FRAs: The Building Blocks of a Swap

You already know what a swap is — two parties trading a fixed cash flow for a floating one. But before we can price one, we need to zoom all the way in to a single screw in the machine. That screw is the FRA: a contract that lets you lock one future interest rate today, on a single notional, for a single period. Master the FRA and a swap stops being a mysterious blob — it becomes a tidy row of FRAs bolted together. And the rate each FRA fairly locks? It’s the forward rate you already met in the yield-curve lessons. So we start there, then build up.

Forward rates: the rate you can lock today for a future period

Pretest first:

Analogy. A spot rate is the price of a trip that starts now. A forward rate is the price, quoted now, of a trip that starts later — like booking next summer’s flight at today’s locked-in fare. The airline can’t make money offering you a future fare that’s inconsistent with today’s prices, and neither can a bank: the forward rate is forced by no-arbitrage from the rates already on the curve.

The definition. A forward rate $f_{1,2}$ is the interest rate, agreed today, for borrowing or lending over a future window — here from year 1 to year 2. It isn’t a forecast and it isn’t a free parameter. It’s locked by the rule that two ways of investing for two years must pay the same:

where $z_1$ and $z_2$ are today’s 1-year and 2-year spot (zero) rates. The left side grows your money at the 2-year rate for two years. The right side grows it at the 1-year rate, then rolls into the forward for the second year. If they differed, you could borrow cheap on one path and lend dear on the other for free money — so the market won’t let them differ.

Worked example — extract the 1-year forward, 1 year out. Take $z_1 = 4\%$ and $z_2 = 5\%$. Solve for $f_{1,2}$:

Working the arithmetic: $1.1025 / 1.04 = 1.06010$, so $f_{1,2} = 0.06010 = 6.01\%$. Notice it lands above both spots — that’s the upward-sloping-curve signature. If the curve had inverted ($z_2 \lt z_1$), the forward would have dropped below both. The forward is a leveraged readout of the curve’s slope, not its level.

The FRA: a forward contract whose underlying is an interest rate

Pretest:

The definition. A forward rate agreement (FRA) is an over-the-counter (privately negotiated, not exchange-traded) contract that locks a single future period’s interest rate, called the FRA rate or strike $K$, on an agreed notional $N$. No principal changes hands — the notional is just the size on which interest is calculated. It’s a forward contract like any other, except the underlying isn’t a barrel of oil or a bushel of wheat; it’s an interest rate.

Who wins which way. There are two sides:

• The buyer / long is, economically, a future borrower. They’ve locked their cost at $K$, so they gain if the reference rate rises above $K$ — the market got expensive, but they pay the cheaper locked rate.
• The seller / short is a future lender. They’ve locked their return at $K$, so they gain if the reference rate falls below $K$ — the market got stingy, but they still earn the higher locked rate.

It’s a zero-sum bet on where one future rate prints relative to $K$. The reference rate $r_{\text{ref}}$ is some agreed benchmark (historically LIBOR, now rates like SOFR or EURIBOR), observed at the start of the contract period — the fixing.

The notation. FRAs are quoted as ”$a$ x $b$”, both numbers being months from today: the rate period starts at month $a$ and ends at month $b$. So a 3x9 FRA starts in 3 months, ends in 9, and locks the 6-month rate that fixes 3 months from now. A 1x4 locks a 3-month rate, one month out. The gap $b - a$ is the length of the rate you’re fixing.

FRA settlement: cash today, discounted because the interest is owed later

Pretest:

The mechanics. Here’s the wrinkle that trips everyone up. The interest an FRA references would, on a real loan, be paid at the end of the period. But an FRA cash-settles a single net payment at the start of the period, right after the rate fixes. Paying early means you must discount the interest difference back to the settlement date. The settlement to the long (buyer) is:

where $N$ is the notional, $K$ the strike, $r_{\text{ref}}$ the realized reference rate, and $\tau$ the day-count fraction — the period length as a fraction of a year (e.g. a 6-month period is roughly $\tau = 0.5$). The numerator is the plain interest difference; the denominator $1 + r_{\text{ref}}\,\tau$ is the discount factor that drags that end-of-period amount back to the start. If $r_{\text{ref}} \gt K$, the number is positive and flows to the long; if $r_{\text{ref}} \lt K$, it’s negative and the long pays.

Worked example. A treasurer buys a 6x12 FRA on a $10 million notional at a strike of $K = 5\%$, fixing a 6-month rate. Six months later the rate fixes at $r_{\text{ref}} = 6\%$. Day-count $\tau = 0.5$.

• Raw interest difference (what an actual loan would settle at period-end): $$N\,(r_{\text{ref}} - K)\,\tau = 10{,}000{,}000 \times (0.06 - 0.05) \times 0.5 = 10{,}000{,}000 \times 0.005 = 50{,}000,$$ i.e. $50,000.
• Discount factor (settle now, not in 6 months): $$1 + r_{\text{ref}}\,\tau = 1 + 0.06 \times 0.5 = 1.03.$$
• Settlement paid to the long at the start of the period: $$\frac{50{,}000}{1.03} = 48{,}543.69,$$ i.e. $48,543.69.

So the long pockets $48,543.69 up front. Why not the full $50,000? Because that $50,000 was an end-of-period figure; receiving it six months early is worth a touch less, so we present-value it. The treasurer is exactly compensated: the cheaper-than-market locked rate saved $50,000 of future interest, and $48,543.69 today is $50,000 in six months at 6%.

A swap is a strip of FRAs

Pretest:

The central insight. Here’s the payoff for all that FRA grinding. Take a multi-period swap and slice it period by period. Each floating period is, economically, a single FRA — one future reference rate exchanged against a fixed rate, on the notional, for that window. The fair fixed rate for that period alone is simply its forward rate, read off today’s curve. So a swap is a strip of FRAs: a row of single-period rate locks, one per payment date, stapled together.

Now the trick that turns a strip of FRAs into a swap. In a swap, both sides want one constant fixed rate across all periods, not a different forward rate every period. So we replace the jagged sequence of forward rates with a single fixed rate — the swap rate — chosen so the deal is still fair overall. That swap rate is a discount-factor-weighted average of the forward rates: the periods don’t count equally, because later cash flows are discounted more, so nearer forwards carry more weight.

The swap rate (~5.2% here) isn’t any single forward — it’s the level that, applied flat to every period, has the same present value as paying each period’s own forward. Lock each period at its forward (a strip of FRAs) or lock all periods at the blended swap rate (a swap): same total value today, by no-arbitrage.

When to use it

Use an FRA when you have exactly one future interest-rate exposure to neutralise — a single loan rolling in three months, a deposit maturing in six. Use the forward rate as the fair strike to set it at, and remember the settlement is discounted to the period start, not paid in full at period end. Reach for the strip-of-FRAs view the moment you face a multi-period exposure: that’s a swap, and decomposing it into its component FRAs is how you’ll price it, risk it, and hedge it. The FRA is the atom; the swap is the molecule.

Putting it together

A forward rate is the interest rate you can lock today for a future period, pinned by no-arbitrage so that rolling the near spot into the forward reproduces the far spot’s compounded return — pushing the forward above both spots on an upward curve. An FRA crystallises one such rate into a binding OTC contract: it locks a strike $K$ on a notional for a single window (quoted “$a$x$b$” in months), the long gains when the reference rate rises above $K$, and it cash-settles up front at the discounted value $N(r_{\text{ref}} - K)\tau / (1 + r_{\text{ref}}\tau)$ — discounted precisely because the referenced interest belongs to the period’s end. And the grand reframing: a swap is a strip of FRAs, each period’s fair rate being its forward, with the single swap rate a discount-factor-weighted average of those forwards. Hold that picture and a swap is no longer a black box — it’s Lego.

Next — pricing a swap as the difference of two bonds — we cash in the strip-of-FRAs insight and value the whole thing as a fixed-rate bond minus a floating-rate bond, solving for the swap rate that makes them balance at inception.