Discussions of crude oil’s forward curve start the same way — plot F/S, call it contango or backwardation. Contango is F > S, and backwardation is the reverse F < S. This ratio construction is a mistake, and it is the same PLI (profit level indicator) mistake I have spent three decades flagging in transfer pricing.
The textbook cost-of-carry inventory is:
F(t,T) = S(t)\cdot e^{kT}where k = (r + u - y), r is the risk-free interest rate, u is storage cost, and y is the convenience yield — the premium for holding physical barrels rather than a paper claim on them.
Take logs and regress \ln\!\left(F/S\right) on time to delivery T, and you have built a ratio-dependent variable. Ratios of a linear relationship are biased estimators of the underlying parameters — I have shown this elsewhere (the harmonic-mean bias theorem). The correction here is the same: estimate level variables.
The theory becomes testable only after we linearize the equation. By expanding the exponential to first order, we find:
e^{kT} ≈ 1 + kTThis leads to the approximation:
F ≈ S(1 + kT) = S + kST
The carry term is linear in S and T through the product ST, rather than treating S and T separately. This distinction is often overlooked in naive specifications.
A first-order expansion of the cost-of-carry inventory gives a linear model in levels:
F(i) = a + b_1 S(i) + b_2\bigl[S(i)\cdot T(i)\bigr] + \varepsilon(i)
Theory disciplines both regression coefficients: b₁ should not differ from 1.0 (dollar-for-dollar spot pass-through), and b₂ recovers the annualized net carry rate (r + u − y) directly — no division, no harmonic mean lurking in the residual.
Estimate the futures price on the WTI or Brent on any given date — a handful of maturities is enough — report \hat{b}_2 \pm \mathrm{SE} at the 68% confidence interval, and you have an honest, falsifiable read on whether traders are pricing storage cost or limited supply. That is the difference between describing a curve and testing a theory.
References
Brennan, M.J. (1958). The supply of storage. American Economic Review, 48(1), 50–72. Extended and tested Working’s theory of storage.
Fama, E.F., & French, K.R. (1987). Commodity futures prices: Some evidence on forecast power, premiums, and the theory of storage. Journal of Business, 60(1), 55–73.
Fama, E.F., & French, K.R. (1988). Business cycles and the behavior of metals prices. Journal of Finance, 43(5), 1075–1093. Tested whether futures prices are unbiased predictors of spot prices.
Gibson, R., & Schwartz, E.S. (1990). Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance, 45(3), 959–976. A stochastic convenience yield model for crude oil; treats y as a mean-reverting state variable rather than a constant, estimated via Kalman filter.
Litzenberger, R.H., & Rabinowitz, N. (1995). Backwardation in oil futures markets: Theory and empirical evidence. Journal of Finance, 50(5), 1517–1545. Links backwardation frequency in oil markets to production/real-option incentives.
Schwartz, E.S. (1997). The stochastic behavior of commodity prices: Implications for valuation and hedging. Journal of Finance, 52(3), 923–973. Three-factor extension (spot price, convenience yield, interest rate).
Working, H. (1949). The theory of price of storage. American Economic Review, 39(6), 1254–1262. Formalized the convenience yield as compensation for holding inventory.