CFA Level III · Level III Core

Swaps, Forwards, and Futures Strategies: formula sheet

Full chapter guide

Key formulas

Swap duration: Swap duration (receive fixed) = Duration of fixed leg − Duration of floating leg; For pay fixed, the sign reverses. The floating leg duration is usually a fraction of a year, close to the time to the next reset.
Notional for target duration: Notional = [(Target duration − Current duration) ÷ Swap duration] × Portfolio value; Always enter the swap duration as the receive-fixed figure (fixed leg minus floating leg). Then a positive notional means receive fixed and a negative notional means pay fixed.
Money duration balance: Portfolio value × Current duration + Notional × Swap duration = Portfolio value × Target duration; This is the same relationship as the notional formula, just rearranged. It is a good check.
Direction rule: Receive fixed → duration up. Pay fixed → duration down.; Floating-leg duration is near zero, so the fixed leg drives the result.
Notional conversion at initiation: Foreign notional = Domestic notional ÷ spot rate (domestic per foreign); Domestic notional = Foreign notional × spot rate; Fixes the amount exchanged at start and returned at maturity. Check the quote direction first.
Periodic interest on each leg: Interest = notional × rate × (days ÷ day-count basis); Each leg uses its own notional, rate and day count. For annual periods, interest = notional × rate.
Converting a liability: Receive the currency of the existing debt, pay the currency you want; Receipts cover debt service. What you pay becomes your new liability.
Converting an asset: Pay the currency of the existing asset, receive the currency you want; The asset's income funds the payments. What you receive becomes your new asset exposure.
Value of a currency swap to a party: Value = PV of currency A leg received (converted at spot) − PV of currency B leg paid; Discount each leg at its own currency's rates, then convert at the current spot rate.
Net payment on a swap period (equity receiver, other leg is a fixed or floating rate): Net to equity receiver = Notional × (Equity return − Rate on other leg × period fraction); Use this only when the other leg is a fixed or floating rate. Use the same period for both legs. If the result is negative, the equity receiver pays the net amount. If the other leg is a bond or another equity return, use Notional × (equity return − other asset return).
Equity leg payment: Equity leg = Notional × (S_end − S_start) ÷ S_start (+ dividends for a total return swap); Use the index level or price change over the swap period. A negative return means the equity receiver pays that amount on the equity leg.
Fixed-rate leg per period: Fixed leg = Notional × Fixed rate × (days ÷ 360 or per-period fraction); Use the day count or period fraction given in the question.
Exposure change in a switch: New exposure to an asset = Old exposure to that asset + notional of that asset's return received − notional of that asset's return paid; Apply it one asset class at a time. In a switch from equities to bonds, equity exposure falls by the notional of the equity return paid, and bond exposure rises by the notional of the bond return received. Express every exposure as a percentage of portfolio value, then compare with the target.
Effective exposure with a swap and cash: Effective equity exposure = Physical equity + Swap notional received − Swap notional paid; The swap notional is not a cash outlay. The cash or bonds stay in the portfolio.
Payer swaption: Right to pay fixed, receive floating; gains when swap rate > exercise rate; Hedge for a future floating-rate borrower or a view that rates will rise.
Receiver swaption: Right to receive fixed, pay floating; gains when swap rate < exercise rate; Hedge for a future fixed-income buyer or a view that rates will fall.
Payoff at exercise (per period): Payer: max(0, market swap rate − exercise rate) × notional × period fraction; Receiver: max(0, exercise rate − market swap rate) × notional × period fraction; Value at expiry is the present value of these payments over the swap tenor.
Synthetic forward swap (pay fixed): Long payer swaption + short receiver swaption, same exercise rate, expiry and tenor; Reverse the positions to get a receive-fixed forward swap.
Terminating a swap with a swaption: Pay-fixed swap → buy receiver swaption; receive-fixed swap → buy payer swaption; The new swap, if exercised, offsets the original swap's cash flows.
Net gain and breakeven vs premium: Net gain = payoff at exercise − premium paid (with time value of money). Approximate breakeven swap rate: payer = exercise rate + annualised premium; receiver = exercise rate − annualised premium; Buyer's maximum loss is the premium; writer's gain is capped at the premium. The breakeven is an approximation, with the premium expressed as an annual rate over the swap tenor.
Contracts to change equity beta: N = [(β_T − β_S) ÷ β_f] × (S ÷ (f × multiplier)); β_T is target beta, β_S is current portfolio beta, β_f is the futures beta (often 1 for an index future on the same index), S is portfolio value, f is futures price. Positive N is buy, negative N is sell.
Contracts to move equity allocation: N = (Target equity − Current equity) ÷ (f × multiplier); Work in money amounts. The money gap divided by the contract value gives the number of contracts. Apply a futures beta adjustment only if the question explicitly gives a futures beta that differs from the asset class. In that case β_f scales the money exposure each contract delivers, so divide the result by β_f. If β_f = 1, it drops out. A negative N means sell futures, which cuts equity exposure synthetically while the holdings stay in place. A positive N means buy futures to increase equity exposure. Pair with the bond leg below for the other side of the shift.
Bond leg of an allocation shift: BPV_T − BPV_P = (Target bond value × MD_T − Current bond value × MD_P) × 0.0001, then N_f = (BPV_T − BPV_P) ÷ BPV_f; Size the bond leg from the target and current bond allocations. MD_T and MD_P are the modified durations of the target and current bond exposures. If only the money moved is given, treat it as signed: positive into bonds, negative out of bonds. Then BPV_T − BPV_P = Money moved × MD × 0.0001, where MD is the duration of the exposure being added or removed. Divide by BPV_f, the BPV of one bond futures contract, to get contracts. Positive means buy bond futures. Negative, for a fall in bond exposure, means sell.
Contracts for bond exposure (BPV method): N_f = (BPV_T − BPV_P) ÷ BPV_f; BPV is basis point value, the money change per 1 bp. BPV_T is the target portfolio BPV and BPV_P is the current portfolio BPV. BPV_f is the BPV of one futures contract, found from the cheapest-to-deliver (CTD) bond's BPV divided by its conversion factor. All BPVs are for the same yield move. Positive means buy bond futures.
Synthetic cash: Equity portfolio + short futures ≈ risk-free asset; Fully hedged equity (beta of zero) earns about the risk-free rate. Short futures are sized so total beta is 0.
Synthetic equity: Cash + long futures ≈ equity portfolio; Buy futures with notional equal to the cash held. Cash earns the risk-free rate and the futures give the equity return.
Portfolio BPV: BPV = modified duration × market value × 0.0001; Use the same unit for market value throughout. The result is the value change for a 1 basis point yield change.
Futures BPV: BPV of futures ≈ BPV of CTD bond (per contract notional) ÷ conversion factor; Use the CTD's price and modified duration, and apply them to the contract's par amount.
Number of contracts: N = (BPV target − BPV portfolio) ÷ BPV futures; Negative means sell contracts to cut duration. Positive means buy contracts to add duration.
Target BPV: BPV target = target modified duration × market value × 0.0001; For a full hedge, set target BPV to zero.
Cost to deliver (CTD test): Cost = bond quoted price − (futures price × conversion factor); Choose the bond with the lowest cost. Add accrued interest consistently if the exam gives it.
Futures price from CTD: Futures price ≈ [CTD full price × (1 + r)^T − accrued interest at expiry − FV of coupons] ÷ CF; The usual no-arbitrage relationship using the CTD. Use the exam's inputs exactly.
Contracts to equitize cash (equity index futures): N = [(β_target − β_current) ÷ β_futures] × [Value ÷ (Futures price × Multiplier)]; For pure cash, β_current = 0. For pure cash with futures beta about 1, N = target beta × Cash ÷ (Futures price × Multiplier). Round to the nearest whole contract.
Contracts for a bond pre-investment (BPV method): N = (BPV_target − BPV_current) ÷ BPV_futures; For pure cash, BPV_current = 0. A positive N means buy (long) futures.
Basis point value: BPV = Modified duration × Market value × 0.0001; Use the value of the position being covered. For futures, BPV per contract is usually given or is derived from the cheapest-to-deliver bond.
Synthetic index fund: Cash + Long index futures ≈ Index exposure; Cash earns the risk-free rate. The futures price embeds the risk-free rate net of dividends, so the combined return approximates the index return if the futures are fairly priced.
Direction of the hedge: Expected inflow → long futures; Expected outflow of assets to be sold → short futures; Reverse the position when the cash flow occurs, and trade the actual securities.

Quick revision

Duration change needs a position sized on basis point value (BPV). For swaps: notional = (target BPV − current BPV) ÷ swap BPV per unit of notional.
Receive-fixed swap raises portfolio duration; pay-fixed lowers it.
Receive-fixed, pay-floating = long fixed-rate bond + short floater; swap duration = fixed-leg duration − floating-leg duration. Pay-fixed, receive-floating is the opposite: swap duration = floating-leg duration − fixed-leg duration.
Futures contracts for duration use the same BPV approach: contracts = (BPV_target − BPV_portfolio) ÷ BPV_futures. A positive result means buy; a negative result means sell.
A payer swaption gives the right to pay fixed; a receiver swaption gives the right to receive fixed.
Currency swaps exchange notional at start and end, plus interest in each currency.
A forward hedge locks the exchange rate; the hedge ratio sets how much exposure is covered.
Equity swaps can add or remove equity exposure without trading the stock.
Equity futures overlay: contracts = (target beta − current beta) ÷ futures beta × portfolio value ÷ futures value.
Pre-investing uses long futures or swaps to gain exposure now, then unwinds as cash arrives.
Derivatives hedges are imperfect: name basis risk, counterparty risk and margin or liquidity needs.
In essays, answer exactly what the command word asks and only the number of responses requested.

Common mistakes

Choosing the wrong position, for example pay fixed to lengthen duration. Fix: Remember that receive fixed means long a fixed-rate bond, so duration goes up. Pay fixed means short it, so duration goes down.
Using the fixed-leg duration alone as the swap duration when the question gives a floating-leg duration. Fix: If the question gives a floating-leg duration, subtract it. Use zero only when the question says to ignore it.
Netting interest across the two legs as in an interest rate swap. Fix: Interest is in two currencies, so show each leg separately. Only same-currency flows can offset.
Forgetting the exchange of notionals at start and end. Fix: Always list the initial exchange and the final return of principal at the agreed rate.
Reversing who pays and who receives the equity leg Fix: Remember that the equity receiver is long equity. Draw two arrows and label the party who receives the equity return.
Treating the notional as cash paid at the start Fix: State that notionals are not exchanged. Only net periodic payments move, so the portfolio keeps its physical assets.
Mixing up payer and receiver by the floating leg instead of the fixed leg. Fix: Always name the fixed leg: payer pays fixed, receiver receives fixed.
Buying a receiver swaption to hedge future floating borrowing. Fix: A borrower is hurt by higher rates, so buy a payer. Rates up means payer gains.
Getting the sign wrong and buying futures to reduce beta. Fix: Always compute target minus current. Negative means sell. Check against intuition: reducing risk means selling.
Forgetting the futures multiplier. Fix: Contract value = price × multiplier. Underline the multiplier in the question before calculating.

Exam tips

Read the command word. 'Calculate' needs a number, with the position stated. 'Justify' needs a reason tied to the client or the rate view.
A correct number typed on its own earns full credit for a calculation. Only the number of responses asked for is evaluated, in the order given, so give exactly what is asked and no extra answers.
In item sets, check whether the answer options include the right size but the wrong position. Direction errors are a common trap.
If the question gives the floating-leg duration, subtract it. If it says to treat the floating leg as zero, do so.
Link to liability-driven cases: a fund with a long liability duration may need to receive fixed to close the gap.
Draw a small arrow diagram of the two legs. It prevents pay and receive errors and takes seconds.
Read the command word. 'Calculate' needs the number, 'Justify' needs a reason tied to the client's objective.
Check the quote convention before any conversion. This is the most common loss of marks.