DER 7 Pricing and Valuation of Interest Rate and Other Swaps

How a swap breaks down into a series of forwards, how the fixed swap rate is set, and how a swap gains or loses value over its life. Check-yourself items are study aids.

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.

Series of FRAs versus a standard interest rate swap

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

Check yourself

Name the two features that separate a series of FRAs from a standard interest rate swap.

The FRA settles at the start of its interest period while the swap settles at the end of each period, and the series of FRAs uses a different fixed rate every period while the swap uses one constant fixed rate throughout.

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.

Coupons, prices, yields, and zero rates

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.

Figure 1: Par swap rate against the spot and forward curves

The fixed swap rate sits as a flat line, an internal rate of return on the rising forward curve. Some individual period exchanges have positive present value and others negative, but they net to zero at inception.

Example 1 · Worked

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%.

Solve for the three-year par swap rate, s₃.

Solution. Discount the forward rates at their zero rates to get the present value of the floating leg, then set it equal to the fixed leg written as a common s₃:

Dividing, s₃ = 0.111017 divided by 2.800545 = 3.9641%. This rate is, in effect, the internal return the forward curve earns: a party would be indifferent between paying it and receiving the forward rates, or the reverse. In the first year a fixed-rate receiver would collect a net 1.5681% (= 3.9641% minus 2.396%) on the notional, showing that individual periods need not net to zero even though the whole contract does.

Example 2 · Worked

At the trade date, Ace observes zero rates of 2.2727%, 3.0323%, and 3.6355% over three periods.

Find the three-period par swap rate, s₃.

Solution. First convert the zero rates into forward rates: IFR of 2.2727%, 3.7975%, and 4.8525%. Then discount those forward rates at the zero rates and set the floating and fixed present values equal:

So s₃ = 0.10159 divided by 2.81819 = 3.605%. The common error is to discount the zero rates by the forward rates, or to discount zero rates by zero rates; the correct routine discounts the implied forward rates at the zero rates.

Check yourself

True or false: over a swap’s life, the MRR is what internally discounts its forward rates to break even.

False. That role belongs to the fixed swap rate: it is the single rate that internally discounts the forward stream to break even across the swap’s life.

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.

Example 3 · Worked

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.

Work out Esterr’s all-in quarterly interest cost when the MRR sets at 3.75% and, separately, at 1.25%.

Solution. When MRR sets at 3.75%, Esterr receives 3.75% and pays 2.05% on the swap, a net receipt of 1.70%, while the loan costs 5.25% (= 3.75% + 1.50%). The net expense is 5.25% paid minus 1.70% received = 3.55%. When MRR sets at 1.25%, Esterr pays a net 0.80% on the swap (2.05% minus 1.25%) and pays 2.75% on the loan (= 1.25% + 1.50%), for a net expense of 2.75% + 0.80% = 3.55%. Whatever the MRR, the cost locks at 3.55% (the 2.05% swap rate plus the 1.50% spread), roughly CAD2,218,750 each quarter (= 3.55% times CAD250 million times 0.25).

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.

Cash bond positions versus interest rate swaps

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.

Check yourself

A portfolio manager expects interest rates to fall and wants to gain without buying bonds. Which swap position fits, and why?

Receive fixed, pay floating. The fixed receipts resemble owning a fixed-coupon bond, so the position records a mark-to-market gain as rates fall, the same way a long bond would.

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.

Example 4 · Worked

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%.

What does the swap settle at for the investor at the six-month date?

Solution. As the fixed-rate payer, apply the settlement formula: (0.25% minus 1.12%) times EUR100 million times 0.5 = negative EUR435,000. Because the MRR set below the fixed rate, the fixed-rate payer owes the net amount at period end. Dropping the 0.5 period would wrongly return EUR870,000, and flipping the sign would describe the counterparty, not the investor.

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.

Example 5 · Worked

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.

Implied forward rates on Esterr’s swap at inception

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%

Find the first two periodic settlements, and state the direction of Esterr’s value immediately after the second settlement.

Solution. Period 1 is (0.5% minus 2.05%) times CAD250 million times 0.25 = negative CAD968,750. Period 2 is (0.74% minus 2.05%) times CAD250 million times 0.25 = negative CAD818,750. Esterr expects to pay net through period 7, where the MRR of 1.94% is still below 2.05%, and to receive net from period 8 onward, where 2.18% exceeds 2.05%. Having paid out in the early periods, the present value of Esterr’s remaining floating receipts now exceeds the present value of its remaining fixed payments, so on this upward-sloping curve Esterr carries a mark-to-market gain.

Example 6 · Worked

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.

Find the first settlement value, then give the direction of value (a) as time passes with rates unchanged and (b) if forward rates fall.

Solution. The first settlement, in six months, is (2.38% minus 0.71%) times GBP200 million times 0.5 = GBP1,670,000 received. (a) With time passing on an unchanged, upward-sloping curve, the present value of the floating payments Fyleton makes rises above the present value of the fixed payments it receives, so the receive-fixed position takes a mark-to-market loss. (b) A fall in forward rates lowers the present value of the floating leg below the fixed leg, giving a mark-to-market gain, exactly as a long fixed-rate bond gains when rates decline.

A swap read as two bonds

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.

Check yourself

What two components make up a swap’s worth on a given settlement date?

That day’s settlement amount, added to the discounted value of every settlement still to come.

Overview

Swaps as a Series of Forwards

Solving for the Par Swap Rate

From zero rates to forward rates

The par swap rate condition

Why Issuers and Investors Prefer Swaps

An issuer transforming a liability

An investor adjusting exposure

Swap Value and Price Over the Contract's Life

The periodic settlement value

How value moves with time and with rates