Estimated study time: 60 minutes
Content:
Forwards and futures are linear derivatives where the payoff is proportional to the difference between the underlying asset price at expiration and the agreed forward price. At CFA Level 2, the analytical focus is on no-arbitrage pricing, basis risk, and the use of futures for hedging and portfolio management. A forward contract obligates the buyer to purchase and the seller to deliver a specified asset at a specified price (the forward price) on a future date. Futures contracts are standardized exchange-traded equivalents with daily mark-to-market (variation margin) and clearing house interposition. The futures price is not the same as the expected future spot price — it is determined by no-arbitrage conditions from the current spot price.
The cost of carry model determines the no-arbitrage forward price. For an asset with no income and no carrying costs: F_0 = S_0 * (1 + r)^T (discrete) or F_0 = S_0 * e^(rT) (continuous). For assets with cash income (dividend yield q for equities or coupon yield for bonds): F_0 = (S_0 - PV of cash income) * (1 + r)^T, or using yield: F_0 = S_0 * e^((r-q)T). For assets with storage costs (commodities): F_0 = (S_0 + PV of storage costs) * (1 + r)^T. For currency forwards (covered interest rate parity): F_0 = S_0 * ((1 + r_d) / (1 + r_f))^T. If the actual forward price differs from the theoretical no-arbitrage price, cash-and-carry arbitrage (buy spot, sell forward) or reverse cash-and-carry (sell spot, buy forward) can be executed until prices converge.
Futures pricing includes an additional adjustment for the daily mark-to-market feature. The futures price equals the forward price when interest rates are deterministic (non-random). When interest rates are stochastic and correlated with the underlying asset, futures prices may systematically differ from forward prices. For positively correlated assets (e.g., equity futures when rates and equities both rise together), futures prices are slightly higher than forward prices because gains on the futures position can be invested at higher rates. This futures-forward price difference is typically small for short-dated contracts and is often ignored in practice.
Basis risk arises when the underlying asset of the futures contract differs from the actual position being hedged. Basis = Spot Price - Futures Price. At expiration, basis converges to zero (or to the delivery cost if physical delivery). Prior to expiration, basis can fluctuate, creating uncertainty in hedged positions. In a cross-hedge (hedging one asset with futures on a different but correlated asset — e.g., hedging jet fuel with crude oil futures), basis risk is the primary residual risk. The optimal hedge ratio for a cross-hedge using futures is: N* = -beta * (Portfolio Value / Futures Contract Value) = -(rho * sigma_S / sigma_F) * (S / F) * (Portfolio Size / Contract Size), where rho is the correlation between the spot and futures price changes, sigma_S and sigma_F are their respective standard deviations.
Futures are used for portfolio management in several key ways tested at Level 2. Equity futures allow rapid adjustment of portfolio beta without trading the underlying stocks: N = (Target Beta - Current Beta) / Beta_futures * (Portfolio Value / Futures Contract Value). Bond futures allow adjustment of portfolio duration: N = ((Target Duration - Current Duration) / Futures Duration) * (Portfolio Value / Futures Price). These adjustments are often used by portfolio managers who want tactical exposure changes without incurring the transaction costs and market impact of trading the underlying securities. Futures can also be used to create synthetic cash (selling equity futures to convert an equity portfolio into a near-cash position) or to equitize cash (buying equity futures to gain equity market exposure on a cash position while investing the cash in T-bills).
Key Terms:
Quiz Questions:
Q1. A stock index currently trades at 4,000. The risk-free rate is 5% per year and the dividend yield on the index is 2% per year. What is the no-arbitrage 6-month futures price?
A) F = 4,000 * (1.05)^0.5 = 4,000 * 1.02470 = 4,098.80. B) F = 4,000 * e^((0.05-0.02)*0.5) = 4,000 * e^0.015 = 4,000 * 1.01511 = 4,060.46. C) F = 4,000 * (1.05 - 0.02)^0.5 = 4,000 * 1.01489 = 4,059.57. D) F = (4,000 - dividend income) * (1.05)^0.5; need the dividend amount to calculate.
Answer: B — Using the continuous compounding cost of carry model for an index with continuous dividend yield q: F = S_0 * e^((r-q)*T) = 4,000 * e^((0.05-0.02)*0.5) = 4,000 * e^0.015 = 4,000 * 1.01511 = 4,060.46. The dividend yield reduces the cost of carry because the index holder receives dividends that offset the financing cost. In discrete form: F = 4,000 * (1.05/1.02)^0.5 ≈ 4,059, consistent with Option B.
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Q2. A US firm will receive GBP 5,000,000 in 3 months from a UK customer. The current USD/GBP spot rate is 1.25. The 3-month USD interest rate is 4% per year and the 3-month GBP interest rate is 2% per year. To hedge the currency risk, the firm should:
A) Buy GBP forward to lock in the receipt. B) Sell GBP forward at the no-arbitrage forward rate: F = 1.25 * (1.04/1.02)^(0.25) = 1.25 * 1.00490 = 1.2561 USD/GBP. C) Buy USD forward to lock in the future USD receipts. D) Enter a GBP interest rate swap.
Answer: B — The US firm will receive GBP in 3 months and wants to lock in the USD equivalent — it needs to sell GBP forward (agree to exchange GBP for USD at a fixed rate in 3 months). The no-arbitrage forward rate using CIP: F = S_0 * (1 + r_USD)^T / (1 + r_GBP)^T = 1.25 * (1.01)/(1.005) = 1.25 * 1.00498 = 1.2562 USD/GBP. (Using quarterly rates: r_USD quarterly = 1%, r_GBP quarterly = 0.5%.) The firm sells GBP 5M forward at 1.2562, locking in $6,281,000 regardless of where the spot rate ends up in 3 months.
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Q3. A portfolio manager has a $100M equity portfolio with a beta of 1.20. She wants to reduce the portfolio's beta to 0.80 for the next month due to market concerns. The S&P 500 futures contract has a price of 4,000 and a contract multiplier of $250 (each point = $250 in value, so one contract = $1,000,000). How many futures contracts should she sell?
A) N = (1.20 - 0.80) / 1.0 * ($100M / $1M) = 0.40 * 100 = 40 contracts to sell. B) N = (0.80 - 1.20) / 1.0 * ($100M / $1M) = -40 contracts (sell 40 contracts). C) N = (Target Beta - Portfolio Beta) / Futures Beta * (Portfolio Value / Contract Value) = (0.80 - 1.20)/1.0 * $100M/$1M = -40; sell 40 contracts. D) Both B and C are correct; sell 40 S&P 500 futures contracts.
Answer: D — Both B and C express the same correct answer. N = ((Target Beta - Current Beta) / Futures Beta) * (Portfolio Value / Contract Value) = ((0.80 - 1.20) / 1.0) * ($100,000,000 / $1,000,000) = (-0.40) * 100 = -40. The negative sign indicates selling. Selling 40 S&P 500 futures reduces the portfolio's systematic market exposure (beta) from 1.20 to approximately 0.80, achieving the desired risk reduction without liquidating portfolio positions. Contract value = 4,000 * $250 = $1,000,000.
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Q4. A commodity trader observes that the current spot price of gold is $1,900/oz and the 6-month futures price is $1,950/oz. The risk-free rate is 5% per year. Storage costs for gold are 0.5% per year. Is there an arbitrage opportunity?
A) No arbitrage; $1,950 is the correct futures price. B) Yes, the futures are underpriced: theoretical F = $1,900 * (1.05 + 0.005)^0.5 = $1,900 * (1.055)^0.5 = $1,900 * 1.02711 = $1,951.51; actual $1,950 < theoretical, so buy futures, sell gold short. C) Yes, the futures are overpriced: theoretical F = $1,900 * (1.05 * 0.5 + 0.005 * 0.5) = $1,900 * 0.5275 = $1,002.25 (wrong calculation). D) No arbitrage because gold convenience yield offsets the overpricing.
Answer: B — Theoretical F = S_0 * (1 + r + storage)^T = $1,900 * (1 + 0.05 + 0.005)^0.5 = $1,900 * (1.055)^0.5 ≈ $1,900 * 1.02711 = $1,951.51. The actual futures price ($1,950) is slightly below the theoretical price ($1,951.51). The futures appear very slightly underpriced — in practice, this difference ($1.51) would be consumed by transaction costs and might not be a practical arbitrage. However, theoretically, a reverse cash-and-carry (sell spot gold, invest proceeds, buy futures) would lock in a small profit. In practice, gold has minimal convenience yield so the cost of carry model applies closely.
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Q5. A fixed income manager has a bond portfolio valued at $50M with a modified duration of 6.5 years. She wants to increase duration to 8.0 years using Treasury bond futures. The cheapest-to-deliver bond has a duration of 10 years and the futures price is $98,000 per contract (standard $100,000 face value). How many futures contracts should she buy?
A) N = (8.0 - 6.5) / 10 * ($50M / $98,000) = 1.5/10 * 510.2 = 0.15 * 510.2 = 76.5 ≈ 77 contracts. B) N = (8.0 - 6.5) * $50M / ($98,000 * 10) = $75M / $980,000 = 76.5 ≈ 77 contracts. C) N = (Target Duration - Current Duration) * (Portfolio Value) / (Futures Duration * Futures Price) = (8.0-6.5) * $50M / (10 * $98,000) = $75M/$980,000 = 76.5 ≈ 77 contracts. Buy 77 contracts. D) Both A, B, and C are equivalent formulations yielding 77 contracts.
Answer: D — All three formulations in A, B, and C are mathematically equivalent expressions of the duration adjustment formula. The standard formula is: N = [(MD_target - MD_portfolio) / MD_futures] * [Portfolio Value / Futures Price] = [(8.0 - 6.5) / 10.0] * [$50,000,000 / $98,000] = 0.15 * 510.2 = 76.5, rounded to 77 contracts purchased. Buying 77 long Treasury bond futures increases the portfolio's effective duration from 6.5 to approximately 8.0 years without trading the underlying bonds.
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