A Forward Is a Carrying Cost, Not a Forecast

A riskless zero-coupon bond trades at 100 and the riskless rate is 5%. Is its six-month forward price above or below 100? Then change one thing: keep the bond riskless, but let it pay a continuous coupon of 8%. Now which side of spot is the forward on?

The first answer is 102.531512, a premium. The second is 98.511194, a discount. The two together kill a habit that most people carry without noticing, which is the idea that a forward price is a guess about the future.

Where the reflex comes from

Ask someone why a forward should sit above spot and you get some version of “money grows, so the price later is higher.” That sentence has two problems. It treats the forward as a forecast, and it forgets that the person holding the asset in the meantime is collecting whatever the asset pays. Both problems disappear once you stop guessing and build the trade.

The ledger that fixes the price

Today: borrow $S$ at the rate $r$, buy the asset, and agree to deliver it at date $T$ for a price $F$ fixed now. At $T$: hand over the asset, collect $F$, repay $Se^{r(T-t)}$, and keep everything the asset paid you while you held it. Nothing in that sequence is uncertain, so the whole thing has to be worth zero, and if the income the asset throws off has present value $P$:

Quote anything above that and the trade above is free money. Quote anything below and you run it backwards, selling the asset short and lending the proceeds. There is no probability in equation (1), no expected return, and no view. An explicit version of this ledger closes to zero at the fair forward and to a strictly signed profit at any other quote, checked across 192 parameter sets.

When the income arrives as a continuous yield $\rho$on the asset’s own value rather than as a lump, the present value of the leakage scales with the asset and equation (1) tightens into a single exponential:

Only one comparison decides the sign

Read equation (2) again and notice how little it depends on. The tenor scales the effect but cannot change its direction, and the spot level cannot either, because it multiplies rather than adds. What is left is one comparison:

A coupon exactly equal to the rate leaves $F = S$ for every tenor, which is the only case in which the forward curve is genuinely flat. Differentiating gives $dF/d\rho = -(T-t)F < 0$ and $dF/dr = +(T-t)F > 0$, so each effect is monotone and neither can overtake the other by accident. The sign rule was checked at seven coupon yields either side of the crossover and in every one of the 192 parameter sets.

What the reflex costs at the screen

Wrong answers in this corner of the market are not merely wrong, they are priced. Suppose the coupon is rich and you quote the forward at spot, on the theory that a forward cannot be below the cash price. Run the ledger against your own quote. It leaves 1.518802 per hundred of riskless profit to whoever takes the other side, and the forward itself is off by 1.488806. The difference between those two numbers is the interest on the profit, which is a good sanity check on both.

A bond pays lumps, not a stream

The continuous yield in equation (2) is a modelling convenience that makes the sign visible in one symbol. An actual bond pays cash on dates. Take a coupon of 4 paid at three months, with delivery still at six. Its present value is $4e^{-0.0125} = 3.950311$, so equation (1) gives

Three basis points from the continuous answer, which is reassuring and also slightly misleading, because the agreement is an accident of this calendar rather than a property of the two conventions. Move the coupon date to nine months, past delivery. Now no coupon reaches the holder of the spot at all, $P = 0$, and the forward is back to 102.531512, a full premium on a bond whose coupon is well above the rate. Equation (2) still insists on 98.51. The two answers are 4.020318 apart, which is roughly the coupon itself.

So the honest form of the answer is that a rich coupon flips the forward below spot when a coupon date falls inside the window. That is a question about the calendar as much as about the coupon rate, and it is the first thing a continuous-yield model smooths away.

Where equation (2) stops working

The quantity $S$ in the ledger is the full price, including anything accrued. Bond markets quote clean prices, with the accrual stripped out, so feeding a screen price straight into equation (1) puts the answer off by the accrual and in the direction that makes a discount look larger than it is.

Nothing here is a credit story. The bond was riskless by assumption, so the flip from premium to discount is pure carry. A defaultable bond’s forward carries a spread that equation (2) knows nothing about, and adding a spread to $r$ is not a repair, because the risk sits in the delivered asset rather than in the financing. Financing itself can also move: a bond that can be borrowed cheaply in the repo market carries at less than the general rate, which shifts the effective $r$ in equation (2) without touching anything else in it.

The last limitation is the one hiding behind the word forecast. Under a constant rate the distinction between carry and prediction has no teeth, because a riskless zero really will be worth 102.531512 in six months. Let rates move randomly and the carry number is still the arbitrage-free forward, but it stops equalling the expected future spot price, and the gap is a covariance between the asset and the discount factor. That same covariance is why a futures price and a forward price on the same asset stop agreeing once rates are stochastic. The arithmetic in this article is exact inside its model, and the model’s deterministic rate is the assumption doing the most work.

Sources and further reading

• The idea equation (2) is built from Cost of carry
• The distinction the article turns on Forward price
• Why a ledger that closes to zero fixes a price Rational pricing
• The convention that makes clean prices dangerous here Accrued interest
• Why forwards and futures separate once rates move Futures contract

Every number was checked twice: symbolically, as a carry factor whose sign follows $r - \rho$ and whose derivatives in $r$ and $\rho$ point opposite ways, and numerically, as a cash-and-carry ledger that closes to zero at the fair forward and to a strictly signed profit at every other quote across 192 parameter sets.