Hedging a Rate Book with Swaps

This is the lesson the whole topic has been building toward. You learned what a swap is (two legs trading fixed for floating), how to price and value one off the swap curve, and how a bond’s DV01 measures its dollar sensitivity to rates. Now we fuse those: a swap is just another instrument with a DV01, so you can bolt it onto a portfolio to cancel the portfolio’s rate risk. A trading desk’s bond inventory, a bank’s loan book, a pension fund’s liabilities — all of them are giant bets on rates that nobody meant to make. The swap is the surgical tool that turns that accidental bet to zero. By the end you’ll size a hedge, pick the right side, and — crucially — know exactly which risks the hedge doesn’t kill.

A swap has a DV01

Analogy. A swap is a tug-of-war between a fixed rope and a floating rope. The floating rope is elastic — it stretches to match the market every reset, so it barely pulls when rates move. The fixed rope is rigid steel: locked payments that gain or lose value exactly like a bond as rates shift. So when you ask “how much does this swap move per basis point?”, you’re really asking about the fixed leg.

The definition. DV01 (dollar value of a basis point) is the change in an instrument’s value per 1 bp rate move:

The floating leg of a vanilla swap resets to the prevailing rate each period, so right after a reset it’s worth roughly par and has near-zero duration. All the sensitivity sits in the fixed leg, which is an annuity (the coupons) plus a notional repayment at maturity — i.e. it looks just like a par bond of the same tenor. Hence:

Which way does it lean? A swap has two sides, and the sign of its DV01 flips depending on which you take:

• Receiver swap (receive fixed, pay floating): you own the fixed-rate stream, so you behave like you’re long a bond — positive duration, you gain when rates fall.
• Payer swap (pay fixed, receive floating): you’re short the fixed stream, so you behave like you’re short a bond — negative duration, you gain when rates rise.

Approximate DV01 by tenor. DV01 grows with tenor, because a longer fixed leg means more discounted cash flows reaching further out. Here are rough magnitudes per $1 million notional (around a 3–4% rate environment — exact figures depend on the curve):

Notice the 10-year carries roughly 4.5× the DV01 of the 2-year: doubling-and-then-some per decade of tenor, because duration compounds. This is why desks quote hedges in terms of DV01 per million — it lets you convert any target risk into a notional.

The hedge ratio

The recipe. To neutralise a book to first order, you want the total DV01 — book plus swap overlay — to land at zero. Solve for the swap notional:

then choose payer or receiver so the swap’s DV01 carries the opposite sign to the book’s. Size first, sign second.

Worked example. A bond desk holds long-duration assets with a book DV01 of $50,000 per bp — meaning the book loses $50,000 for every 1 bp rise in rates (long bonds → hurt by rising rates). A 10-year swap has a DV01 ≈ $850 per $1 million notional. Step through it:

1. Size the hedge.

   $$N_{\text{swap}} = \frac{\text{DV01}_{\text{book}}}{\text{DV01 per million}} = \frac{50{,}000}{850} \approx 58.8 \text{ million in notional}.$$

   (The \$ sign sits in the prose around the formula, never inside the math.)

2. Pick the side. The book is long bonds (it loses when rates rise), so we need an overlay that gains when rates rise — that’s a payer swap (pay fixed, receive floating). A receiver swap would add more long-duration risk, doubling down instead of hedging.

3. Check the net. Book: −$50,000 per bp. Payer swap of $58.8 mm: $+58.8 \times 850 \approx +49{,}980$, i.e. +$49,980 per bp. Net DV01 ≈ $0 — a 1 bp move now barely scratches the combined position.

So the desk enters a $58.8 million 10-year payer swap, and its parallel-rate risk is gone. The visual below shows the same idea as a balance scale: the book’s DV01 on one pan, the swap overlay’s on the other — level when they match.

Who hedges which way

The hedge direction isn’t arbitrary — it falls straight out of the shape of your balance sheet. Two classic institutions, two opposite trades:

The bank (asset-liability management, ALM). A bank holds long-duration assets — fixed-rate loans and mortgages — funded by short-duration liabilities — deposits that reprice quickly. When rates rise, the loan portfolio loses value and the deposit funding gets more expensive: a double squeeze. The bank is effectively long duration, so it enters payer swaps — receive floating (which moves with its deposit cost) and pay fixed (locking the spread). The payer swap’s “gain when rates rise” exactly counters the bank’s “lose when rates rise.”

The pension / insurer (liability-driven investing, LDI). A pension promises payouts decades out, so its liabilities are extremely long-duration — often far longer than the bonds it can buy. When rates fall, the present value of those promises balloons, opening a funding hole. To plug it, the fund receives fixed in long-dated swaps, which behaves like owning a very long bond — gaining when rates fall, right alongside the liability. This extends asset duration to match the liability without having to find scarce 30-year+ physical bonds.

The residual risks

A DV01-matched hedge is excellent — and not a force field. It zeroes out one specific risk and leaves several others standing. Know them or get blindsided:

1. Curve risk (twists and steepening). A single-tenor swap assumes rates move in a parallel shift — every point on the curve up or down by the same amount. But curves twist: the 2-year can rise while the 30-year falls (a flattener), or vice versa. If your book’s risk is concentrated at the 5-year but you hedged with a 10-year swap, a steepening leaves you exposed even though your total DV01 reads zero. Fix: hedge with swaps at multiple tenors, matching key-rate DV01s (the DV01 attributable to each maturity bucket) rather than one lumped number.

2. Basis risk. Your book might earn an index the swap doesn’t reference. A loan tied to Prime hedged with a SOFR swap leaves you exposed to the spread between Prime and SOFR drifting. The two move together — mostly — but “mostly” is where basis risk lives. Fix: hedge with the matching index where possible, or accept and size the residual basis.

3. Convexity. DV01 itself isn’t constant — it changes as rates move (that’s convexity, the second-order term). A hedge that’s perfectly DV01-matched today drifts out of balance after a large rate move, because the book’s DV01 and the swap’s DV01 change by different amounts. Fix: re-hedge dynamically — rebalance the swap notional as rates move and as time passes (the book and swap both season).

Putting it together

A swap is just another instrument with a DV01, concentrated in its fixed leg, which behaves like a par bond of the same tenor — so a receiver swap is long-duration (gains when rates fall, like owning a bond) and a payer swap is short-duration (gains when rates rise, like shorting one). To hedge a book you compute $N_{\text{swap}} = \text{DV01}_{\text{book}} / \text{DV01 per unit notional}$, then pick the side whose DV01 carries the opposite sign to the book’s. A bank with long fixed-rate assets pays fixed (payer); a pension with long-dated liabilities receives fixed (receiver, LDI). And the catch that separates pros from amateurs: a DV01 match neutralises parallel shifts only — curve, basis and convexity risk all survive, so real desks hedge multiple key-rate buckets and re-balance dynamically. That’s the full arc of this topic: from a single swap’s two legs, to pricing and valuing it off the curve, to wielding it as the surgical instrument that flattens an entire rate book’s risk.

That’s every teaching lesson in the topic. Next up is the final exam — one shot, one question at a time, no going back. Bring everything: swap mechanics, pricing off the curve, valuation, and the DV01 hedging you just mastered.