8.7 Physical Commodities
A difference between financial markets and physical commodities markets is the fungibility of what is traded. Japanese yen, Eurodollar futures, and shares of IBM are fungible. Coffee, oil, and electricity are not. Because of transmission costs, electricity in Quebec is worth a different price than electricity in Massachusetts. Because of marketing and perceived quality differences, coffee grown in Nicaragua commands a different price than coffee grown in Colombia. A value-at-risk measure for a commodities portfolio may need to address different qualities, origins, or delivery locations.
In many commodities markets, futures contracts are used as a benchmark for pricing spot or forward contracts. A future is for a specific quality, origin, and/or point of delivery. Spot or forward transactions for other qualities, origins, and/or points of delivery trade at spreads to the future.
Assume cash valuation and measure value-at-risk as 1-day 90% USDvalue-at-risk. An international coffee wholesaler trades arabica coffee from its US headquarters as well as local offices around the world. Although coffee transacts in various currencies, each office hedges foreign exchange risk locally. Accordingly, our firm-wide value-at-risk measure does not consider foreign exchange risk. All prices are in USD, and all transactions mature within a year. We shall construct a primary mapping 1P = θ(1R) of the form
[8.80]
Exhibit 8.19 indicates the specific arabicas traded by our firm.
Although the value-at-risk measure distinguishes between growths of coffee, it does not distinguish between delivery points. This is because the cost of shipping coffee is small compared to its value. This does not mean that coffee at one location is fungible with that at another—you can’t deliver coffee at Rotterdam while it is afloat off the Horn of Africa. A value-at-risk measure can only address market risk. Liquidity issues must be addressed with alternative means, such as careful scheduling of deliveries.
Because of the vagaries of shipping, forward contracts typically specify a delivery month, with delivery acceptable on any day during that month. For this reason, the precise day counts and discounting of our earlier examples will not be evident in this example.
NYBOT coffee futures are benchmarks for arabica coffees. Coffee forwards are priced at a spread to the first future maturing either during or subsequent to the forward’s delivery month. That future’s expiration month is called the forward’s cover month. Cover months corresponding to specific delivery months are indicated in Exhibit 8.20.
There are two forms of forward contracts: fixed and to-be-fixed. Both forms specify quantity, quality, delivery point, and delivery month. A fixed contract indicates a fixed price. A to-be-fixed contract indicates a spread to be paid over the benchmark future. The actual price is determined on a subsequent fixing date by adding that spread to the benchmark future’s settlement price on the fixing date. For this example, all prices are USD per pound of delivered coffee.
The practical difference between fixed and to-be-fixed contracts is that a fixed contract offers absolute price exposure to the physical coffee. A to-be-fixed contract offers, until its fixing date, exposure to only the spread between the physical coffee and the benchmark future.
A forward contract may provide some flexibility in the actual growth to be delivered. Such flexibility is called an optional provision. Standard optional provisions are
- 2 Horsemen: Seller’s option to deliver coffee from Guatemala or Mexico.
- 3 Horsemen: Seller’s option to deliver coffee from 2 Horsemen or El Salvador.
- 4 Horsemen: Seller’s option to deliver coffee from 3 Horsemen or Honduras.
- 5 Horsemen: Seller’s option to deliver coffee from 4 Horsemen or Costa Rica.
- 6 Horsemen: Seller’s option to deliver coffee from 5 Horsemen or Nicaragua.
Optional provisions trade at spreads to benchmark futures just as growths do. Indeed, a forward on 4 Horsemen is similar to a forward on Guatemalan. The only difference is that, at delivery, the seller has more flexibility in the actual coffee delivered.
Our firm also holds NYBOT futures to hedge fixed forward contracts. Each NYBOT future is for 37,500 pounds of coffee.
To conform with our firm’s accounting system, contracts are aggregated by delivery month. The following discussion speaks of 1-month, 2-month, … , 12-month contracts. These correspond to contracts for delivery in the current month, next month, etc. If today is October 28, 1-month contracts will deliver prior to November 1; 2-month contracts will deliver during November; etc.
As in our foreign exchange example of Section 8.5, it is convenient to split forwards into “legs” and model each as a separate asset. For a fixed forward, three legs correspond to:
- component of coffee value due to the spread over its benchmark future,
- component of coffee value due to the price of the benchmark future, and
- discounted fixed purchase price to be paid on delivery.
The value of the forward is the sum of the first two less the third. For a to-be-fixed forward, there are two legs:
- component of coffee value due to the spread over its benchmark future, and
- discounted fixed spread to be paid on delivery.
The value of the forward is the first less the second.
Along these lines, we specify
[8.81]
The first component, 1S SpreadLeg, provides the discounted values of spreads on a pound of coffee. There are 30 growths of coffee and 12 delivery months out to a year, so it has 360 components:
[8.82]
The second component, 1S SpreadLegOpt, is similar. It provides the discounted values of spreads on optional provisions for a pound of coffee. There are 5 optional provisions and 12 delivery months, so it has a total of 60 components:
[8.83]
The third component, 1S FutureLeg, provides, for each delivery month, the discounted value of the corresponding benchmark future’s settlement price. For example, suppose today is June 10. Then 
corresponds to October delivery. December is the cover month for October, so represents the time-1 December futures price discounted from October back to time 1. 1S FutureLeg has 12 components corresponding to 12 delivery months out to a year:
[8.84]
The fourth component, 1S CashFlowLeg, represents the discounted values of a USD. Maturities are monthly out to a year:
[8.85]
These four components are for modeling the various “legs” of forward contracts. For modeling direct holdings in futures, 1S Future represents per-pound accumulated values of the first six nearby futures. Because futures settle daily, they have 0 market value at time 0. Their accumulated values at time 1 are simply the margin payments for the futures at that time. To keep our units consistent, 1S Future represents margin payments on a per-pound basis:
[8.86]
Finally, the last component, 1S SpreadLegPhy, represents the spreads to the nearby future at which physical coffee already held by the firm is valued:
[8.87]
Together, the foregoing assets can represent any instrument held by our firm. Suppose today is October 29, 2001, and consider a fixed forward to pay USD 70,000 for 100,000 pounds of Brazilian coffee to be delivered during January 2002. This is represented as
[8.88]
A fixed forward to pay USD 74,000 for 95,000 pounds of 4 Horsemen during February 2002 is represented as
[8.89]
A to-be-fixed forward to pay a spread of –.03 USD per pound for 1.6MM pounds of Mexican coffee to be delivered during April 2002 is represented as
[8.90]
Taking into account the fact that each NYMEX future is for 37,500 pounds of coffee, a short position in 14 March futures is represented as
[8.91]
A physical position comprising 700,000 pounds of Zimbabwe coffee is represented as
[8.92]
In this manner, all instruments held by the portfolio are represented in terms of 1S. Summing coefficients for each component, we obtain the portfolio’s holdings ω and define
[8.93]
Our selection of key factors 1Ri is driven as much by practical issues of data availability as by pricing theory. We model futures’ settlement prices, spreads for individual growths, and interest rates. Futures prices for various nearbys define a term structure. Conceivably, spreads for each growth might also vary by maturity. As a practical matter, individual growths do not trade in sufficient volume for such spread term structures to be discernable. Market participants track cash spreads—spreads between cash prices and the first nearby future. Spreads for other maturities are treated as equal to these. This practice is common in a number of commodities markets.
We specify
[8.94]
where
[8.95]
[8.96]
and
[8.97]
We define the mapping 1S = φ(1R) by component vectors. Since the timing of forward deliveries is uncertain, we do not try to achieve a false sense of precision in our discounting. We simply discount from the last day of each delivery month using 1-month Libor for month 1 contracts, 2-month Libor for month 2 contracts, etc. Set
[8.98]
[8.99]
[8.100]
where di is actual days from time 1 to the end of the delivery month; 
, 
and 
are applicable spreads, Libor rates, and futures settlement prices.
We model optional provision spreads as the minimum of applicable growth spreads. This is a crude but effective solution, which treats the optional provision’s value as its intrinsic value:
[8.101]
As an example, for a 3 Horsemen contract,
[8.102]
For futures,
[8.103]
Finally, we define:
[8.104]
