Investment Analysis 20, 22, 23, 26

Forward Contract

A deferred-delivery sale of an asset at a stipulated price agreed upon today, with no money changing hands until the delivery date. It protects both buyer and seller from future price fluctuations.

Futures Contract

A standardized, exchange-traded contract calling for delivery of a commodity on a specified maturity date at an agreed-upon futures price (paid at maturity). Differs from a forward by being standardized, exchange-traded, and marked to market daily.

Futures vs. Forward — Key Differences

Futures are standardized (size, grade, delivery date) and trade on a centralized exchange with daily marking to market and a clearinghouse guarantee. Forwards are private, customizable agreements with no daily settlement and no exchange guarantee.

Standardization (in futures contracts)

The exchange specifies contract size, acceptable grade, delivery dates, and place/means of delivery. This sacrifices flexibility but creates liquidity by concentrating trading on a small set of contracts.

Margin (in futures)

A good-faith deposit posted by each trader (long and short) to guarantee contract performance. Typically 5%-15% of contract value; higher for more volatile assets. May be satisfied with interest-earning securities (e.g., T-bills).

Long Position

The trader who commits to PURCHASING the commodity on the delivery date. Said to 'buy' a contract. Profits when the spot price rises above the original futures price.

Short Position

The trader who commits to DELIVERING the commodity on the delivery date. Said to 'sell' a contract. Profits when the spot price falls below the original futures price.

Profit at Maturity (Long)

Profit per unit = Spot price at maturity (P_T) − Original futures price (F_0). Multiplied by contract size for total dollar profit. Can be negative (unlike a call option).

Profit at Maturity (Short)

Profit per unit = Original futures price (F_0) − Spot price at maturity (P_T). The mirror image of the long position's profit.

Settlement Price

A representative trading price during the last few minutes of trading on a given day, used to mark contracts to market.

Open Interest

The number of contracts outstanding (counting either long OR short positions, not both). Begins at zero when contracts are first listed and grows as more contracts are entered.

Spot Price

The actual current market price of the commodity for immediate delivery.

Futures vs. Call Option (key distinction)

A long futures position has unlimited downside (mirrors spot price one-for-one). A call option's loss is limited to the premium paid. Also, a futures contract is not 'purchased' — it is entered into with no initial cash flow because the futures price adjusts to make PV = 0.

Four Broad Categories of Futures

(1) Agricultural commodities, (2) Metals and minerals (including energy), (3) Foreign currencies, and (4) Financial futures (fixed-income securities and stock market indexes).

Prediction Markets

Specialized 'futures' markets where payoffs depend on event outcomes (e.g., elections, box-office receipts, Olympic hosts). Contract prices can be interpreted as risk-neutral probabilities of the event.

Forward Market in Foreign Exchange

A massive over-the-counter network of banks and brokers that negotiate currency forwards with customizable size and date — distinct from formal futures exchanges. London alone trades >$3 trillion of currency per day.

Section 22.2 — Trading Mechanics

Term / Question

Definition / Answer

Clearinghouse

An intermediary that becomes the seller to every long and the buyer to every short, netting its position to zero. It guarantees performance, eliminating the need for credit checks between traders.

Reversing Trade

Closing an existing futures position by entering an offsetting trade (e.g., a long enters the short side of an identical contract). The clearinghouse nets positions to zero, eliminating the obligation at maturity.

Frequency of Actual Delivery

Less than 1% to 3% of futures contracts result in actual delivery; the rest are reversed before maturity. When deliveries occur, they typically use warehouse receipts.

Marking to Market

The daily settling-up process by which gains and losses on futures positions are credited or debited to traders' margin accounts each day, rather than waiting until maturity.

Initial Margin

The deposit (typically 5%-15% of contract value) that both long and short traders post when initiating a position. Higher for more volatile underlying assets.

Maintenance Margin

A critical (lower) value below which the margin account cannot fall. If breached, the trader receives a margin call to replenish funds or reduce the position.

Margin Call

A demand that a trader add funds to a margin account that has fallen below the maintenance margin, or else reduce the position to one supported by remaining funds.

Long-trader profit (held from time 0 to time t)

F_t − F_0 — the change in the futures price over the holding period.

Short-trader profit (held from time 0 to time t)

F_0 − F_t — the negative of the long position's profit.

Convergence Property

On the contract's maturity date, the futures price must equal the spot price (F_T = P_T). Otherwise, arbitrageurs would buy from the cheaper source and sell in the more expensive market.

Total Profit Held to Maturity (long)

Sum of daily settlements = F_T − F_0 = P_T − F_0 (using convergence). Mirrors the payoff of a forward contract, which is why futures and forwards are often treated as nearly interchangeable.

Cash Settlement

Settlement by delivering a cash amount equal to the value the underlying attains at maturity (e.g., S&P 500 index futures), rather than physically delivering the asset. Used when delivery is impractical.

E-mini S&P 500 Contract

A widely traded stock index futures contract calling for delivery of $50 × the value of the S&P 500 index. Cash-settled: e.g., S&P at 4,000 → $200,000 notional.

Quality Adjustments (delivery)

When agricultural goods of different grades may be delivered, exchanges apply premiums/discounts to the agreed price to account for the quality difference.

Commodities Futures Trading Commission (CFTC)

The U.S. federal regulator of futures markets. Sets capital requirements for member firms, authorizes new contracts, and oversees daily trading-record maintenance.

Price Limits

Daily caps set by the exchange on how much a futures price may move from one day to the next. Often eliminated in the final month before maturity. Considered ineffective protection against true equilibrium-price changes.

Taxation of Futures Gains/Losses

Because of mark-to-market, gains/losses are realized gradually and taxed at year-end on cumulated profits regardless of whether the position is closed. As a general rule: 60% are treated as long-term capital gains, 40% as short-term.

Section 22.3 — Futures Markets Strategies

Term / Question

Definition / Answer

Speculator

Uses futures to PROFIT from anticipated price movements. Goes long if expecting price increases, short if expecting price decreases.

Hedger

Uses futures to PROTECT against unfavorable price movements (insulates a position from price risk), not to speculate on direction.

Why Speculators Use Futures (vs. spot)

(1) Lower transaction costs than direct trading. (2) Greater LEVERAGE — futures require margin much smaller than the underlying asset's value, magnifying percentage returns.

Futures Leverage Example

If margin = 10% of contract value, a 1.39% rise in the underlying produces a 13.9% gain on margin — a 10-to-1 ratio of percentage changes. Leverage = 1 / margin %.

Short Hedge

A SHORT futures position taken to lock in the SALES PRICE of an asset the hedger plans to sell (e.g., an oil distributor or a wheat farmer).

Long Hedge

A LONG futures position taken to lock in the PURCHASE PRICE of an asset the hedger plans to buy (e.g., a petrochemical processor or a miller).

Total Proceeds of a Hedged Position

Spot Revenue + Futures Profit = P_T + (F_0 − P_T) = F_0. The hedged result is independent of the eventual spot price — equal to the original futures price.

Cross-Hedging

Hedging a position using futures on a DIFFERENT but related asset (e.g., using S&P 500 index futures to hedge an actively managed equity portfolio). Used when no exact futures contract exists.

Sources of Risk in Cross-Hedging

The hedge is imperfect because the hedged asset and the futures' underlying asset are not identical — their returns are correlated but not perfectly matched (basis risk + tracking error).

Basis

Basis = Spot Price − Futures Price. Must equal zero on the maturity date (by convergence). Before maturity, it can be positive or negative.

Basis Risk

The risk that, BEFORE maturity, the spot price and futures price do not move in perfect lockstep — meaning a hedge held less than to maturity may not perfectly cancel gains and losses.

Speculating on the Basis

A strategy of betting on changes in the spot-futures price difference rather than on price direction. A long-spot/short-futures position profits when the basis narrows.

Calendar Spread

Long a futures contract of one maturity AND short a contract on the same commodity with a different maturity. Profits if the spread between the two futures prices moves in the expected direction.

Why Hedgers Tolerate the Hedge's 'Cost'

They give up potential upside in exchange for ELIMINATING price risk; the result is a known, locked-in price equal to the original futures price.

Section 22.4 — Futures Prices

Term / Question

Definition / Answer

Spot-Futures Parity Theorem

F_0 = S_0 (1 + r_f) − D, or equivalently F_0 = S_0 (1 + r_f − d), where d = D/S_0 is the dividend yield. Establishes the no-arbitrage relationship between today's spot and futures prices.

Cost-of-Carry Relationship

Another name for spot-futures parity. The futures price reflects the spot price plus the NET cost of carrying the asset: time value of money (r_f) MINUS yield received (d).

Net Cost of Carry

r_f − d (per period). The differential the futures price must add to (or subtract from) the spot price to compensate for deferred delivery.

Multi-Period Parity Formula

F_0 = S_0 (1 + r_f − d)^T, where T is the number of periods to contract maturity. Differential grows with maturity.

Implication when r_f < d

Futures prices are LESS than the spot price, and longer maturities have lower futures prices (consistent with 2021's S&P listing, where T-bill rate ~0.1% < dividend yield ~2%).

Implication when r_f > d

Futures prices EXCEED the spot price, and longer maturities have higher futures prices (typical for low-yield or non-yielding assets like gold).

Arbitrage Strategy if F_0 is Too HIGH

Borrow $S_0; buy the stock; short the futures contract. Cash flow at maturity = F_0 − S_0(1 + r_f) + D > 0 risk-free profit equal to the mispricing.

Arbitrage Strategy if F_0 is Too LOW

Short the stock (receive S_0 and lend it); buy/long the futures contract. Cash flow at maturity = S_0(1 + r_f) − D − F_0 > 0, again locking in the mispricing risk-free.

Parity for Other Asset Classes

Same logic applies: gold futures use d = 0 (no yield); bond futures use coupon income in place of dividends.

Spread Parity (between two maturities)

F(T_2) = F(T_1) (1 + r_f − d)^(T_2 − T_1). The relative pricing of futures with different maturities depends on the net cost of carry between them.

Index Arbitrage

A trading strategy that exploits violations of spot-futures parity in stock index futures. (Discussed in Chapter 23.)

Why S&P Dividend Yield Is Seasonal

Annualized dividend yield on broad indexes is fairly stable but has regular peaks and troughs by month. January and April tend to be low; May tends to be high. The relevant month's yield should be used in parity calculations.

Forward vs. Futures Price (Theory)

If marking to market favors the long (positive correlation between futures prices and interest rates), the futures price > forward price. Negative correlation → futures < forward. Differences are usually negligible EXCEPT for long-term fixed-income contracts.

Why Forward Pricing Formulas Apply to Futures

Parity theorems strictly apply to forwards (proceeds realized at maturity). Because the covariance between most futures prices and interest rates is small, the formulas are accurate approximations for futures too.

Section 22.5 — Futures Prices vs. Expected Spot Prices

Term / Question

Definition / Answer

Expectations Hypothesis

F_0 = E(P_T). The futures price equals the expected spot price at maturity. Implies zero expected profit to either side. Ignores risk premiums; only valid in a world of certainty.

Normal Backwardation (Keynes & Hicks)

Producers (natural short hedgers) bid the futures price BELOW the expected spot price to entice speculators to take long positions. F_0 < E(P_T), and the futures price rises over the contract's life.

Contango

Theory that natural hedgers are PURCHASERS (long hedgers), willing to pay a premium to lock in their cost. They bid the futures price ABOVE the expected spot: F_0 > E(P_T).

Net Hedging Hypothesis

A compromise: F_0 < E(P_T) when short hedgers outnumber long hedgers (and vice versa). The 'strong side' must pay a premium to entice speculators to take the opposite side.

Modern Portfolio Theory View

Refines risk-premium analysis by focusing on SYSTEMATIC (not total) risk. F_0 = E(P_T) × (1 + r_f) / (1 + k), where k is the required return on the asset.

F_0 vs. E(P_T) — Modern View

If k > r_f (positive beta asset), F_0 < E(P_T) — long position earns expected profit for bearing systematic risk. If k < r_f (negative beta), F_0 > E(P_T).

When is F_0 an Unbiased Estimate of E(P_T)?

When the asset has ZERO systematic risk (k = r_f, i.e., zero beta). Then no risk premium is required and the futures price equals the expected spot price.

Why Long Holders Earn Expected Profit on Positive-Beta Assets

The long futures profit = P_T − F_0 inherits positive systematic risk from P_T. Diversified speculators only take this risk if compensated, so F_0 is bid below E(P_T).

Why Short Holders Accept Expected Loss on Positive-Beta Assets

The short profit = F_0 − P_T has NEGATIVE systematic risk and acts as a hedge in a diversified portfolio. Investors will accept lower expected return for the diversification benefit.

Key Formulas — Quick Reference

Term / Question

Definition / Answer

Long Profit at Maturity

Profit = P_T − F_0 (per unit) × contract size

Short Profit at Maturity

Profit = F_0 − P_T (per unit) × contract size

Margin Call Trigger (corn example)

Loss that drives the margin account from initial margin (10%) to maintenance margin (5%) — i.e., a price drop of about 5% of contract value.

Spot-Futures Parity (single period)

F_0 = S_0 (1 + r_f) − D = S_0 (1 + r_f − d)

Spot-Futures Parity (T periods)

F_0 = S_0 (1 + r_f − d)^T

Spread Parity

F(T_2) = F(T_1) (1 + r_f − d)^(T_2 − T_1)

Modern Portfolio Theory Futures Price

F_0 = E(P_T) × (1 + r_f) / (1 + k)

Basis

Basis = Spot Price − Futures Price (must = 0 at maturity)

Hedged Total Proceeds (short hedge)

P_T + (F_0 − P_T) = F_0 (independent of spot)

Cash Settlement Profit (index)

S_T − F_0, multiplied by index multiplier (e.g., $50 for E-mini S&P 500)

Currency forward market

Informal network of banks and brokers for custom-size, custom-date FX contracts. No daily marking-to-market. Counterparty risk matters — traders must be creditworthy.

Currency futures market

Formal exchanges (CME International Monetary Market, LIFFE). Standardized contract size and maturities (four per year); daily marking-to-market; clearinghouse guarantees performance so counterparty identity is irrelevant.

Counterparty risk

Possibility that a trading partner may not be able to make good on its obligations under the contract if prices move against it. Present in forward markets; eliminated in futures markets by the clearinghouse.

Direct exchange rate quote

Number of dollars required to purchase one unit of foreign currency (e.g., $1.3546/£). Used by convention for the British pound and the euro, and by all currency futures listings.

Indirect exchange rate quote

Units of foreign currency required to purchase $1 (e.g., ¥111.07/$). Used by convention for most currencies except the pound and euro.

Spot exchange rate

Exchange rate for immediate delivery of currency.

Interest rate parity theorem (covered interest arbitrage)

F0 = E0 × [(1 + rUS)/(1 + rUK)]^T, where E0 is the current direct spot rate ($/£) and F0 is the forward price. Violations create riskless arbitrage, so enforcement is tight in well-functioning markets.

IRP numerical example

If rUS = 4%, rUK = 1%, and E0 = $1.30/£, the 1-year forward price is F0 = 1.30 × 1.04/1.01 = $1.3386/£.

Intuition behind forward discount

If rUS > rUK, money grows faster in U.S. dollars — but the dollar must be depreciating (F0 > E0) by just enough in the forward market to offset the interest rate advantage. Otherwise, arbitrage.

Covered interest arbitrage strategy

If IRP is violated: borrow in the lower-yield currency, lend in the higher-yield currency, and take an offsetting futures position. Net proceeds = E0(1 + rUS) − F0(1 + rUK), risk-free and zero net investment.