Forwards, futures and swaps

Level 1 Plain words

For every reader: the idea in plain words, no formulas.

A promise to trade later

A wheat farmer plants in spring and sells in autumn. A bakery buys wheat all year. Both worry about the autumn price: the farmer fears it will fall, the baker that it will rise. They can agree in spring that the baker will buy 5,000 bushels in October at a price set now. That agreement is a forward contract. Neither pays anything today. In October the farmer delivers and the baker pays the agreed price, whatever the market price has done.

Whoever agrees to buy holds the long side; whoever agrees to sell holds the short side. If the market price in October is above the agreed price, the buyer gains (they pay less than the going rate) and the seller loses the same amount. One side's gain is the other side's loss, dollar for dollar.

Hedgers and speculators

People who use these contracts to remove a risk they already carry are hedgers: the farmer, the baker, an airline that will buy jet fuel, a company that will receive euros. People who take on the risk hoping to profit are speculators. Markets need both, because hedgers on one side rarely match hedgers on the other in size and timing.

Futures: forwards with a referee

A futures contract is a forward with standard terms that trades on an exchange. A clearing house stands between buyer and seller, so neither has to trust the other. To make that safe, the clearing house asks both sides to post margin, a deposit, and settles gains and losses every day. If your position loses money today, cash leaves your account tonight. If it loses too much, you must top up the next morning or the broker closes the position.

That daily settlement makes futures safer for the system and harder on the holder. A hedge can be right in the end and still demand large amounts of cash along the way.

Swaps: trading one stream of payments for another

Think of a swap as a string of forwards bundled together. In the most common one, an interest rate swap, one side pays a fixed rate on an agreed amount and the other pays a floating rate that resets with the market. A company with a floating-rate loan that wants predictable payments can enter a swap where it pays fixed and receives floating. Its floating receipts cover the loan's floating payments, and it is left with a fixed cost.

Level 2 How it works

For readers who want the mechanism, with the diagrams.

Beginner path stops here

The payoff of a forward

Write $K$ for the agreed price and $S_T$ for the market price on delivery. The long side gains $S_T - K$, which can be positive or negative; the short side gains $K - S_T$. Plotted against $S_T$, the long's payoff is a straight line rising at 45 degrees and crossing zero at $K$. Nothing is paid at the start because $K$ is set so that the contract is worth zero on the day it is signed. Level 3 shows how.

Margin and daily settlement

The exchange sets an initial margin, the deposit needed to open a position, and a lower maintenance level. Every evening it marks each position to the day's settlement price and moves the gain or loss between accounts. That daily flow is variation margin. If an account falls below maintenance, the holder gets a margin call and must bring it back up to the initial level.

Here is one long crude oil futures contract on 1,000 barrels, bought at $80.00, with an initial margin of $8,000 and a maintenance level of $6,000. Exchanges set the real figures and change them often; these are round numbers for the example.

On day 2 the balance drops below $6,000 and the holder must add $3,000. By day 4 the price has recovered to $81. The holder deposited $11,000 in all and has $12,000 in the account, a gain of +$1,000, which matches the $1 rise on 1,000 barrels. A holder who could not find the cash on day 2 would have been closed out at a loss and missed the recovery.

Hedging with futures: an airline

An airline expects to buy 1,000,000 barrels' worth of fuel in six months and fears a price rise. It buys crude oil futures on 1,000,000 barrels at $80. If oil rises to $100, its fuel costs $100.0 million, but the futures gain +$20.0 million, so the net cost is $80.0 million. If oil falls to $60, the fuel costs $60.0 million and the futures lose −$20.0 million; the net cost is again $80.0 million. The hedge removes the gain from falling prices along with the loss from rising ones.

Two things spoil the neat result. First, jet fuel and crude oil do not move one for one. If jet fuel's premium over crude widens by $5 a barrel while the hedge sits in crude, the net cost rises to $85.0 million. That leftover risk is basis risk, and it is why Level 3 sizes the hedge with a ratio instead of a one-for-one match. Second, the margin flows hit cash before the fuel bill does. An airline hedged against a rise that sees oil collapse pays margin on the futures now and saves on fuel only later.

Interest rate swaps

In a plain swap on $10 million, one side pays a fixed 4 % a year and the other pays a floating rate. In US dollar markets the floating leg now references SOFR, the Secured Overnight Financing Rate, which replaced dollar LIBOR after its panel ended on 30 June 2023. Only the net difference changes hands. If the floating rate for a period is 4.5 %, the fixed payer receives 0.5 % of $10 million for that period.

Swaps let a borrower or investor change the rate exposure of what they own without selling it. A bank with fixed-rate mortgages and floating-rate deposits can pay fixed and receive floating to match the two. A pension fund with long-dated promises to retirees can receive fixed to add duration without buying bonds outright. Other common swaps follow the same pattern: currency swaps exchange interest and principal in two currencies; total return swaps pay one side the full return on a stock or index in exchange for a floating rate; commodity swaps fix the price of a stream of deliveries.

Interest rate and Treasury futures

Futures on interest rates let a borrower or a bond fund change its rate exposure in minutes. A three-month SOFR futures contract on CME quotes 100 minus the expected average rate for its quarter, so a rate of 4.25 % shows as 95.75. If rates rise, the price falls, which keeps the usual rule that a long futures position gains when the price rises. A string of these contracts across coming quarters traces the market's expected path of short rates, and traders read the Fed's next moves from them.

Treasury futures deliver a government bond chosen by the seller from a list of eligible issues, each adjusted by a conversion factor. The seller picks whichever bond is cheapest to deliver, so the futures price tracks that bond. Fund managers use Treasury futures to lengthen or shorten duration without trading the bonds themselves, as Level 3 shows.

The risk drivers of forwards, futures and swaps

Clearing houses

After 2008, regulators pushed standard swaps onto central clearing houses, which already stood behind every futures trade. A clearing house nets each member's positions, collects margin and keeps a default fund. Clearing removes most counterparty risk between members, at the cost of more margin calls in volatile markets. The Dodd–Frank Act of 2010 required clearing of standard swaps in the United States, and Chapter 17 covers the credit side.

Level 3 The math

For Series 65 and CFA readers: formulas and worked examples.

Series 65 path stops here

Forward pricing

Take a stock at $S$ that pays a continuous dividend yield $q$, and a risk-free rate $r$. Someone who wants the stock in $T$ years can buy it now, borrowing $S$ at $r$, and collect dividends along the way. Or they can sign a forward. Both routes must cost the same, or traders would buy the cheap one and sell the dear one. That gives:

In words, the forward price is the spot price grown at the risk-free rate less the income the asset pays, for the time to delivery.

For a commodity, storage costs add to the rate and a convenience yield subtracts from it, which gives the cost-of-carry formula of Chapter 13. For a currency, the foreign interest rate plays the role of the dividend yield, which gives covered interest parity in Chapter 14.

A forward is worth zero when signed. Later its value to the long side is the spot value of the asset less the present value of the agreed price:

In words, the value of a long forward equals the asset's price net of the income it will pay before delivery, minus the present value of what the long will pay at delivery.

Marking to market

A futures contract resets its value to zero every day: the day's gain or loss is paid in cash, and the contract restarts at the new settlement price. Writing $F_t$ for the settlement price on day $t$, the cash paid to the long side on that day is:

In words, the variation margin each day equals the change in the settlement price multiplied by the number of units the contract covers.

The margin table in Level 2 applies this rule day by day. Summed over the life of the trade, the daily payments add up to the final price minus the starting price, the same total a forward would pay at the end. The timing differs, and timing is what can sink a hedger who is short of cash.

The minimum-variance hedge ratio

If the price of what you hedge moves with standard deviation $\sigma_S$, the futures with $\sigma_F$, and the two have correlation $\rho$, the number of futures units per unit of exposure that leaves the least variance is:

In words, the best hedge ratio is the correlation between the two prices times the ratio of their volatilities.

Hedging a bond portfolio with futures

A fund manager holds $50 million of bonds with a duration of 6 and expects rates to rise. Selling the bonds would be slow and costly. Selling Treasury futures does the same job in minutes. The number of contracts that brings the portfolio's duration to a target sets the dollar duration of the futures against the dollar duration of the bonds:

In words, the number of futures equals the change in duration wanted, times the portfolio's value, divided by the futures contract's duration times its value; a negative number means sell.

A swap as two bonds

A receive-fixed swap on notional $N$ behaves like owning a fixed-rate bond and owing a floating-rate note of the same size. A floating-rate note resets to the market rate each period, so it trades near par. The fixed-rate bond does not. So the swap's value comes almost entirely from the fixed side.

The fixed rate that makes a new swap worth zero is the par swap rate. With discount factors $D_1, \ldots, D_n$ for annual payment dates:

In words, the par swap rate equals one minus the final discount factor, divided by the sum of all the discount factors.

After rates move, the swap is worth the gap between the new market rate and the fixed rate, paid every year on the notional, discounted at the new rate. For a pay-fixed swap with a flat curve and annual payments:

In words, the value to the fixed payer equals the rise in the swap rate, times the notional, times the annuity factor at the new rate.

Level 4 History, analysed

For readers who want past episodes run through the Level 3 math.

Metallgesellschaft, 1993

In the early 1990s a US subsidiary of the German firm Metallgesellschaft sold customers fixed-price oil products for up to ten years ahead, and hedged with short-dated futures that it rolled every month, a stack-and-roll hedge. When oil prices fell in late 1993, the fixed-price contracts gained value on paper, but the futures lost money in cash every day. The curve was also in contango, so each roll cost money. The parent's board closed the positions in December 1993 with losses reported at more than a billion dollars.

Christopher Culp and Merton Miller argued in 1995 that the hedge was sound and that closing it locked in the loss; others, including Antonio Mello and John Parsons, argued that the maturity mismatch made it a funding bet. The Level 3 margin rule explains why the argument mattered: a long-dated position hedged with short-dated futures can be economically balanced and still run out of cash.

LTCM, 1998

LTCM ran convergence trades: it bought securities that looked cheap against near-identical ones and sold the dear ones, often through swaps and futures, expecting the gap to close. Each trade carried little risk on its own. The fund made them large with leverage. Roger Lowenstein's account, When Genius Failed (2000), puts its capital at about $4.7 billion at the start of 1998 against about $125 billion of assets on its balance sheet, before counting derivatives.

The arithmetic of leverage tells the story. Assets of $125 billion on $4.7 billion of equity is leverage of 26.6 to 1. A fall of 1 % in the value of the assets costs 26.6 % of the equity. A fall of 3.8 % wipes the equity out. After Russia defaulted on 17 August 1998, investors fled to the safest and most liquid securities everywhere at once. Nearly all of LTCM's trades were bets that such gaps would narrow, so they widened together. Its many "uncorrelated" positions turned out to share one risk factor, liquidity, and their correlations went toward one, as Chapter 3 describes.

The Fed did not lend its own money. It brought the creditors together because a forced sale of LTCM's positions would have hit the same banks' own books, and their swap contracts with LTCM would have needed replacing in a falling market.

UK gilts and liability-driven investment, 2022

UK defined-benefit pension schemes owe payments decades ahead, so the value of their promises rises when long interest rates fall. To hedge, many used liability-driven investment (LDI) funds that held gilts and interest rate swaps on borrowed money, often through repurchase agreements. Leverage let a scheme hedge more of its liabilities while keeping part of its assets in stocks.

A stylised fund shows the problem. Say it holds gilt exposure three times its own capital, with a duration of 20 years. A 100 basis-point rise in yields cuts the exposure by −20 % and the fund's capital by −60 %. A rise of 167 bp erases it. Long gilt yields rose by more than a percentage point within a few days after 23 September. Each fund had to post collateral at once or cut its hedges, and the quickest way to raise cash was to sell gilts, which pushed yields higher still and triggered the next round of calls. The pension promises themselves had become cheaper, so the schemes were not insolvent; they were short of cash, the same problem Metallgesellschaft faced. The Bank's purchases stopped the spiral, and the Bank later asked LDI funds to hold larger buffers.