Interest rate risk in the trading book · RiskAtoms Academy

By the end you will be able to

1Compute the DV01 of bonds and swaps and add them into a book's parallel rate risk.
2Split rate risk by key rate, and estimate the P&L of parallel shifts, twists and butterflies.
3Recognise a curve position, size it to be DV01-neutral, and say which curve moves it is exposed to.
4Use the second-order Taylor term (convexity) to judge when DV01 is not enough.

Why it matters

RiskAtoms Bank's Rates book loses £23,043 for every basis point that rates rise in parallel. On day 205 of its history, a surprise policy move pushed the 2-year rate up 25bp and the 10-year up 15bp. The book made +£223,683. One number for interest rate risk is not enough: a desk has to know where on the curve its risk sits.

Core explanation

Rate risk along the curve

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

Position	Size	Value today	Risk factors
Rates
10-year gilt (£100m face)	£100m face, 4.25% coupon, 10 years	£98.80m	10y rate
2-year swap, pay fixed (£300m)	£300m notional, pay 4.00% fixed, 2 years	£0.00m	2y rate
Credit
Corporate bonds (£80m)	£80m, 5.35% coupon, 5 years	£80.00m	2y and 10y rates, IG spread

DV01 for each position

DV01 (L1.4.08: Duration, L2.5.04: Sensitivities) is value × modified duration × 0.0001. The gilt's modified duration is 8.059 on £98.80m, so it loses £79,625 for each 1bp rise in the 10-year rate.

The swap is worth zero, but it still has a DV01. Paying fixed and receiving floating is the same as being short a 2-year 4% fixed-rate bond and long a floating-rate note (L1.7.04: Interest rate swaps). The floating note barely moves with rates; the fixed bond has modified duration 1.886. Being short it, the desk gains £56,583 for each 1bp rise in the 2-year rate.

Signs in this module

DV01 is quoted positive for a long bond. Risk reports instead show the signed P&L for a 1bp rise, so that positions can be added: the gilt is −£79,625 per bp, the swap +£56,583. From here on, “per bp” means that signed figure.

Added up, the Rates book is −£23,043 per bp of parallel move. The corporate bonds in the Credit book carry rate risk too (−£34,303 per bp), so the whole desk is −£57,346 per bp.

Key rates: where the risk sits

A parallel number hides the fact that the swap's risk is at 2 years and the gilt's at 10. Key rate sensitivities keep them apart: bump one curve point at a time.

Position	2-year (per bp)	10-year (per bp)	Parallel (per bp)
10-year gilt (£100m face)	£0	−£79,625	−£79,625
2-year swap, pay fixed (£300m)	+£56,583	£0	+£56,583
Corporate bonds (£80m)	−£21,440	−£12,864	−£34,303
Rates book (gilt + swap)	+£56,583	−£79,625	−£23,043
Whole desk	+£35,143	−£92,489	−£57,346

The Rates book gains when 2-year rates rise and loses when 10-year rates rise. It is a curve position: it profits if the curve flattens (short rates up relative to long rates) and loses if it steepens. The modest parallel number makes it look small; the curve exposure is not.

Shifts, twists and butterflies

Curve moves come in three basic shapes. A parallel shift moves every point equally. A twist moves the ends in opposite directions: a steepener or a flattener. A butterfly moves the middle against the ends. Using the key rate sensitivities:

Curve move	Rates book (key rates)	Rates book (full)	Whole desk (full)
Parallel +10bp	−£230,426	−£227,431	−£569,523
Steepener (2y −10, 10y +10)	−£1,362,082	−£1,359,090	−£1,273,273
Flattener (2y +10, 10y −10)	+£1,362,082	+£1,365,103	+£1,279,404
Day 205: 2y +25, 10y +15	+£220,190	+£223,683	−£501,023

A 10bp parallel rise costs the Rates book £227,431; a 20bp twist moves it by about £1.37m. If the trader wanted only the curve view, she would size the swap to cancel the gilt's DV01: £422m instead of £300m (DV01-neutral). A neutral flattener earns from a twist and nothing from a small parallel move.

A butterfly needs a third point. The book's corporate bonds depend on the 5-year rate, but the 5-year rate is not a factor: it is read off the line between 2 and 10 years (L3.2.02: Risk factors and position mapping). If the wings rise 10bp and the 5-year point falls 10bp, the model prices the bonds as if their rate rose 10bp (−£342,092); repriced at their own rate they make +£343,978. That gap is a risk the factor set cannot see.

Which shape is it?

When DV01 is not enough

DV01 is a slope, good for small moves. For large ones, add the second-order term: convexity (L1.4.09: Convexity). The gilt has convexity 77.3. For a 100bp rise, duration alone predicts −£7.96m; adding convexity gives −£7.58m; full revaluation gives −£7.59m.

Yield move	Duration only	Duration + convexity	Full revaluation	Error after convexity
-200bp	+£15.925m	+£17.452m	+£17.564m	+£111,894
-100bp	+£7.963m	+£8.344m	+£8.358m	+£13,596
-25bp	+£1.991m	+£2.014m	+£2.015m	+£208
+25bp	−£1.991m	−£1.967m	−£1.967m	−£205
+100bp	−£7.963m	−£7.581m	−£7.594m	−£12,867
+200bp	−£15.925m	−£14.398m	−£14.499m	−£100,217

The formal version

One yield, two terms (M-B03: Taylor expansion). Expand the value around today's yield $y$ and use the definitions $D_{\text{mod}} = -V'/V$ and $C = V''/V$ :

\begin{aligned} V(y + \Delta y) &= V(y) + V'(y)\,\Delta y + \tfrac12 V''(y)\,(\Delta y)^2 + \tfrac16 V'''(y)\,(\Delta y)^3 + \cdots \\ \Delta V &\approx -D_{\text{mod}}\,V\,\Delta y + \tfrac12\,C\,V\,(\Delta y)^2 \\ \text{DV01} &= D_{\text{mod}}\,V \times 0.0001 \end{aligned}

The error after two terms is of order $(\Delta y)^3$ : double the move and it grows about eightfold, which the table shows (about £12,867 at 100bp, £100,217 at 200bp).

Many key rates. Let the value depend on key rates $r_1, \dots, r_K$ . To first order, with $\text{KR01}_k$ the P&L per bp rise in $r_k$ and moves in bp:

\begin{aligned} \Delta V &\approx \sum_{k=1}^{K} \text{KR01}_k\,\Delta r_k \\ \text{parallel } (\Delta r_k = s): \quad \Delta V &\approx s \sum_k \text{KR01}_k = s \times \text{(parallel P\&L per bp)} \\ \text{twist } (\Delta r_2 = -\tfrac{t}{2},\ \Delta r_{10} = +\tfrac{t}{2}): \quad \Delta V &\approx \tfrac{t}{2}\,(\text{KR01}_{10} - \text{KR01}_2) \\ \text{KRD}_k &= -\frac{1}{V}\,\frac{\partial V}{\partial r_k}, \qquad \sum_k \text{KRD}_k = D_{\text{mod}} \end{aligned}

For the Rates book, a 20bp steepener gives $10 \times (-79{,}625 - 56{,}583) = -1{,}362{,}082$ . The corporate bonds' key rate durations are 2.680 (2-year) and 1.608 (10-year); they add up to 4.288, the bonds' modified duration (4.288). The gilt's are 0 and 8.059. The swap's value is zero, so durations mean nothing for it; its KR01s are what count.

DV01-neutral sizing. Choose the swap notional $N$ so that $N \times \text{KR01}_2^{\text{swap}}/\text{£}300\text{m} + \text{KR01}_{10}^{\text{gilt}} = 0$ : $N = \text{£}300\text{m} \times 79{,}625 / 56{,}583 = \text{£}422.2\text{m}$ .

Try it

Move the level, the slope and the belly of the curve. Watch the Rates book's P&L swing with the slope far more than with the level, and watch the belly reach the bonds without the model noticing.

Try itShift, twist and bend the curve

Level (both points)0bp

Slope (10y minus 2y)0bp

Belly: 5y off the line0bp

2-year move 0.0bp, 10-year move 0.0bp.

TodayAfter the move5-year point (not a book factor)

Position	Key-rate estimate	Full revaluation (model)	With the belly
10-year gilt	£0	£0	£0
2-year swap (pay fixed)	£0	£0	£0
Corporate bonds (5y)	£0	£0	£0
Total	£0	£0	£0

Rates book (gilt + swap), full£0

Rates book, key-rate estimate£0

Missed by the model (belly)£0

The model maps the bonds onto 2y and 10y, so a belly move is invisible to it. The last column reprices the bonds at their own 5-year rate.

Worked example

The desk's rate risk, and day 205

From DV01s to a curve move

Step 1 of 6

1
The gilt
Price 98.797 at a 4.40% yield (4.25% semi-annual coupon, 10 years). Modified duration 8.0595. $\text{DV01} = 8.0595 \times \text{£}98{,}796{,}986 \times 0.0001 = \text{£}79{,}625$ .

Regulatory lens · looking ahead

The Basel standardised approach measures trading-book rate risk as key rate sensitivities at ten tenor points per currency, from 3 months to 30 years, with set correlations between tenors, plus a curvature charge for options (M4.3: Market risk capital (FRTB) and CVA). Interest rate risk in the banking book (loans, deposits and their hedges) has its own framework with six prescribed curve shocks, including parallel, steepener and flattener scenarios (M4.5: Liquidity and IRRBB regulation). Statistical models of the whole curve, such as principal components, come in Level 6: Advanced Risk Modelling.

Spark Atoms

Spark AtomThree shapes explain most of it

In 1991 Robert Litterman and José Scheinkman showed that three factors, which they called level, steepness and curvature, explained the great majority of the variation in US Treasury returns. Those are the shift, the twist and the butterfly of this lesson, found by statistics rather than chosen by hand.

Spark AtomSeptember 2022

After the UK government's fiscal statement of 23 September 2022, long-dated gilt yields rose by more than a percentage point in a few days. Pension funds using liability-driven investment faced collateral calls on their swaps, and on 28 September the Bank of England announced temporary purchases of long-dated gilts to restore orderly markets. Moves that size are where DV01 alone stops being a good guide.

Key takeaways

DV01 = value × modified duration × 0.0001. A pay-fixed swap gains when rates rise; risk reports show the signed P&L per bp so positions add.
Key rate sensitivities show where on the curve the risk sits; they add up to the parallel figure.
A book with a small parallel DV01 can carry a large twist exposure: RiskAtoms Bank's Rates book is a flattener.
Butterflies need at least three curve points; a factor set with two cannot see them, a gap to track.
For large moves add convexity, the second Taylor term; the remaining error grows with the cube of the move.

Reaction

Check what stuck

Question 1 · objective 1

A desk holds £50m face of a bond priced at 102 with modified duration 6. What is its DV01 (the loss for a 1bp rise), in £?

£

Question 2 · objective 4

A bond has modified duration 7 and convexity 60. Estimate its percentage price change for a 150bp rise in yield, in %, with both terms.

%

Question 3 · objective 2

A book makes +£40,000 per bp rise in the 2-year rate and −£90,000 per bp rise in the 10-year. What is its P&L if the curve steepens with 2y −15bp and 10y +15bp, in £?

£

Question 4 · objective 1

The desk pays fixed on a £300m 2-year swap. What happens when the 2-year rate rises 1bp?

AIt loses, like a long bondBIt gains, because paying fixed is like being short a fixed-rate bondCNothing, because the swap was struck at parDIt gains only if the 10-year rate rises too

Question 5 · objective 3

The book's only rate factors are the 2-year and 10-year swap rates. What does a butterfly move do to its measured risk?

AIt shows up as a parallel shiftBIt cannot be expressed: with two points there is no belly to moveCIt shows up as a twistDIt doubles the DV01

Glossary

DV01: The change in value for a 1 basis point fall in yields (positive for a long bond). Risk reports usually show the signed P&L for a 1bp rise instead.
Key rate sensitivity (KR01): The P&L for a 1bp rise in one key rate (one point of the curve), all other key rates unchanged. They add up to the parallel DV01.
Key rate duration: A key rate sensitivity as a duration: −(1/V) × the P&L per unit rise in that key rate. Key rate durations add up to the modified duration.
Parallel shift: Every point of the curve moves by the same number of basis points.
Twist: Short and long rates move in opposite directions (or by different amounts), so the curve steepens or flattens.
Steepener: A move in which long rates rise relative to short rates; a steepener position gains from it.
Butterfly: The middle (belly) of the curve moves against the two ends (wings), so the curve's bend changes.
Convexity: The curvature of value against yield: the second-order term that makes a plain bond gain more than duration predicts when yields fall and lose less when they rise.
DV01-neutral: Sized so that the positions' DV01s cancel: no P&L for a small parallel move, leaving only the curve exposure.

Deep Dives & Origins

Deep Dives

B. Tuckman and A. Serrat, Fixed Income Securities: Tools for Today's Markets, 3rd ed., Wiley, 2011, the chapters on key rate and multi-factor risk. Intermediate.
L. Martellini, P. Priaulet and S. Priaulet, Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies, Wiley, 2003. Intermediate.

Origins

T. S. Y. Ho, “Key Rate Durations: Measures of Interest Rate Risks”, Journal of Fixed Income 2(2), 1992.
R. Litterman and J. Scheinkman, “Common Factors Affecting Bond Returns”, Journal of Fixed Income 1(1), 1991.
Bank of England, “Bank of England announces gilt market operation”, news release, 28 September 2022.
Basel Committee on Banking Supervision, Basel Framework, MAR21: Standardised approach: sensitivities-based method (general interest rate risk), Bank for International Settlements.

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