Both Quoted at 5 Percent, and the Forward Is Worth Two Basis Points More

A futures settles up every day, so its fair price is a plain expectation, while a forward settles once, so its fair price is a discounted expectation renormalised. The difference between the two is exactly the covariance of the discount factor with the contract price divided by the expected discount factor, and for a deposit contract quoted as 100 minus the rate both fall together, so the fair forward price is 95.019999 against the futures' 95.000000. Long the forward and short the futures is worth 0.0195 points at inception, the gap reaches 39.48 basis points at ten years, and on an asset whose price rises with rates the whole answer reverses.

A six-month futures and a six-month forward, both written on the same three-month deposit rate, are both quoted at 5 percent. Same underlying, same expiry, same number on the screen. Do you prefer one, and if so which way do you trade?

Take the forward. The two contracts are not worth the same amount, the gap here is two basis points, and the reason has nothing to do with the underlying and everything to do with when cash changes hands.

Two settlement schedules, two prices

A forward pays once, at expiry. A futures is settled up every day, so the profit or loss on it is collected or paid as it happens and then financed or invested at whatever the short rate is that day. That difference in timing shows up as a difference in what a fair price means.

The two fair prices

Write $S$ for the contract price at expiry and $D$ for the discount factor to expiry, both random. Under the pricing measure the fair futures price is the plain expectation $F_{\text{fut}} = \mathbb{E}[S]$ , because daily settlement makes the contract worth zero afresh every day. The fair forward price is the discounted expectation renormalised, $F_{\text{fwd}} = \mathbb{E}[DS]/\mathbb{E}[D]$ , because the single payment has to be discounted along the path that produced it.

Both of those were recovered in the checks below from zero-value conditions rather than assumed: a futures entered at 95 has zero expected margin while one entered at 94.5 is worth half a point and one at 95.5 is worth minus half a point, and a forward is worth exactly zero at $\mathbb{E}[DS]/\mathbb{E}[D]$ and nothing else.

The difference is one covariance

Subtract the two definitions and the algebra is immediate:

F_{\text{fwd}} - F_{\text{fut}} = \frac{\mathbb{E}[DS]}{\mathbb{E}[D]} - \mathbb{E}[S] = \frac{\mathbb{E}[DS] - \mathbb{E}[D]\,\mathbb{E}[S]}{\mathbb{E}[D]} = \frac{\operatorname{Cov}(D, S)}{\mathbb{E}[D]}

(1)

Everything else in this article is a discussion of the sign of one covariance. The two contracts agree in price when the discount factor and the contract price are uncorrelated, and they disagree otherwise, in the direction that covariance points.

Why a deposit contract has a positive one

Put the smallest model on it that carries the mechanism. One period of length $T$ , and under the pricing measure the rate comes out at 7 percent or 3 percent with probability one half each, so its mean is the quoted 5 percent. A deposit contract is quoted as 100 minus the rate, and the discount factor is $e^{-LT}$ :

\begin{array}{lcc} & L = 7\% & L = 3\% \\ S = 100 - 100L & 93 & 97 \\ D = e^{-LT} & 0.965605 & 0.985112 \end{array}

(2)

Both rows fall from right to left. When the rate rises the contract price falls, and the discount factor falls too, so the two move together and the covariance is positive. With two equally likely states it is a quarter of the product of the two gaps, $\operatorname{Cov}(D,S) = (D_1 - D_2)(S_1 - S_2)/4 = +0.01950652$ , and that is what (1) turns into a price difference.

Fig. 1 — The positive covariance, drawn. Nothing here is about futures; it is about two quantities that happen to fall together.

Two basis points, and the trade

Evaluate both prices at $T = 0.5$ :

F_{\text{fut}} = \tfrac{93 + 97}{2} = 95.000000, \qquad F_{\text{fwd}} = \frac{0.965605 \times 93 + 0.985112 \times 97}{0.965605 + 0.985112} = 95.019999

(3)

The gap is 0.02 of price, two basis points, and it matches $\operatorname{Cov}(D,S)/\mathbb{E}[D] = 0.0199993$ to eight decimals. So the honest forward price is above the honest futures price, and both contracts are being offered at 95. A long forward struck at 95 is worth $+0.0195$ points at inception, which is a contract being given away. Long the forward and short the futures at the same quoted 5 percent is worth $+0.019507$ points per hundred of notional on day one, and the reverse package is worth strictly less than nothing, so the direction is not a matter of taste.

The verbal version of (1) is worth keeping alongside the algebra, because it is what gets said out loud. The futures pays its losses in cash on the days when rates have gone up, which are exactly the days when funding those losses is expensive, and it collects its gains on the days when cash is cheap to invest. Nobody takes that schedule for free, so the futures has to be cheaper.

How big the gap gets

Two basis points at six months is inside most bid-ask spreads, and a candidate who stopped at the sign would be right and unpersuasive. The gap is roughly linear in maturity over this range and quadratic in the rate uncertainty:

1.00\,\text{bp at }3\text{m}, \quad 2.00 \text{ at } 6\text{m}, \quad 4.00 \text{ at } 1\text{y}, \quad 7.996 \text{ at } 2\text{y}, \quad 19.93 \text{ at } 5\text{y}, \quad 39.48 \text{ at } 10\text{y}

(4)

Halving the spread between the two rate states, from 7 and 3 down to 6 and 4, takes the six-month gap from 1.9999 to 0.5000 basis points, a factor of four for a factor of two, which is what a covariance built out of a product of two proportional gaps has to do. Kill the rate uncertainty entirely and the gap is exactly zero: no dispersion in the discount factor, no covariance, no difference.

Fig. 2 — The same mechanism at every maturity. Two basis points is a curiosity; forty is a position.

It is not a fact about futures

The most useful check on the whole argument is to break it deliberately. Run the same model on a contract whose price rises with rates instead of falling. Now the two gaps in (2) have opposite signs, the covariance turns negative, and the futures becomes the dearer contract: the gap measures minus 2.00 basis points and the correct trade reverses.

So the answer is never prefer forwards. It is prefer whichever contract the covariance in (1) favours, and for a deposit-rate contract quoted as 100 minus the rate it happens to be the forward. Getting this right in an interview means saying which correlation you are relying on before you say which contract you want.

What the two prices assume

The clean statement $F_{\text{fut}} = \mathbb{E}[S]$ is a modelling result, not a definition, and it needs margin to be settled continuously and financed at the realised short rate, with no default on either side and no cost to posting collateral. Real margin is posted daily rather than continuously, earns something other than the theoretical short rate, and a large mark against you can be a funding event rather than a bookkeeping entry.

The two-state model in (2) is the smallest object that carries the mechanism and the exact covariance form of (1) is exact for it. A realistic term-structure model changes the size of the adjustment and not the sign, because the sign came from the observation that the price of a deposit contract and the value of a dollar at expiry fall together. That observation survives any model in which rates are the only thing moving.

One boundary that is easy to miss: the whole argument is stated under the pricing measure, where the rate has a mean of exactly the quoted 5 percent. If your view is that rates will be somewhere else, that is a different trade and a different article. The gap in (3) is available to somebody with no view on rates at all, which is what makes it worth two basis points rather than an opinion.

Sources and further reading

The contract settled every day — Futures contract
The contract settled once — Forward contract
The quantity that equation (1) computes — Covariance
The change of measure hiding in $\mathbb{E}[DS]/\mathbb{E}[D]$ — Forward measure
The adjustment (4) tabulates, under its market name — Convexity (finance)

Both fair prices were recovered from zero-value conditions rather than assumed, the identity in (1) was confirmed symbolically and then to eight decimals against the direct computation, the maturity profile in (4) was checked for monotonicity, and the sign was deliberately flipped on a control asset whose price rises with rates.