargeable to the contract. Nonlabor items for all manner of material, travel, subsistence, and other things are charged at actual cost, or actual cost plus a small percentage uplift for administrative handling.

The T&M contract is somewhat of a hybrid, having FP rates but CP materials. In addition, the total charges for labor are wholly dependent on how much of what kind of labor is actually used in the contract. The T&M contract shares the CP problem of no limit to the dollar liability of the project. The contractor need only provide a "best effort" toward the SOW. There is no fixed or incentive fee. The fee is fixed on a labor rate, but the total fee paid is dependent on what labor rates are employed and how much of each is used in the performance of the tasks on the work breakdown structure.

#### Time and Materials Example 1

Scenario: Two labor categories are chargeable to the contract, developer at \$75/hour and tester at \$50/hour; 50 hours of the former and 100 hours of the latter are employed on the work breakdown structure. In addition, \$500 in tool rental and \$300 in training expense are chargeable to the contract. A 2% fee is assessed for material handling.

 Q: What is the contractor paid at the end of services rendered?

 A: Contractor paid = \$75 * 50 + \$50 * 100 + (\$500 + \$300) * 1.02 = \$9,566

The supplier is at some risk on T&M contracts, but compared to FFP, the supplier's risk is pretty minimal. The supplier's risk is in the difference between the actual salaries and benefits paid to the employees and the standard rates charged in the contract. If salaries are higher than the standard rates for the specific individuals provided, the supplier will lose money. Consider the following example.

#### Time and Materials Example 2

Scenario: For the T&M scenario given above, assume that two developers, Tom and Mary, are provided for 25 hours each. Including benefits, Mary makes \$76/ hour, more than the standard rate, and Tom makes \$65/hour, less than the standard rate. Each billable hour by Mary is a loss for the supplier, but each billable hour by Tom is profitable. Susan is provided as the tester and her salary is \$48/hour, again profitable at the standard rate.

 Q: What is the contractor's ROC on this deal, considering only the labor (time) component?