DER 1 – Pricing and Valuation of Forward Commitments
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A forward commitment is a binding agreement to transact at a price fixed today. The family has three members: forwards, futures and swaps. A forward contract obliges one party to buy and the other to sell an asset on a stated future date at a price agreed when the contract is struck. Its payoff is linear, so both parties carry genuine upside and genuine downside. A swap is the same idea repeated: a promise to transact at a set price or rate on a series of future dates. The technique for handling a swap is therefore to build a portfolio of simpler instruments, usually a pair of bonds, whose cash flows match the swap, and then let the law of one price supply the answer.
Practitioners reach for these instruments constantly. A private wealth manager uses equity index futures and swaps to hedge a concentrated, low tax basis holding without triggering a sale. A defined benefit plan manager uses interest rate swaps to close a duration gap. A university endowment overlays total return swaps and bond futures to rebalance and to tilt asset allocation. A corporate treasurer swaps a floating-rate bond issue into fixed. A market strategist reads VIX futures and inflation swaps to extract what the market expects volatility and inflation to be.
Pricing and valuation are two different tasks
The vocabulary matters and candidates lose marks on it. Pricing a forward commitment means determining the fixed price or fixed rate written into the contract at initiation. Valuation means determining what an existing contract is worth, expressed in currency units, at some later moment. Pricing produces a number such as a forward price of 105 or a swap rate of 1.2968%. Valuation produces a number such as a gain of 6.2729 per unit or a swap value of positive three million euros. Almost every forward commitment is priced so that its value at initiation is zero.
Think like an arbitrageur
Every result in this reading is derived from the behaviour of arbitrageurs, who are the price setters in forward commitment markets. The arbitrageur follows two rules without exception. Rule one: do not use your own money. Rule two: do not take any price risk. Working under those constraints, the arbitrageur borrows or lends, buys or sells short, and assembles a package whose future value is known with certainty. If that package can be assembled for nothing today and is guaranteed to pay something positive later, the market has mispriced something.
Two principles underpin the whole exercise. The law of one price says that two investments delivering identical future cash flows in every possible future state must sell for the same price today. Violate it and a trader buys the cheap version, sells the dear version, and books a gain with no risk and no capital. The value additivity principle says that the value of a portfolio is the sum of the values of its parts, which is what allows a swap to be decomposed into two bonds and valued piece by piece.
Four assumptions run through everything that follows. Instruments that replicate or offset the position can be identified and can actually be bought. Frictions in the market are taken to be nil. A short sale is permitted and the seller keeps full use of the proceeds. Everyone may borrow and lend at one known risk-free rate. The cash flow sign convention is equally simple: money the arbitrageur receives carries a positive sign and money paid out carries a negative sign, so an initial outlay of 100 euros enters as −100.
Carry costs and carry benefits
The engine of the reading is the carry arbitrage model, sometimes called the cost-of-carry model or the cash-and-carry model. The name is literal. To offset a short forward position, an arbitrageur buys the underlying today with borrowed money and carries it to the expiration date, at which point it can be sold or delivered. Carrying is not free, and it is not always costly either. Some underlying assets throw off cashflows while they are held and some do not, and the model has to account for both.
| Item | Notation | Examples | Effect on the forward price |
|---|---|---|---|
| Carry costs | CC0, CCT | Storage, insurance, waste and spoilage for physical goods; essentially zero for financial assets | Raises it |
| Financing cost | r | Interest forgone or paid on the money tied up in the spot asset | Raises it |
| Carry benefits | CB0, CBT | Dividends, bond coupon payments, interest earned on a foreign currency holding | Lowers it |
Costs of any kind increase the burden of holding the asset through time, so they are added. Benefits reduce that burden, so they are subtracted. Commodities are not covered in this reading, but the storage and insurance side of the table is worth remembering because it explains why a gold forward behaves differently from an equity forward.
Holding a financial asset generates no direct storage cost, yet it is not free either: money withdrawn from the bank to buy the asset stops earning interest. That opportunity cost is the financing cost, and it is already inside the future value operator in F0 = FV(S0). Storage and insurance on a physical asset are conceptually the same thing, simply charged by a warehouse instead of by a lender.
Forwards and futures do the same economic job. One party is legally obliged to sell and the other legally obliged to buy a specified asset at an agreed price on a specified future date. The difference is institutional. A futures contract is exchange traded and standardised, which makes it liquid and creditworthy but inflexible. A forward contract is negotiated bilaterally, which makes it flexible but exposes each side to the other.
| Futures | Forwards |
|---|---|
| Traded on an exchange | Negotiated directly between the two counterparties |
| Standardised dates and deliverable assets | Dates and deliverables customised to the client |
| Performance guaranteed by a clearinghouse | Each side bears the credit risk of the other |
| Initial value of zero | Initial value of zero, typical but not required |
| Initial margin set by the exchange, adjusted daily for gains and losses; a balance falling below the maintenance level triggers a call for more funds or closure of the position | Margin arrangements, if any, are whatever the two parties negotiate |
| Daily settlement resets the contract price to the market price and the contract value to zero | Value may build up or drain away between settlement dates, or right through to maturity if no early settlement is required |
A CME gold futures contract illustrates the standardisation: a fixed size of 100 ounces, a defined list of deliverable gold, and a short menu of maturity dates. Since the financial crisis, best practice for over-the-counter contracts has moved toward daily settlement and margin arrangements that look much like the exchange model.
Notation
Let St be the spot price of the underlying at any time t, meaning the cash price for immediate delivery. Let Ft be the forward price at time t and ft the futures price at time t. The subscript 0 marks initiation and the subscript T marks expiration. The initial forward price F0 is set so that the contract is worth nothing to either side on day one, and it does not change over the life of the contract. What does change is the price at which a newly created contract with the same underlying and the same expiration would be struck. The gap between that new price and the old fixed price is where value comes from. Vt denotes the value of a forward at time t and vt the value of a futures contract.
Convergence
As expiration approaches, a newly created forward or futures contract becomes indistinguishable from a spot transaction. At expiration the three prices are one and the same:
The intuition is easy. The price agreed today for gold delivered in a year may be nothing like the spot price of gold, but the price agreed today for gold delivered one hour from now must be almost exactly the spot price. Arbitrage enforces the result. If at maturity the forward or futures price sat above spot, a trader would sell the contract, borrow to buy the asset, deliver it, repay the loan and pocket the difference. The selling pressure drives the contract price down. If instead the futures price sat below spot, the trader would short sell the asset, invest the proceeds at the risk-free rate, buy the futures, take delivery at expiration and use the asset to close the short. The profit is the short-sale price less the futures price, after allowing for financing and carrying costs.
Before expiration the two prices may differ slightly from each other as well as from spot. Counterparty default risk, or correlation between the underlying price and interest rates that affects the financing of daily settlement, can drive a small wedge between them. For most purposes it is safe to treat Ft and ft as equal before expiration.
The difference between the spot price and the futures price is the basis, and convergence means the basis shrinks toward zero as the maturity date nears. Which side the contract price converges from is decided by carry, and this is the single most useful picture in the reading.
Value at expiration, and the value of a futures contract during the day
The profit on any completed transaction is the sale price less the purchase price, and a forward contract is no exception. A long forward locks in a purchase price of F0, and at expiration the asset can be sold in the spot market for ST. So:
Futures behave differently in the interim because of daily settlement. During the trading day, before the contract is marked to market, the accumulated value of a long futures position is the multiplier times the change in the futures price since the previous settlement:
Once the settlement runs, the gain or loss is swept into the margin account and the contract price is reset to the settlement price, so the futures value returns to zero. This is why a futures position generates a stream of small realised cash flows while a forward position accumulates an unrealised claim.
Three unrelated positions, each testing a different piece of the mechanics above.
−VT = £10,000,000 − £9,800,000 = £200,000.
The manager profits because the portfolio was sold at a price above what the market would pay at expiration.
vt = 100 × ($1,310 − $1,300) = +$1,000.
Note that the investor never paid the full $130,000 (= 100 × $1,300); only a margin deposit was required. Immediately after marking to market the $1,000 gain is credited to the margin account and the contract price is reset to the new settlement price, so vt = 0.
Start with the simplest possible underlying: an asset that pays nothing and costs nothing to store. The arbitrageur who is short a forward contract wants to neutralise the price risk. The way to do it is to buy the asset now with borrowed funds and hold it until the forward expires.
Trace the cash flows. At time 0 the asset costs S0, an outflow, and the loan brings in S0, an inflow, so the net is zero. Entering the forward also costs nothing, since a newly struck forward has zero value. At time T the asset is worth ST, the loan must be repaid at FV(S0), and the short forward is worth F0 − ST. Add the three together:
ST − FV(S0) + (F0 − ST) = F0 − FV(S0).
The ST terms cancel, which is the whole point: the package has no price risk. Since it cost nothing to assemble, it must be worth nothing at expiration too, or the arbitrageur has money for free. Setting the expression to zero gives the central pricing result:
Seeing the risk before the hedge, and after
Take a non-dividend-paying stock with S0 = 100, r = 5% and T = 1, and allow the price at expiration to move to either 90 or 110. Buying one share with a borrowed 100 produces a net cash flow of zero today. At expiration the loan repayment is 100(1.05) = 105, so the position pays 110 − 105 = +5 in the up state and 90 − 105 = −15 in the down state. Two different outcomes means price risk remains.
Now add the third leg and sell a forward at F0 = 105. In the up state the short forward is worth 105 − 110 = −5, so the total is 110 − 105 − 5 = 0. In the down state the short forward is worth 105 − 90 = +15, so the total is 90 − 105 + 15 = 0. The proceeds of the sale plus the forward payoff always come to 105, precisely the amount owed on the loan. There is no uncertainty left, which is why the forward price must be 105 and nothing else.
When the forward is too expensive
Suppose the market quotes F0 = 106 while the model says 105. The quoted price is too high, so sell it. The three simultaneous steps are: sell the forward contract, borrow the funds, and buy the underlying. Step two enforces rule one, and step three enforces rule two.
In the up state the short forward pays 106 − 110 = −4, and the total is 110 − 105 − 4 = +1. In the down state the short forward pays 106 − 90 = +16, and the total is 90 − 105 + 16 = +1. The profit is +1 whatever the stock does, so it is risk free and it was obtained with none of the arbitrageur own capital. Discounting it back, the short forward struck at 106 is worth
V0 = PV[F0 − FV(S0)] = (106 − 105) ÷ (1 + 0.05)1 = 0.9524 at time 0.
When the forward is too cheap
Now suppose the market quotes F0 = 104. The steps reverse: buy the forward contract, sell the underlying short, and lend the short-sale proceeds. Note the convention: when the arbitrageur needs to sell the underlying it is always assumed to be a short sale, because an asset already sitting in inventory would not appear as an investment in the analysis. Selling a derivative, by contrast, is simply selling and never short selling.
The 100 lent grows to 105. A long forward at 104 is worth 90 − 104 = −14 in the down state and 110 − 104 = +6 in the up state. Totals: +105 − 14 − 90 = +1 and +105 + 6 − 110 = +1. The same guaranteed +1. Because the underlying is sold short rather than carried, this mirror image is called reverse carry arbitrage.
Put the two cases together and the market discipline is obvious. When F0 exceeds FV(S0), traders sell forwards and buy the asset, pushing the forward price down and the spot price up. When F0 falls below FV(S0), traders buy forwards and short the asset, pushing the forward price up and the spot price down. Both roads lead to F0 = FV(S0), and at that point no arbitrage profit exists.
An Australian stock that pays no dividends trades at A$63.31. The annual Australian interest rate is 2.75% with annual compounding.
F0 = FV(S0) = 63.31(1 + 0.0275)0.25 = A$63.7408.
Notice how little the quarter of a year of interest adds: about 43 Australian cents on a A$63 stock.
F0 = 63.31(1 + 0.0225)0.25 = A$63.6632.
The forward price is directly related to the interest rate, because the interest rate is the cost of financing the spot position. This relationship holds generally so long as the underlying itself is not driven by interest rates, which is a caveat that matters when the underlying is a bond.
Pricing is done once, at initiation. Valuation is needed repeatedly afterwards: for mark-based accounting, for collateral calls, and simply to know whether a position is making or losing money if it were exited early.
The trick is the same as before. Suppose a long forward was bought at F0 at time 0, expiring at time T. At time t, part way through, consider selling a new forward at the prevailing price Ft, again expiring at time T. The long position is worth ST − F0 at expiration and the new short position is worth Ft − ST. Together they are worth Ft − F0, which contains no ST term at all. The combined position is fully hedged, so its time T payoff is certain and can be discounted at the risk-free rate.
An equivalent form is often quicker, because a spot price is easier to observe than a forward price:
The second follows from the first. Under annual compounding Ft = St(1 + r)(T−t), so PV[Ft] = St(1 + r)(T−t) ÷ (1 + r)(T−t) = St. Keep both in your toolkit. The first version is preferable when market frictions make the observed forward price drift from its theoretical level, since it uses the price actually quoted. The second version is more intuitive and needs only the spot price.
The short position is the negative of the long position throughout: −Vt = PV[F0 − Ft], or equivalently PV[F0] − St.
One point of method deserves emphasis. These transactions do not have to be executed. Valuation asks what the position would fetch, not what it did fetch. Holding a liquid asset does not require selling it to know its market price, and the same logic applies here.
At time 0 an investor entered a one-year long forward contract at F0 = 105. Nine months later, at t = 0.75, the underlying stock trades at 110 and the interest rate is 5%.
Ft = FV(St) = 110(1 + 0.05)0.25 = 111.3499.
Then discount the difference against the original price of 105:
Vt = PV[Ft − F0] = (111.3499 − 105) ÷ (1 + 0.05)0.25 = 6.2729.
The alternative route gives the same figure without ever computing Ft:
Vt = St − PV[F0] = 110 − [105 ÷ (1 + 0.05)0.25] = 6.2729.
Note that the value of 6.2729 is not the same as the raw difference of 5 between the spot price of 110 and the contract price of 105. The extra comes from the three months of financing cost embedded in the new forward price.
Most underlying assets that matter to an investor are not inert. Stocks pay dividends, bonds pay coupons, and a foreign currency holding earns foreign interest. Each of these is a benefit to whoever is carrying the asset, and the forward price must reflect it.
Let CB denote carry benefits, with CB0 the present value at time 0 and CBT the future value at time T. Let CC denote carry costs on the same convention. The complete pricing equation, which is also called the cost of carry model or future-spot parity, is:
For financial assets the explicit carry cost is zero, which simplifies the equation to F0 = FV(S0) − FV(benefits). For a stock paying a dividend D before the forward expires:
There is a timing subtlety here that trips people up. The stock price is compounded over the full life of the contract, from 0 to T. The dividend is compounded only from the date it is received to expiration. If a dividend arrives at time t and is held to time T, its future value is D(1 + r)T−t, not D(1 + r)T. The two future value calculations therefore use different exponents.
A US stock trading at $1,000 will pay a $10 dividend in two months. The US interest rate is 5% with annual compounding.
F0 = FV(S0) − FV(D) = 1,000(1 + 0.05)3/12 − 10(1 + 0.05)1/12
= $1,012.2722 − $10.0407 = $1,002.2315.
The general equation gives the same answer. Carry costs are zero, and the present value of the dividend is CB0 = 10 ÷ (1 + 0.05)2/12 = $9.919. Then:
F0 = FV(1,000 + 0 − 9.919) = 990.081 × (1 + 0.05)3/12 = $1,002.23.
F0 = 1,000(1 + 0.05)3/12 − 10(1 + 0.05)2/12 = $1,012.2722 − $10.0816 = $1,002.1906.
The move is small, about four cents, but the direction is what the examiner is testing.
Valuation is unaffected in form. The value of a long position is still the present value of the difference in forward prices. Carry benefits and carry costs simply feed into both forward prices, since Ft = FV(St + CCt − CBt) and F0 = FV(S0 + CC0 − CB0). Nothing new has to be memorised.
An investor took the long side of a twelve-month forward on a productive asset, agreeing a price of ₡1,000. Seven months on, the asset itself changes hands at ₡1,050. The present value of the cost to store, insure and maintain the asset for the next five months is ₡4.00, and the asset will generate income over the next five months with a present value of ₡28.00. Assume annual compounding throughout and a risk-free rate of 2%.
Step 1. Price a new forward today. Add the carry cost and subtract the carry benefit before compounding:
Ft = FV(St + CCt − CBt) = (1,050 + 4 − 28)(1 + 0.02)5/12 = ₡1,034.50.
Step 2. Discount the difference in forward prices.
Vt = PV[Ft − F0] = (₡1,034.50 − ₡1,000) ÷ (1 + 0.02)5/12 = ₡34.22.
Observe that the income of 28 is much larger than the storage cost of 4, so on net the carry cash flows pull the new forward price down and reduce the gain on the long position.
Continuous compounding
When the underlying is a broad index it is awkward to model dividends as a series of discrete payments arriving on scattered dates from hundreds of constituents. A dividend index point measures the quantity of dividends attributable to a particular index and is a genuinely useful number for arbitrage trading, but the standard simplification is to assume dividends accrue continuously. For an index with many components that assumption is not unreasonable. The carry arbitrage model then takes its continuous form:
The sign logic is unchanged. If CB exceeds rc the exponent is negative and the forward price sits below spot, which is the lower branch of Figure 1. If storage and insurance costs are added for a physical asset, they push the exponent up and the forward price rises above spot. Getting the compounding convention right matters: market interest rates are inferred from market prices, and applying a quoted rate under the wrong convention introduces errors that can be large enough to change an answer choice.
Equity forwards and futures are the cleanest application of the model, because the only carry cash flow is the dividend. The three examples below cover the three cases that appear on exams: a continuous dividend yield on an index, a single discrete dividend on a named stock, and the valuation of an existing position.
The continuously compounded dividend yield on the EURO STOXX 50 is 3% and the current index level is 3,500. The continuously compounded annual interest rate is 0.15%.
f0 = 3,500 × e(0.0015 + 0 − 0.03)(3/12) = 3,475.15.
The index falls by about 25 points over three months, which is roughly a quarter of the 2.85% annual net carry drag. This is the lower branch of the convergence picture: dividends exceed financing costs, so the futures price approaches spot from below.
Nestlé common stock trades for CHF70 and pays a CHF2.20 dividend in one month. The Swiss one-month risk-free rate is 1.0% on an annual compounding basis. The stock goes ex-dividend on the same day the single stock forward contract expires, and that contract expires in one month.
F0 = FV(S0) + FV(CC0) − FV(CB0) = 70(1 + 0.01)1/12 + 0 − 2.20 = CHF67.86.
The dividend of CHF2.20 dominates one month of interest on CHF70, which is a few rappen, so the forward price sits well below the spot price.
Nine months ago an investor took the long side of a forward on a stock, with a twelve-month tenor and a contract price of 102, so a quarter of a year remains. The stock now changes hands at 110 and the annually compounded interest rate is 5%. As a variant, imagine the same investor had instead taken the long side of a twelve-month futures contract at 102, and that the futures quote today stands at 112.35. No carry cash flows arise on this underlying.
Ft = FV(St) = 110(1 + 0.05)0.25 = 111.3499.
Vt = PV[Ft − F0] = (111.3499 − 102) ÷ (1 + 0.05)0.25 = 9.2366,
or by the alternative route,
Vt = St − PV[F0] = 110 − 102 ÷ (1 + 0.05)0.25 = 9.2366.
The value exceeds the raw gap of 8 between the spot price of 110 and the contract price of 102, because the interest cost of carrying the stock pushes the new forward price up to 111.35.
For decades the reference rate under most interest rate derivatives was Libor, the London Interbank Offered Rate, compiled by the British Bankers Association. From 2008 regulators and market participants came to suspect that certain banks were manipulating the daily quotes, and the eventual consequence was the replacement of Libor by a new market reference rate in 2021. Among the replacements are SOFR, a Secured Overnight Financing Rate published by the New York Federal Reserve, and SONIA, a Sterling Overnight Index Average run by the Bank of England. The curriculum uses the generic label MRR for all of them.
The spot market first
Notation for the spot deposit market:
- Lm is the MRR spot rate, set at time 0, for a deposit of m days.
- NA is the notional amount, the funds initially deposited.
- NTD is the number of total days in a year used for interest, which is 360 in the MRR market.
- tm is the accrual period, the fraction of a year for an m-day deposit, so tm = m ÷ 360.
- TA is the terminal amount repaid when the deposit is withdrawn.
Take a 90-day deposit of $50,000 with dollar MRR quoted at 2%, so L90 = 0.02 and t90 = 90 ÷ 360 = 0.25. Then TA = $50,000 × [1 + 0.02(90/360)] = $50,250 and the interest is $250. The add-on convention is the opposite of a discount basis, where the price today is discounted from a known amount paid at maturity.
What a forward rate agreement is
A forward rate agreement is an over-the-counter forward contract whose underlying is an interest rate on a deposit. It has two sides. The fixed-rate payer is long the FRA and is simultaneously the floating-rate receiver. The fixed-rate receiver is short the FRA and is simultaneously the floating-rate payer. Naming one side automatically names the other, and candidates who lose track of this lose the sign of the answer.
The long side gains when rates rise. If the floating rate ends up above the contracted rate, the long can be thought of as having secured borrowing below market, and it receives a payment. A firm that plans to borrow in the future and fears rising rates is the natural long. A bank hoping to lock in a lending rate is the natural short. Since no cash changes hands at initiation, the FRA rate is set at the level that makes the contract worth zero on day one.
FRAs are labelled X by Y. In a 3 × 9 FRA, the 3 says the contract expires in three months. The underlying is implied by the gap between the numbers: nine minus three is six, so the payoff depends on six-month, meaning 180-day, MRR observed when the FRA expires. The rate itself is derived from the relationship between nine-month and three-month spot MRR at initiation, and a long 3 × 9 FRA replicates going long a nine-month deposit and short a three-month deposit. Although the labels are quoted in months, the arithmetic uses days on the assumption of 30 days per month.
Two points about the underlying are worth stating plainly. It is not a financial asset and not even a financial instrument; it is an interest payment. And the two counterparties need not have any deposit in the spot market at all. The MRR spot market is only the benchmark from which the payoff is computed, in the same way that a stock index futures trader need hold no shares.
Advanced set, advanced settled
Two settlement conventions exist. Advanced set means the reference rate is fixed at the start of the underlying deposit period, which is the FRA expiration date at time h. This convention is almost universal, because issuers and buyers want to know the rate while they hold the position. The distinction is what happens next. Settled in arrears means the interest is paid at time h + m, when the underlying deposit matures, exactly like an ordinary bank deposit. Advanced settled means the payment is made at time h, so the FRA expires and settles at the same moment.
FRAs are almost always advanced set, advanced settled. Interest rate swaps and interest rate options are normally advanced set, settled in arrears. That single difference explains the discount factor in the settlement formula:
Read the signs from the position. The long receives floating, so Lm carries a positive sign, and pays fixed, so FRA0 carries a negative sign. The short is the reverse. The divisor 1 + Dmtm discounts the payoff because the rate on which the payoff is based is obtained from a spot market that settles in arrears, while the FRA itself pays early. The money arrives m days sooner than the interest it stands in for, so it must be discounted for those m days. It is commonly assumed that Dm equals Lm, but this is a convenience rather than a requirement.
In 30 days a UK company expects to place a bank deposit of £10,000,000 for 90 days at 90-day MRR set 30 days from today. The company fears a fall in interest rates. Its adviser recommends a 1 × 4 FRA, expiring in 30 days and based on 90-day MRR. The company enters a £10,000,000 notional 1 × 4 receive-fixed FRA on an advanced set, advanced settled basis, so it is the short side. The appropriate discount rate for the settlement cash flows is 2.40%. After 30 days, 90-day sterling MRR is 2.55%.
TA = 10,000,000 × [1 + 0.0255(0.25)] = £10,063,750,
so the interest is £63,750.
NA × {[FRA0 − Lm]tm} ÷ [1 + Dmtm]
= 10,000,000 × {[0.0260 − 0.0255](0.25)} ÷ [1 + 0.0240(0.25)]
= £1,242.54.
The short pays floating, so it benefits from rates ending below the contracted rate. Note that in this problem Dm of 2.40% is not equal to Lm of 2.55%.
= 10,000,000 × {[0.0250 − 0.0255](0.25)} ÷ [1 + 0.0240(0.25)] = −£1,242.54.
The sign flips because the floating rate of 2.55% now exceeds the contracted 2.50%. The short is paying floating and therefore suffers when rates rise. The company still gets 2.55% on its actual deposit, so the FRA has done its job of fixing the combined outcome.
Pricing the FRA
The FRA rate is the implied forward rate between the FRA expiration date and the underlying maturity date. Two spot rates therefore determine it, and pricing an FRA is no different in spirit from pricing any other forward.
Recall the simple two-period case. With simple interest, [1 + y(1)][1 + F(1)] = [1 + y(2)]2, where y(1) and y(2) are the one-period and two-period yields and F(1) is the forward rate for the second period. Borrowing along the two-year path must cost exactly what borrowing for one year and rolling into the one-year forward costs. If the one-year spot rate is 3% and the two-year spot rate is 4%, then F(1) = ([1 + 0.04]2 ÷ [1 + 0.03]) − 1 = 0.0501. Any other forward rate would let an arbitrageur borrow along one path and lend along the other.
Written out for the FRA, the no-arbitrage condition is [1 + Lhth][1 + FRA0tm] = [1 + LTtT], and solving for the annualised rate gives:
An example: if 180-day MRR is 2.0% and 90-day MRR is 1.5%, the 3 × 6 FRA rate is
{[1 + 0.02(180/360)] ÷ [1 + 0.015(90/360)] − 1} ÷ (90/360) = 0.024907, or 2.49%.
A quick sanity check is available whenever the shorter period is exactly half the longer one. Solve the arithmetic average (1/2)(1.5%) + (1/2)X = 2.0% to get X = 2.5%. That approximation is always biased slightly high, so the true answer should land a little under 2.5%, and 2.49% duly does. Use it to catch a mis-keyed calculator entry.
Assume a 30/360 day count throughout. For questions 2 and 3, use the following US dollar MRR spot rates: one-month 2.48%, three-month 2.58%, six-month 2.62% and one-year 2.72%.
FRA0 = {[1 + 0.0175(270/360)] ÷ [1 + 0.015(180/360)] − 1} ÷ (90/360)
= [(1.013125 ÷ 1.0075) − 1] ÷ 0.25 = 0.022333, or 2.23%.
FRA0 = {[1 + 0.0258(90/360)] ÷ [1 + 0.0248(30/360)] − 1} ÷ (60/360)
= [(1.00645 ÷ 1.00207) − 1] ÷ 0.1667 = 0.026220, or 2.62%.
Take care with the accrual period in the divisor. It is 60/360, the length of the underlying deposit, not the 90 days that appear in the numerator.
Valuing an existing FRA
The method is by now familiar: offset the position and discount the difference. Suppose a long 3 × 6 FRA was struck at FRA0 = 2.49% with tm = 90/360. Thirty days later, at time g, the contract has 60 days left to expiration, so it is now effectively a 2 × 5 FRA, and the market rate on an FRA expiring in 60 days based on 90-day MRR is FRAg = 2.59%.
Go short a new FRA at 2.59%. The floating legs cancel exactly, since [Lm − FRA0] + [FRAg − Lm] = [FRAg − FRA0]. What remains is receiving fixed at 2.59% and paying fixed at 2.49%, a spread of 10 basis points applied for 90/360 of a year on the notional amount. Discount that certain cash flow from time T back to time g:
Note carefully that the discounting runs over T − g, from the maturity of the underlying deposit back to the valuation date, not merely to the FRA expiration date.
At time 0 an investor entered a receive-floating 6 × 9 FRA with a notional amount of C$10,000,000. Six-month Canadian dollar MRR was 0.628% and nine-month MRR was 0.712%, and the 6 × 9 FRA was quoted in the market at 0.877%. After 90 days have passed, three-month Canadian dollar MRR is 1.25% and six-month MRR is 1.35%, and the six-month rate is used as the discount rate.
Step 1. Reset the clock. The FRA originally expired in 180 days, so it now expires in h − g = 180 − 90 = 90 days. The underlying originally matured in 270 days, so it now matures in T − g = 270 − 90 = 180 days. The offsetting contract is therefore a 3 × 6 FRA on 90-day MRR.
Step 2. Price the new FRA.
FRAg = {[1 + 0.0135(180/360)] ÷ [1 + 0.0125(90/360)] − 1} ÷ (90/360)
= [(1.006750 ÷ 1.003125) − 1] ÷ 0.25 = 0.014455, or 1.445%.
Step 3. Discount the rate difference. Discounting runs over T − g = 180 days at 1.35%:
Vg = 10,000,000 × {[0.01445 − 0.00877](90/360)} ÷ [1 + 0.0135(180/360)]
= C$14,105.
The position has gained about 57 basis points of rate on a quarter of a year on ten million dollars, which is roughly C$14,200 before discounting, and C$14,105 after.
The carry arbitrage model applies unchanged to bonds, but three market conventions complicate the arithmetic and every one of them shows up in exam questions.
Clean prices, dirty prices and accrued interest
In many countries a bond is quoted without the interest that has built up since the last coupon date. That quotation is the clean price. What the buyer actually hands over is the dirty price, which adds accrued interest. Futures quotation conventions normally follow whatever the underlying cash market does. Accrued interest is computed by linear interpolation:
After 60 days, a 3% semi-annual coupon bond with par of 1,000 has accrued interest of AI = (60/180) × (30/2) = 5. Accrued interest is expressed in currency units rather than percent, and NTD depends on the coupon frequency: semi-annual coupons on a 360-day year give NTD = 360 ÷ 2 = 180.
Conversion factors and the cheapest to deliver
A bond futures contract usually lets the seller choose among several deliverable bonds. Since those bonds carry different maturities and coupons, they trade at different prices, so the exchange applies a conversion factor to put them on a comparable footing. The Chicago Mercantile Exchange describes the factor as roughly the decimal price that one dollar of par would fetch on a security yielding 6% to maturity, so the CF rescales every deliverable to a 6% coupon benchmark. Other exchanges use different conventions.
The adjustment is approximate rather than exact, which is why a cheapest-to-deliver bond emerges. Among the eligible bonds, the seller delivers whichever one is least expensive to buy in the open market relative to what the contract pays for it.
The pricing equation
Let B0 be the quoted bond price at time 0 and AI0 the accrued interest at time 0, so the full spot price is S0 = B0 + AI0. The carry benefits are the coupons paid between time 0 and expiration, so CB0 = PVCI, the present value of coupon interest over the contract horizon, with future value FVCI. There are no carry costs. Where multiple bonds are deliverable, the market quotes a futures price Q0 and the economically meaningful adjusted price is F0 = Q0 × CF.
Because the futures contract settles against the quoted bond price, without accrued interest, the profit at expiration on a long position is vT = BT − F0, equivalently (ST − AIT) − F0. Rearranging the equilibrium condition F0 + AIT = FV[B0 + AI0 − PVCI] and solving for the quoted price gives the working formula:
What disequilibrium looks like
The equation is easiest to trust once you have watched it fail. Consider a three-month forward, so T = 0.25, on a bond quoted at 107.00% of par with no coupon due before the contract expires.
| Item | Symbol | Value |
|---|---|---|
| Quoted bond price | B0 | 107.00 |
| Present value of coupon interest | PVCI | 0 |
| Accrued interest at time 0 | AI0 | 0.07 |
| Accrued interest at time T | AIT | 0.20 |
| Quoted futures price | Q0 | 135.00 |
| Conversion factor | CF | 0.80 |
| Adjusted futures price | F0 = Q0 × CF | 108.00 |
| Rate for discounting and compounding | r | 0.20% |
The full spot price is S0 = 107 + 0.07 = 107.07. Carrying that bond to expiration costs FV[B0 + AI0 − PVCI] = (107 + 0.07 − 0)(1.002)0.25 = 107.12. Meanwhile the futures market lets the arbitrageur lock in a sale at the adjusted price of 108.00, and a seller of a bond also collects the accrued interest of 0.20 at that date, so the total received is 108.20. The contract is overpriced by 108.20 − 107.12 = 1.08.
The arbitrage is therefore to sell the overpriced futures contract, borrow the funds, and buy the underpriced deliverable bond. At maturity the bond is delivered against the futures, the loan is repaid, and 1.08 per bond remains. Its present value at time 0 is 1.08 × (1.002)−0.25 = 1.0795.
The equilibrium quoted price follows directly:
Q0 = (1 ÷ 0.8) × {(1 + 0.002)0.25(107 + 0.07) − 0.20 − 0.0} = 133.65.
Equivalently, the equilibrium adjusted price is FV(S0) − AIT − FVCI = 107.12 − 0.2 − 0 = 106.92, and the quoted 135.00 corresponds to an adjusted 108.00, so the gap of 108.00 − 106.92 = 1.08 is the same profit seen from the other direction. Any quoted price above 133.65 leaves an arbitrage available. Had the mispricing run the other way, with the full bond price exceeding the adjusted futures price plus accrued interest, the arbitrageur would short sell the bond, lend the proceeds and buy the futures, which is the reverse carry version of the trade.
Euro-bund futures traded on Eurex have a contract value of €100,000, and the underlying is long-term German government debt with 8.5 to 10.5 years to maturity. The underlying 2% coupon German bund, paying semi-annually, is quoted at €108 and carries accrued interest of €0.083, reflecting 15 days since the last coupon. The futures contract matures in one month, that is 30 days. At expiration the bund will have accrued interest of €0.25. No coupon is due before the futures contract expires. The one-month risk-free rate is 0.1% and the conversion factor is 0.729535.
AI0 = (15/180) × (2%/2) = €0.083 and AIT = (45/180) × (2%/2) = €0.25.
The accrued interest at expiration uses 45 days because 15 days had already elapsed and the contract runs a further 30.
Q0 = [1 ÷ 0.729535] × [(1 + 0.001)1/12(108 + 0.083) − 0.25 − 0] = €147.82.
The same answer via the adjusted price:
F0 = FV(S0) − AIT − FVCI = (1 + 0.001)1/12(108 + 0.083) − 0.25 − 0 = 107.84,
and Q0 = F0 ÷ CF = 107.84 ÷ 0.729535 = 147.82.
The conversion factor of 0.729535 is well below one, which is why the quoted price of about 148 looks so far above the bond price of 108. Because the futures contract is marked to market, its value is essentially the price change since the previous settlement and resets to zero once that settlement is taken.
Valuing a bond forward
Without daily settlement a forward accumulates value, and the valuation rule is the one used throughout. Buying a bond forward at F0 and later selling a new one at Ft to the same expiration leaves no price risk, since the two positions at expiration are (BT − F0) and (Ft − BT), which sum to Ft − F0. So Vt = PV[Ft − F0] once again.
A quick illustration: a long forward at F0 = 119.12 and a short forward at Ft = 119.92, with one month to expiration and a discount rate of 0.5%, gives
Vt = (119.92 − 119.12) ÷ (1 + 0.005)1/12 = 0.7997.
One month ago an investor purchased five euro-bund forward contracts with two months to expiration, each with a contract notional value of €100,000, at a price of 145 quoted as a percentage of par. The contracts now have one month left. The annualised one-month risk-free rate is 0.1% and the current forward price is 148.
Vt = PV[Ft − F0] = (148 − 145) ÷ (1 + 0.001)1/12 = 2.99975.
That figure is per €100 of par, because the forward price is quoted as a percentage of par. Scaling to five contracts of €100,000 each:
0.0299975 × €100,000 × 5 = €14,998.75.
The undiscounted gain would have been €15,000, so the whole present value effect is about €1.25. When rates are near zero and the remaining term is short, discounting barely moves the answer, but it is still the correct method.
Forwards against futures, summarised
For every market in this reading the same two expressions do the work: F0 = FV(S0 + CC0 − CB0) for pricing, and Vt = PV[Ft − F0] for valuation. Financing costs and carry costs push the forward price up because they increase the burden on the arbitrageur carrying the asset; carry benefits push it down because they relieve that burden. Futures prices are found with the same model. Futures values differ only because daily marking to market resets them to zero at the end of every trading day.
A swap is an agreement to exchange a series of cash flows on periodic dates. An interest rate swap is an FRA repeated: where an FRA hedges a single period of interest rate risk, a swap hedges many. The cash flows come from multiplying a notional amount by a rate, one fixed and one floating. In a plain vanilla, pay-fixed and receive-floating swap, the fixed-rate payer pays a fixed rate on the notional and receives a floating rate, referenced to MRR, on the same notional.
Only the net amount changes hands. If the floating rate Si in a period exceeds the fixed rate FS, the fixed-rate payer receives the difference. If the floating rate falls below the fixed rate, the fixed-rate receiver collects instead.
The replicating portfolio
A pay-fixed, receive-floating swap is economically identical to being short a fixed-rate bond and long a floating-rate bond. If both bonds are struck at par, the initial cash flows offset and the par repayments at maturity cancel, leaving exactly the swap coupons. Put another way, the swap rate is the fixed rate at which the present value of all expected floating payments equals the present value of all expected fixed payments.
Bonds are the natural replicating instruments, but they are not the only ones. A swap can also be built from a strip of forwards, though standardised futures rarely line up exactly with a swap schedule. It can even be built from options, since a long call and a short put at the same strike, set equal to the swap fixed rate, reproduce the payoff of a pay-fixed swap. In every case the procedure is identical: match the cash flows with marketable instruments and let the law of one price supply the value.
Transforming a liability
Suppose REB, Inc. finds it easy to issue fixed-rate bonds but, after studying the interest rate sensitivity of its assets, would rather have a floating-rate liability. REB issues the fixed-rate bond and makes fixed payments to bondholders. It then enters a receive-fixed, pay-floating swap. The two fixed streams cancel and REB is left paying floating, which is a synthetic floating-rate bond. The same trick, applied with a currency swap, manages currency and interest rate exposure together, and applied with an equity swap it manages equity exposure.
Legs, accrual periods and day counts
The floating leg cash flow in period i is Si = APFLT × rFLT,i, and the fixed leg is FS = APFIX × rFIX, where AP is the accrual period, computed as accrued days divided by total days in the year. The two dominant day count methods are 30/360, which treats every month as 30 days and every year as 360, and ACT/ACT, which uses actual days over the actual 365 or 366. A single swap can carry different frequencies and different day counts on its two legs. The floating rate is advanced set and settled in arrears, so it is fixed at the start of each period and paid at the end.
With equal accrual periods on both legs the net payment simplifies:
Put the fixed leg at 5%, let the floating leg fix at 5.2%, and take an accrual period of 30 days against a 360-day year. The receive-fixed party then owes (30/360) × (0.05 − 0.052) = −0.000167 per unit of notional. On a notional of £100 million that is a payment of £16,700 from the receive-fixed party to the pay-fixed party. Only this single net amount moves. Had the fixed rate exceeded the floating rate, the sign would be positive and the money would flow the other way.
Present value factors and the swap rate
Valuing the fixed leg means discounting each coupon at the appropriate spot rate. The discount factor for the cash flow at date i is:
| Days to maturity | US dollar spot rate (%) | Present value of US$1 |
|---|---|---|
| 90 | 2.10 | 0.994777 |
| 180 | 2.25 | 0.988875 |
| 270 | 2.40 | 0.982318 |
| 360 | 2.54 | 0.975229 |
| Sum | 3.941199 |
For instance the 90-day factor is 0.994777 = 1 ÷ [1 + 0.0210(90/360)], and the 360-day factor is 0.975229 = 1 ÷ [1 + 0.0254(360/360)].
With those factors in hand, a fixed 4% bond paying quarterly on par of 1 has a quarterly coupon of 0.01 and a value of
VFIX = 0.01(3.941199) + 0.975229(1) = 1.014641, or 101.464% of par.
Now set the fixed bond value equal to 1, which is what a floating-rate bond is worth on a reset date, since its coupon is reset to the discount rate and a bond whose coupon equals its yield trades at par. Solving for the fixed rate gives the swap pricing equation:
Price a five-year, MRR-based interest rate swap with annual resets on a 30/360 day count. The estimated present value factors are as follows.
| Maturity (years) | Present value factor |
|---|---|
| 1 | 0.990099 |
| 2 | 0.977876 |
| 3 | 0.965136 |
| 4 | 0.951529 |
| 5 | 0.937467 |
0.990099 + 0.977876 + 0.965136 + 0.951529 + 0.937467 = 4.822107.
The final cash flow on the implied bond is the fifth coupon plus par, so the fifth factor is used twice: once inside the sum in the denominator, applied to that coupon, and once in the numerator, applied to par. With annual coupons the accrual period is 360/360 = 1. Therefore:
rFIX = [(1 − 0.937467) ÷ 4.822107] × (360 ÷ 360) = 0.012968, or about 1.30%.
Valuing a swap after initiation
Once rates move, an existing swap acquires value. The receive-fixed party is long a fixed bond and short a floating bond, so it gains when rates fall and the fixed bond trades above par. The pay-fixed party is the mirror image and gains when rates rise.
The valuation method matches the FRA method. Offset the original swap with a new swap at the current rate. The floating legs cancel at every settlement date, and what remains is a stream of certain differences between the old and new fixed cash flows, each discounted by its own present value factor:
In this equation n counts the cash flows still to come from time t, which is generally fewer than the number used to price the swap at inception. The sign convention is fixed by the positive FS0 term: the equation as written gives the value to the party that originally agreed to receive fixed. Valuation between payment dates requires further adjustments that are not covered here.
An investor holds a receive-fixed position, on a notional of €100,000,000, in a MRR-based swap of seven-year original tenor that resets once a year. It was struck two years ago at a contract fixed rate of 2.0%. The present value factors are the same five figures used above, summing to 4.822107, and the current equilibrium fixed swap rate is close to 1.30%.
VSWAP,t = €100,000,000 × (0.02 − 0.013) × 4.822107 = €3,375,475, closest to €3,375,000.
Rates have fallen since the swap was struck, and the receive-fixed party is locked into collecting 2.0% while the market now pays only 1.3%. That 70 basis point advantage, applied to €100 million for five years and discounted, is worth roughly €3.4 million.
−VSWAP,t = €100,000,000 × (0.013 − 0.02) × 4.822107 = −€3,375,000.
A swap is a zero sum contract between the two counterparties, so one side gains precisely what the other loses.
A currency swap is a contract to exchange future interest payments denominated in different currencies. One party is long a bond in one currency and short a bond in another. The pricing and valuation machinery carries over from interest rate swaps, with three features to keep in mind.
- Notional amounts are usually exchanged both at initiation and at expiration, unlike an interest rate swap where the notional never moves.
- The two legs are paid in different currency units, so they are not netted. Both payments are made in full.
- Each leg can be fixed or floating independently of the other.
Pricing a fixed-for-fixed currency swap means solving for three variables: two fixed rates, one in each currency, and one notional amount, given the notional in the other currency. The value of the swap is the difference between two bonds, with the second converted into the units of the first:
For the swap to be worth nothing at inception the two bond values must match, so Va = S0Vb, and therefore the notionals must satisfy NAa = S0 × NAb. That single condition determines the second notional amount.
Watching the cash flows cancel
Consider an at-market ten-year swap in which one party receives fixed US dollars and pays fixed euros. The US dollar bond has an annual coupon of US$30 and par of US$1,150, and the euro bond has an annual coupon of €9 and par of €1,000. Both trade at par, and the spot rate is US$1.15 per euro, so US$1,150 ÷ 1.15 = €1,000 and the initial values match. To offset the swap, short sell the US dollar bond and buy the euro bond.
| Position | Time 0 | Time 1, at $1.50/€ | Time 2, at $1.10/€ | Time 10, at $1.20/€ |
|---|---|---|---|---|
| Receive-fixed US dollars, pay-fixed euros swap | 0 | +16.5 | +20.1 | −30.8 |
| Short sell the US dollar bond | +1,150 | −30 | −30 | −1,180 |
| Buy the euro bond | −1,150 | +13.5 | +9.9 | +1,210.8 |
| Net | 0 | 0 | 0 | 0 |
The swap cash flow at time 1 is +$30 − ($1.5/€) × €9 = +$16.5, at time 2 it is +$30 − ($1.1/€) × €9 = +$20.1, and at time 10 it is ($30 + $1,150) − ($1.2/€) × (€9 + €1,000) = −$30.8. The exchange rates chosen for the future dates are arbitrary; any values at all would leave the net row at zero, which is exactly what makes the replication valid.
Because the fixed rate in each currency does not depend on the notional amounts, each rate is found the same way as an ordinary swap rate, using that currency own term structure:
Viewing the swap as a pair of fixed-rate bonds has a large practical advantage: every foreign exchange consideration is pushed into the initial exchange rate, so no forecasting of future currency transactions is needed. It also generalises. A fixed-for-floating currency swap is simply a fixed-for-fixed currency swap combined with a plain interest rate swap in the second currency. A floating leg does not need pricing at all, because it is worth par on any reset date.
A US company needs to borrow A$100 million for one year for its Australian subsidiary. It issues US dollar bonds in an equivalent amount and enters a one-year currency swap with quarterly resets on a 30/360 day count, exchanging notional amounts at initiation and at maturity. At initiation the company receives the Australian dollar notional and pays the US dollar notional; at expiration the flows reverse. The agreed spot rate is A$1.140 per US$1. Interbank spot rates and their present value factors are as follows.
| Days to maturity | A$ spot rate (%) | PV of A$1 | US$ spot rate (%) | PV of US$1 |
|---|---|---|---|---|
| 90 | 2.50 | 0.993789 | 0.10 | 0.999750 |
| 180 | 2.60 | 0.987167 | 0.15 | 0.999251 |
| 270 | 2.70 | 0.980152 | 0.20 | 0.998502 |
| 360 | 2.80 | 0.972763 | 0.25 | 0.997506 |
| Sum | 3.933870 | 3.995009 |
For example A$0.993789 = 1 ÷ [1 + 0.0250(90/360)] and US$0.999251 = 1 ÷ [1 + 0.00150(180/360)].
rAUD = [(1 − 0.972763) ÷ 3.933870] × (360 ÷ 90) = 0.027695, or 2.7695%.
rUSD = [(1 − 0.997506) ÷ 3.995009] × (360 ÷ 90) = 0.002497, or 0.2497%.
The Australian rate is more than ten times the US rate, which mirrors the underlying term structures.
A$100,000,000 = (A$1.14/US$1) × NAb, so NAb = A$100,000,000 ÷ (A$1.14/US$1) = US$87,719,298, that is about US$88 million.
FS in Australian dollars = A$100,000,000 × (90/360) × 0.027695 = A$692,375.
FS in US dollars = US$87,719,298 × (90/360) × 0.002497 = US$54,759.
Both payments are made in full every quarter, in their own currencies, with no netting.
Valuing a currency swap
Two sources of risk drive the value: the interest rate term structure in each currency, and the exchange rate between them. At time t the value of a receive-currency-a, pay-currency-b swap is again the difference between two bonds, with the second converted at the current spot rate St:
The equation looks forbidding and is not. The first term is the present value of the inflows to the party receiving currency a: the quarterly interest payments plus the terminal notional, all discounted on the new term structure. The second term is the present value of that party outflows in currency b, converted into currency a at the new spot rate. The value to the counterparty is simply the negative. The one thing to establish before writing anything down is which currency is currency a, and the answer is dictated by how the exchange rate is quoted.
Continue the previous swap. The fixed rates were 2.7695% for Australian dollars and 0.2497% for US dollars, the US dollar notional was US$87,719,298, and the initial spot rate was A$1.14 per US$1. Sixty days have now passed and the market has moved.
| Days to maturity | A$ spot rate (%) | PV of A$1 | US$ spot rate (%) | PV of US$1 |
|---|---|---|---|---|
| 30 | 2.00 | 0.998336 | 0.50 | 0.999584 |
| 120 | 1.90 | 0.993707 | 0.40 | 0.998668 |
| 210 | 1.80 | 0.989609 | 0.30 | 0.998253 |
| 300 | 1.70 | 0.986031 | 0.20 | 0.998336 |
| Sum | 3.967683 | 3.994841 |
The spot exchange rate is now A$1.13 for US$1.
The periodic rates are 0.00692375 per quarter in Australian dollars (2.7695% ÷ 4) and 0.00062425 per quarter in US dollars (0.2497% ÷ 4).
VCS = A$100,000,000 × [0.00692375(3.967683) + 0.986031]
− 1.13 (A$/US$1) × US$87,719,298 × [0.00062425(3.994841) + 0.998336]
= A$2,145,167.
The first bracket values the dealer long Australian dollar bond: four quarterly coupons discounted at the new factors, plus the terminal A$100 million discounted at the 300-day factor. The second values the short US dollar bond and converts it at the new spot rate.
−VCS = −A$2,145,167 × (1 US$ ÷ 1.13 A$) = −US$1,898,378.
The same answer can be reached by re-labelling. If the exchange rate is quoted the other way round, as US$0.885 per A$1, then currency a is the US dollar and the firm is the party receiving currency a, since the swap gives it the equivalent of a long US dollar bond:
VCS = US$87,719,298 × [0.00062422(3.994841) + 0.998336]
− (1 US$ ÷ A$1.13) × A$100,000,000 × [0.00692381(3.967683) + 0.986031]
= −US$1,898,410, the small difference from −US$1,898,378 arising from rounding in the periodic rates.
Two forces made the swap a liability to the US firm. The Australian dollar strengthened, from A$1.14 to A$1.13 per US dollar, so every US dollar now buys fewer Australian dollars for making payments. And the Australian term structure fell, so the above-market Australian dollar rate the firm agreed to pay is now worth more in present value terms as an obligation.
An equity swap is an over-the-counter contract in which two parties exchange a series of cash flows, one of which is determined by an equity. The other side pays either a variable series driven by a different equity or a rate, or a fixed series. The purpose is to convert the return on an equity investment into some other return, and the appeal is that a party can obtain exposure to an equity or an index without owning a single share, or hedge such an exposure for a defined period.
Three structures appear:
- Receive-equity return, pay-fixed. The cash flow is NA(equity return − fixed rate).
- Receive-equity return, pay-floating. The cash flow is NA(equity return − floating rate).
- Receive one equity return, pay another. The cash flow is NA(equity return a − equity return b).
The third is not a separate animal. Glue together a swap that receives equity a against a fixed leg and a second swap that pays equity b against the identical fixed leg. Every fixed payment nets out, and what survives is the equity-for-equity structure. Three practical nuances are worth noting. The reference for the equity leg can be a single stock, a published index or a custom portfolio. The equity leg may be defined with or without dividends. And every complication that applies to the fixed or floating leg of an interest rate swap applies here too.
The equity leg cash flow is Si = NAERE, where RE is the periodic equity return as specified in the contract. The fixed leg is FS = NAE × APFIX × rFIX, exactly as for an interest rate swap.
An investor entered a receive-equity-index, pay-fixed swap with quarterly resets on a 30/360 day count and a notional amount of €5,000,000. The fixed leg is 1.6% annualised, paid quarterly, which is 0.4% per quarter.
€5,000,000 × (90/360) × (0.160 − 0.016) = €5,000,000 × (0.040 − 0.004) = €180,000.
The investor receives 4% and pays 0.4% for the quarter, netting 3.6% on €5 million.
€5,000,000 × (90/360) × (−0.240 − 0.016) = €5,000,000 × (−0.060 − 0.004) = −€320,000.
When the equity leg is negative the receive-equity party pays twice over: it pays away the equity loss and it pays the fixed rate. This is a real source of liquidity risk. A party using an equity swap to hedge a holding it does not want to sell can face a substantial cash outflow in precisely the quarter its portfolio has fallen.
Replication and pricing
To keep the analysis tractable, dividends are ignored on the understanding that the equity leg assumes all dividends are reinvested in the equity position. Arbitrage transactions for an equity swap that excludes dividends are extremely complex and beyond the scope here.
Mechanically, the equity leg is produced by selling the equity position on each reset date and reinvesting the original notional amount NAE, so that whatever is left over is the cash flow required by the swap. If the position has grown, the excess above NAE is sold off; if it has shrunk, additional shares are bought to restore the position to NAE.
Consider a manager holding a large position in a stock expected to underperform, who prefers not to sell for liquidity or tax reasons and instead enters a receive-fixed, pay-equity swap. Offset it by buying NAE of the equity and short selling a fixed-rate bond whose coupon equals the fixed swap payment. The net cash flow is zero at every intermediate date. At the final date the equity position returns both the last periodic return and the sale proceeds NAE, so the terminal flows cancel only if the bond par value equals the initial equity notional, or the difference is financed. In equilibrium:
Setting that to zero and taking NAE = Par = 1 delivers a pleasing result: the fixed rate on an equity swap is found from exactly the same formula as the fixed rate on a comparable interest rate swap, even though the two sets of future cash flows look nothing alike.
Return to the five-year, annual reset, 30/360, MRR-based swap used earlier, with present value factors of 0.990099, 0.977876, 0.965136, 0.951529 and 0.937467. Assume an annual reset, MRR-based floating-rate bond trading at par. The comparable interest rate swap fixed rate was found to be 1.2968%.
0.990099 + 0.977876 + 0.965136 + 0.951529 + 0.937467 = 4.822107,
and rFIX = [(1 − 0.937467) ÷ 4.822107] × (360 ÷ 360) = 0.012968.
Nothing about the equity entered the calculation. The fixed leg is priced against the term structure alone, because the equity leg, like a floating leg, is worth par at each reset by construction. This is a fast mark on an exam if you recognise it.
Valuing an equity swap after initiation
Valuation follows the interest rate swap template with one substitution. Instead of adjusting a floating-rate bond for the last observed floating rate, adjust the notional amount of equity for the movement in the equity price since the last reset:
Six months ago an investor entered a receive-fixed, pay-equity five-year annual reset swap with the fixed leg on a 30/360 day count. At initiation the fixed swap rate was 1.5%, the equity traded at 100 and the notional amount was 10,000,000. All spot interest rates have now fallen to 1.2%, giving a flat term structure, and the equity trades at 105. The par value of the implied bond equals NAE.
Step 1. Value the implied fixed-rate bond on the new flat 1.2% curve. Each factor is 1 ÷ (1 + years × 0.012); for instance the 1.5-year factor is 1 ÷ (1 + 3 × (0.012/2)) = 0.982318.
| Date (years) | Present value factor | Fixed cash flow | Present value |
|---|---|---|---|
| 0.5 | 0.994036 | 150,000 | 149,105 |
| 1.5 | 0.982318 | 150,000 | 147,348 |
| 2.5 | 0.970874 | 150,000 | 145,631 |
| 3.5 | 0.959693 | 150,000 | 143,954 |
| 4.5 | 0.948767 | 10,150,000 | 9,629,981 |
| Total | 10,216,019 |
VEQ,t = 10,216,019 − [(105/100) × 10,000,000] − 0 = −283,981.
Both moves worked against this investor. Rates fell, which lifted the fixed bond to 102.16% of par and would on its own have been a gain, but the equity rose 5%, and the investor is paying the equity return. The equity effect dominates.
VEQ,t = 10,216,019 − [(102.1602/100) × 10,000,000] − 0 = 0.
In other words the swap breaks even when the equity has appreciated by the same percentage as the implied fixed bond, which is the general condition for a receive-fixed, pay-equity swap to be at market.