FMP 20 – Swaps

A swap is an over-the-counter contract under which two parties agree to trade cash flows on a series of future dates. What each side owes turns on the future level of a variable, an interest rate, an exchange rate, an equity price or a commodity price, which market participants call a risk factor. A forward contract is the simplest member of the family, with a single exchange date. The package as a whole is usually worth zero when it is struck, although the exchanges inside it carry offsetting positive and negative values.

Fixed for floating on a principal that never moves

In a plain vanilla interest rate swap one leg pays a fixed rate while the other pays a floating reference rate, SOFR and SONIA being the usual choices. Party A might contract to hand Party B 2% fixed, quarterly compounded, on USD 100 million for three years, receiving three-month SOFR on that same amount over the same period, with interest traded every three months. The USD 100 million is the notional principal. It never changes hands, and its only job is to scale the interest calculation.

Where the floating leg is built from realised overnight rates the contract is an overnight indexed swap, or OIS, and these keep gaining ground as Libor is retired. A Libor rate for a period is known at the start of that period, while an OIS rate stays unknown until the period has ended, because it is compounded from daily rates as they arrive. Either way the rate settles an exchange at the end of the period. An OIS of a year or less usually produces one exchange at maturity; longer contracts pay quarterly. Figures from the Bank for International Settlements put interest rate swaps at the top of the over-the-counter derivatives market.

Take a hypothetical two-year OIS that Apple and Citigroup initiate on March 8, 2022. Apple undertakes to pay a fixed 3% per annum, quarterly, on a $100 million notional principal, while Citigroup pays three-month SOFR, the floating reference rate, on that same amount. Apple is the fixed-rate payer and Citigroup the floating-rate payer. Assume quarterly compounding and set day count and holiday conventions aside. Two years of quarterly exchanges give eight exchange dates.

The first falls on June 8, 2022. Apple pays $750,000, a quarter of the 3% rate on the $100 million, and Citigroup pays the rate compounded from the daily SOFR fixings running from March 8, 2022 to June 8, 2022, assumed here to be 2.2% a year, which is 0.55% over three months and so $550,000. Netted, Apple hands over $200,000. On September 8, 2022 the fixed payment is again $750,000 while SOFR has reached 2.6% a year, worth 0.65% over the quarter, so the floating payment is $650,000 and the net from Apple is $100,000.

Source: chapter scenario. The rate gap and running total columns are derived here.

The rate gap is the floating rate less the 3% fixed rate, and each net payment is a quarter of that gap applied to the principal. A gap of -0.20% on Dec 8, 2022 means $50,000 paid; a gap of +0.10% on Mar 8, 2023 means $25,000 received. Apple pays out for three quarters and collects thereafter, the running total reaching zero on December 8, 2023 and $200,000 by the end.

Those figures are approximations, because dividing an annual rate by four ignores how many days a quarter contains. Floating reference rates such as SOFR carry an actual/360 quotation basis. Take the first floating payment, based on 2.2%: 92 days separate March 8, 2022 from June 8, 2022, so the amount really paid is

somewhat above the rounded USD 550,000 shown in the table. The fixed rate of 3% carries a convention too, and the candidates are actual/365 and 30/360. Under 30/360 the fixed cash flows in the table are correct, leaving aside holidays and weekends. Under actual/365 the first fixed payment becomes

rather than the USD 750,000 in the table. So quoted fixed and floating rates are not directly comparable, since one applies over 360 days and the other over a whole year, and the small gaps compound across eight exchanges.

When a payment date is a weekend or a holiday

Some rule has to say what happens when a scheduled date is not a business day. The commonest is the next business day convention, moving payment to the first business day after the specified date, while the preceding business day convention moves it back. The modified following business day convention rolls forward, but reverts to the preceding business day whenever that would carry the payment into a different month.

The confirmation

The agreement covering an over-the-counter derivatives transaction is called a confirmation. It sets the payment dates, the method of calculation, the day count conventions in force and whose holiday calendars apply, which is why two swaps quoted at the same rate can settle different amounts.

Where both sides of a standard interest rate swap are financial institutions, clearing through a central counterparty is compulsory. Each side then faces the CCP and posts initial margin and variation margin, and a party that is not a CCP member clears through one that is. With an end user on one side the swap can be cleared bilaterally, the confirmation stating that a master agreement already entered into governs it. The International Swaps and Derivatives Association supplies much of that documentation.

Large financial institutions make markets in interest rate swaps, quoting a rate at which they will pay fixed and a higher one at which they will receive fixed. The table shows quotes a dealer might post where exchanges fall every three months. Averaging the bid and the ask gives the swap rate, and the final column, the gap between them in basis points, is where the dealer earns its living.

Source: chapter quote table. The spread column is derived here.

Read the two-year row as a pair of standing offers. The dealer will enter a swap paying 2.95% against a floating rate, and one receiving 2.97% against a floating rate, the particular reference rate being left unspecified for generality. A corporate treasurer accepting the five-year bid on a notional principal of USD 100 million receives 3.08% and pays floating for five years.

That leaves the dealer holding the opposite exposure. Should another end user arrive wanting to pay fixed at five years, the two positions offset and the spread is kept; otherwise the dealer lays the risk off with another financial institution. The spread widens from two basis points at two years to six at thirty.

Which rates are used for discounting

Valuing a swap requires a risk-free discount rate. Treasury rates look like an obvious candidate but sit too low to serve, so the convention is to derive risk-free rates from OIS rates. An OIS of a year or less has a single exchange at the end, so its fixed rate directly estimates the zero-coupon risk-free rate for that horizon. Longer contracts pay quarterly, so their fixed rates are par yields, and the bootstrap method strips zero-coupon rates from that sequence. Libor was the pre-crisis proxy; interbank overnight rates have taken over.

Interest rate swaps let a company transform an asset or a liability, rewriting its interest character without reopening the underlying contract. A floating-rate loan becomes a fixed-rate liability, a fixed-rate loan becomes a floating-rate liability, and a fixed-rate bond becomes floating-rate income. The net rate is read off the three legs together.

Suppose a borrower has taken USD 10 million of five-year debt carrying a fixed 3.5%, compounded quarterly, with interest paid quarterly. Accepting 3.08%, the five-year bid, produces three sets of cash flows: 3.5% paid on the borrowings, the floating rate paid under the swap, and 3.08% received under the swap, all on USD 10 million. Netting gives floating plus 3.5% less 3.08%, which is floating plus 0.42%.

Now take a two-year borrowing of USD 10 million priced at three-month floating plus 30 basis points, so the rate is observed quarterly and interest 0.3% above it is settled three months later. Accepting 2.97%, the two-year ask, leaves the borrower paying floating plus 30 basis points on the loan, receiving floating under the swap and paying 2.97%, for a fixed 3.27%.

Source: chapter examples, arranged here as a single comparison.

Assets work the same way with the signs reversed. The ten-year investor receives 2.8% and the floating rate and pays 3.18%, ending at floating minus 38 basis points; the three-year investor receives floating minus 0.1%, pays floating and receives 3.06%, ending at 2.96%.

One long-standing explanation for the swap market starts from the claim that highly creditworthy companies find fixed-rate borrowing much easier to arrange. Consider the five-year rates offered to Company A, with a high credit rating, and Company B, with a lower one, both involving quarterly payments.

Source: chapter rate table. The final row is derived here.

The gap differs between the two markets. In the fixed market the advantage held by Company A runs to 1.4%, while in the floating market it narrows to 0.8%, which is why Company A is said to hold a comparative advantage in fixed-rate borrowing and Company B in floating-rate borrowing.

A swaps trader would say to borrow where the comparative advantage lies and then swap into the market you actually want. Suppose Company A wants floating-rate borrowing, Company B wants fixed-rate borrowing, and the two deal directly: Company B pays X% fixed to Company A, and Company A pays floating back.

Two non-financial companies of this sort are unlikely to find each other directly, so in practice a swaps trader at a financial institution stands between them, running one swap with each side. Suppose the bank pays Company A 3.68% fixed and receives 3.72% from Company B, with the floating legs running the other way in each swap.

Company A pays 3.5% on its borrowings, receives 3.68% and pays floating, ending at floating minus 0.18%, an improvement of 0.28% on the floating plus 0.1% available directly. Company B pays floating plus 0.9%, pays 3.72% and receives floating, ending at 4.62%, again 0.28% better than 4.9%. The bank keeps 0.04%, and 0.28% twice plus 0.04% is the same 0.6% as before.

Why the gain is largely illusory

The weakness is that Company B is not actually being handed a fixed rate. Take the fixed-rate offer and 4.9% is locked in for five years. Take the floating-rate offer and only the reference rate floats freely: the 0.9% spread over it is committed three months at a time, and at each reset it becomes whatever the market then makes of the creditworthiness of Company B.

Suppose that by year two Company B is paying floating plus 2% instead of floating plus 0.9%. The swap then leaves it paying 3.72% plus floating plus 2% minus floating, or 5.72%, worse than the 4.9% it could have fixed at the outset. Company A faces no such uncertainty, borrowing at floating minus 0.18% for five years on funding fixed for the whole term. So swapping floating into fixed delivers a fixed-rate liability only while the borrower keeps the same spread over the reference rate.

At inception a swap is worth zero or close to it, bid-ask spreads leaving it slightly positive to the dealer posting the quotes and slightly negative to the client accepting one. As rates move it acquires value, and two routes lead there. The first treats the swap as a pair of bonds held at once.

The fixed bond is the stream of fixed coupons plus the notional principal at maturity, discounted at risk-free rates. The floating bond is worth exactly the notional principal immediately after a reset date, since a bond paying the current market rate prices at par. Reverse the subtraction for a payer of fixed.

A practical variant compares the swap against a new one known to be worth zero. Consider a swap struck some time ago with two years left, on which 4% fixed is received against floating paid quarterly, on USD 100 million of principal. Treat the dealer quotes above as today’s market: a new two-year swap prices at 2.96%, the mid-market rate, so a swap paying 2.96% and receiving floating is worth zero today.

The method carries over to any floating rate for which dealer quotes exist, with discount rates taken from OIS quotes, and it needs one market number rather than a full forward curve. Where the remaining life falls between two quoted maturities, interpolate: at 2.5 years the fixed rate is half of (2.96% + 3.075%), namely 3.0175%.

The second route dismantles the swap into its individual exchanges. Every exchange settles one floating rate on one future date against a fixed rate, which is what a forward rate agreement does, so a swap with eight exchanges left is a portfolio of eight FRAs worth the sum of their values.

Valuing a single FRA follows one rule: assume the forward rate will be realised. Read the forward rate for the period off today’s curve, treat it as the floating rate that will apply, compute the net exchange, then discount at the risk-free rate for that payment date. The route needs a whole forward curve but prices each period separately.

A fixed-for-fixed currency swap trades a fixed interest rate in one currency for a fixed interest rate in another. Two principal amounts are specified, one per currency, and unlike an interest rate swap these principals really are exchanged: opposite in direction to the interest payments at the start, and in the same direction at the end.

Take a swap where USD interest of 5% on a USD 10 million principal is received, against euro interest of 4% on a principal of 8 million euros paid away, both annually compounded and exchanged yearly for four years. Each year the USD leg carries 0.05 x USD 10 million, USD 0.5 million, and the euro leg 0.04 x 8 million euros, 0.32 million euros.

Source: chapter cash flow table. The final column is derived here.

The two principals imply an exchange rate of 10 divided by 8, or 1.25 USD per euro, and the final column values every exchange at that rate. The initial exchange is then worth nothing and each later one a constant 0.1 million USD, the 1% gap between the coupon rates applied to the USD principal. That difference is the price of the exchange rate risk.

Transforming borrowings and investments

The same swap converts borrowings at 5% on USD 10 million into borrowings at 4% on 8 million euros. The opening exchange turns the USD principal into a euro principal, and each later one supplies the USD interest and, at maturity, the USD principal needed to service the original loan, against euro payments. Reversed, an 8 million euro investment earning 4% becomes a USD 10 million investment earning 5%.

Comparative advantage was largely illusory for interest rate swaps because floating spreads reset. For currency swaps a genuine version can exist, created by tax: a company facing a low marginal tax rate in France while its marginal rate in the United States is high may prefer to borrow in USD, take the interest deduction where it is worth most, and swap into euros. Two variants complete the family, a floating rate in one currency against a fixed rate in another, and floating against floating across currencies. Each leg is valued in its own currency.

Each of the two cash flow streams in a swap is called a leg, so the contract above carries one leg in USD and one in euros. Currency swaps are designed to be worth close to zero at negotiation, and afterwards each leg is valued separately. With principal and interest in currency X traded for principal and interest in currency Y, and the answer wanted in currency X, four steps follow: present value each side in its own currency, convert the currency Y figure at the current spot rate, then subtract it from the currency X figure.

That the two routes agree is no coincidence: they rearrange the same cash flows, and the forward rates come from the two discount curves used in the bond calculation. A currency swap also concentrates far more risk in the final exchange, since the principals move at maturity.

The exchanges inside a swap admit almost any definition. One variation lets the notional principal change on a pre-determined schedule, which suits a company whose borrowings will grow through a multi-year project, and fixed can still be swapped for floating on the varying principal.

An equity swap trades a fixed return against whatever a pre-specified equity delivers on the notional principal. The parties might agree that the return on USD 10 million placed in the S&P 500 in each successive six-month window across five years is swapped for what USD 10 million earns at a fixed interest rate of 4%. The index is normally the total return index, with dividends reinvested.

A commodity swap periodically trades a set quantity of a commodity at a pre-determined fixed price for the same quantity at the floating price ruling at the time, which makes it equivalent to a portfolio of forward contracts. A volatility swap trades, at the end of each period, the historical volatility observed over that period against a pre-determined fixed volatility, both scaled by the notional principal, and a variance swap does the same with the variance rate, the square of volatility.

A credit default swap works as insurance against a default by a company. The protection buyer pays a periodic fixed premium to the seller over an agreed period, and if the reference entity defaults the seller pays out; otherwise nothing is owed. An index credit default swap names a portfolio of companies instead, compensating the buyer for defaults by any member.

Some structures, at the exotic swap end of the market, become overly complicated. The 5/30 swap defined by Bankers Trust and sold to several clients in 1993 tied its payments in an intricate way to three things at once: the 30-day commercial paper rate, the price of a 30-year Treasury bond, and a five-year Treasury bond yield. The Spanish bank Santander in 2007 defined swaps whose payments turned on whether a three-month reference rate stayed within a 2% to 6% range. Clients in both cases were alleged not to have understood the risks. Bankers Trust itself went to Deutsche Bank for USD 10.1 billion in 1998.

Credit risk in a swap position

A swap creates credit exposure once it has moved into positive value for one side. If Company X runs into financial difficulty and defaults at a moment when Company Y holds outstanding swaps against it that are in the money, Company Y takes a loss, though exposures are netted across transactions rather than assessed one contract at a time.

Between financial institutions, interest rate swaps and index credit default swaps have to clear through a central counterparty, which requires initial margin and variation margin from both sides and greatly reduces credit risk, leaving the separate question of a default by the CCP itself. Other swaps clear bilaterally, and regulation still requires both kinds of margin: variation margin passes directly between the parties, while initial margin sits with a third-party trustee.

No collateral requirement applies between a financial institution and a non-financial company, so where nothing is posted, or a transaction is only partly collateralised, the exposure must be monitored carefully and expected credit losses belong in the initial pricing. Credit value adjustment is the downward adjustment a bank makes to a derivative price for expected losses from a counterparty default; debit value adjustment is the upward adjustment reflecting a possible default by the bank itself. Both need estimates of default probabilities.