# Forwards, Futures and Swaps Concept Card: [Course, exposure or hedging programme] > A concept card for the linear derivatives taught in an MBA derivatives and risk management elective: forward > pricing by cost of carry, futures and daily settlement, hedging with the minimum-variance hedge ratio (Johnson 1960; > Ederington 1979), stock index futures and beta, and interest rate and currency swaps valued from a discount curve. > It builds on two core cards and does not re-teach them: mba/mba-options-real-options-card (payoffs, no-arbitrage and > replication) and mba/mba-fx-interest-parity-card (covered interest parity, the currency forward). It is for > second-year MBA students, candidates preparing for treasury, sales and trading or corporate banking interviews, and > managers who sign off a hedging programme. Worked example A is fully solved with Excel formulas, B is half solved > and Exercise C is left to you; self-test answers sit in a separate key. Source basis: the original works cited in > each section, the Basel Committee and central bank pages on benchmark rates (checked October 2026) and standard > textbooks named as further reading; all wording, notation and examples are original. Northfield Outdoor Co. and > Corvane Robotics are fictional, and every price, rate and contract size is an *Example*, not a market quote. > Replace every [bracketed] value and every row marked *Example*. This is a learning aid, not investment, accounting > or legal advice. ## Document control | Field | Value | |---|---| | Document ID | [MBA-DRV-FWD-001] | | Course or project | [Derivatives elective / treasury project / case] | | Learner | [Your name] | | Exposure or underlying | [Northfield aluminium purchases (fictional) / Corvane term loan (fictional)] | | Rate conventions | [Continuous for carry examples; annual compounding for swap zero rates; ACT/360 for money market] | | Data sources and timestamps | [Exchange settlement prices, swap curve source, time of capture] | | Excel version used for checks | [Microsoft 365 / Excel 2021] | | Reviewed by | [Study group member, tutor or mentor] | | Last updated | YYYY-MM-DD | | Version | [0.1] | ## Contents 1. When to use it 2. The idea in plain language 3. Formulas and notation 4. Assumptions and limits 5. Benchmark rates after LIBOR (dated facts) 6. How managers use it in practice 7. Worked example A (fully solved) 8. Worked example B (half solved) 9. Exercise C (unsolved) 10. Connections to other cards 11. Common mistakes and misconceptions 12. Self-test questions 13. My notes and next steps ## When to use it **One-sentence rule:** use a forward, future or swap when you want to **lock in** a price or rate you will pay or receive later, giving up the upside to remove the downside; use an option (mba/mba-options-real-options-card) when you want to keep the upside and are willing to pay a premium for it. | Situation | Instrument | Typical tool on this card | |---|---|---| | A known foreign currency payment or receipt on a known date | Currency forward | Covered interest parity (mba/mba-fx-interest-parity-card) | | Commodity purchases (metal, fuel, cotton) over the next months | Commodity futures or forwards | Cost of carry, minimum-variance hedge ratio | | An equity portfolio whose market exposure must be reduced for a period | Stock index futures | Beta-adjusted number of contracts | | A floating-rate loan whose interest cost must be fixed | Interest rate swap (pay fixed) | Discount factors, par swap rate | | Fixed-rate debt that the firm wants to turn floating | Interest rate swap (receive fixed) | Same, with the sign reversed | | Debt raised in one currency to fund assets in another | Cross-currency swap | Two discount curves and the spot rate | | Pricing a quote from a bank or checking a dealer's offer | Any of the above | No-arbitrage fair value, then compare | ## The idea in plain language A **forward** is an agreement today to buy or sell an asset on a later date at a price fixed today. Nothing is paid at the start. Because the buyer could instead borrow, buy the asset now and hold it until the date, the forward price must equal the spot price plus the **cost of carry**: interest on the money tied up, plus storage, minus any income the asset pays while it is held (dividends, coupons, or the "convenience" of having a physical commodity on hand). If the forward price were higher, a trader would buy spot, sell forward and pocket the difference without risk (cash and carry); if lower, the reverse trade (reverse cash and carry) would pay. This argument goes back to Keynes (1923) for currencies and is the same no-arbitrage logic as put-call parity. A **future** is a standardised forward traded on an exchange. Two features change how it behaves. First, it is **marked to market daily**: each day gains and losses are paid in cash through a clearing house, so the contract is reset to zero value every evening and counterparty risk is replaced by margin. Second, it is standardised (size, quality, delivery months), so a hedger rarely finds a contract that matches the exposure exactly. The gap between the spot price of what you actually buy and the futures price you hedge with is called the **basis**, and the risk that it moves is **basis risk**. When interest rates are known in advance, forward and futures prices for the same date are equal (Cox, Ingersoll and Ross 1981); when rates move with the asset price, daily settlement makes them differ slightly. A **swap** is a series of forwards packaged together. In a plain interest rate swap one party pays a fixed rate and the other pays a floating rate on the same notional amount, which itself is never exchanged. A company with a floating-rate loan that pays fixed in a swap ends up paying a fixed rate overall. At inception the fixed rate is set so the swap is worth zero: that is the **par swap rate**. Later, when rates move, the swap gains or loses value like a bond: the party paying fixed gains when rates rise. A **cross-currency swap** exchanges interest and principal in two currencies and is priced from two discount curves and the spot exchange rate. Linear instruments are cheap to enter (no premium) but not free: they carry a funding and margin burden, a counterparty or clearing exposure, and an **opportunity loss** when the market moves your way. The hedger's question is never "will this hedge make money?" but "does this hedge reduce the variance of the cash flows I care about, at an acceptable cost?" ## Formulas and notation **Forward price by cost of carry** (continuous compounding): $$ F_0 = S_0\, e^{(r + u - q - y)T} $$ Special cases: a non-income asset $F_0 = S_0 e^{rT}$; an index with dividend yield $q$: $F_0 = S_0 e^{(r-q)T}$; a commodity with storage cost $u$ and convenience yield $y$: $F_0 = S_0 e^{(r+u-y)T}$; a currency (foreign rate $r_f$ plays the role of $q$): $F_0 = S_0 e^{(r - r_f)T}$, which is covered interest parity in continuous form. With a known cash income $I$ (present value) instead of a yield: $F_0 = (S_0 - I)e^{rT}$. **Value of an existing long forward** with delivery price $K$ and current forward price $F_t$: $$ f_t = (F_t - K)\, e^{-r(T-t)} $$ **Futures daily settlement** for a long position of $m$ units: gain on day $t$ $= m\,(F_t - F_{t-1})$. If the margin balance falls below the maintenance margin, the holder must top it up to the **initial** margin. **Minimum-variance hedge ratio and number of contracts** (Johnson 1960; Ederington 1979): $$ h^* = \rho\, \frac{\sigma_S}{\sigma_F}, \qquad N^* = \frac{h^* Q_A}{Q_F}, \qquad \text{effectiveness} = \rho^2 $$ **Stock index futures**, to move a portfolio's beta from $\beta$ to $\beta^*$: $$ N = (\beta - \beta^*)\, \frac{V_A}{F_0 \times \text{multiplier}} $$ (a positive $N$ means sell futures). **Interest rate swap** with annual payments, discount factors $D_i = 1/(1+z_i)^{i}$ from zero rates $z_i$: $$ c_{par} = \frac{1 - D_n}{\sum_{i=1}^{n} D_i}, \qquad V_{\text{pay fixed}} = N \left[ 1 - \left( c \sum_{i=1}^{n} D_i + D_n \right) \right] \text{ on a reset date} $$ The bracket is the value of a fixed-rate bond with coupon $c$; on a reset date the floating leg is worth par, so the pay-fixed party holds "long a floating-rate note, short a fixed-rate bond". Equivalently, $V_{\text{pay fixed}} = N\,(c_{par,new} - c)\sum D_i$. | Symbol | Meaning | |---|---| | $S_0$, $F_0$ | Spot price today; forward or futures price today for delivery at $T$ | | $K$ | Delivery price agreed in an existing forward | | $r$ | Risk-free rate, continuously compounded, for maturity $T$ | | $q$, $r_f$ | Dividend yield of an index; foreign interest rate for a currency | | $u$, $y$ | Storage cost and convenience yield as continuous rates | | $I$ | Present value of known cash income during the life of the forward | | $T$, $t$ | Delivery date and today, in years | | $\sigma_S$, $\sigma_F$, $\rho$ | Standard deviation of spot and futures price changes over the hedge horizon, and their correlation | | $h^*$ | Minimum-variance hedge ratio (futures units per unit of exposure) | | $Q_A$, $Q_F$ | Size of the exposure and size of one futures contract, in the same units | | $\beta$, $\beta^*$ | Current and target portfolio beta (mba/mba-capm-wacc-card) | | $V_A$ | Market value of the equity portfolio | | $z_i$, $D_i$ | Zero rate and discount factor for year $i$ | | $c$, $c_{par}$ | Fixed rate of an existing swap; par swap rate today | | $N$ | Swap notional, or number of futures contracts (context makes it clear) | ## Assumptions and limits - **No arbitrage, with access to borrowing and storage.** Cash-and-carry pricing needs someone who can borrow near the risk-free rate, buy the asset and store it. For commodities that cannot be stored cheaply (power, some agricultural products) or are in short supply, forward prices can sit well below full carry; the gap is the convenience yield. - **Forward equals futures** only if interest rates are deterministic (Cox, Ingersoll and Ross 1981). For short-dated contracts the difference is negligible; for long-dated interest rate futures it is not, and dealers apply a convexity adjustment. - **The minimum-variance ratio is estimated from history.** $\rho$ and the volatilities change, especially in stress. Re-estimate on the hedge horizon (monthly changes for a monthly hedge), test it out of sample and round to whole contracts. - **Tailing the hedge.** Because futures settle daily, gains can be reinvested; a precise hedge scales $N^*$ by the ratio of the spot value to the futures value or by a discount factor. For short horizons the correction is small. - **Liquidity of margin.** A hedge that is right in economic terms can still create cash calls that a firm cannot meet. Metallgesellschaft's 1993 losses on rolling short-dated oil futures against long-dated delivery commitments are the standard warning: the debate between Culp and Miller (1995) and Mello and Parsons (1995) is about whether the hedge was wrong or the funding plan was. - **Swaps on a reset date, flat notional, annual payments** in the formulas above. Real swaps have different frequencies and day counts on the two legs, payment lags, compounded overnight floating rates and collateral agreements that decide which curve discounts the cash flows. The method is the same; the bookkeeping is heavier. - **Accounting is separate.** A good economic hedge can create earnings volatility unless it qualifies for hedge accounting under IFRS 9 or ASC 815, which requires documentation at inception (finance/hedge-designation-documentation). ## Benchmark rates after LIBOR (dated facts) As of October 2026, the following was confirmed on the official pages (check them before you rely on it): | Rate | Currency | What it is | Administrator and source | |---|---|---|---| | SOFR (Secured Overnight Financing Rate) | USD | A broad measure of the cost of borrowing cash overnight against US Treasury collateral (repo); published each business day at about 8:00 a.m. New York time | Federal Reserve Bank of New York, newyorkfed.org/markets/reference-rates/sofr | | SONIA (Sterling Overnight Index Average) | GBP | The risk-free overnight sterling rate that replaced sterling LIBOR | Bank of England, bankofengland.co.uk | | €STR (euro short-term rate) | EUR | Published each TARGET2 business day from transactions on the previous day; first published on 2 October 2019 | European Central Bank, ecb.europa.eu | LIBOR has ended. According to the Bank of England's transition page: 24 of the 35 LIBOR settings stopped from 1 January 2022; the remaining panel-bank US dollar settings ceased after the end of June 2023; temporary "synthetic" sterling settings ran until the end of March 2023 (one and six months) and the end of March 2024 (three months); and the last synthetic US dollar settings were published on 30 September 2024. New swaps reference overnight risk-free rates compounded over each period. Older contracts move to these rates through fallback clauses, industry protocols or legislation; which route applies is a legal question for the contract and counsel. For a learner, the practical effect is twofold. First, a floating leg now usually pays an overnight rate **compounded in arrears**, so the payment is known only at the end of the period, not at the start as with LIBOR. Second, the overnight rates are nearly free of bank credit risk, so a borrower's loan spread over SOFR is wider than its old spread over LIBOR for the same credit; compare all-in costs, not spreads. ## How managers use it in practice - **Lock in a budget rate.** Treasurers buy currency forwards for committed supplier payments so that the budget is set at the forward rate (mba/mba-fx-interest-parity-card, Worked example A there). The forward rate is not a forecast; it reflects the interest differential. - **Hedge commodity inputs with futures and accept basis risk.** A manufacturer rarely buys the exact grade and location of the exchange contract. It sizes the hedge with $h^*$ and reports effectiveness ($\rho^2$) to the board. - **Fix the interest cost of floating-rate debt.** Pay-fixed swaps are the most common corporate derivative. The decision is about the share of debt to fix and for how long, set in the treasury policy (finance/treasury-policy). - **Adjust portfolio beta quickly and cheaply.** Asset managers and pension funds sell index futures to cut equity exposure during a transition instead of selling the shares (mba/mba-portfolio-theory-card). - **Separate the funding currency from the asset currency.** Cross-currency swaps let a firm borrow where it is cheapest and convert the debt into the currency of its cash flows. - **Why hedge at all?** In perfect markets shareholders can hedge for themselves, so corporate hedging adds no value (the Modigliani and Miller logic of mba/mba-capital-structure-card). It adds value when cash-flow volatility is costly: convex taxes and expected distress costs (Smith and Stulz 1985), managers' undiversified exposure (Stulz 1984), and above all when costly external finance would force the firm to cut good investments in bad years (Froot, Scharfstein and Stein 1993). Hedge the cash flows that fund the investment plan, not every price that moves. - **Govern the programme.** Write limits, approved instruments, separation of dealing and confirmation, and a "no speculation" rule into policy (finance/market-risk-limit-framework). Most corporate derivative losses come from positions larger than the exposure, not from pricing errors. ```mermaid flowchart TD A["Identify the exposure: what moves, when, how much"] --> B{"Is the amount and date known?"} B -->|"Yes, committed"| C{"Keep the upside?"} B -->|"No, forecast only"| D["Hedge a share of the forecast, layer over time"] C -->|"No, lock it in"| E{"Standard exchange contract fits?"} C -->|"Yes, pay a premium"| F["Option or collar: see options card"] E -->|"Yes or close enough"| G["Futures: size with h-star, plan margin liquidity"] E -->|"No, custom date or amount"| H["OTC forward with a bank"] A --> I{"Interest rate on floating debt?"} I -->|"Fix it"| J["Pay-fixed swap at the par rate"] I -->|"Leave floating"| K["Document the decision and the limit"] D --> L["Check hedge accounting and policy limits"] F --> L G --> L H --> L J --> L L --> M["Report effectiveness, margin and mark-to-market monthly"] ``` ## Worked example A (fully solved) **A1. Stock index forward and cash-and-carry arbitrage (*Example*).** An equity index stands at 5,000. The nine-month risk-free rate is 4.0 percent and the index dividend yield is 1.5 percent, both continuous. An investor also holds a long forward agreed earlier at 4,950 for the same date. | Item | Value | Excel | |---|---|---| | Fair forward $F_0 = 5000\, e^{(0.04-0.015)\times 0.75}$ | 5,094.63 | `=5000*EXP((4%-1.5%)*0.75)` | | Value of the old long forward $(F_0 - K)e^{-rT}$ per index unit | 140.36 | `=(5094.63-4950)*EXP(-4%*0.75)` | | A dealer quotes 5,150: profit at $T$ per unit from cash and carry | 55.37 | `=5150-5094.63` | | Same profit in today's money | 53.73 | `=55.37*EXP(-4%*0.75)` | The cash-and-carry trade: borrow to buy the index basket at 5,000, reinvest the dividends in the basket, sell forward at 5,150, and deliver at $T$. The cost of the basket carried to $T$ is 5,094.63, so 55.37 per unit is locked in. Such gaps close quickly in liquid index markets; in practice transaction costs and borrowing spreads define a band within which no arbitrage pays. **A2. Commodity forward and convenience yield (*Example*).** Aluminium (a teaching price, not a quote) trades at USD 2,400 per tonne spot. Storage and insurance cost 2.0 percent a year, the six-month rate is 4.0 percent, both continuous. The six-month forward trades at USD 2,420. | Item | Value | Excel | |---|---|---| | Full-carry forward $2400\,e^{(0.04+0.02)\times 0.5}$ | 2,473.09 | `=2400*EXP((4%+2%)*0.5)` | | Implied convenience yield $y = r + u - \ln(F/S)/T$ | 4.34% | `=4%+2%-LN(2420/2400)/0.5` | The market price is below full carry, so holders of physical metal value having it on hand at 4.34 percent a year. Nobody can arbitrage this by selling metal they do not hold, which is why the gap persists. A forward below full carry is a sign of a tight physical market. **A3. Minimum-variance hedge for Northfield (*Example*).** Northfield Outdoor Co. will buy 2,000 tonnes of aluminium for tent poles and cookware over the next month. It hedges with a futures contract of 25 tonnes (an *Example* size) whose grade and delivery point differ from its supplier contract. From 36 months of history, the monthly change in its supplier price has a standard deviation of 3.2 percent, the futures price 3.0 percent, and their correlation is 0.92. | Item | Value | Excel | |---|---|---| | Hedge ratio $h^* = 0.92 \times 0.032/0.030$ | 0.9813 | `=0.92*0.032/0.030` (or `=SLOPE(SpotChanges,FutChanges)` on the raw data) | | Contracts $N^* = 0.9813 \times 2000/25$ | 78.51, so buy 79 | `=ROUND(0.9813*2000/25,0)` | | Hedge effectiveness $\rho^2$ | 0.8464 | `=0.92^2` (or `=RSQ(SpotChanges,FutChanges)`) | | Monthly standard deviation of cost, unhedged (at USD 2,400 per tonne) | USD 153,600 | `=0.032*2400*2000` | | Same, hedged at $h^*$ | USD 60,199 | `=SQRT(0.032^2+h^2*0.03^2-2*h*0.92*0.032*0.03)*2400*2000` | | Same, naive hedge $h = 1$ | USD 60,259 | same formula with h = 1 | The hedge removes about 61 percent of the standard deviation (and 84.64 percent of the variance). The remaining risk is basis risk, which no number of contracts can remove. Because $h^*$ is close to one here, the naive hedge is almost as good; when the spot and futures volatilities differ more, the gap grows. The chart shows the standard deviation of the hedged cost (as a percent of the price) against the hedge ratio: a parabola with its minimum at $h^*$. ```vega-lite { "$schema": "https://vega.github.io/schema/vega-lite/v5.json", "description": "Monthly standard deviation of the hedged aluminium cost against the hedge ratio, sigma_S 3.2%, sigma_F 3.0%, rho 0.92 (fictional example)", "width": 380, "height": 220, "data": {"values": [ {"h": 0.0, "sd": 3.200}, {"h": 0.1, "sd": 2.926}, {"h": 0.2, "sd": 2.658}, {"h": 0.3, "sd": 2.398}, {"h": 0.4, "sd": 2.148}, {"h": 0.5, "sd": 1.913}, {"h": 0.6, "sd": 1.698}, {"h": 0.7, "sd": 1.512}, {"h": 0.8, "sd": 1.367}, {"h": 0.9, "sd": 1.278}, {"h": 1.0, "sd": 1.255}, {"h": 1.1, "sd": 1.304}, {"h": 1.2, "sd": 1.415}, {"h": 1.3, "sd": 1.577}, {"h": 1.4, "sd": 1.775} ]}, "layer": [ {"mark": {"type": "line", "point": true}, "encoding": { "x": {"field": "h", "type": "quantitative", "title": "Hedge ratio (futures tonnes per tonne of exposure)"}, "y": {"field": "sd", "type": "quantitative", "title": "Std dev of hedged cost (% of price, monthly)"} }}, {"mark": {"type": "rule", "strokeDash": [4, 4]}, "encoding": {"x": {"datum": 0.9813}}} ] } ``` **A4. Daily settlement of a futures position (*Example*).** Northfield buys 10 contracts (250 tonnes) at USD 2,410. The broker requires an initial margin of USD 30,000 and a maintenance margin of USD 22,000 for the position. Settlement prices for the next five days are 2,385, 2,350, 2,362, 2,330 and 2,371. | Day | Settlement price | Daily gain (USD) | Cumulative gain | Balance before call | Margin call | |---|---|---|---|---|---| | 0 | 2,410 | | | 30,000 | | | 1 | 2,385 | -6,250 | -6,250 | 23,750 | 0 | | 2 | 2,350 | -8,750 | -15,000 | 15,000 | 15,000 | | 3 | 2,362 | 3,000 | -12,000 | 33,000 | 0 | | 4 | 2,330 | -8,000 | -20,000 | 25,000 | 0 | | 5 | 2,371 | 10,250 | -9,750 | 35,250 | 0 | Excel, with day in column A, settlement price in B, daily gain in C, cumulative gain in D, balance in E and call in F: daily gain `=(B3-B2)*250`; balance `=E2+F2+C3` (previous balance plus previous call plus today's gain); call `=IF(E3<22000,30000-E3,0)`. Check: final balance 35,250 = initial margin 30,000 + calls 15,000 - cumulative loss 9,750. On day 2 the balance fell below maintenance, so Northfield paid USD 15,000 to restore the **initial** margin, not the maintenance level. The economic result is the same as a forward (a loss of USD 39 per tonne so far), but the cash moved on day 2. That is the liquidity cost of futures, and the reason a treasury plan includes a margin buffer. **A5. Pay-fixed interest rate swap for Corvane (*Example*).** Corvane Robotics has a four-year USD 50m term loan at SOFR plus 2.50 percent, with annual resets (a simplification). The board wants a fixed cost. The zero rates (annual compounding) from the SOFR swap curve are 3.80, 3.65, 3.60 and 3.62 percent for years 1 to 4. | Item | Value | Excel | |---|---|---| | Discount factors $D_1$ to $D_4$ | 0.963391, 0.930811, 0.899333, 0.867412 | `=1/(1+B2)^A2` filled down | | Four-year par swap rate $(1 - D_4)/\sum D_i$ | 3.6217% | `=(1-E5)/SUM(E2:E5)` | | All-in fixed cost: swap rate + loan spread | 6.1217% | `=3.6217%+2.5%` | | Annual interest on USD 50m | USD 3,060,837 | `=50000000*6.1217%` | One year later, just after a reset, rates have risen: the zero rates for 1 to 3 years are 4.20, 4.10 and 4.05 percent. | Item | Value | Excel | |---|---|---| | New three-year par rate | 4.0533% | same formula on the new discount factors | | PV of the remaining fixed leg (as a bond with coupon 3.6217%) | USD 49,402,129 | `=50000000*(3.6217%*SUM(F2:F4)+F4)` | | **Value of the swap to Corvane (pays fixed)** | **USD 597,871** | `=50000000-49402129` | | Check: $N\,(c_{par,new} - c)\sum D_i$ | USD 597,871 | `=50000000*(4.0533%-3.6217%)*SUM(F2:F4)` | Corvane's swap is now an asset because it pays 3.62 percent while a new swap would cost 4.05 percent. The gain offsets the higher interest it pays on the loan: that offset is the hedge working. If rates had fallen, the swap would be a liability, and Corvane would pay more than new borrowers, which is the opportunity cost of having fixed. ## Worked example B (half solved) A pension fund holds a USD 20m equity portfolio with a beta of 1.2 against index X, whose futures trade at 5,000 points with a multiplier of USD 50 per point. Steps 3 and 4 use a second index, Y, which stands at 6,000; its six-month rate is 3.5 percent and its dividend yield 2.0 percent, both continuous. | Step | Item | Value | Excel | |---|---|---|---| | 1 | Contracts to remove all market exposure (beta to zero) | 96.00, sell | `=1.2*20000000/(5000*50)` | | 2 | Contracts to cut beta to 0.6 | [ ] | `=(1.2-0.6)*20000000/(5000*50)` | | 3 | Six-month fair forward on index Y | [ ] | `=6000*EXP((3.5%-2%)*0.5)` | | 4 | Value of an old long forward on index Y at 5,950 for the same date | [ ] | `=(F-5950)*EXP(-3.5%*0.5)` | | 5 | Does selling futures remove the stock-specific risk of the portfolio? | [ ] | | | 6 | What return should the fully hedged portfolio earn, roughly? | [ ] | | ## Exercise C (unsolved) 1. **Carry arbitrage.** Gold trades at USD 2,000 an ounce (teaching price). The one-year rate is 4.0 percent and storage costs 0.5 percent a year, both continuous. A dealer quotes a one-year forward at USD 2,150. What is the fair forward? Describe the arbitrage trade and the profit per ounce at the delivery date. 2. **Swap value.** A company pays fixed on a three-year USD 30m swap with annual payments. Zero rates at inception are 3.0, 3.2 and 3.4 percent (annual compounding). Compute the discount factors and the par swap rate. One year later, just after a reset, the one- and two-year zero rates are both 2.5 percent. What is the swap worth to the company? 3. **Hedge design.** Northfield buys diesel for its own delivery trucks, and the only liquid futures contract nearby is on heating oil. Write which three numbers you would estimate to size the hedge, and why the hedge can never be perfect. ## Connections to other cards | Card or template | Connection | |---|---| | mba/mba-options-real-options-card | No-arbitrage and replication; options when you want to keep the upside | | mba/mba-fx-interest-parity-card | The currency forward is the carry formula with the foreign rate as the "dividend" | | mba/mba-fixed-income-card | Zero rates, discount factors, duration; a swap is long one bond and short another | | mba/mba-value-at-risk-card | Measures the risk left after hedging; VaR of the hedged and unhedged position | | mba/mba-portfolio-theory-card | Beta and diversification behind index futures hedges | | mba/mba-capm-wacc-card | Beta estimation used for the number of index futures | | mba/mba-capital-structure-card | Why hedging can add value: distress costs, debt capacity, costly external finance | | mba/mba-derivatives-risk-guide | The elective that sequences this card; mini-case on Northfield's hedging programme | | finance/fx-interest-rate-hedging-policy | How a board sets hedge ratios, tenors and instruments | | finance/hedge-designation-documentation | Documentation needed for hedge accounting | | finance/irrbb-policy-and-report | Interest rate risk in a bank's balance sheet, where swaps are the main tool | ## Common mistakes and misconceptions - **"The forward rate is the market's forecast."** It is the spot price plus carry, set by arbitrage. Whether it predicts the future spot price is a separate, empirical question (for currencies, often badly). - **Forgetting income or storage in the carry.** An index forward without the dividend yield, or a commodity forward without storage, is mispriced. - **Topping up to maintenance margin.** A margin call restores the initial margin. - **Counting margin as a cost.** Margin is collateral that earns interest; the economic cost is the funding and the liquidity strain, not the deposit itself. - **Hedging one for one by habit.** The variance-minimising ratio is $\rho\,\sigma_S/\sigma_F$; when the exposure is less volatile than the future or poorly correlated, $h = 1$ overhedges. - **Estimating $h^*$ on the wrong horizon.** Daily data for a three-month hedge understates the correlation of three-month changes for many commodities; match the data frequency to the hedge horizon where you can. - **Judging a hedge by its own profit or loss.** A hedge that loses money while the exposure gains has done its job. Judge the combined position. - **Valuing a swap from the fixed rate alone.** A swap is worth zero at inception and moves with the whole curve; use discount factors. - **Ignoring the floating leg's convention after LIBOR.** Compounded SOFR is known only at period end; payment systems and cash forecasts must allow for that. - **Overhedging forecast exposures.** Hedging 100 percent of a forecast that does not happen creates a speculative position. Hedge a declining share of later periods (layering). ## Self-test questions 1. Write the cost-of-carry formula for an index, a commodity and a currency, and say what plays the role of the "dividend" in each. 2. In A1, describe the reverse trade if the dealer had quoted 5,040 instead of 5,150. 3. Why can a forward price sit below full carry for a commodity but not for a financial asset held for investment? 4. What does daily settlement change for a hedger, and what does it not change? 5. In A3, why is the variance reduction larger than the standard deviation reduction? 6. A futures position is long 20 contracts of 100 units. The price moves from 80.00 to 78.50. What is the gain or loss? 7. A portfolio of USD 50m has a beta of 0.8. Index futures trade at 4,000 with a multiplier of 250. How many contracts to raise the beta to 1.1, and do you buy or sell? 8. Why is the floating leg of a swap worth par on a reset date? 9. In A5, what would have happened to the value of Corvane's swap if rates had fallen, and what does that mean for the hedge? 10. Give two reasons from the corporate finance literature why hedging can raise firm value, with authors. 11. What changed for a floating leg when USD LIBOR was replaced by compounded SOFR? ## My notes and next steps | Item | My entry | |---|---| | Confidence with forwards, futures and swaps (1 = lost, 5 = could teach it) | [ ] | | Margin account and swap valuation rebuilt in Excel without notes (date) | YYYY-MM-DD | | An exposure I found in my own work or a case, and the instrument I would use | [ ] | | Questions for my professor, tutor or study group | [ ] | | Mistakes logged in mba/mba-error-log | [Date and short description] | | Flashcards made in mba/mba-concept-flashcards | [Card IDs] | | Next review date (spacing: +2 days, +1 week, +1 month) | YYYY-MM-DD | | Next card to study | [mba/mba-value-at-risk-card / mba/mba-fixed-income-card] | ## Sources and further reading - Keynes, J. M. (1923). *A Tract on Monetary Reform*. Macmillan. - Johnson, L. L. (1960). The theory of hedging and speculation in commodity futures. *The Review of Economic Studies*, 27(3), 139-151. - Black, F. (1976). The pricing of commodity contracts. *Journal of Financial Economics*, 3(1-2), 167-179. - Ederington, L. H. (1979). The hedging performance of the new futures markets. *The Journal of Finance*, 34(1), 157-170. - Cox, J. C., Ingersoll, J. E., and Ross, S. A. (1981). The relation between forward prices and futures prices. *Journal of Financial Economics*, 9(4), 321-346. - Stulz, R. M. (1984). Optimal hedging policies. *Journal of Financial and Quantitative Analysis*, 19(2), 127-140. - Smith, C. W., and Stulz, R. M. (1985). The determinants of firms' hedging policies. *Journal of Financial and Quantitative Analysis*, 20(4), 391-405. - Froot, K. A., Scharfstein, D. S., and Stein, J. C. (1993). Risk management: coordinating corporate investment and financing policies. *The Journal of Finance*, 48(5), 1629-1658. - Culp, C. L., and Miller, M. H. (1995). Metallgesellschaft and the economics of synthetic storage. *Journal of Applied Corporate Finance*, 7(4), 62-76. - Mello, A. S., and Parsons, J. E. (1995). Maturity structure of a hedge matters: lessons from the Metallgesellschaft debacle. *Journal of Applied Corporate Finance*, 8(1), 106-121. - Federal Reserve Bank of New York. Secured Overnight Financing Rate (SOFR), newyorkfed.org (checked 2026-10-28). - Bank of England. Transition to sterling risk-free rates from LIBOR, bankofengland.co.uk (checked 2026-10-28). - European Central Bank. Euro short-term rate (€STR), ecb.europa.eu (checked 2026-10-28). - Textbooks (further reading, by name only): Hull, *Options, Futures, and Other Derivatives*; McDonald, *Derivatives Markets*; Berk and DeMarzo, *Corporate Finance*; Brealey, Myers and Allen, *Principles of Corporate Finance*. - Open resource: OpenStax, *Principles of Finance* (2022), free online at https://openstax.org (check the licence on the book page). ## Answer key **Worked example B.** (2) 48.00 contracts, sell. (3) $6000\,e^{0.015\times 0.5}$ = 6,045.17. (4) $(6045.17 - 5950) e^{-0.0175}$ = 93.52. (5) No: index futures remove market (beta) risk only; the stock-specific risk of the portfolio remains. (6) Roughly the risk-free rate, plus or minus the stock-specific (alpha and residual) return, because a zero-beta position earns the risk-free rate under the CAPM. **Exercise C.** (1) Fair forward $2000\,e^{0.045}$ = 2,092.06. The quote of 2,150 is too high: borrow 2,000, buy gold, pay storage, sell forward at 2,150, deliver; profit 57.94 per ounce at the delivery date. (2) Discount factors 0.970874, 0.938946, 0.904562; par swap rate 3.3911 percent. After one year, with $D_1 = 1/1.025$ and $D_2 = 1/1.025^2$, the swap is worth USD -515,246 to the company (`=30000000*(1-(3.3911%*SUM(D1:D2)+D2))`): a liability, because the company pays 3.39 percent while new two-year swaps pay less. Board sentence: "Rates fell, so our fixed rate now costs more than the market rate; the swap shows a USD 0.5m liability, offset by the lower floating interest we pay on the loan." (3) The standard deviation of monthly changes in Northfield's diesel price, the standard deviation of monthly changes in the heating oil futures price, and their correlation; the hedge cannot be perfect because diesel and heating oil prices, delivery points and taxes differ, so the correlation is below one (basis risk). **Self-test.** 1. Index $F = Se^{(r-q)T}$, dividend yield $q$; commodity $F = Se^{(r+u-y)T}$, convenience yield $y$ (storage adds); currency $F = Se^{(r-r_f)T}$, the foreign interest rate $r_f$. 2. The quote 5,040 is below fair value 5,094.63: sell (or short) the basket, invest the proceeds at the risk-free rate, buy forward at 5,040; gain 54.63 per unit at $T$, if the basket can be borrowed or is already held. 3. A financial asset held for investment can be sold short or sold from inventory to arbitrage a low forward; a commodity held for consumption cannot easily be lent or sold by those who need it, so a low forward persists. 4. It changes the timing of cash (daily gains and losses, margin calls) and replaces counterparty risk with margin; it does not change the total gain or loss, which is the price change times the size. 5. Variance falls by $\rho^2$ = 84.64 percent; standard deviation falls by $1 - \sqrt{1 - \rho^2}$ = 61 percent. Standard deviation is the square root of variance, so the proportional fall is smaller. 6. $20 \times 100 \times (78.50 - 80.00)$ = a loss of 3,000. 7. $(0.8 - 1.1) \times 50{,}000{,}000/(4000 \times 250)$ = -15, so buy 15 contracts. 8. At a reset date the next payment is set at the current market rate, so the floating leg is like a floating-rate note that pays the market rate, which is worth par. 9. Its value would have become negative, a liability. The hedge still worked: Corvane pays a lower floating rate on the loan, and the swap loss offsets that saving, leaving the all-in cost at about 6.12 percent. 10. Costly external finance and underinvestment (Froot, Scharfstein and Stein 1993); convex taxes and expected distress costs (Smith and Stulz 1985); managers' undiversified risk (Stulz 1984). 11. The floating rate is now an overnight risk-free rate compounded over the period, known only at the end of the period and with almost no bank credit premium, so loan spreads are wider and payment calculations come later. ## Revision history | Version | Date | Author | Change summary | |---|---|---|---| | 0.1 | YYYY-MM-DD | [Your name] | First version from the template | | 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced the aluminium data with my case exposure and re-estimated the hedge ratio | ## Standard and licence note Based on the cost-of-carry and hedging theory in the original works cited above (Keynes 1923; Johnson 1960; Black 1976; Ederington 1979; Cox, Ingersoll and Ross 1981) and on corporate hedging theory (Stulz 1984; Smith and Stulz 1985; Froot, Scharfstein and Stein 1993), as taught in MBA derivatives electives; benchmark-rate facts come from the Federal Reserve Bank of New York, the Bank of England and the European Central Bank as checked in October 2026. Check the current official pages before you rely on them. The explanations, notation, worked examples and layout are original; no textbook text, published case or exam question is reproduced. Every number was computed in Python and is shown with the matching Excel formula. The template is not affiliated with or endorsed by any business school, exchange, central bank or publisher, and it is not investment, accounting or legal advice. Template licence: MIT.