DER 7 Pricing and Valuation of Interest Rate and Other Swaps
A swap is a firm commitment under which two parties trade a stream of future cash flows. A forward, by contrast, arranges a single exchange of value on one later date. The most common swap exchanges fixed interest payments for floating payments, and although this lesson works through interest rate swaps, the same logic carries over to any underlying on which a stream of cash flows is swapped.
The link to forwards is direct. A forward rate agreement (FRA) settles one cash flow equal to the difference between a fixed rate fixed at inception and a market reference rate (MRR) observed later, applied to a notional over a set period. That single settlement behaves like a one-period swap. Stringing several such exchanges together, one per period, produces something very close to a swap: in both structures the net of a fixed rate against the realized MRR drives each cash settlement on the notional.
Two differences matter. First, an FRA settles at the start of its interest period, whereas a standard swap settles at the end of each period. Second, and more important for pricing, the term structure gives a different implied forward rate for each future period, so a series of FRAs carries a different fixed rate at every maturity. A standard interest rate swap instead applies one constant fixed rate across all of its periods. What the two share is a symmetric payoff, no cash changing hands at inception, and exposure to counterparty credit risk on both sides.
A fixed-rate payer, on either an FRA or a swap, comes out ahead when the MRR sets above the agreed fixed rate, collecting the net amount the floating-rate payer then owes; the fixed-rate payer pays out when the MRR sets below the fixed rate.
| Feature | Series of FRAs | Standard interest rate swap |
|---|---|---|
| Settlement timing | Start of each interest period | End of each interest period |
| Fixed rate | Differs by maturity (each period’s forward rate) | One constant rate for every period |
| Payoff profile | Symmetric | Symmetric |
| Cash at inception | None | None |
| Counterparty credit risk | Yes | Yes |
Pricing a swap means finding its fixed rate. That rate, called the par swap rate, is the one fixed rate whose fixed leg carries the same present value as the expected floating leg, so the contract is worth zero to both sides at inception. The floating cash flows are not yet known, but at the trade date they are expected to equal the implied forward rates for each period, and those forward rates come from the zero (spot) curve.
From zero rates to forward rates
An implied forward rate is the break-even reinvestment rate between a shorter zero rate and a longer zero rate: the rate for a future period at which no riskless profit is available. It is recovered from the no-arbitrage relation below.
Consider three newly issued annual fixed-coupon government bonds whose prices imply the zero rates listed below.
| Years to maturity | Annual coupon | PV (per 100 FV) | YTM | Zero rate |
|---|---|---|---|---|
| 1 | 1.50% | 99.125 | 2.3960% | 2.3960% |
| 2 | 2.50% | 98.275 | 3.4068% | 3.4197% |
| 3 | 3.25% | 98.000 | 3.9703% | 4.0005% |
Applying the relation gives the spot one-year rate and the forward rates that begin in one and two years:
One way to lock in a multi-period exposure worth zero at inception is a series of FRAs struck at these three different forward rates. A swap instead replaces them with a single fixed rate held constant over all three periods.
The par swap rate condition
The starting point is the par bond rate: the coupon PMT at which a fixed-coupon bond prices to par (100) when its cash flows are discounted at the zero rates.
A swap exchanges a constant fixed rate for the expected floating rates and returns no principal, so the par bond rate is adapted into the par swap rate: set the present value of the floating leg (the forward rates) so that it matches the fixed leg (the constant rate), with each period discounted at its own zero rate.
The three-bond example gives zero rates of 2.3960%, 3.4197%, and 4.0005%, from which the implied forward rates are 2.396%, 4.4536%, and 5.1719%.
At the trade date, Ace observes zero rates of 2.2727%, 3.0323%, and 3.6355% over three periods.
Financial intermediaries tend to manage rate exposure period by period with FRAs or short-term interest rate futures. Issuers and investors usually prefer swaps, because a single contract matches a whole stream of asset or liability cash flows and spares them the work of arranging and tracking many separate forwards at many different rates.
An issuer transforming a liability
An issuer with a floating-rate liability can turn it into a fixed cost by paying fixed on a swap whose dates line up with the loan. The floating receipts on the swap offset the floating loan interest, and what remains is a fixed outflow equal to the swap rate plus the loan spread.
Esterr Inc. holds a CAD250 million floating-rate loan priced at three-month Canadian MRR plus a 150 basis point spread, with 3.5 years left and quarterly payments. It adds a CAD250 million swap on which it pays a fixed quarterly 2.05% and receives three-month MRR on matching dates. Every interest period runs 0.25 year.
An investor adjusting exposure
Because a swap delivers periodic fixed cash flows through maturity, it behaves much like a fixed-coupon bond of the same maturity, which makes it a handy substitute for a cash bond position. Receiving fixed resembles holding a long fixed-rate bond; paying fixed resembles a short one. Swaps are typically more liquid than the underlying bonds, so active managers often reach for them to shift interest rate exposure rather than trading the bonds directly.
| Instrument and position | Higher interest rates | Lower interest rates |
|---|---|---|
| Long fixed-rate bond | Loss | Gain |
| Receive-fixed swap | Loss | Gain |
| Short fixed-rate bond | Gain | Loss |
| Pay-fixed swap | Gain | Loss |
The pairing is exact: a receive-fixed swap tracks a long bond, and a pay-fixed swap tracks a short bond. A manager who expects rates to fall can therefore receive fixed on a swap, instead of buying bonds, to capture a gain as rates decline.
The price of a swap is its fixed rate, the par swap rate solved for earlier. Its value is what the contract is worth at a point in time. Like other forwards, a swap is worth zero at inception once transaction and counterparty credit costs are set aside. Unlike a plain forward, which settles once at maturity, a swap settles repeatedly, with a final settlement on the last date.
The periodic settlement value
Recall that a long forward is worth the spot price minus the forward price at maturity. For a swap, the current MRR plays the part of the spot price and the fixed rate plays the part of the forward price. For the fixed-rate payer, each period settles at the difference below.
The value of the swap on any settlement date is that current settlement amount plus the present value of every remaining future settlement. For this lesson the MRR is assumed to set at the start of each period, on the same schedule and day count as the fixed rate, with the net exchanged at the end of the period.
An investor takes the fixed-payer side of a 10-year EUR100 million swap struck at 1.12% versus six-month EUR MRR, which fixes today at 0.25%.
How value moves with time and with rates
Two forces move a swap’s value: the simple passage of time and shifts in the forward curve. For the party that pays fixed and takes floating, the swap posts a mark-to-market gain whenever the present value of the floating receipts tops the present value of the fixed payments, and a loss when the ranking flips. A rise in expected forward rates lifts the present value of the floating leg while the fixed rate stays put, so it favors the fixed-rate payer and hurts the fixed-rate receiver.
Return to Esterr’s 3.5-year CAD250 million swap: pay 2.05% fixed, receive three-month MRR. Assume the forward curve stays exactly as it was at inception, with the per-period implied forward rates below.
| Period | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| IFR | 0.50% | 0.74% | 0.98% | 1.22% | 1.46% | 1.70% | 1.94% | 2.18% | 2.43% | 2.67% | 2.91% | 3.15% | 3.39% | 3.78% |
Fyleton Investments enters a five-year, receive-fixed GBP200 million swap at 2.38% semiannually against six-month GBP MRR to lengthen the duration of its portfolio. Initial six-month GBP MRR sets at 0.71%, and the forward curve slopes upward.
Paying fixed and receiving floating is equivalent to holding a floating-rate note that pays the MRR while having issued a bond that pays the swap coupon. At the outset the two bond prices offset, and at maturity the two principals repay each other, so all that survives is the swap of a fixed coupon for a floating one. That is why the pay-fixed side behaves like a short bond and trims duration, while the receive-fixed side behaves like a long bond and extends it.