Discussions of oil’s forward curve start the same way — plot \frac{F}{S}, call it contango or backwardation. This ratio construction is a mistake, and it is the same mistake I have spent three decades flagging in transfer pricing.
The textbook cost-of-carry identity is:
F(t,T) = S(t)\cdot e^{(r+u-y)T}where r is the risk-free 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(\frac{F}{S}\right) on maturity 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 formally elsewhere (the harmonic-mean bias theorem). The fix here is the same fix: stay in levels.
A first-order expansion of the cost-of-carry identity gives a linear model entirely in levels:
F(i) = a + b_1 S(i) + b_2\bigl[S(i)\cdot T(i)\bigr] + \varepsilon(i)
Theory disciplines both coefficients: b₁ should not differ from 1.000000 (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 it on the WTI or Brent term structure 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 the market is pricing storage cost or scarcity. 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.
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.
Gibson, R., & Schwartz, E.S. (1990). Stochastic convenience yield and the pricing of oil contingent claims. Journal of Finance, 45(3), 959–976.
Litzenberger, R.H., & Rabinowitz, N. (1995). Backwardation in oil futures markets: Theory and empirical evidence. Journal of Finance, 50(5), 1517–1545.
Schwartz, E.S. (1997). The stochastic behavior of commodity prices: Implications for valuation and hedging. Journal of Finance, 52(3), 923–973.
Working, H. (1949). The theory of price of storage. American Economic Review, 39(6), 1254–1262.