# Futures returns and P&L

### Abstract

Exchange-listed futures play an important part in many active strategies; they offer leverage, efficient access to illiquid physical markets, and risk management tools. In this paper, we illustrate the issues and procedures related to incorporating futures in our strategy (e.g. contract specifications, margin requirements, contract size, leverage, mark-to-market (MTM) etc.). Finally, we define returns on margin and compute strategy's P&L and max P&L draw throughout the holding period.

### Overview

From our perspective, a futures contract for a given delivery month is similar to any other security with the exception of limited lifetime (i.e. expiry). Furthermore, at any given time, the exchange lists several contracts for different delivery months of the same underlying. The returns series for these contracts are strongly correlated and usually the liquidity is highest at front-contract. As time approaches expiry date of the front contract, the trading volume switches to the next nearest expiry contract (i.e. rollover); futures participants primarily seek exposure and are unwilling to take delivery so they rollover their position.. The actual mechanism for rolling forward can vary significantly from a simple switch (i.e. price jump) on a given day before or on expiry date, to a more gradual rollover (i.e. continuous price) before expiry.

### Raw Data

For our analysis, we will use the front-month contract with simple rollover mechanism. Our series may experience price jump on rollover days, but for daily returns, we are only concerned with prices at market open-close and do not hold a position overnight, thus the analysis of this data set is immune from day to day price jumps.

### Trading Strategy

We will use S&P E-mini future contract listed on Chicago Mercantile Exchange (CME) under (ES). The futures are traded in two trading sessions: day-session on trading floor (pit), and after-hours electronic trading session. In this paper, we will assume the following strategy:

- Underlying asset – S&P 500 E-minis.
- Entry decision - we buy (go long) the futures contract on market open using a Market-on-open order (MOO).
- Exit decision – we close the futures position by taking an offsetting position (unwind) in the same future market at market close (MOC)

### Underlying future

- Exchange - CME
- Contract Months - Quarterly cycle (March, June, September and December)
- Contract Size – 50
- Expiry – 3rd Friday of the contract month
- Minimum fluctuation – 0.25 Index point ($12.50/contract)

### Performance Bond/Margin Requirement

Entering a position (long/short) in the futures is theoretically free; however, the listing exchange –through your broker – requires us to post cash (or cash equivalent securities) into your account as collateral.

Later on, the futures position is marked-to market at least once per day , and profit or loss on the day of a position is then paid to or debited from the account. If the margin posted in the margin account falls below the minimum margin requirement (the minimum amount to be collateralized in order to keep an open position), the broker or exchange issues a margin call, and as a result, we either have to increase the margin that they have requested, or they (exchange) can close out their position.

### Futures returns

Entering into a future position is not free, but, given the margin requirement and the size of position, it is highly leveraged. To calculate the futures returns, we define the following term – Return on Margin:

$${P\&L}_t=(S_t^{close}-S_t^{open}) \times Z$$

$$R_t=\frac{(S_t^{close}-S_t^{open})} {M_t^{open}} \times Z$$

Where

- $R_t$: Return on Margin
- ${P\&L}_t$: P&L for the holding period per contract
- $S_t^{close}$: Future price at position close (e.g. market close)
- $S_t^{open}$: Future price at position open (e.g. market open)
- $Z$: Contract size
- $M_t^{open}$: Initial margin requirement per contract

The futures prices are not continuous, but rather discrete. For example, the ES contract has a minimum price change of 0.25 Index point. In this paper, we will ignore the discrete nature of price change.

We assume that positions are marked at market close, so we won’t receive any margin call throughout the day. For risky markets, it is plausible that positions are marked-to-market several times per day, and margin calls can be issued throughout the day.

### Sample Data

We used ES front contract prices between May 1st, 2008 and May 29th, 2009. Furthermore, we chose the day-trading session for liquidity purposes. Furthermore, we assume zero transaction cost (i.e. no broker fees, zero slippage and no price impact).

As shown, the strategy’s daily returns fluctuate widely, and, on several days, we lose more than our initial margin.

Next, we perform a series of statistical test for the distribution:

The returns are serially correlated and exhibit ARCH effect.

### Strategy P&L

The P&L’s we are concerned with are the closing daily P&L and the maximum drawdown P&L. The closing P&L is the realized return should we hold the position to closing.

- Assuming we have trading capital of \$24,752 and the initial margin of one S&P500 E-mini contract is \$6,188.
- The strategy’s P&L average is not statistically different from zero, but it is symmetrically distributed and possesses fat-tails. The standard deviation is relatively high given the size of the trading capital.

The maximum unrealized loss experienced in one holding period is a widely used measure of the riskiness of a strategy. For our purposes, we compute it as follow:

- Long position
- $$P\&L_t=(S_t^{LO}-S_t^{open})\times Z \times K$$
- $$\%DD=\frac{(S_t^{LO}-S_t^{open})\times Z}{M}$$

- Short position
- $$P\&L_t=(S_t^{open}-S_t^{HI})\times Z \times K$$
- $$\%DD=\frac{(S_t^{open}-S_t^{HI})\times Z} {M}$$

Where:

- $\%DD$: Maximum percentage loss
- $K$: Number of contracts
- $Z$: Contract Size
- $M$: Initial Margin requirement

The P&L drawdown has an average of loss o $3,086, but, on few days, it can wipe-out 77% of the account value. If this happens to a proprietary trader, a trading risk manager may force him/her out of the position.

### Data Analysis

The descriptive statistics above suggest a mixture of ARMA model (serial correlation) for the mean and with GARCH/EGARCH (ARCH effect) process for volatility.

For the sake of our discussion here, we will use a plain EGARCH model, but with GED leptokurtic innovations. The model does not capture the serial correlation, but does represent the volatility to some extent[FN].

### Other Considerations

In our earlier discussion, we ignored the transaction costs and we assumed we invested all our capital in the strategy, which implies we can buy/sell fractional shares.

**1. Transaction cost ** The transaction cost refers to brokerage fees to work the buy/sell order on our behalf. The fee structure varies significantly across brokers, traded asset, order type (market versus limit order), order size, and any special agreement your firm has with their primary broker. For future contract, brokerage fees are higher than simple equity shares. Check your broker fees schedule.

**2. Non-fractional shares ** Aside from mutual fund accounts, a trader can only buy whole number of contracts. Why do we care? The initial margin requirement for a future contract can be large. You will need to round down the number of contracts when we compute the P&L and computed weighted returns.

### Conclusion

Prior to any analysis, we should understand the structure of the underlying contract. Define the strategy, compute returns and P&L. In our discussion, we assumed a daily holding period, ES front contract as trading instrument, and specified when we long/short the underlying security. Finally, we compute the P&L at close, minimum and maximum value within holding period and compare it with our risk limits.

Our proposed strategy loses 2.1% of our capital daily and has a volatility of 19.8%. The large returns and volatility are due primarily to the leveraged-nature of futures contracts.

This is not a great strategy, so we’ll try to improve it in following white papers.