The Theory​ of Normal Backwardation. Docendo Disco, Scribendo Cogito.

Featured Image by Mr. Timothy Wilson Powers


After re-reading Mr. Eric Chang’s paper in The Journal of Finance, “Returns to Speculators and the Theory of Normal Backwardation,”  I thought I would begin journaling my notes on the subject myself beacuse I learn by teaching, think by writing.

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Part 1 of a 2 part series.

Let’s dive in.

“The trend is your friend, until the end when it bends.” -Ed Seykota

First, in regards to the Theory of Normal Backwardation, which I will refer to here and fore after as just the “Theory.” To wit (Heisenberg), the challenge here with this idea is that we don’t expect the forward price of a commodity to equal the expected future spot price, this is what the Theory tells us about the relationship between the forward and spot price of a futures contract.

Moreover, this is the essential formula.

And we can see the forward price is the expected future spot price but with the additional multiplication of an exponent of a risk premium, denoted by (r – k). The difference between the risk-free-rate and the discount rate. Therefore, we deduce that the forward price is a biased estimate of this expected future spot price based on the risk premium.

The main takeaway from the Theory is that on average, hedgers in the market will be taking short positions in forward contracts. Why? Thinking of the supply and demand effects on a producer of a commodity is that they expect to sell the said commodity, may it be Wheat, Soybeans, Corn, what have you, in the future at the current spot price when planted or greater. To guarantee this future events livelihood, they need to enter into a short position in a forward contract to hedge their price risk.

Taking the other side of this trade is the speculators, taking long positions in the forward contracts. Their motivations are the alpha-generating effects of the price fluctuations in the price of commodities.

To reiterate, the central premise of the Theory is that on average this is the standard goings-on in the marketplace. When the hedgers or the producers of the commodities plan to sell their goods to market, they will naturally need to take short forward positions to mitigate their downside price risk.

Taking an example to make the above points more concrete; say in the cash market the spot price of a Corn No. 2 yellow. Cent. III. bu-BP, U contract in the cash market is $3.2150 per bushel. And we notice based upon the forward curve that Corn No. 2 yellow forward contract is trading at $3.935 per bushel.

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You’ll also notice that the forward price is higher than the current spot price, this is an upward sloping curve, and that is called contango. Both rates, the spot and forward prices we observe today about part of the forward curve. Take note that we do not see the expected future spot price, which is what today other market participants and we are hoping the y/y spot price to be. However, we can not observe this price today and will have to wait a year’s time to find out is the actual value. Ergo, the difference between the forward price, which is observed, albeit, we can enter into a contract today for it and what we expect the spot price to be in a year. The differences between the forwards price and the futures price reflect the Theory of Normal Backwardation, where the forward price is lower than the expected future spot price.

True, given the following assumptions:

Spot (S0) = $3.2150

Riskless (r) = 1.65 %

Time = 1 yr

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Then applying the most commonly used framework for the existence of a structural risk premium Capital Asset Pricing Model (CAPM) to the commodity.

According to the CAPM, the expected rate of return of a given security is the risk free rate (1.65%) plus the market risk premium times the beta of the security, where the market risk premium s the excess return of the market over the risk-free rate, and the beta of the security measures the sensitivity of the security to the market ( or measures the systematic risk).

-Frank Fabozzi, The Handbook of Commodity Investing

Equity risk premium (ERP)*excess return of the market = 6%

Commodity beta = 0.50 * this commodity has a positive systemic risk or positive risk with the correlation to the market.

Commodity discount rate (k) = *this is the application of the CAPM, where: the riskless rate of (1.65%+( 0.50 *6.0%) )= (k) of 4.65%. Given that the commodity discount rate is greater than the risk-free rate, denotes that this commodity has an apparent risk. To wit, 3% higher risk premia over the risk-free rate.

Risk premium = ( 𝛋 – 𝙧 ) or ( 4.65%-1.65%)= 3%

Then we can express the spot function as an application to that of the risk premium in the underlying commodity. The expected future spot price is a function of the forward price. Therefore, the implied future spot price ($3.93* 𝛜 ^( risk premium (3%) = $4.0496

Further, we can compute the expected future spot price by applying the commodity discount rate:

Spot (S0) of 3.2150 times Euler’s constant[ 𝛜 ] (continuous compounding) ^ raised by [ 𝛋 ] the commodity discount rate of 4.65% times the time period [𝙏 ] of 1.

Therefore, 3.2150* 𝛆 ^ (4.65% *1) = $3.3680 as our expected future spot rate, arrived at by intuitively thinking about continuously compounding the current spot price of Corn No. 2 yellow by our commodity discount rate of 4.65% over a years time.

Now the Implied Forward spot price uses the risk-free rate instead of the discount rate. Therefore, our spot price of $3.2150 * 𝛜 ^ (1.65%* 1yr) = $3.2684. (e.g., $3.2150 spot price continuously compounded over a year’s time at the risk-free rate of 1.65%.)

Another way to look at or state what the expected future spot price is, would the to think of the expected future spot price is a function 𝙁(𝛘) of the risk premium. Saying that if that entering into a forward contract in the underlying commodity is riskless, the speculator deserves no profit for taking a riskless position. Given a scenario where the commodity has zero systemic risks, then and only then, when the risk premium for the underlying commodity equals zero,  do we expect the expected future spot price to equal the observed forward price.

Zooming back out to examine the positions of both market participants, the speculators, and hedgers. Recalling that the Theory of Normal Backwardation starts with the idea that the hedgers on average need to be short forward contracts, so if that’s true, they will need to trade with the speculators who are long the forward curve in that commodity.

Given that the spot price of Corn No. 2 yellow is $3.2150 today, what would entice us to enter into a long position in a forward contract if we are only looking to generate alpha?

If we calculate that the expected future spot price to be $3.3680 applying the discount rate (risk premia) to our CAPM, could someone entice you to enter into a long position in a forward contract where the delivery price was $3.3680? The answer being a hopeful no because your expected profit is zero while you’re assuming risk as denoted by the commodity beta of 0.50 or positive systemic risk.

More to the point, a zero profit because if the forward price is $3.3680, then you’re going to pay that in order to receive a commodity that’s also worth $3.3680. This would be a non-economic decision and in order for you to be enticed to take a long position on a forward contract, you need an expected profit.

Therefore, given our assumption of the of the expected future spot price one year hence, you agree to purchase the commodity in the future with a delivery price of $3.93 ( as observed on the forward curve for December 2019) and let’s assume further that the expected spot price in one year is realized at $3.3680, then you will pay your agree upon the price of $3.93 and receive a spot commodity worth $5.1183. (+) 𝞟 $1.1883.

Subsequently, as the speculator, you enter into a long forward contract expecting to profit the difference between the current spot of ($3.2150)  and the observed spot price in the future of ($3.93) = (+) $0.715 cents or the cost of carry is 4.49%. Not knowing whether the expected future spot price calculated above will be realized or not.

The assumed profit of (=) $0.715 is the compensation you will receive for assuming the positive systemic risk on the underlying commodity.

In culmination, the key emphasis of the Theory of Normal Backwardation is that in order to entice you as a speculator to enter into a long position on the forward contract and assume the systematic risk of the underlying commodity, the forward contract needs to be priced below the expected future spot price so that the speculator receives a positive profit to compensate the assumed risk.

Additionally, it’s important to note that given the symmetrical pricing of the forward contract the hedger loses on their short positions because they’re the only goal is to mitigate or to hedge against the price risk when they bring the underlying commodity to market for sale.

  • R.W.N II


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