The Trade That Sees the Curve and Not the Level

You think the gap between two-year and ten-year yields is going to widen. You have no view at all on whether rates in general go up or down, and you would rather not be paid or punished for being lucky about that. What position gives you the gap and nothing else?

One leg is a bet on the level

Start with the instinct: if the short end is going to look good relative to the long end, buy the short end. Put 400 dollars into a note of modified duration 2. Now let the curve steepen exactly as you predicted, but let it steepen with everything rising: the two-year yield up 25 basis points, the ten-year up 75. The gap widened by half a point. Your view was correct, and the position lost money:

Run the same steepening with rates falling instead and the same trade makes two dollars. The sign of the outcome is decided by the one thing you had no opinion about. A single bond is a position on the level of rates that happens to sit at a maturity.

Two legs, and the level drops out identically

Equation (1) is the whole of the first-order story: a bond’s exposure to its own yield is carried by one number.

Now split every yield move into a piece the two legs share and a piece specific to each. Write $\Delta y_{\text{short}} = m + a$ and $\Delta y_{\text{long}} = m + b$, where $m$ is whatever the whole curve does together. Give both legs the same dollar duration $K$, and go long the short maturity against a short in the long maturity:

The common part is gone. Not small, not zero on average, absent: it entered both legs with the same coefficient and opposite signs, so it cancels as an identity. What is left is the dollar duration times the change in the gap.

The sizing, and the four dollars

Take 400 dollars of a duration-2 note against 100 dollars of a duration-8 bond. Then $2 \cdot 400 = 800$ and $8 \cdot 100 = 800$: four to one by market value, one to one by the number that matters. Half a point of widening pays $800 \times 0.0050 = 4$ dollars, and it pays exactly four in every level scenario. Short down 25 with long up 25, short up 25 with long up 75, short down 75 with long down 25: four dollars each time. A parallel move of 50 basis points nets exactly zero, and so do parallel moves of 25, 100, 300 and 1000, all verified in exact rational arithmetic with no floating point anywhere.

What “matched” does not mean

A version of this trade circulates in which the two legs are matched on duration and on market value. With two bonds that is impossible, and the reason is one line of algebra. Equal market value means $P_{\text{short}} = P_{\text{long}} = V$. Equal dollar duration means $D_{\text{short}} V = D_{\text{long}} V$. Both at once forces $D_{\text{short}} = D_{\text{long}}$, and two bonds of equal duration leave no curve trade to put on.

The position above is 400 long against 100 short, so 300 dollars of cash goes in and the net market value is 300, not zero. Making it zero needs a third leg that carries no duration: borrow the 300, or fund it in the repo market, or short 300 dollars of an overnight instrument. Then market value nets to zero, dollar duration still nets to zero, and the two-bar picture is unchanged, because the third leg has no bar to draw. The picture is matched on dollar duration, and on nothing else.

What “cancels” actually means

Equation (2) is a statement about the first-order term, and the first-order term is not the price. To see the difference, drop duration entirely and reprice from cash flows. Take a real two-year and a real ten-year, both paying a 4% coupon and yielding 4%, so both priced at par. Their durations come out of the cash flows as 1.8861 and 8.1109, which sizes the trade at 4.3004 units of the short leg against one of the long, and puts 811.09 of dollar duration on each side.

On a full reprice a parallel shift does not net to zero. At 50 basis points it leaves 0.0702 against a twist profit of 4.0377, which is 1.7% of the thing you came for. At 200 basis points it leaves 1.0483, which is a quarter of the profit and no longer ignorable.

What makes that honest rather than fatal is how it scales. Halve the shift and the residual falls by roughly four: measured at 3.82, 3.91 and 3.95 across shifts of 200, 100, 50 and 25 basis points. Dying like the square of the move is the signature of a second-order term, and this one has a name. It is the convexity difference between the legs, because a ten-year bond is far more convex than a two-year and matching durations does nothing whatsoever about that.

The twist side converges from the other direction. Its full reprice tracks the first-order prediction more and more closely as the move shrinks: 4.037733 against 4.055448 at 50 basis points, 0.405365 against 0.405545 at 5, and 0.040553 against 0.040554 at half a basis point. With the level moving as well, the reprice gives 3.8418 when rates rise and 4.0960 when they fall, against a first-order 4.0554. So the level is still in the P&L. It is in the small term, worth a few percent at realistic move sizes, and the correct claim is that it cancels to first order rather than that it cancels.

Where the trade stops being what it looks like

A curve move is not two numbers. Writing it as a level plus a slope is a projection onto two factors, and real moves have a third: a butterfly, in which the middle of the curve moves against both ends, shifts the two legs in a way no single spread describes. The position has an exposure to that shape and equation (2) cannot see it, because $a$ and $b$ were defined as whatever was left after removing the common part.

Two smaller assumptions carry weight as well. Duration is a derivative with respect to a yield, which presumes each leg has one yield to differentiate against; two bonds of the same maturity and different coupons have different yields, and matching their durations is not the same as matching their sensitivity to any single curve factor. And nothing here prices carry or roll-down, so a steepener that is right and slow can still be a losing trade on financing alone.

The claim worth defending in an interview is narrow and true. The position is neutral to the level of rates at first order, which is why a correct view on the gap gets paid regardless of what the level does, and 0.0702 at 50 basis points is the price of the word “first”.

Sources and further reading

• The single number equation (1) compresses a bond into Bond duration
• The term that survives the cancellation in Fig. 2 Bond convexity
• What equation (2) is really doing to the move Linear approximation
• The object being decomposed into a level and a slope Yield curve
• The neighbouring technique built from the same arithmetic Immunization

The first-order claims were checked symbolically, with the parallel part carried as an explicit symbol so the cancellation is visible rather than numeric, and then in exact rational arithmetic over nine curve moves. The second-order claims come from a full cash-flow repricing of two real bonds that never uses duration at all.