One formula spreads the contract sum over the programme, sets the monthly funding peak, and shows how much a front- or back-loaded programme moves the interest bill.
Spread the contract sum along a cumulative S-curve, then difference it month by month. With the symmetric curve 3x² − 2x³, where x is the share of the programme elapsed, a 26,301,102 contract over 20 months certifies only 15.6 per cent in the first five months and peaks at 1,966,007 a month, 7.48 per cent of the contract, half as much again as the 1,315,055 a straight line gives. The shape of the curve changes the client's interest bill by up to 624,647.
Worked in full in Construction Cost Control by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A construction cash flow forecast answers two questions for the client and the funder: how much will be certified each month, and how much will be drawn and paying interest at each point. Spreading the contract sum evenly answers neither well, because construction value builds slowly through enabling works and substructure, runs fastest through frame and envelope, and tails off through fit-out and commissioning. An S-curve captures that shape with one formula.
| Input | Value |
|---|---|
| Contract sum | 26,301,102 |
| Programme, months | 20 |
| Retention on interim certificates | 5% |
| Payment of each certificate | One month after valuation |
| Interest on drawn funds, simple | 7.5% |
The contract sum is the one built up layer by layer in how to calculate preliminaries and OH&P in a contract sum. Any forecast should be rebuilt from the contractor's priced programme once it exists; the curve is the tool for the months before it does, and the check on it afterwards.
Cumulative value at month m = contract sum × (3x² − 2x³), with x = m ÷ programme
Month 5: x = 0.25, 3 × 0.0625 − 2 × 0.015625 = 15.6%, so 4,109,547 cumulative
Excel, with the month in A2 and the programme in a cell named T: =Contract*(3*(A2/T)^2-2*(A2/T)^3); the month's value is this row less the row above.
This curve is the cumulative distribution of a symmetric beta distribution. It starts flat, is steepest exactly at mid-programme, and flattens into completion. Its peak monthly rate is 1.5 divided by the number of months, 7.50 per cent here, against 5.00 per cent for a straight line.
| Month | Cumulative % | Cumulative value | Month's value | Net payment |
|---|---|---|---|---|
| 2 | 2.8% | 736,431 | 545,748 | 518,460 |
| 4 | 10.4% | 2,735,315 | 1,137,523 | 1,080,647 |
| 6 | 21.6% | 5,681,038 | 1,571,491 | 1,492,916 |
| 8 | 35.2% | 9,257,988 | 1,847,652 | 1,755,270 |
| 10 | 50.0% | 13,150,551 | 1,966,007 | 1,867,707 |
| 12 | 64.8% | 17,043,114 | 1,926,556 | 1,830,228 |
| 14 | 78.4% | 20,620,064 | 1,729,297 | 1,642,833 |
| 16 | 89.6% | 23,565,787 | 1,374,233 | 1,305,521 |
| 18 | 97.2% | 25,564,671 | 861,361 | 818,293 |
| 20 | 100.0% | 26,301,102 | 190,683 | 181,149 |
Months 10 and 11 both certify 1,966,007, the peak. The middle ten months carry 68.8 per cent of the value, and the first and last five 15.6 per cent each. At completion 1,315,055 of retention is still held, released under the contract's terms as worked in how to calculate retention.
The symmetric curve is a mirror image about mid-programme: what it lags the straight line in the first ten months it gains in the last ten. The average balance drawn, and therefore the interest over the programme, is identical: 1,483,547 on 47.5 per cent average drawn. What the S-curve changes on a symmetric programme is the monthly funding peak, which a facility's drawdown limits and an equity call schedule must accommodate. What changes the interest is skew.
| Curve | Formula | Done by month 10 | Peak month's value | Average drawn | Interest |
|---|---|---|---|---|---|
| Straight line | x | 50.0% | 1,315,055 | 47.5% | 1,483,547 |
| Symmetric S | 3x² − 2x³ | 50.0% | 1,966,007 | 47.5% | 1,483,547 |
| Front-loaded S | 6x² − 8x³ + 3x⁴ | 68.8% | 2,333,401 | 57.5% | 1,795,870 |
| Back-loaded S | 4x³ − 3x⁴ | 31.2% | 2,333,401 | 37.5% | 1,171,223 |
A front-loaded programme, typical where heavy substructure or a large frame package comes early, costs 312,324 more interest than the symmetric case; a back-loaded one, typical where fit-out and services dominate, costs 312,324 less. Between the two extremes the difference is 624,647, 2.4 per cent of the contract sum, from the same price and the same programme length. The front-loaded peak of 2,333,401 a month is 8.87 per cent of the contract, landing in month 7, not month 10.
Using a straight line for the finance budget and then discovering the monthly peak when the first large certificate lands. The second mistake is fitting the curve and never re-fitting it: once actual valuations come in, compare the cumulative certified with the curve each month. A project running below its curve is either late or front-loading its claims less than expected, and the cost report should say which, as an earned value comparison does in CPI, SPI and EAC.
Forecast with cumulative value = contract × curve(month ÷ programme) and difference it. On 26,301,102 over 20 months the symmetric curve peaks at 1,966,007 a month, 1.5 times the straight line, and skewing it either way moves the interest bill by 312,324. Construction Cost Control follows a contract from its sum to the final account and the client's appraisal around it, and the free workbooks for this case carry the appraisal that this funding profile feeds.
A simple symmetric one is cumulative share = 3x^2 - 2x^3, where x is months elapsed divided by programme months. On a 26,301,102 contract over 20 months, month 5 is 15.6 per cent and month 10 is 50.0 per cent. Back-loaded and front-loaded versions are 4x^3 - 3x^4 and 6x^2 - 8x^3 + 3x^4.
On a symmetric S-curve the peak month is 1.5 divided by the number of programme months, so 7.50 per cent of the contract on a 20-month job, against 5.00 per cent on a straight line. On the illustrative 26,301,102 contract that is 1,966,007 a month; a front-loaded programme peaks higher, at 2,333,401 in month 7.
Only if it is skewed. A symmetric S-curve and a straight line both average 47.5 per cent drawn on the illustrative contract, and both cost 1,483,547 of interest at 7.5 per cent. A front-loaded curve costs 1,795,870 and a back-loaded one 1,171,223, a range of 624,647.
This article is one calculation from Construction Cost Control. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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