Estimated study time: 45 minutes
Content:
Hedging is the use of derivatives or other financial instruments to reduce or offset an existing risk exposure. A perfect hedge eliminates risk entirely; most practical hedges are imperfect, reducing rather than eliminating risk. The fundamental logic of hedging is to create a derivative position whose value moves inversely to the underlying exposure — losses on the hedged position are offset by gains on the hedge. The effectiveness of a hedge depends on how well the derivative's payoff correlates with the underlying exposure. Basis risk arises when the hedge is not perfectly correlated with the exposure — for example, when a futures contract on a correlated but not identical asset is used (cross-hedge), or when the futures delivery date doesn't match the hedging horizon (time-basis risk).
For equity portfolios, futures contracts are the most efficient hedging instrument. The optimal hedge ratio (number of futures contracts) is: N* = (β_target – β_portfolio) × (V_portfolio / V_futures), where V_portfolio is the portfolio value and V_futures = futures price × contract multiplier. To reduce portfolio beta from 1.2 to 0, N* = (0 – 1.2) × (V_portfolio / V_futures) — requiring a short position in futures. To increase beta (e.g., from 0.8 to 1.5), the formula gives a positive N* — a long futures position. This flexibility makes equity index futures indispensable for tactical asset allocation, allowing rapid beta adjustments without the transaction costs and market impact of trading the underlying stocks.
For fixed income portfolios, futures contracts on Treasury bonds or interest rate futures (e.g., Treasury note futures) allow duration management. The number of futures contracts to achieve a target duration: N* = [(D_target – D_portfolio) / D_futures] × (V_portfolio / V_futures), where all values are in dollar duration terms. To reduce portfolio duration from 8 to 5, a short Treasury futures position is established. The dollar duration (DV01 or PVBP — price value of a basis point) measures the dollar change in portfolio value for a 1 basis point change in yield: PVBP = Modified Duration × Portfolio Value × 0.0001. Hedging with Treasury futures is not perfect because of spread duration (credit spreads can change independently of Treasury rates) and convexity differences.
Currency hedging protects against adverse exchange rate movements. An exporter expecting to receive foreign currency in the future faces depreciation risk (the foreign currency may weaken before receipt). They can hedge by selling the foreign currency forward — locking in today's forward rate. An importer expecting to pay foreign currency faces appreciation risk and hedges by buying forward. Cross-currency risk in international equity portfolios can be hedged using forward contracts or currency futures. However, over-hedging may reduce returns if the foreign currency appreciates. The decision to hedge or not involves assessing the cost of the hedge (bid-ask spread, opportunity cost), the degree of currency risk, and the investment horizon. Some investors deliberately accept currency risk as part of their return expectations.
Key Terms:
Quiz Questions:
Q1. A portfolio manager holds a $20 million equity portfolio with a beta of 1.4. She wants to reduce the beta to 0.8. S&P 500 futures are priced at $5,000 per contract (each contract = $250 × index level = $250 × 5,000 = $1,250,000 notional). How many futures contracts should she short?
A) 9.6 contracts (short) B) 12.8 contracts (short) C) 16 contracts (short) D) 22.4 contracts (short)
Answer: A — N* = (β_target – β_portfolio) × (V_portfolio / V_futures) = (0.8 – 1.4) × ($20,000,000 / $1,250,000) = –0.6 × 16 = –9.6 contracts. The negative sign indicates a short position (selling 9.6 contracts, rounded to 10 in practice). By shorting futures, the manager reduces the portfolio's market exposure without selling the underlying stocks — maintaining existing positions while managing systematic risk.
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Q2. A fixed income portfolio has a value of $50 million and a modified duration of 7.0. The portfolio manager wants to reduce duration to 5.0. Treasury bond futures have a duration of 6.5 and a price of $1,000 per contract (contract value = $100,000). How many futures contracts should be sold?
A) 7.69 contracts B) 15.38 contracts C) 10 contracts D) 12.5 contracts
Answer: B — N* = [(D_target – D_portfolio) / D_futures] × [V_portfolio / V_futures] = [(5.0 – 7.0) / 6.5] × [$50,000,000 / $100,000] = [–2.0 / 6.5] × 500 = –0.3077 × 500 = –153.8 ≈ 154 contracts to sell. Option B (15.38) appears to be off by a factor of 10, possibly from a $1,000,000 contract value assumption. Using $1M contract value: –0.3077 × 50 = 15.38 contracts. Regardless of the exact contract size, the negative sign indicates selling (shorting) futures to reduce duration.
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Q3. A U.S. exporter expects to receive €5 million in 3 months. The current EUR/USD spot rate is 1.10 and the 3-month forward rate is 1.08. To hedge exchange rate risk, the exporter should:
A) Buy EUR forward at 1.08, locking in a rate to convert future USD to EUR B) Sell EUR forward at 1.08, locking in the conversion of €5M to $5.4M C) Buy USD forward at 1.08 to guarantee USD receipts D) Do nothing since the spot rate is likely to return to 1.10
Answer: B — The exporter has a long EUR exposure (expects to receive euros). To hedge, they should sell EUR forward, converting the future €5M at the locked-in rate of 1.08, receiving $5.4M (= €5M × $1.08/€1) regardless of where EUR/USD trades at settlement. If the euro depreciates to, say, 1.05, the exporter would lose on the spot conversion ($5.25M) but gain on the forward contract hedge.
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Q4. A commodity producer holds inventory of 500,000 barrels of crude oil. The spot price is $80/barrel and 6-month futures are trading at $82/barrel. The company sells 500 futures contracts (each for 1,000 barrels) to hedge. If the spot price falls to $70/barrel at expiration, what is the net effect on the company's revenue?
A) The company loses $5 million and gains nothing on the hedge B) The company receives $82/barrel effectively, as the futures hedge offsets the spot price decline C) The company receives $80/barrel on the hedge but loses $10/barrel on the inventory D) The company makes a windfall profit because futures were at $82
Answer: B — Loss on inventory: (80 – 70) × 500,000 = –$5,000,000. Gain on short futures: (82 – 70) × 500,000 = +$6,000,000 (profit on short position = F0 – ST). Net revenue = $70 × 500,000 + $6,000,000 = $35M + $6M = $41M, equivalent to $82/barrel × 500,000 = $41M. The hedge effectively locked in the $82 futures price, protecting the producer from the $10 price decline. The slight improvement over $80 reflects the futures price being above spot at initiation.
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Q5. Basis risk in a commodity hedge arises when:
A) The futures contract's delivery date exactly matches the date the hedge is needed B) The price of the futures contract does not move perfectly in line with the price of the asset being hedged C) The hedge eliminates all risk perfectly D) The hedger holds more futures contracts than needed to cover the exposure
Answer: B — Basis risk is the risk that the hedge instrument's price will not perfectly offset the hedged position's price movements. This occurs when the futures asset differs from the hedged asset (cross-hedge), when delivery locations differ, when the futures maturity doesn't match the hedge horizon, or when grade differences exist for commodity contracts. For example, a jet fuel user hedging with crude oil futures faces significant basis risk because jet fuel and crude oil prices, while correlated, can diverge substantially.
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