Hedge Ratios For Specified Derivative Exposures

Why Hedge Ratios Require Careful Unit Matching

A hedge is a position in a derivative intended to reduce the effect of an adverse change in an underlying price or rate. The derivative position must have the right direction and the right size. A business that will buy a commodity is hurt by a price increase; a business that will sell it is hurt by a price decrease. Those exposures call for opposite derivative positions.

A hedge ratio translates the exposure into a number of derivative contracts. The calculation is only meaningful after identifying what one unit of exposure means, what one contract covers, and how the derivative value changes when the underlying changes. A ratio without units—or a correct ratio applied in the wrong direction—can leave the exposure unhedged or make it larger.

Definition: A hedge ratio is the signed size of a derivative position chosen to offset a stated exposure. A positive signed position denotes a long position and a negative signed position denotes a short position. The number of contracts is the absolute position divided by the exposure represented by one contract, when the contract units are directly comparable.

The examples below use specified prices, contract terms, and rate sensitivities. Unless an example says otherwise, assume the derivative and the exposure refer to the same underlying and settlement date, contract units match the stated exposure, and there are no transaction costs, taxes, or financing effects. These assumptions make the hedge calculation transparent; if the underlying, timing, or price response differs, the offset may not be exact.

The Signed Sensitivity Method

For a small change in an underlying price or rate, suppose the value of the exposure changes by $a$ dollars per unit change, and one derivative contract changes in value by $b$ dollars per the same unit change. If $n$ is the signed number of contracts, the combined change is approximately $a+nb$ per unit. Setting this change to zero gives the sensitivity-based hedge ratio.

a+nb=0 \qquad\Longrightarrow\qquad n=-\frac{a}{b}

The minus sign is essential: the derivative position must respond in the opposite direction to the exposure. The units in the numerator and denominator must also match. For example, dollars per basis point divided by dollars per basis point per contract gives contracts. Dollars per dollar divided by dollars per unit of some different measure does not.

For a direct, linear exposure to $Q$ units of an asset, a forward or futures contract covering $M$ units gives a particularly simple ratio: use $Q/M$ contracts in the direction opposite the exposure. This is exact under the matching assumptions above. For an option, the payoff is generally nonlinear in the underlying price, so a single sensitivity-based ratio need not offset the exposure at every possible final price. A stated protection objective, such as protecting a specified number of shares below a strike, may instead determine the contract count directly.

Derivation: Under the stated linear-sensitivity model, a one-unit increase in the underlying changes the exposure by $a$ dollars. It changes each derivative contract by $b$ dollars, so $n$ signed contracts change by $nb$ dollars. The total change is $a+nb$. A locally neutral hedge requires this total to equal zero; solving for $n$ yields $n=-a/b$. This result does not promise a perfect hedge when sensitivities change with the price, the underlying differs, or settlement dates do not match.

Hedging With Forwards and Futures

A forward or futures position can offset a known future purchase or sale when its underlying and contract quantity match the exposure. A buyer facing an uncertain future purchase cost typically takes a long position: the derivative gains value when the purchase price rises. A seller facing an uncertain future sale price typically takes a short position: the derivative gains value when the sale price falls.

For a forward contract on $M$ units with delivery price $F_0$, the long position’s payoff at expiration is $M(S_T-F_0)$, where $S_T$ is the underlying spot price at expiration. The short position’s payoff is $M(F_0-S_T)$. For multiple contracts, multiply the relevant payoff by the number of contracts. A futures position has the same cumulative price-change exposure in this simplified matched setting, although futures gains and losses are settled through margin over time.

Worked Example: Locking a Future Sale Price With Forwards

A hypothetical biofuel producer expects to sell 15,000 gallons in three months. It wants to protect against a price decline and can enter short forward contracts, each covering 1,000 gallons, at $F_0=\$2.65$ per gallon. At expiration, the spot price is $S_T=\$2.40$. Find the hedge ratio and the combined proceeds. Assume the entire expected sale occurs at expiration and the forward settles for the difference between the contract and spot prices.

Method. The producer is exposed to a falling sale price, so it takes a short forward position. The number of contracts is the gallons exposed divided by gallons per contract:

\frac{15{,}000\text{ gallons}}{1{,}000\text{ gallons per contract}} =15\text{ short contracts}.

The physical sale brings in:

15{,}000\times \$2.40=\$36{,}000.

The short forwards gain the delivery price less the expiration spot price, multiplied by the total contracted quantity:

15{,}000\times(\$2.65-\$2.40) =15{,}000\times \$0.25 =\$3{,}750.

Combined proceeds are $\$36{,}000+\$3{,}750=\$39{,}750$. Dividing by 15,000 gallons gives $\$2.65$ per gallon, the contracted price. Check: the spot sale is $\$0.25$ below the forward price, and the short hedge gains exactly $\$0.25$ per gallon. A common mistake is to take a long forward because the producer owns the biofuel; the relevant fact is that the producer will sell it and is hurt by a price decline.

Worked Example: Hedging a Future Purchase With Futures

A hypothetical manufacturer expects to buy 12,000 units of a raw material in six months. It uses long futures contracts covering 1,000 units each, entered at $\$48$ per unit. At expiration, the spot price is $\$53$. Calculate the hedge ratio and net purchase cost. Assume the futures price converges to the spot price at expiration, the material purchase occurs then, and the cumulative futures gain is received without financing or margin effects.

Method. The manufacturer loses when the material price rises, so it goes long futures. The contract count is:

\frac{12{,}000\text{ units}}{1{,}000\text{ units per contract}} =12\text{ long contracts}.

The physical purchase costs $12{,}000\times \$53=\$636{,}000$. The futures gain is the price increase per unit times the total units covered:

12\times1{,}000\times(\$53-\$48) =12{,}000\times \$5 =\$60{,}000.

Subtracting the futures gain from the purchase cost gives a net cost of $\$636{,}000-\$60{,}000=\$576{,}000$, or $\$48$ per unit. Check: the physical price rose by $\$5$ per unit and the long futures position gained the same $\$5$ per unit. The common direction error is to short futures because the manufacturer will eventually pay for the material. A buyer is harmed by a price increase, so the hedge must gain when prices rise.

Hedging Interest-Rate Sensitivity

A rate exposure can be translated into dollars per basis point. One basis point is $0.0001$ in decimal rate form. If a borrower owes principal $L$ for a fraction $t$ of a year and interest is calculated simply for that period, a rate increase of one basis point increases interest cost by $Lt(0.0001)$. A floating-rate borrower is therefore hurt by a rate increase and needs a position that gains when rates rise.

In a problem that states a futures contract’s gain per basis point directly, use that sensitivity as given. It is not necessary to infer it from a contract’s quoted price. Keep the same time horizon and basis-point unit in both sensitivities, then divide the exposure’s dollar sensitivity by the per-contract dollar sensitivity.

Worked Example: Matching a Borrower’s Rate Exposure

A hypothetical borrower owes $\$4{,}000{,}000$ for six months. Interest is calculated simply on an annualized rate of 5.20% for one-half year. A short futures contract gains $\$40$ for each basis-point increase in the relevant annualized rate over this same six-month horizon. Determine the hedge ratio and the added interest cost if the rate rises by 75 basis points. Assume the stated futures sensitivity applies over this horizon and there are no other costs.

Method. The borrower is short the economic exposure to rates: higher rates increase its cost. Because a short futures contract gains when rates rise, the hedge position is short.

First calculate the borrower’s added cost per basis point:

\$4{,}000{,}000\times\frac{1}{2}\times0.0001 =\$200\text{ per basis point}.

The units match: both the borrower’s exposure and each contract’s gain are dollars per basis point. Thus the required number of short contracts is:

\frac{\$200\text{ per basis point}} {\$40\text{ per basis point per contract}} =5\text{ short contracts}.

A 75-basis-point increase is a decimal rate increase of $75\times0.0001=0.0075$. The additional borrowing cost is:

\$4{,}000{,}000\times\frac{1}{2}\times0.0075 =\$15{,}000.

The five short contracts gain $5\times75\times\$40=\$15{,}000$, so the specified added cost is offset completely. Check: the per-contract gain over 75 basis points is $\$3{,}000$, and $5\times\$3{,}000=\$15{,}000$. A common error is to divide by a contract’s gain for the whole 75-basis-point move and then apply the 75-basis-point move again. Either compare both sensitivities per basis point or compare both total changes over 75 basis points, but do not mix the two.

Option Hedge Ratios and Net Cash Flows

An option hedge has two separate sizing questions: how much underlying exposure is protected, and what payoff the option provides in each price scenario. A put gives its holder the right, but not the obligation, to sell at its strike price $K$; its payoff per underlying unit at expiration is $\max(K-S_T,0)$. A call gives its holder the right to buy at the strike; its payoff per unit is $\max(S_T-K,0)$. The payoff is not the same as profit because the option buyer has paid a premium.

If the goal is to protect $Q$ shares with options that each cover $M$ shares, $Q/M$ contracts provide one option for each exposed share. That count does not mean the entire stock position is insured at all prices: a put only pays below its strike, and the premium reduces the net value of the hedge. Include the premium when calculating total profit or loss.

Worked Example: Sizing a Put Hedge and Including Its Premium

An investor owns 1,000 shares purchased for $\$52$ per share. To protect those shares below a strike, the investor buys puts with a strike of $\$48$, a premium of $\$0.90$ per share, and a 100-share contract multiplier. At expiration, the stock price is $\$44$. Find the number of contracts and the total profit or loss, including the premium. Assume the puts are held to expiration and ignore transaction costs.

Method. The investor wants one put for each owned share. The contract count is:

\frac{1{,}000\text{ shares}}{100\text{ shares per contract}} =10\text{ put contracts}.

The initial share cost is $1{,}000\times\$52=\$52{,}000$. The expiration value of the shares is $1{,}000\times\$44=\$44{,}000$. Since the stock is $\$4$ below the strike, each share has a put payoff of $\$48-\$44=\$4$, so the total put payoff is:

10\times100\times \$4=\$4{,}000.

The premium paid is $10\times100\times\$0.90=\$900$. The complete profit or loss is ending share value plus put payoff, less premium and initial share cost:

\$44{,}000+\$4{,}000-\$900-\$52{,}000 =-\$4{,}900.

Without the puts, the share loss would have been $\$44{,}000-\$52{,}000=-\$8{,}000$. The hedge improves the result by $\$3{,}100$, which is the $\$4{,}000$ payoff less the $\$900$ premium. Check: the net loss is $\$8{,}000-\$4{,}000+\$900=\$4{,}900$. A common mistake is to report the $\$4{,}000$ put payoff as the investor’s profit; that ignores both the share loss and the premium.

Common Hedge-Ratio Traps

Reversing the direction. A future buyer generally needs a long price hedge; a future seller generally needs a short price hedge. Decide who is harmed by the adverse move before choosing the derivative position.
Mixing units. A sensitivity in dollars per basis point must be compared with a sensitivity in dollars per basis point per contract. For example, $\$100$ per basis point divided by $\$30$ per basis point per contract is $10/3$ contracts, not 2.5. The units permit the division, and the arithmetic gives approximately 3.333 contracts.
Ignoring contract multipliers. A contract may cover 100 shares, 1,000 gallons, or another specified amount. Divide the exposure by the contract quantity, not by one contract as though it covered one underlying unit.
Forgetting option premiums. An option payoff is the amount received at expiration; profit must also include the premium paid. Keep those amounts separate until combining the cash flows.
Assuming a ratio is a perfect hedge in every setting. Matching quantities only eliminates the stated exposure under the model assumptions. If the derivative and exposure have different underlying prices, dates, or sensitivities, an offset may be incomplete. The calculation here is not a model for that residual risk.

The central exam technique is to write the exposure’s change in dollars, write one contract’s change in dollars on the same basis, and solve for the signed position. For options, also state what price region creates a payoff and account for the premium. This disciplined setup is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the hedge direction and size, then check the combined cash flows.

A hypothetical jeweler expects to buy 7,500 ounces of silver at expiration. Silver futures cover 2,500 ounces per contract. The spot price rises by $\$1.20$ per ounce by expiration, and the futures position gains the same amount per ounce. How many contracts are needed, and what is the futures gain that offsets the added purchase cost? (Answer: 3 long contracts; $\$9{,}000$ gain and $\$9{,}000$ added purchase cost.)
A hypothetical exporter will receive €1,200,000 in three months and sells euros forward at $\$1.08$ per euro. The expiration spot rate is $\$1.02$ per euro. How many short contracts are needed if each covers €100,000, and what are the combined dollar proceeds? (Answer: 12 short contracts; $\$1{,}296{,}000$ combined proceeds.)
A borrower has a $\$2{,}500{,}000$ loan for nine months, with simple interest for the period. The annualized rate increases by 40 basis points. Each short futures contract gains $\$15$ per basis point over the same horizon. Find the theoretical number of short contracts and the residual added cost if only 12 contracts are used. (Answer: $\$7{,}500$ added cost; 12.5 short contracts theoretically; $\$300$ residual added cost with 12 contracts.)
An investor owns 1,000 shares bought at $\$52$ each and buys puts with a $\$48$ strike, a $\$0.90$ premium per share, and a 100-share multiplier. At expiration the stock is $\$44$. Find the number of puts and the total profit or loss, including premiums. (Answer: 10 puts; a loss of $\$4{,}900$.)

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