Eduzan

Learning Hub

Eduzan

Eduzan / 07 Derivatives

DER 1 – Pricing and Valuation of Forward Commitments

Worked examples are fully visible. Check-yourself items are study aids you can reveal one at a time.

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.

What carrying an asset costs and what it pays

Item	Notation	Examples	Effect on the forward price
Carry costs	CC₀, CC_T	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, CB_T	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 against forwards

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 S_t be the spot price of the underlying at any time t, meaning the cash price for immediate delivery. Let F_t be the forward price at time t and f_t 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. V_t denotes the value of a forward at time t and v_t 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 convergence property.

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 F_t and f_t 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.

Figure 1: Which side the forward price converges from

When the cost of financing and storing the asset outweighs what the asset pays you for holding it, the forward price sits above spot and falls to meet it. When dividends or coupons outweigh the financing cost, the forward price sits below spot and rises to meet it. Either way the two prices are identical at expiration.

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 S_T. So:

Value of a long forward at expiration. The short position is the mirror image, F₀ − S_T.

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:

N_f is the standard contract size set by the exchange.

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.

Example 1 · Worked

Three unrelated positions, each testing a different piece of the mechanics above.

1. In January a fund manager agrees to sell a bond portfolio in May at a forward price of F₀ = £10,000,000. At expiration the spot value of the portfolio is £9,800,000. What is the value of the manager position at expiration?

Solution. The manager is short the forward, so the expiration value is the locked-in sale price less the spot price:
−V_T = £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.

2. An industrial firm needs platinum for automotive components. On 10 March it enters a long forward to buy 4,500 ounces on 10 September at $900 per ounce. By May the September forward price of platinum has moved to $1,100. Is the position worth something, and what would the sign have been had the price moved to $800 instead?

Solution. At $1,100 the firm has locked in a purchase price of $900 on an asset now priced at $1,100, so the long forward has positive value. At $800 the firm would still be obliged to pay $900 for something worth $800, so the long forward value would be negative. The firm cannot walk away in either case, which is exactly why forward value has to be tracked and collateralised.

3. An investor is long one June gold futures contract with a multiplier of N_f = 100 ounces. The contract settled yesterday at $1,300 per ounce. During today the price of June gold rises to $1,310. What is the value of the contract before settlement, and what is it immediately after?

Solution. Before settlement:
v_t = 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 v_t = 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 S_T, the loan must be repaid at FV(S₀), and the short forward is worth F₀ − S_T. Add the three together:

S_T − FV(S₀) + (F₀ − S_T) = F₀ − FV(S₀).

The S_T 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:

Annual compounding. Under continuous compounding the same result is F₀ = S₀e^r_cT.

Figure 2: The three legs of a carry arbitrage

All three transactions are executed simultaneously, not one after another. The asset price at expiration cancels out of the total, which is what makes the package riskless and what pins the forward price to the future value of the spot price.

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.

Example 2 · Worked

An Australian stock that pays no dividends trades at A$63.31. The annual Australian interest rate is 2.75% with annual compounding.

1. What is the equilibrium three-month forward price?

Solution. Here S₀ = A$63.31, r = 2.75% with annual compounding, and T = 0.25 of a year. With no cash flows on the underlying the forward price is simply the future value of spot:
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.

2. The interest rate immediately falls 50 basis points to 2.25%. What happens to the three-month forward price?

Solution. It decreases. Recomputing:
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.

Why the forward price is not a forecast

Nothing in the derivation used an opinion about where the stock is heading. The inputs were the current spot price, the interest rate, the time to expiration and the absence of arbitrage. An investor convinced the stock will double still cannot bid the forward price above FV(S₀), because any such bid would be arbitraged away by someone who simply buys the stock and carries it. The future value operator is a financing calculation, not a mathematical expectation of the spot price at time T. The two are routinely confused, and the confusion is worth a mark or two on exam day.

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 F_t, again expiring at time T. The long position is worth S_T − F₀ at expiration and the new short position is worth F_t − S_T. Together they are worth F_t − F₀, which contains no S_T 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.

Value of a long forward at time t, the present value of the difference in forward prices.

An equivalent form is often quicker, because a spot price is easier to observe than a forward price:

The same value expressed from the current spot price.

The second follows from the first. Under annual compounding F_t = S_t(1 + r)^(T−t), so PV[F_t] = S_t(1 + r)^(T−t) ÷ (1 + r)^(T−t) = S_t. 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: −V_t = PV[F₀ − F_t], or equivalently PV[F₀] − S_t.

Figure 3: Valuing a forward by offsetting it

The offsetting contract runs to the same expiration date as the original. Because both settle against the same spot price, that price drops out and what is left is the difference in the two forward prices, discounted from expiration back to the valuation date.

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.

Example 3 · Worked

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%.

What is the value of the existing forward contract, which now has three months to run?

Solution. With S_0.75 = 110, r = 5% and T − t = 0.25, first find the forward price a new contract would carry today:
F_t = FV(S_t) = 110(1 + 0.05)^0.25 = 111.3499.

Then discount the difference against the original price of 105:
V_t = PV[F_t − F₀] = (111.3499 − 105) ÷ (1 + 0.05)^0.25 = 6.2729.

The alternative route gives the same figure without ever computing F_t:
V_t = S_t − 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 CB_T 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:

The future value of the underlying adjusted for carry cash flows.

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.

Example 4 · Worked

A US stock trading at $1,000 will pay a $10 dividend in two months. The US interest rate is 5% with annual compounding.

1. What is the equilibrium three-month forward price?

Solution. S₀ = $1,000, r = 5%, T = 0.25. The dividend arrives after two months and therefore earns interest for one month before the contract expires:
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.

2. Suppose instead the dividend is paid after one month. What happens to the three-month forward price?

Solution. It decreases. An earlier dividend earns interest for an extra month, so its future value at expiration is larger, and a larger benefit is subtracted:
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 F_t = FV(S_t + CC_t − CB_t) and F₀ = FV(S₀ + CC₀ − CB₀). Nothing new has to be memorised.

Example 5 · Worked

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%.

What is the value of the forward position with five months remaining?

Solution. The inputs are F₀ = 1,000, S_t = 1,050, CC_t = 4, CB_t = 28, t = 7 months, T − t = 5 months and r = 2%.

Step 1. Price a new forward today. Add the carry cost and subtract the carry benefit before compounding:
F_t = FV(S_t + CC_t − CB_t) = (1,050 + 4 − 28)(1 + 0.02)^5/12 = ₡1,034.50.

Step 2. Discount the difference in forward prices.
V_t = PV[F_t − 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:

Here r_c, CC and CB are all continuously compounded rates.

The sign logic is unchanged. If CB exceeds r_c 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.

Example 6 · Worked

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%.

What is the three-month futures price under the carry arbitrage model?

Solution. Carry costs are zero for a stock index. The carry benefit is the 3% continuous dividend yield, and it exceeds the financing cost of 0.15%, so before doing any arithmetic you know the futures price must be below the spot index level.

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.

Example 7 · Worked

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.

What is the one-month forward price?

Solution. S₀ = 70, r = 1.0%, T = 1/12. Because the dividend is received exactly at expiration, no compounding is applied to it, so FV(CB₀) = 2.20 = CB_T.

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.

Example 8 · Worked

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.

1. What is the value of the existing forward contract?

Solution. F₀ = 102, S_0.75 = 110, r = 5% and T − t = 0.25. The new forward price is:
F_t = FV(S_t) = 110(1 + 0.05)^0.25 = 111.3499.

V_t = PV[F_t − F₀] = (111.3499 − 102) ÷ (1 + 0.05)^0.25 = 9.2366,
or by the alternative route,
V_t = S_t − 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.

2. A dividend payment is announced for a date between the valuation date and expiration. Assuming the announcement leaves the current stock price unchanged, what happens to the value of the existing long forward?

Solution. It decreases. The old forward price of 102 is locked and cannot move. A newly imposed dividend is a carry benefit, so it lowers the new forward price F_t. Value is the discounted difference F_t − F₀, so a lower F_t means a lower value for the existing long contract.

3. After marking to market, what is the value of the futures contract?

Solution. 0.00. Futures are marked to market daily, so any profit or loss is paid or received at each settlement. Once settlement has occurred, the equity futures value is zero. The figure of 112.35 tells you the size of the cash that has already changed hands, not the value of an outstanding claim.

Why the futures answer is zero but the forward answer is not

Both contracts have made the same money. The difference is where that money sits. The futures holder has already received it in cash through the margin account, day by day, so nothing is owed and the contract value resets to zero at every settlement. The forward holder has received nothing; the gain exists only as a claim on a counterparty at expiration, so it must be carried on the books as a positive value and, in practice, collateralised. This is also why the two instruments can price slightly differently when the underlying is correlated with interest rates: the timing of the futures cash flows starts to matter, because the gains are reinvested and the losses financed at rates that move with the underlying.

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:

L_m 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.
t_m is the accrual period, the fraction of a year for an m-day deposit, so t_m = m ÷ 360.
TA is the terminal amount repaid when the deposit is withdrawn.

Interest is added on at the end, the add-on convention, so the interest paid is NA × L_mt_m.

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:

Receive-floating. The receive-fixed side is the same expression with FRA₀ − L_m in the numerator.

Read the signs from the position. The long receives floating, so L_m carries a positive sign, and pays fixed, so FRA₀ carries a negative sign. The short is the reverse. The divisor 1 + D_mt_m 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 D_m equals L_m, but this is a convenience rather than a requirement.

Example 9 · Worked

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%.

1. What interest will actually be paid at maturity on the bank deposit itself?

Solution. This is an ordinary deposit, unconnected to the FRA. With m = 90 and L₉₀ = 2.55%:
TA = 10,000,000 × [1 + 0.0255(0.25)] = £10,063,750,
so the interest is £63,750.

2. If the FRA was priced at FRA₀ = 2.60%, what payment is received on settlement?

Solution. Here m = 90, t_m = 90/360 and h = 30. Use the receive-fixed formula:
NA × {[FRA₀ − L_m]t_m} ÷ [1 + D_mt_m]
= 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 D_m of 2.40% is not equal to L_m of 2.55%.

3. If instead the FRA had been priced at FRA₀ = 2.50%, what payment is received?

Solution. Same formula, new fixed rate:
= 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.

Figure 4: The two routes that must cost the same

Lending for the whole period at the long spot rate must produce the same terminal amount as lending to the FRA expiration date and rolling the proceeds forward at the FRA rate. That equality is the entire content of the pricing formula.

Written out for the FRA, the no-arbitrage condition is [1 + L_ht_h][1 + FRA₀t_m] = [1 + L_Tt_T], and solving for the annualised rate gives:

L_T reaches the underlying maturity; L_h reaches the FRA expiration.

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.

Example 10 · Worked

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%.

1. Canadian dollar six-month MRR is 1.5% and nine-month MRR is 1.75%. What is the 6 × 9 FRA fixed rate?

Solution. L₁₈₀ = 1.50% and L₂₇₀ = 1.75%. The FRA expires in 180 days and the underlying deposit runs 90 days from there to day 270.
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%.

2. Given the four US dollar spot rates, how many distinct FRA rates can be computed?

Solution. Six. Every ordered pair of the four maturities produces one forward rate: 1 × 3, 1 × 6, 1 × 12, 3 × 6, 3 × 12 and 6 × 12. Counting them is faster than trying to remember a formula, and the count of six is simply the number of ways to choose two maturities from four.

3. What is the 1 × 3 FRA fixed rate?

Solution. L₃₀ = 2.48% and L₉₀ = 2.58%. The FRA expires in 30 days and the underlying runs the remaining 60 days:
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 t_m = 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 FRA_g = 2.59%.

Go short a new FRA at 2.59%. The floating legs cancel exactly, since [L_m − FRA₀] + [FRA_g − L_m] = [FRA_g − 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:

Value at time g of a long, receive-floating FRA. The short position is the negative of this.

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.

Example 11 · Worked

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.

What is the value of the original receive-floating FRA after 90 days?

Solution. Rates have risen, and a receive-floating position gains when rates rise, so expect a positive answer.

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.
FRA_g = {[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%:
V_g = 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:

NAD is days since the last coupon, NTD is total days in the coupon period, n is coupons per year and C is the annual coupon amount.

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.

The adjusted futures price is the future value of the full spot price less the coupons collected along the way.

Because the futures contract settles against the quoted bond price, without accrued interest, the profit at expiration on a long position is v_T = B_T − F₀, equivalently (S_T − AI_T) − F₀. Rearranging the equilibrium condition F₀ + AI_T = FV[B₀ + AI₀ − PVCI] and solving for the quoted price gives the working formula:

The conversion factor adjusted, that is quoted, futures price.

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.

Inputs for a mispriced bond futures contract

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	AI_T	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) − AI_T − 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.

Example 12 · Worked

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.

What is the equilibrium quoted euro-bund futures price?

Solution. Collect the inputs: T = 1/12, CF = 0.729535, B₀ = 108, FVCI = 0, r = 0.1%,
AI₀ = (15/180) × (2%/2) = €0.083 and AI_T = (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) − AI_T − 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 F_t to the same expiration leaves no price risk, since the two positions at expiration are (B_T − F₀) and (F_t − B_T), which sum to F_t − F₀. So V_t = PV[F_t − F₀] once again.

A quick illustration: a long forward at F₀ = 119.12 and a short forward at F_t = 119.92, with one month to expiration and a discount rate of 0.5%, gives

V_t = (119.92 − 119.12) ÷ (1 + 0.005)^1/12 = 0.7997.

Example 13 · Worked

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.

What is the value of the euro-bund forward position?

Solution. Both forward prices are given, so no carry calculation is needed:
V_t = PV[F_t − 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 V_t = PV[F_t − 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 S_i 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.

Figure 5: A receive-fixed swap decomposed

The coupon dates on the two bonds match the settlement dates on the swap and their maturity matches the swap expiration. Because the value of a portfolio is the sum of its parts, the swap can be valued by valuing each bond separately.

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 S_i = AP_FLT × r_FLT,i, and the fixed leg is FS = AP_FIX × r_FIX, 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:

Net cash flow to the receive-fixed party. Reverse the signs for the pay-fixed party.

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:

Present value factors from a term structure

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

V_FIX = 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:

One minus the final present value factor, divided by the sum of the factors, scaled by the accrual period.

Example 14 · Worked

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.

Present value factors for the five-year swap

Maturity (years)	Present value factor
1	0.990099
2	0.977876
3	0.965136
4	0.951529
5	0.937467

What is the fixed rate of the swap?

Solution. First add the factors:
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:
r_FIX = [(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:

Value of a receive-fixed swap. The pay-fixed value is the negative of this, NA(FS_t − FS₀) times the same sum.

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.

Example 15 · Worked

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%.

1. What is the swap worth to the party receiving fixed?

Solution. Seven years less two years leaves five annual cash flows, which is why the same five factors apply. With annual resets AP = 360/360 = 1, so the cash flows per unit of notional are FS₀ = 2.0% and FS_t = 1.3%.

V_SWAP,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.

2. What is the swap worth to the party paying fixed?

Solution. Exactly the negative:
−V_SWAP,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:

S₀ is the spot exchange rate quoted as units of currency a per unit of currency b.

For the swap to be worth nothing at inception the two bond values must match, so V_a = S₀V_b, and therefore the notionals must satisfy NA_a = S₀ × NA_b. 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.

Currency swap offset with a pair of bonds, in US dollars

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.

Example 16 · Worked

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.

Spot rates and present value factors at initiation

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)].

1. What are the annual fixed swap rates for Australian dollars and US dollars?

Solution. Apply the swap rate formula separately in each currency, using that currency own factors, and annualise with 360/90 since the resets are quarterly.

r_AUD = [(1 − 0.972763) ÷ 3.933870] × (360 ÷ 90) = 0.027695, or 2.7695%.
r_USD = [(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.

2. What is the US dollar notional amount?

Solution. From NA_a = S₀ × NA_b, with currency a being the Australian dollar:
A$100,000,000 = (A$1.14/US$1) × NA_b, so NA_b = A$100,000,000 ÷ (A$1.14/US$1) = US$87,719,298, that is about US$88 million.

3. What are the quarterly fixed payments on each leg?

Solution. Each payment is the notional times the accrual period times the annual fixed rate.

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 S_t:

The first bracket is a long bond in currency a; the second is a short bond in currency b, expressed in currency a.

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.

Example 17 · Worked

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.

Spot rates and present value factors 60 days later

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.

1. What is the current value of the swap to the swap dealer, expressed in Australian dollars?

Solution. The exchange rate is quoted as A$1.14 per US$1, so currency a is the Australian dollar. The dealer receives quarterly interest in Australian dollars, so the dealer is the party receiving currency a and the equation as written gives the dealer value.

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).

V_CS = 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.

2. What is the current value of the swap to the US firm, expressed in US dollars?

Solution. The firm value is the negative of the dealer value, converted at the current spot rate:
−V_CS = −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:
V_CS = 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 S_i = NA_ER_E, where R_E is the periodic equity return as specified in the contract. The fixed leg is FS = NA_E × AP_FIX × r_FIX, exactly as for an interest rate swap.

Example 18 · Worked

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.

1. The equity index returns 4.0% for the quarter, not annualised. What is the swap cash flow?

Solution. The equity return is already a quarterly figure while the fixed leg is quoted annually, and mixing the two conventions is the classic error here. Using S_i − FS = NA_E × AP × (R_E − r_FIX):
€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.

2. The equity index instead returns −6.0% for the quarter. What is the swap cash flow?

Solution. Same expression, negative equity return:
€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 NA_E, so that whatever is left over is the cash flow required by the swap. If the position has grown, the excess above NA_E is sold off; if it has shrunk, additional shares are bought to restore the position to NA_E.

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 NA_E 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 NA_E, 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 NA_E = 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.

Example 19 · Worked

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%.

What is the fixed interest rate on a EURO STOXX 50 equity swap with these same terms?

Solution. 1.2968%, identical to the interest rate swap. The sum of factors is
0.990099 + 0.977876 + 0.965136 + 0.951529 + 0.937467 = 4.822107,
and r_FIX = [(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:

V_FIX(C₀) is the time t value of a fixed-rate bond carrying the original coupon; S_t is the current equity price and S_t−1 the price at the last reset.

Example 20 · Worked

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 NA_E.

1. What is the current fair value of the equity swap?

Solution. The first reset has not yet been passed, so five cash flows remain, falling at 0.5, 1.5, 2.5, 3.5 and 4.5 years from today. A fixed rate of 1.5% on 10,000,000 gives 150,000 per settlement, with the final settlement adding the 10,000,000 par.

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.

Present value of the fixed leg

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

Step 2. Adjust for the equity. The equity has risen from 100 to 105, so the pay-equity leg has grown by 5%:
V_EQ,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.

2. At what equity price would the swap be worth zero?

Solution. The fixed leg is worth 102.16% of par, computed as (10,216,019 ÷ 10,000,000) × 100. The swap is worth zero when the equity leg matches it, so a price of 102.1602 does it:
V_EQ,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.

Check yourself

1. A non-dividend-paying share trades at 80 and the annual rate is 4% with annual compounding. What is the six-month forward price, and what happens to it if the rate rises to 5%?

F₀ = 80(1.04)^0.5 = 81.58. At 5% it becomes 80(1.05)^0.5 = 81.98, so the forward price rises. The forward price is directly related to the financing rate because the rate is the cost of carrying the spot position. Nothing here reflects a view on where the share is going.

2. A long forward struck at 105 is being valued three months before expiration. The spot price is 110 and the rate is 5%. A colleague says the position is worth 5. What has the colleague missed?

The correct value is 6.2729, not 5. Value is the discounted difference between the new forward price and the old one, and the new forward price is 110(1.05)^0.25 = 111.3499, not 110. The colleague compared the spot price against the forward price, which mixes two different dates. The equivalent shortcut is S_t − PV[F₀] = 110 − 105/(1.05)^0.25, which discounts the old forward price rather than leaving it undiscounted.

3. An index has a continuous dividend yield of 2.5% and the continuously compounded rate is 1.0%. Will the futures price lie above or below the index level, and why?

Below. The exponent is (0.010 − 0.025)T, which is negative, so the futures price is the index level multiplied by a number less than one. Carry benefits exceed carry costs, so the holder of the index is compensated for holding it and the forward buyer must be compensated in turn through a lower price. The futures price then converges up to spot as expiration approaches.

4. A firm plans to borrow in six months and fears rising rates. Should it be long or short an FRA, and which leg does it pay?

It should be long the FRA, which makes it the fixed-rate payer and the floating-rate receiver. If rates rise, the floating rate received exceeds the fixed rate paid and the FRA pays out, offsetting the higher cost of the actual borrowing. The naming is the trap: long an FRA sounds like a bet on prices rising, but the underlying is a rate, so the long gains when rates rise.

5. Why is the FRA settlement amount divided by a discount factor, when a futures payoff is not?

Because FRAs are advanced set and advanced settled. The payoff is based on a rate drawn from a spot deposit market that pays interest in arrears, at time h + m, but the FRA itself pays at time h. The money arrives m days early, so it is discounted for those m days. If the FRA settled in arrears the divisor would disappear. A futures contract needs no such adjustment because it settles daily against the current market price.

6. A bond futures contract has a conversion factor of 0.75. The equilibrium adjusted price is 96. What is the equilibrium quoted price, and why do the two differ so much?

Q₀ = F₀ ÷ CF = 96 ÷ 0.75 = 128. The conversion factor exists because several bonds may be delivered against one contract, and they trade at different prices. Scaling each deliverable to what it would be worth at a 6% yield to maturity puts them on a comparable footing. The quoted price is therefore a scaled number, not the price of any actual bond, and the economically meaningful figure is F₀ = Q₀ × CF. Because the adjustment is only approximate, one bond ends up cheapest to deliver.

7. Two years into a receive-fixed interest rate swap struck at 3.0%, the current equilibrium swap rate for the remaining term is 3.8% and the present value factors for the remaining four annual payments sum to 3.65. On a notional of 50 million, what is the position worth, and to whom?

V_SWAP,t = 50,000,000 × (0.030 − 0.038) × 3.65 = −1,460,000 to the receive-fixed party, and therefore +1,460,000 to the pay-fixed party. Rates rose, so the party locked into receiving 3.0% while the market now pays 3.8% is losing 80 basis points a year for four more years. The floating legs of the original and offsetting swaps cancel, which is why only the fixed rate difference appears.

End of lesson.

Back to 07 Derivatives