Core Path · L2.6.04

Hedging interest rate risk

12 minHedgingInterest rate risk

Know these first

Math Atoms

By the end you will be able to

  1. 1Use DV01 and duration to size a hedge for a bond portfolio.
  2. 2Explain how a pay-fixed swap hedges a fixed-rate asset, and work out the notional.
  3. 3Estimate the hedged and unhedged profit or loss after a rate move.
  4. 4Explain what a duration hedge leaves behind: convexity, curve twists and the need for collateral.

Why it matters

In 2022 US interest rates rose faster than they had for decades. Banks holding long fixed-rate bonds saw them lose value, and Silicon Valley Bank, with large losses and few hedges, failed in March 2023 (L2.1.07: Landmark failures: SVB and Archegos). Hedging interest rate risk is a matter of one number, DV01, and one contract, the swap.

Core explanation

Make the DV01 add up to zero

When yields rise, bond prices fall. Modified duration tells you by roughly what percentage: a bond with a duration of 8 loses about 8% if its yield rises by one percentage point (L1.4.08: Duration). DV01 says the same in money: how much the holding loses for each basis point (L2.5.04: Sensitivities). DV01s add up across a portfolio, which makes hedging a matter of arithmetic.

The idea. Add a position that gains when rates rise, with the same DV01 as the portfolio. The total DV01 is then close to zero: the position is DV01-neutral (or duration-neutral), and small parallel moves in rates barely change its value. Two tools do this.

Sell bonds or bond futures

A short bond position gains when prices fall. Simple, but a short has to be borrowed through repo (L1.6.04: Repo and securities financing) and its maturity rarely matches the portfolio.

Pay fixed on a swap

A pay-fixed swap (L1.7.04: Interest rate swaps) gains when rates rise. It can be written for any maturity and needs no cash up front, only collateral.

Why a pay-fixed swap is like a short bond. The payer pays a fixed coupon and receives a floating one. Paying fixed coupons is like having issued a fixed-rate bond; receiving floating is like owning a floating-rate note, whose value hardly moves because its coupon resets to the market. So a pay-fixed swap behaves like a short position in a fixed-rate bond of the same maturity, and has about the same DV01. A bank that buys a fixed-rate bond and pays fixed on a matching swap has, in effect, turned the bond into a floating-rate asset: an asset swap.

The mirror image. Pension funds and life insurers have long fixed promises that rise in value when rates fall. They hedge by receiving fixed on long swaps or buying long bonds, often under the name liability-driven investment (L1.2.05: Pension funds).

What a DV01 hedge leaves behind. Predict each answer, then reveal it.

Think firstAfter a DV01-matched hedge

  • Rates jump by 2 percentage points in parallel. Is the hedged position still flat?

  • You hedge a 10-year bond with a 5-year instrument. Ten-year yields rise by 1 point, 5-year yields by only 0.75. What happens?

  • Rates fall sharply. What does the hedged bank gain?

The formal version

We build on L1.4.08: Duration. For a holding of value VV with modified duration DmodD_{\text{mod}}, a change Δy\Delta y in yield gives a percentage change (see the Math Atom on percentages) and a DV01:

ΔVV≈−Dmod ΔyDV01=Dmod×V×0.0001\frac{\Delta V}{V} \approx -D_{\text{mod}}\,\Delta y \qquad \text{DV01} = D_{\text{mod}} \times V \times 0.0001

Add a hedge of size n, each unit with its own DV01, and ask for the total to be zero:

DV01P−n×DV01H=0\text{DV01}_P - n \times \text{DV01}_H = 0
n=DV01PDV01H=DP×VPDH×VHn = \frac{\text{DV01}_P}{\text{DV01}_H} = \frac{D_P \times V_P}{D_H \times V_H}

The minus sign says the hedge must gain when the portfolio loses: sell bonds, or pay fixed. The second form is the duration-based hedge ratio, with VHV_H the value of one unit of the hedge.

The swap. From L1.7.04: Interest rate swaps, on a flat curve at yield yy the floating leg is worth N(1−DT)N(1-D_T) and the fixed leg NKANKA. So

Vpay fixed=N(1−DT)−NKA=N(1−(KA+DT)⏟BK/100)V_{\text{pay fixed}} = N(1 - D_T) - NKA = N\Big(1 - \underbrace{(KA + D_T)}_{B_K/100}\Big)

where BKB_K is the price per 100 of a bond paying coupon KK for the swap's life. The payer is short that bond, so a new swap's DV01 per unit of notional is the DV01 of a par bond of the same maturity, divided by 100.

Conventions here

Annual coupons and annual compounding; a flat curve, so every maturity has the same yield and a new swap's fixed rate equals it. Real swap rates and government bond yields differ (the swap spread), and real curves are not flat; the method is the same.

Try it

It opens on the worked example: $200m of a 10-year 4% bond at 4.5%, hedged with a 10-year swap. Move the parallel shift: the hedged line stays flat. Now add a twist of +50 bp and switch the swap to 2 years. Then set the hedge to 50% and see what half a hedge does.

Try itDV01 hedge sizer

Interest rate swap model →

Pay fixed on a swap of

P&L, $m, against the parallel move bond alone hedged
−2502550-300-200-1000100200300parallel move, bp
Bond DV01$154,364
Swap DV01 per $1m$791
Swap notional$195.08m

At +100 bp

Bond, $m−14.70
Swap, $m+14.70
Net, $m+0.004

Annual coupons, annual compounding, a flat curve at the bond's yield, and a new swap at that rate. The twist adds the stated move for every 10 years of maturity beyond 10 years (and takes it off below), so with a twist of +50 bp a 5-year yield moves 25 bp less than a 10-year yield.

Worked example

A bank hedges its bond portfolio

A US bank holds $200m face value of a 10-year bond with a 4% annual coupon, yielding 4.5%. It funds it with deposits whose rates move with the market, so rising rates would hurt it twice. It decides to hedge.

Size the hedge, then test it

Step 1 of 6
  1. 1

    Price and duration

    Price 96.0436 per 100, so the holding is worth $192.09m. Modified duration 8.0361: a 1-point rise in yield cuts the value by about 8.04%.

Matching DV01 removes almost all the risk of parallel moves. Matching maturity as well removes most of the curve risk. What is left is small: convexity, and the gap between swap rates and bond yields.

Regulatory lens

Banks must measure the interest rate risk in their banking book. Under the Basel Committee's 2016 standards on interest rate risk in the banking book (IRRBB), they report how their economic value of equity and their net interest income would change under six prescribed rate shocks, and supervisors look closely at any bank whose economic value would fall by more than 15% of its Tier 1 capital. Hedges with swaps count in these measures. M3.5: Interest rate risk in the banking book and ALM and M4.5: Liquidity and IRRBB regulation cover them.

Sparks

SparkA hedge that needed cash
Many UK pension schemes hedged their liabilities with leveraged gilt and swap positions. When gilt yields jumped in late September 2022, those hedges lost value quickly and needed collateral at once. Selling gilts to raise cash pushed yields higher still, until the Bank of England stepped in to buy long-dated gilts. The hedges did their job on paper; the cash was the problem (L1.2.05: Pension funds).
SparkHedges that move the market
US mortgage borrowers can repay early when rates fall, so the duration of mortgage-backed securities shrinks as rates fall and grows as they rise. Large holders who keep their portfolios duration-hedged must buy or sell Treasuries or swaps as rates move, and analysts have long linked these flows to sharp moves in long-term US yields.

Key takeaways

  • DV01s add up. To hedge, add a position that gains when rates rise, with the same DV01: n = DV01_P ÷ DV01_H.
  • In duration terms: n = (D_P × V_P) ÷ (D_H × V_H).
  • A pay-fixed swap behaves like a short fixed-rate bond of the same maturity, so it hedges fixed-rate assets.
  • A DV01 hedge works for small parallel moves. Convexity, curve twists and swap spreads leave some risk.
  • Swaps need collateral: a sound hedge can still create a sudden need for cash.

Reaction

Check what stuck

Question 1 · objective 1

A bond portfolio has a DV01 of $45,000. A new pay-fixed swap has a DV01 of $850 per $1m of notional. What notional, in $ millions, makes the position DV01-neutral? Two decimals.

$m

Question 2 · objective 3

A bond has a modified duration of 6.5. Yields rise by 0.8 percentage points. By roughly what percentage does its price change? (Enter a fall as a negative number.)

%

Question 3 · objective 1

A $50m portfolio has a modified duration of 7. You hedge by selling a bond priced at 98 per 100 with a modified duration of 4.5. What face amount, in $ millions, do you sell? Two decimals.

$m

Question 4 · objective 2

A bank has made fixed-rate loans and funds them with deposits whose rates follow the central bank. Which swap reduces its interest rate risk?

Question 5 · objective 4

You hedge a 10-year bond with a 2-year pay-fixed swap sized to the same DV01. Which move hurts the hedged position most?

Glossary

DV01
The change in value, in money, when yields move by one basis point (0.01%).
DV01-neutral
A position whose total DV01 is close to zero, so small parallel moves in yields barely change its value.
Pay fixed
The side of an interest rate swap that pays the fixed rate and receives floating. It gains when rates rise.
Receive fixed
The side of a swap that receives the fixed rate and pays floating. It gains when rates fall.
Asset swap
Buying a fixed-rate bond and paying fixed on a swap of the same maturity, so the holding earns a floating rate.
Parallel shift
A move in which yields of every maturity change by the same amount.
Twist
A move in which short and long yields change by different amounts, so the curve steepens or flattens.
Curve risk
The risk that a hedge at one maturity does not offset an exposure at another, because the curve does not move in parallel.

Deep Dives & Origins

Deep Dives

  • Bruce Tuckman and Angel Serrat, Fixed Income Securities, 4th edition (Wiley, 2022), the chapters on DV01, duration and hedging. Intermediate.

  • Basel Committee on Banking Supervision, Interest rate risk in the banking book (April 2016). Advanced.

Origins

  • Bruce Tuckman and Angel Serrat, Fixed Income Securities, 4th edition, Wiley, 2022.
  • John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021, the chapters on interest rate futures and swaps.
  • Basel Committee on Banking Supervision, Standards: Interest rate risk in the banking book, BIS, April 2016.
  • Board of Governors of the Federal Reserve System, Review of the Federal Reserve's Supervision and Regulation of Silicon Valley Bank, April 2023.
  • Bank of England, Financial Stability Report, December 2022, on liability-driven investment.

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Hedging with options

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Diversification and its limits