What can you back out of a curve that nobody publishes?

A curve is a set of observable prices. Financing rates are observable. Storage rates are quoted in physical markets. Everything else in the identity has to be solved for — and solving for it is how professionals turn a picture of a curve into a statement about the physical world.

The method

Take any two points on the curve and annualise the gap between them:

Implied carry (annualised) = [(F₂ ÷ F₁) − 1] × 12 ÷ months between

Then subtract what you can observe:

Implied convenience yield = financing + storage + insurance − implied carry

Two lines of arithmetic. The discipline is in reading the answer honestly, which comes at the end.

Worked: a contangoed crude curve

M1 $78.00 · M2 $78.55 · M3 $79.05 · M6 $80.30 · M12 $82.10

• M1 → M2: (78.55 ÷ 78.00) − 1 = 0.705% over one month → 8.46% annualised
• M1 → M6: (80.30 ÷ 78.00) − 1 = 2.95% over five months → × 12/5 → 7.08% annualised
• M1 → M12: (82.10 ÷ 78.00) − 1 = 5.26% over eleven months → × 12/11 → 5.73% annualised

Now subtract the observables. Say the reference rate is 5.0% and insurance is 0.5% a year:

• Front of the curve: 8.46 − 5.0 − 0.5 = 2.96% a year of net storage cost. On $78 that is $2.31 a barrel a year, about $0.19 a barrel a month.
• Far end of the curve: 5.73 − 5.0 − 0.5 = 0.23% a year — essentially nothing, about 1.5 cents a barrel a month.

Read what that says. The front of the curve is paying for storage; the back is almost pure financing. The market is pricing a storage cost today and pricing essentially none a year out — a comfortably supplied market with no expectation of storage scarcity on the horizon. Notice how much more that says than "the curve is in contango."

Worked: a backwardated curve

M1 $78.00, M6 $74.20.

(74.20 ÷ 78.00) − 1 = −4.87% over five months → × 12/5 → −11.69% annualised.

A negative implied carry is impossible as a cost, so the residual must be doing the work. With financing 5.0%, a quoted storage rate of 3.0% a year and insurance 0.5%:

Implied convenience yield = 5.0 + 3.0 + 0.5 + 11.69 = 20.2% a year

On a $78 barrel that is about $15.75 a year. The market is pricing the operational value of having a barrel now rather than in six months at roughly a fifth of the barrel annually.

That number is a measurement of physical tightness, and it is far more precise than any paragraph of commentary about it.

The honest part

A residual absorbs everything you did not model. Before quoting an implied convenience yield as if it were data, list what is hiding inside it:

• The risk premium from Unit 2 — not separable, not observable, contested.
• Balance-sheet and credit costs of financing inventory, which are not the same as a policy rate for anyone who is not a government.
• Quality and location differences between the exchange's deliverable grade and the physical barrel or bushel you actually have in mind.
• Measurement error in "spot" — for many commodities there is no clean spot quote at all, and the front future is used as a proxy. That means the front spread is doing double duty as both the input and part of the answer.

So always call it implied, never the convenience yield, and treat the number the way this Academy treats a term-premium estimate: a made-visible assumption, useful precisely because you can see what went into it.

Try it now

1. Rebuild the annualised-carry table above with today's financing in place of the 5.0%. The financing term is the thirty-day SOFR average below, a term average rather than an overnight print. Keep the worked strip, or take a real one from the exchange's public settlement page.

2. Produce your own implied net storage cost at the front of the curve, in currency per unit per month, and compare it with the $0.40 a barrel a month this course has been assuming for crude.
3. Repeat the whole exercise a month later. The interesting output is not the level — it is which direction the residual moved, and whether inventory data moved with it.