The fixed price is a weighted average of the forward curve, and the weights are your own consumption profile and the discount factors.
The fixed price on a commodity swap is the price that makes the present value of the fixed leg equal to the present value of the floating leg: the forward price of each settlement period, weighted by that period's volume and discount factor. On an illustrative 12-month diesel swap with a backwardated curve and volumes rising into the winter, the mid fixed price is 730.18 a tonne, not the 732.50 simple average of the forwards, and the dealer quotes it at 733.18 once a 3.00 margin is added.
Worked in full in Derivatives by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Commodity swaps are how hauliers, airlines, shipping lines and utilities fix fuel costs without touching the physical market. The company keeps buying at the market price, the swap pays it the difference when the monthly average is above the fixed price and charges it when below, and the net cost is the fixed price on the hedged volume. What the company should check before it signs is whether the fixed price it is quoted is a fair weighting of the curve.
| Input | Value |
|---|---|
| Term | 12 months |
| Forward curve, month 1 to month 12 | 760.00 to 705.00 |
| Fall per month (backwardation) | 5.00 |
| Hedged volume, total | 11,600 t |
| Hedged volume, month 1 to month 12 | 800 to 1,300 t |
| Discount rate, flat, annual | 4.5% |
| Dealer margin added to the mid price | 3.00 |
The volumes follow the company's own consumption, heavier in the last quarter. Each month settles in cash on the average of the daily published price for that month, discounted here from month end for simplicity.
At inception the swap is worth zero to both sides, so Σ Vi × (Fi − K) × DFi = 0, which solves to:
K = Σ(Vi × Fi × DFi) / Σ(Vi × DFi)
DFi = 1 / (1 + 4.5%)i/12, from 0.9963 for month 1 to 0.9569 for month 12.
Volume times forward times discount factor sums to 8,255,791. Volume times discount factor sums to 11,306.5. K = 8,255,791 / 11,306.5 = 730.18.
Excel, with forwards in B2:B13, volumes in C2:C13 and discount factors in D2:D13: =SUMPRODUCT(B2:B13,C2:C13,D2:D13)/SUMPRODUCT(C2:C13,D2:D13)
| Month | Forward | Volume, t | Discount factor | V × F × DF | V × DF |
|---|---|---|---|---|---|
| 1 | 760.00 | 800 | 0.9963 | 605,774 | 797.1 |
| 4 | 745.00 | 850 | 0.9854 | 624,027 | 837.6 |
| 8 | 725.00 | 950 | 0.9711 | 668,833 | 922.5 |
| 12 | 705.00 | 1,300 | 0.9569 | 877,033 | 1,244.0 |
| All 12 months | 11,600 | 8,255,791 | 11,306.5 |
Three prices are on the table and only one is right:
The gap between the simple average and the fair price is 2.32 a tonne, or 26,909 over the 11,600 tonnes: about three-quarters of the dealer's whole margin. The dealer's 3.00 is worth 34,800 undiscounted and 33,920 in present value, which is the real cost of the hedge; the rest of the fixed price is just the curve.
Settlement is mechanical. If the month 4 average comes in at 770.00, the swap pays the company (770.00 − 733.18) × 850 = 31,297, which offsets the dearer physical fuel. If it comes in at 700.00, the company pays 28,203. Either way month 4's 850 tonnes cost 623,203 all-in, 733.18 a tonne.
Check the quote, not just the margin. A dealer quoting 735.50 on this profile is quoting the simple average plus 3.00. The margin looks the same; it is in fact 5.32 a tonne, because the volume weighting has been left out.
| Curve | Simple average | Seasonal volumes | Flat volumes, 966.7 t a month |
|---|---|---|---|
| Backwardation, 760 down to 705 | 732.50 | 730.18 | 732.72 |
| Flat at 732.50 | 732.50 | 732.50 | 732.50 |
| Contango, 705 up to 760 | 732.50 | 734.82 | 732.28 |
The volume profile matters only when the curve has a slope, and its sign flips with the slope. Back-loaded volumes on a contango curve raise the fair price to 734.82, so a company whose consumption peaks late in the hedge period pays more than the simple average in a contango market and less in a backwardated one. With flat volumes the only effect left is discounting, worth about 0.22 either way.
A commodity swap's fixed price is a volume- and discount-weighted average of the forward curve, and on a sloped curve it can sit several dollars from the simple average. Ask for the curve, the volume schedule and the discount factors, rebuild K with one SUMPRODUCT, and the dealer's margin is whatever is left over. How a hedge's real costs separate from the curve, and what it does to cash, is set out in the free workbook for this case; the cost-of-carry logic behind commodity forwards is in how to calculate a futures price with the cost of carry.
Because each month carries a different volume and is paid on a different date. On the illustrative backwardated diesel curve, heavier winter volumes fall on the cheapest months, so the fair fixed price is 730.18 a tonne against a 732.50 simple average. On a contango curve with the same volumes it would be 734.82. Only flat volumes on a flat curve give the simple average.
In cash, on the difference between the average published price for the month and the fixed price, times that month's volume. With a fixed price of 733.18 and 850 tonnes, a 770.00 monthly average means the dealer pays the company 31,297; a 700.00 average means the company pays 28,203. Either way the fuel costs 733.18 a tonne all-in.
Rebuild the mid fixed price from the forward curve, the volume schedule and the discount factors, then subtract it from the quote. In the illustrative case the mid is 730.18, so a 733.18 quote carries a 3.00 margin, worth 33,920 in present value on 11,600 tonnes. A quote of 735.50 would carry 5.32.
This article is one calculation from Derivatives. 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.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: corporate finance, valuation and markets → · All 1682 articles →
If this book helped, or didn’t, a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.