Futures and Forwards: Calculating Profit and Loss

1. From Payoff per Unit to Total Profit and Loss

As we saw, the payoff from a single unit of the underlying asset is determined by the difference between the spot price at maturity, $S_T$, and the agreed-upon forward or futures price, $K$.

However, contracts are rarely for a single unit. To find the total profit or loss, we must multiply this per-unit payoff by the total quantity of the asset covered by the contract.

The fundamental formulas are:

• Total Profit/Loss (Long Position) = $(S_T-K)\times \text{Contract Size}$
• Total Profit/Loss (Short Position) = $(K-S_T)\times \text{Contract Size}$

Let's start with a straightforward example involving a forward contract.

2. Calculating Profit and Loss on Forward Contracts

Forward contracts are often customized in the over-the-counter (OTC) market, so the contract size is whatever the two parties agree upon. The calculation is a direct application of the formulas above.

The following reading from a lecture series at HEC Paris business school provides a clear, numerical example of this calculation for both the long and short parties in a forward contract.

Let's break down the logic from that example:

• Parties: Nestlé (Long), Farm (Short)
• Underlying Asset: Corn
• Contract Size: 10,000 bushels
• Forward Price (K): $4.00 per bushel
• Spot Price at Maturity ($S_T$): $3.40 per bushel

Calculation for the Long Position (Nestlé):

• Payoff per bushel = $S_T-K=\$3.40-\$4.00=-\$0.60$ (a loss)
• Total Loss = $-\$0.60\times 10,000\text{ bushels}=-\$6,000$

Calculation for the Short Position (the Farm):

• Payoff per bushel = $K-S_T=\$4.00-\$3.40=+\$0.60$ (a profit)
• Total Profit = $+\$0.60\times 10,000\text{ bushels}=+\$6,000$

As expected from our work with payoff diagrams, this is a zero-sum game: Nestlé's loss is the farm's gain.

3. Calculating Profit and Loss on Futures Contracts

Futures contracts add a layer of detail because they are standardized and traded on exchanges. The key parameters for calculation are the contract size (the fixed amount of the underlying asset in one contract) and the tick size (the minimum price movement).

The CME Group, one of the world's largest derivatives exchanges, has produced an excellent video explaining exactly how to calculate profit and loss using a real-world WTI crude oil futures contract.

Let's summarize the WTI crude oil example from the video to ensure the steps are clear:

1. Identify Contract Details:

   • Position: Long 1 WTI futures contract
   • Contract Size: 1,000 barrels
   • Tick Size: $0.01 per barrel
   • Entry Price (K): $53.60
   • Exit Price ($S_T$): $54.00

2. Calculate the Price Move:

   • Price difference = $S_T-K=\$54.00-\$53.60=+\$0.40$ per barrel.

3. Calculate the Total Profit:

   • Method A (Directly using total contract value):
     • Profit per barrel = $0.40
     • Total Profit = Profit per barrel $\times$ Contract Size = $\$0.40\times 1,000=\$400$
   • Method B (Using Ticks, as shown in the video):
     • Value of 1 tick = Tick Size $\times$ Contract Size = $\$0.01\times 1,000=\$10$
     • Number of ticks moved = Price difference / Tick Size = $\$0.40/\$0.01=40$ ticks
     • Total Profit = Value of 1 tick $\times$ Number of ticks moved = $\$10\times 40=\$400$

Both methods yield the same result. Losses are calculated in exactly the same way. If the price had fallen to $53.00, the loss would have been $(\$53.00-\$53.60)\times 1,000=-\$600$.

Example: A Short Futures Position

Now, let's apply this to a short position. We can use another example from the HEC Paris PDF, this time concerning S&P 500 ESG Index Futures.

Here's the breakdown:

• Position: Short 10 E-mini S&P 500 ESG futures contracts
• Contract Size: $500 \times$ the Index Value
• Entry Price (K or $F_{0,T}$): 316 index points
• Exit Price ($S_T$): 300 index points

Calculation for the Short Position:

• Profit per index point = $K-S_T=316-300=16$ points
• Profit per contract = $16\text{ points}\times \$500/\text{point}=\$8,000$
• Total Profit = $\$8,000/\text{contract}\times 10\text{ contracts}=\$80,000$

This demonstrates a profitable short trade, where the trader correctly anticipated a fall in the market.

4. Application: The Role of P&L in Hedging

These profit and loss calculations are not just for speculation. Their primary purpose in corporate finance is hedging, where a company uses a derivative to offset a risk it faces in its normal business operations. The profit or loss from the futures contract is designed to cancel out a loss or profit in the physical market.

The article "Using Futures for Hedging" provides an excellent example of an oil refiner using a short futures position to lock in a selling price.

In Scenario 1 of that example, the oil price falls to $55.

• The company receives less for its physical oil than it hoped.
• However, its short futures position generates a profit of $27.5 million.
• The profit from the futures contract effectively compensates the company for the fall in the physical oil price, allowing it to achieve a net selling price close to the original futures price of $60.50.

This shows the practical importance of being able to accurately calculate the profit or loss on a futures position as a core component of a risk management strategy.

Conclusion

In this lesson, we have moved from the theoretical structure of payoff diagrams to the practical arithmetic of calculating profits and losses.

Key Takeaways:

• The total profit or loss on a forward or futures contract is the per-unit payoff multiplied by the contract size.
• For a long position, the profit is $(S_T-K)\times \text{Size}$.
• For a short position, the profit is $(K-S_T)\times \text{Size}$.
• Futures contracts require attention to standardized contract sizes and tick values to arrive at the final dollar amount.
• These calculations are fundamental to both speculation and, more importantly for corporate finance, hedging business risks.

We have now solidified our understanding of linear derivatives. In our next lesson, we will introduce a new type of derivative with a fundamentally different risk profile: options. You will see how options create asymmetric, non-linear payoffs, allowing users to cap their downside risk while retaining upside potential. This will require us to construct and interpret entirely new kinds of payoff diagrams.