The Field That Earns Nothing Has the Higher Forward Price

Two properties are worth a million each, one an empty field and the other a beach collecting admission. The six-month forward is 1,020,000 for the field and 990,000 for the beach, and the gap is exactly the income the forward buyer never collects. Today's spot already capitalises every future admission, which is why income enters the forward as a subtraction, and why a carrying cost on the field would only widen the gap.

Two pieces of property are for sale today and both cost exactly one million. One is an empty field in the desert that produces nothing. The other is a stretch of beach that charges people to walk onto it. Agree today to buy one of them in six months, with payment and delivery both at that date. Which contract has the higher price?

The field. Its six-month forward price is 1,020,000 against the beach's 990,000, and the 30,000 gap is exactly the admission money the forward buyer never gets to collect. The answer is not a subtlety of pricing conventions, and it does not depend on anyone's view of property prices.

Two identical spots, one of which earns

The premise is doing more work than it looks like it is, so it is worth pausing on. Both properties are worth a million today. The beach's admission revenue is not a secret; anyone can see it, and today's buyer is paying for it. So the fact that the two spots are equal already tells you the market has priced the income in, and the beach must be giving something back elsewhere, in acreage or location or whatever else.

With that granted, the two contracts differ in one respect only. Buying the beach spot today gets you six months of admissions before the forward buyer even takes possession. Buying it forward gets you none of them. A price that ignored that difference would be handing out free money, and the next section shows how much.

Why the beach feels like the answer

The reflex is that the earning asset is the better asset, so it should cost more in every market it trades in. That reasoning is correct about the spot, where a claim on future revenue genuinely does lift the price, and it is a straight import of that intuition into a market where it does not apply.

A forward price is not an opinion about value. It is the answer to a narrower question: given that I can buy the thing today, what must I be paid at the later date to be indifferent? A seller who buys the asset now and holds it collects whatever it produces in the meantime. The income shows up on the seller's side of the ledger, so it comes off the price.

The forward from a portfolio, not a forecast

Cash and carry

To deliver an asset at time $T$ for a price fixed today, borrow the spot price $S$ , buy the asset, hold it, and use any income it produces to reduce the debt. At delivery the debt owed is $S(1+r\tau) - I$ , where $I$ is the income accumulated to $T$ . That debt is the forward price, because any other price makes the whole operation a free lunch.

Six months at four percent a year, applied as simple interest, is a carry factor of 1.02. For the field there is no income, so the debt at delivery is the whole financed amount:

F_{\text{field}} \;=\; 1{,}000{,}000 \times 1.02 \;=\; 1{,}020{,}000

(1)

For the beach the same financing applies, and the 30,000 of net admission collected along the way pays part of the debt down before delivery:

F_{\text{beach}} \;=\; 1{,}000{,}000 \times 1.02 \;-\; 30{,}000 \;=\; 990{,}000

(2)

Fig. 1 — The same carry lifts both. Only one of them hands part of it back, and the amount handed back is the income itself.

Suppose someone quoted the beach forward at 1,020,000, the same as the field's, on the grounds that the two spots are equal. Sell that forward. Borrow a million, buy the beach, run it for six months and collect the 30,000. At delivery, hand over the beach, receive 1,020,000, and repay the 1,020,000 owed. The 30,000 is yours, and nothing about the beach's price in six months entered the calculation.

Where the income goes

The gap between the two forwards is not approximately the income. It is the income, exactly. With a common spot $S$ , a common carry factor $1+a$ and an income $I$ accumulated to delivery:

\bigl(S(1+a) - I\bigr) - S(1+a) \;=\; -I

(3)

The spot cancels, the carry cancels, and what remains carries the sign of the income and nothing else. So the ordering is strict for any strictly positive income, however small, and the size of the gap is a fact about the income rather than about the rate or the price.

A forward price is also not a forecast, and equation (3) is where that becomes visible. Nothing in the derivation mentioned the terminal price of either property. Running the arbitrage above against a mispriced quote returns the same profit at every terminal price, which is what makes it riskless rather than merely attractive.

Fig. 2 — Checked at 2,001 terminal prices, the profit does not move. A forward price is the level that removes the free lunch, not a prediction of where the price will be.

The general statement, and the compounding convention

The simple-interest factor of 1.02 was chosen to keep the arithmetic visible, and nothing rests on it. Under continuous compounding the two forwards become

F_{\text{field}} = S e^{r\tau}, \qquad F_{\text{beach}} = S e^{r\tau} - I \;<\; F_{\text{field}}

(4)

and the inequality is the same inequality. Only the size of the carry moves, from 20,000 to 20,201 at four percent over half a year, while the 30,000 of income sits untouched on the other side.

The timing of the income does not change the direction either, and it changes the size in the direction that strengthens the answer. The figures above assume the 30,000 arrives at delivery. If instead it arrives steadily across the six months, each instalment can be reinvested until delivery, and its accumulated value is larger: an evenly received 30,000 has an average holding period of a quarter of a year, so it accumulates to 30,300 at four percent. The forward falls further and the gap widens.

The one assumption that could reverse it

What the problem grants silently is that neither property costs anything to hold. That is the load-bearing assumption, and it is worth spelling out because carrying costs enter the forward with the opposite sign to income.

Costs are money the seller spends while holding the asset, so they get added to the debt at delivery rather than subtracted from it. The full statement is that the forward price is the carried spot minus the accumulated net income, income less costs:

F \;=\; S(1+a) \;-\; \underbrace{\bigl(I - C\bigr)}_{\text{net income}}

(5)

A property tax on the field is a cost with no income beside it, so its net income is negative and its forward moves up, further above the beach's. On that side the conclusion only gets stronger. What would reverse the ordering is the beach costing enough to run, with upkeep and staffing that exceed the admissions by more than the field's tax bill. Then the beach's net income is the lower of the two and the beach takes the higher forward.

So the answer to the question as asked is the field, and the reason is worth carrying in the general form. The comparison is never between which asset earns more; it is between which asset delivers less net cash to its holder over the life of the contract. An empty field delivers zero, and zero beats a positive number in this particular race.

Sources and further reading

Wikipedia: Forward price and Cost of carry, which together give equation (5).
Wikipedia: Rational pricing, for the replication argument that makes the price an equality rather than an estimate, and Present value.
John Maynard Keynes, A Tract on Monetary Reform (1923), where the relation between a spot price, a forward price and the cost of carrying is set out for currencies.

Both forwards were derived twice before publication, once from the carry formula and once from a replicating portfolio built cash flow by cash flow, so the ordering is not assumed by the method. The ordering was then checked across 3,280 combinations of rate, income and compounding convention, and the arbitrage profit against a mispriced quote was confirmed identical at all 2,001 terminal prices tested.