Tn5 Futures Risk Premium

Document Date: November 2, 2006 An Introduction To Derivatives And Risk Management, 7th Edition Don Chance and Robert Brooks Technical Note: Futures Risk Premiums, Ch. 9, p. 309 This technical note supports the material in the Asset Risk Premium Hypothesis section of Chapter 9 Principles of Pricing Forwards, Futures, and Options on Futures. We explore here the relationship between futures prices and expected spot prices under various assumptions about market microstructure.

Insights are supported with

observations from the gold, copper and natural gas futures markets. Futures prices and expected spot prices We explore the relationship between spot asset prices and futures or forward contracts when arbitrage is and is not possible. We do not distinguish between forward or futures markets here. Asset Price in Secondary Market The price of the spot asset ( S 0 ) could be represented as the present value of the expected future asset value ( E 0 (S T ) ) adjusted for any holding costs (future value of storage, insurance and such), any benefits (convenience yield, dividends, and such) (all carry costs are denoted θ ), and a spot asset risk premium ( φ S,0 (S T ) ), expressed as:

S0 = E 0 (ST ) − θ − φS, 0 (ST ) Using a standard supply and demand graph, we illustrate this result in Figure 1.

Figure 1. Asset Supply and Demand Curves

Price

Asset Supply

E0[PA] PA Asset Demand QA

Quantity

Insights: •

The clearing price and quantity are determined by asset supply and demand in the secondary asset market

•

The asset price is the present value of expected future cash flows, discounted at a risk adjusted discount rate

Net Hedging: A Digression

It is reasonable to assume that hedgers are willing to pay a futures risk premium to reduce their risk. Figure 2 illustrates the net hedging theory when net hedgers are long. The phrase net hedger is used because in any futures market there are typically hedgers on both the long and short side of the market. One can add up all the hedgers that are long and all the hedgers that are short. Subtracting these two totals, one can determine whether a particular futures contract has net long or net short hedgers. These net hedgers are long the futures contracts and are willing to contract at futures prices above the expected future spot price. Net speculators provide the hedging protection for a futures risk premium above the expected future spot price. This market is said to be in normal IDRM7e, © Don M. Chance and Robert-Brooks

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contango as the futures price is above the unobservable expected future spot price. The figure below illustrates the case where net hedgers are long and speculators demand a risk premium. Figure 2. Net Positions of Hedgers and Speculators when Net Hedgers are long: Normal Contango

Futures Price Equilibrium Futures Price

Net Speculators φ f ,0

Net Hedgers

Expected Future Spot Price

Short Contracts

Long Contracts

Recall that hedgers are willing to pay a premium to reduce their risk. If net hedgers are short the futures contract they are willing to hedge at futures prices below the expected future spot price. Net speculators will provide the hedging protection for a futures risk premium below the expected future spot price. This market is said to be in normal backwardation as the futures price is below the unobservable expected future spot price, illustrated in Figure 3.

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Figure 3. Net Positions of Hedgers and Speculators when Net Hedgers are short: Normal Backwardation

Expected Future Spot Price

Futures Price Equilibrium Futures Price φ f ,0

Net Hedgers

Net Speculators

Short Contracts

Long Contracts

Futures Price in Unarbitraged Derivatives Market

The price of the futures contract on the spot asset could be represented as the expected future asset value adjusted for any holding costs (margin, impact of marking-tomarket and such), any benefits (embedded options and such), and a futures risk premium (which could be positive or negative depending on net hedging demand): f 0 (T ) = E 0 (ST ) − θ − φf ,0 (ST ) Assuming no holding costs or benefits from having a position in the futures market, f 0 (T ) = E 0 (S T ) − φ f ,0 (S T )

Thus, in an unarbitraged market we expect to find that the futures price is a function of the expected future spot asset value adjusted for perhaps a significant futures risk premium. Figure 4 depicts equilibrium in a futures markets where arbitrage is not feasible.

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Figure 4. Equilibrium in Unarbitraged Futures Market

Speculator Supply – Net Short

Price

PF,UA

E0[PA]

PA Hedger Demand – Net Long QF,UA

QA

Quantity

Insights: •

Speculators must earn a positive risk premium to be induced to participate

•

Hedgers are assumed to be net long

•

The difference between the futures price and the expected future asset price is the compensation to the speculator (futures risk premium)

•

Greater hedging demand results in higher compensation to speculators

•

The difference between expected the future spot asset price and today’s asset price depends on both the risk-free rate and any asset risk premiums

Futures Price in Fully-Arbitraged Derivatives Market

Recall that only when the futures market is fully arbitraged, we have f 0 (T ) = S0 + θ . Solving for the future value of costs and benefits and substituting this result into the spot price equation, we have: S0 = E 0 (ST ) − θ − φS,0 (ST ) . IDRM7e, © Don M. Chance and Robert-Brooks

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Thus substituting S0, f 0 (T ) = E 0 (S T ) − φ S, 0 (S T ) .

Thus, the futures price does not equal the expected future asset price. This equation is a direct outcome, however, of assuming the market is arbitrage-free and there are no market imperfections. Every derivatives market has some imperfections and hence this result is not exactly correct. Figure 5 illustrates both the unarbitraged case and the fully arbitraged case. Figure 5. Equilibrium in Unarbitraged and Arbitraged Futures Market

Speculator Supply – Net Short

Price

PF,UA

E0[PA]

PA

PF,A = FV[PA] Hedger Demand – Net Long QF,UA

QA QF,FA

Quantity

Insights: •

A fully arbitraged futures market implies that the futures price is equal to the future value of the asset price, adjusted for the marginal dealer’s cost of funds (“risk-free” interest rate).

•

The difference between the current futures price and the expected future asset price is the asset’s risk premium.

•

The difference between the quantity of long futures positions in a fully-arbitraged market and the quantity of long futures positions in an unarbitraged market represents the net additional supply of short contracts provided by arbitragers

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•

Arbitrageurs, however, are net short futures contracts and will hence buy the same amount of the underlying asset, driving the asset price up, resulting in less demand by arbitrageurs.

•

Assuming the asset market price does not materially change, the difference between the quantity of long futures positions in a fully-arbitraged market and the quantity of long futures positions in an unarbitraged market represents the net societal benefit from satisfying greater hedging demand.

•

There is no futures risk premium in fully arbitraged futures markets.

•

The futures price reflects the asset risk premium (the difference between the expected future asset price and current futures price).

Based on the analysis above, futures markets can be classified into three types, fullyarbitraged, quasi-arbitraged, and un-arbitraged. In a futures market that is fullyarbitraged, both carry arbitrage and reverse-carry arbitrage can be conducted by a wide array of market participants. In a futures market that is quasi-arbitraged, either carry arbitrage or reverse-carry arbitrage can be conducted by a wide array of market participants, but not both. In a futures market that is un-arbitraged, neither carry arbitrage nor reverse-carry arbitrage can be conducted by market participants. Market Classification Illustrated Fully-Arbitraged Market

One characteristic of a fully arbitraged market is the stochastic nature of futures contracts that differ only by maturity. Assuming the carry model, the percentage difference in futures prices of different maturities is: (denoted %TP for percentage term premium) %TP =

f 0 (T + τ ) − f 0 (T ) S0 + θT + τ − (S0 + θT ) = f 0 (T ) S0 + θT

θ = T+τ − 1 θT

.

Notice that the difference in futures prices depends solely on the difference in the maturity and the carry costs. We assume the longer time to maturity, the higher the carry IDRM7e, © Don M. Chance and Robert-Brooks

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costs. Hence, one would expect that at a point in time, the percentage difference in futures prices would be slightly positive and stable. Figure 6 illustrates gold futures contracts. Since 1985 gold futures contracts exhibit the characteristics common for fullyarbitraged markets. In the late 1970s and early 1980s there appeared some evidence that gold futures were not fully arbitraged. For example, at the end of 1979 and in early 1980, there were periods of time when the percentage term premium was large and both positive and negative. A large positive term premium implies a profitable arbitrage trade: enter a long position in the distant contract and a short position in the near contract. When the near contract expires, buy the underlying asset with borrowed money and deliver it when the distant contract expires. A large negative term premium implies a profitable arbitrage trade: enter a short position in the distant contract and a long position in the near contract. When the near contract expires, short sell the underlying asset and lend money and cover the short position when the distant contract expires. Figure 6. Term Premium for Gold Futures 20.00%

15.00%

Percentage Premium

10.00%

5.00%

0.00%

-5.00%

-10.00%

-15.00%

-20.00% Jan-75

Jan-77

Jan-79

Dec-80

Dec-82

Dec-84

Dec-86

Dec-88

Dec-90

Dec-92

Dec-94

Dec-96

Dec-98

Dec-00

Calendar Time

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Quasi-Arbitraged Market

A quasi-arbitrage market is rare to find. One side of the carry arbitrage must be feasible, whereas the other is not. The copper futures market during the late 1980s and mid-1990s fit this classification scheme. Figure 7 illustrates the copper futures market. Notice that apparent ceiling in the percentage term premium. There are large negative term premiums but not large positive term premiums. In the copper market, by the late 1980s traders apparently had entered the copper arbitrage business. Recall, a large positive term premium implies a profitable arbitrage trade: enter a long position in the distant contract and a short position in the near contract. When the near contract expires, buy the underlying asset with borrowed money and deliver it when the distant contract expires. The only requirement is to be able to store copper cheaply, not a difficult task. Short-selling copper, however, is more difficult. By 1997, firms with an inventory of copper apparently entered the reverse carry arbitrage business. Recall a large negative term premium implies a profitable arbitrage trade: enter a short position in the distant contract and a long position in the near contract. When the near contract expires, short sell (or sell out of inventory) the underlying asset and lend money and cover the short position when the distant contract expires. Apparently, by 1997 both sides of the carry arbitrage were feasible, and hence copper graduated to a fully-arbitraged futures market.

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Figure 7. Term Premium for Copper Futures 20.00%

15.00%

Percentage Premium

10.00%

5.00%

0.00%

-5.00%

-10.00%

-15.00%

-20.00% Jan-75

Jan-77

Jan-79

Dec-80

Dec-82

Dec-84

Dec-86

Dec-88

Dec-90

Dec-92

Dec-94

Dec-96

Dec-98

Dec-00

Calendar Time

Un-Arbitraged Market

This type of market is easy to find. Neither side of the carry arbitrage is feasible. The natural gas futures market fits this classification scheme at this time, as illustrated in Figure 8. Notice that the percentage term premium can vary dramatically.

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Figure 8. Term Premium for Natural Gas Futures 20.00%

15.00%

Percentage Premium

10.00%

5.00%

0.00%

-5.00%

-10.00%

-15.00%

-20.00% Jun-93

Jun-94

Jun-95

May-96

May-97

May-98

May-99

May-00

May-01

May-02

May-03

Calendar Time

Valuation models need to be tailored to the category of futures market. A fullyarbitraged market would apply a carry model for valuation purposes. The carry model is solely dependent on the cost of carrying the underlying asset through time and the appropriate discount rate. The quasi-arbitrage market would apply the carry model for valuation purposes at some times and not at others. The un-arbitraged market requires an entirely different approach to valuation. References

Black, F.

“The Pricing of Commodity Contracts.”

Journal of Financial

Economics 3 (1976), 167-179.

French, K. R. “Detecting Spot Price Forecasts in Futures Prices.” The Journal of Business 59 (1983), S39-S54.

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