Economic value of equity sensitivity · RiskAtoms Academy

By the end you will be able to

1Compute a bank's economic value of equity as the present value of its rate-sensitive cash flows, with equity excluded, and its change under a rate shock.
2Estimate the change from the duration of equity, add the convexity term, and say when the approximation fails.
3Explain why EVE leaves equity out and why that makes the structural hedge look like an open position.
4Contrast the EVE and NII readings of the same book and the same shock.

Why it matters

When the whole sterling curve rises by 275 basis points, RiskAtoms Bank earns £254.7m more net interest income next year (L3.6.04: Net interest income sensitivity). Yet the same shock wipes £257.2m off the value of its balance sheet. Both numbers are right. NII asks what the next year's earnings look like; the economic value of equity asks what everything the bank owns and owes today is worth once the new rates are applied to every future cash flow. A bank that watches only one of them can be surprised by the other, as the US regional banks of 2023 were (L3.6.12: Credit spread risk in the banking book and the SVB case).

Core explanation

Valuing the banking book

Illustrative dataRiskAtoms Bank plc's banking book

The same £67bn balance sheet as Module 3.5, on one made-up reporting date, in pounds, with how each line reprices (fixed, floating or administered), its customer rate, a SONIA curve with Bank Rate at 4.00%, a £19.2bn macro hedge on the fixed-rate mortgages and a £16bn structural hedge. Tier 1 capital is taken as £3.5bn. The figures are round illustrative values, not any bank's data. Every lesson in Module 3.6 uses this book, so the figures agree.

This lesson uses the book's EVE: quarterly interest at today's customer rates (commercial margins included) and principal at each item's repricing date, its behavioural slot or its prepayment, discounted on the SONIA zero curve. Betas do not enter EVE; only the prepayment rate changes with the shock.

A trader marks positions to market every day (L3.2.04: Interest rate risk in the trading book). The banking book is not traded, but the same idea applies. Take every rate-sensitive position the bank holds today, list its future cash flows, and discount them on the risk-free curve. Assets less liabilities, plus or minus the hedges, is the economic value of equity (EVE). Revalue the same cash flows on a shocked curve and the difference is ΔEVE.

Three choices define the measure:

Run-off. Only today's positions count, to their repricing dates (a run-off balance sheet). No new lending is assumed, unlike the constant balance sheet of NII.
Discounting. At a risk-free zero curve, here SONIA. The cash flows may include the commercial margin or leave it out; this book includes it.
Equity excluded. Shareholders' equity has no contractual rate or maturity, so it has no cash flows to discount. EVE measures the value of everything else.

The book at today's rates

Part of the book	Present value	Σ t·CF·DF (£m·years)	Duration
Rate-sensitive assets	£65.60bn	82,764	1.26 years
Rate-sensitive liabilities	−£61.53bn	−80,444	1.31 years
Macro hedge (pay fixed £19.2bn)	£291m	−28,562	n/a
Structural hedge (receive fixed £16bn)	−£49m	35,500	n/a
EVE	£4,308.9m	9,258	2.15 years

EVE is £4.31bn, not the £4bn of book equity: the cash flows include future margins, the back-book rates differ from today's, and £1.4bn of assets with no rate (shares and other assets) are left out. What matters is how it changes. The swaps are worth little today, so their duration in years means nothing; their dollar duration Σ t·CF·DF is what they add.

Assets and liabilities have almost the same duration, 1.26 years and 1.31 years. On the balance sheet alone the dollar durations nearly cancel, leaving 2,320: a duration of equity of 0.57 years. The hedges change that. The pay-fixed macro hedge takes 28,562 off; the receive-fixed structural hedge adds 35,500. With both, the duration of equity is 2.15 years, and EVE loses about £0.93m for each basis point the curve rises (the book's DV01).

Parallel up, 275bp

The fixed-rate assets lose £2.19bn and the fixed and slotted liabilities gain £2.11bn, because the bank now owes less in today's money. The macro hedge, paying fixed against the mortgages, gains £756m. The structural hedge, receiving fixed, loses £932m. The net is a fall of £257.2m, 7.35% of Tier 1.

Five largest losers and gainers in parallel up	ΔEVE
Five-year fixed-rate mortgages	−£840.1m
Gilts (fixed coupons, not swapped)	−£543.8m
Structural hedge (receive fixed, five-year ladder): fixed leg at 4.10%	−£340.9m
Structural hedge (receive fixed, five-year ladder): fixed leg at 3.90%	−£273.4m
Two-year fixed-rate mortgages	−£229.5m
Retail fixed-term deposits	+£187.4m
Covered bonds issued	+£226.7m
Senior unsecured bonds	+£333.6m
Macro hedge of five-year fixes (pay fixed): fixed leg	+£641.2m
Personal current accounts	+£1,031.3m

Why equity is excluded, and what it does to the structural hedge

Equity has no repricing date. A bank could pretend it is overnight money, or a ten-year liability, and get very different answers, so the Basel standard takes it out of the calculation altogether: EVE is the value of the balance sheet excluding equity. That choice has a consequence. The bank invests its £4bn of equity, along with £12bn of stable current accounts, in a rolling five-year structural hedge to steady its income (L3.6.10: Hedging the banking book). The current accounts are slotted out over five years in EVE, so they offset their part of the ladder. Equity is not, so the quarter of the ladder that invests it shows up as a naked fixed-rate asset.

The equity quarter of the ladder

In parallel up the whole structural hedge loses £932.2m. The quarter that invests equity accounts for £233.1m of it. Take that part away and the rest of the book loses only £24.2m. Most of the EVE risk the bank reports is the deliberate investment of its own capital.

Dropping the hedge is no cure. Without it, EVE would gain £675m in parallel up but lose £771.6m in parallel down, 22.0% of Tier 1, and the duration of equity would be −6.02 years. NII would also swing far more (L3.6.04: Net interest income sensitivity). With no hedges at all, the book moves −£80.7m and +£45.5m: the mortgages and the funding are close to matched on their own. Each hedge trades one measure against the other, which is why ALCO watches both (L3.6.01: What ALM does).

Sort itMore or less EVE at risk when rates rise?

For each change to RiskAtoms Bank's book, does EVE fall by more or by less in a parallel up shock?

Treasury lengthens the structural hedge from a five-year to a ten-year ladder.
The bank adds £2bn of pay-fixed swaps to its macro hedge.
The bank sells more five-year fixed-rate mortgages, funded overnight.
The bank issues a £1bn seven-year fixed-rate senior bond and holds the cash at the central bank.
Deposit modelling slots core current accounts over three years instead of five.

The formal version

EVE is a sum of discounted cash flows on a continuously compounded zero curve (M-A05: The number e and continuous growth); its sensitivity is a derivative (M-B01: Derivatives). Duration and convexity are as in L1.4.08: Duration and L1.4.09: Convexity, with one simplification that continuous compounding brings.

The derivation in full. For a parallel shift $\Delta$ :

\begin{aligned} \text{EVE}(\Delta) &= \sum_i CF_i\, e^{-(z_i + \Delta)\,t_i} = \sum_i CF_i\, DF_i\, e^{-\Delta t_i} \\ \frac{d\,\text{EVE}}{d\Delta} &= \sum_i CF_i\, DF_i \,\frac{d}{d\Delta} e^{-\Delta t_i} = -\sum_i t_i\, CF_i\, DF_i\, e^{-\Delta t_i} \\ \frac{d^2\text{EVE}}{d\Delta^2} &= \sum_i t_i^2\, CF_i\, DF_i\, e^{-\Delta t_i} \\ \text{EVE}(\Delta) &\approx \text{EVE}(0) + \Delta\,\frac{d\,\text{EVE}}{d\Delta}\bigg|_0 + \frac{\Delta^2}{2}\,\frac{d^2\text{EVE}}{d\Delta^2}\bigg|_0 \\ \Delta\text{EVE} &\approx -\Big(\sum_i t_i\, CF_i\, DF_i\Big)\Delta + \tfrac{1}{2}\Big(\sum_i t_i^2\, CF_i\, DF_i\Big)\Delta^2 \end{aligned}

Here $DF_i = e^{-z_i t_i}$ is today's discount factor. Dividing the first sum by EVE gives $D_E$ , the second gives the convexity $C_E$ . With annual compounding the slope carries an extra $1/(1+y)$ (modified against Macaulay duration); with continuous rates the two durations are the same number.

Dollar durations add. Because the derivative of a sum is the sum of the derivatives, the book's dollar duration is the sum of its parts. Dividing by EVE gives the duration of equity in terms of the parts:

\begin{aligned} D_E\,\text{EVE} &= D_A\,PV_A - D_L\,PV_L + \textstyle\sum_{\text{hedges}} t\,CF\,DF \\ 9{,}258 &= 1.262 \times 65{,}598 - 1.307 \times 61{,}531 + (-28{,}562) + 35{,}500 \\ D_E &= 9{,}258 \div 4{,}308.9 = 2.149\ \text{years} \end{aligned}

Leverage is why it matters. $PV_A$ is 15.2 times EVE, so a gap of a few weeks between $D_A$ and $D_L$ is magnified about 15-fold in $D_E$ .

Where the approximation fails. Duration assumes the cash flows stay fixed. In the book they do not: fixed-rate mortgages prepay more slowly when rates rise (CPR × 0.8) and faster when they fall (× 1.2), which L3.6.08: Optionality in loans takes apart. Non-parallel shocks need the cash flows bucket by bucket, which L3.6.06: The standard shock scenarios does.

Try it

Move the curve and compare the exact revaluation with the duration line and the convexity curve. Then switch off the structural hedge: the duration of equity turns negative and EVE now loses when rates fall. Switch off both hedges and see how close the balance sheet alone is to matched.

Try itShift the curve, revalue the book

Parallel shift of the SONIA curve+275bp

Hedges in the book

Exact ΔEVE (book)Duration estimateDuration + convexity

Exact ΔEVE−£257.2m

As a share of Tier 1 (£3.5bn)7.35%

Duration estimate−£254.6m

Duration + convexity−£234.8m

EVE today£4,309m

Duration of equity D_E2.15 years

DV01 (loss for +1bp)£0.926m

Outlier test (15% of Tier 1)Within

The exact line uses the book's CPR multipliers (0.8 when rates rise, 1.2 when they fall), so prepayment makes the book slightly negatively convex: the exact change sits below the duration line on both sides, while the convexity estimate sits above it. No floor is binding in a parallel shift on this curve. Illustrative book.

Worked example

Parallel up, 275bp, from duration to the exact answer

All figures are from the book, in pounds. The shock adds 275bp to the SONIA zero rate at every tenor.

Estimating and then measuring ΔEVE

Step 1 of 6

1
EVE and its dollar duration today
$\begin{aligned} \text{EVE} &= \textstyle\sum_i CF_i\,DF_i = \text{£}4{,}308.9\text{m} \\ \textstyle\sum_i t_i\,CF_i\,DF_i &= \text{£}9{,}258.2\text{m}\cdot\text{years} \\ D_E &= 9{,}258.2 \div 4{,}308.9 = 2.149\ \text{years} \end{aligned}$

In parallel down the book gains £251.7m of EVE while NII falls by £263.3m: the mirror image. Whether a 7.35% loss is the worst of the six standard shocks, and what the 15% test means, is L3.6.06: The standard shock scenarios.

Regulatory lens

The Basel standard on interest rate risk in the banking book (Basel Framework SRP31, with application guidance in SRP98) asks banks to measure both ΔEVE and ΔNII. The disclosure rules fix how ΔEVE is computed: a run-off balance sheet, instantaneous shocks, own equity excluded, and cash flows discounted at a risk-free zero rate, with or without commercial margins as long as the bank says which. Supervisors compare the largest ΔEVE over six prescribed shocks with 15% of Tier 1 capital (L3.6.06: The standard shock scenarios). SRP98 also describes an alternative that gives equity the profile of the assets hedging it, as a supplement to EVE. The full rules, disclosure templates and the EU and UK versions are in M4.5: Liquidity and IRRBB regulation; behavioural models for deposits and prepayment are in M6.7: Liquidity, ALM and behavioural models.

Spark Atoms

Spark AtomOne duration, not two

With annual rates, Level 1 needed two durations: Macaulay (the average wait) and modified (the price sensitivity), one divided by 1 + y. On a continuously compounded curve the derivative of e^(−Δt) is just −t e^(−Δt), so the average wait and the sensitivity are the same number. That is one reason risk systems discount on continuous zero rates.

Spark AtomMost of the risk is a choice

Of the £257.2m the book loses in parallel up, £233.1m comes from investing equity in five-year swaps, a choice made to steady earnings. A reader who sees only ΔEVE could take a deliberate hedge of NII for an unhedged bet, which is why ALCO reads the EVE figure next to the hedge that causes it.

Key takeaways

EVE is the present value of the rate-sensitive assets less liabilities, plus or minus hedges, on a risk-free curve, for today's positions only; equity is excluded.
On RiskAtoms Bank's book EVE is £4.31bn with a duration of equity of 2.15 years: about £0.93m per basis point.
ΔEVE ≈ −D_E × EVE × Δ, plus ½ × convexity × Δ²; dollar durations of the parts add up. Prepayment that changes with rates makes the exact answer differ.
Parallel up costs £257.2m of EVE, 7.35% of Tier 1, while NII rises £254.7m: the two measures can disagree.
Because equity is excluded, the part of the structural hedge that invests equity counts as an open position: £233.1m of the parallel-up loss.

Reaction

Check what stuck

Question 1 · objective 2

A bank's EVE is £5,000m and its duration of equity is 3 years. Using duration only, what is the change in EVE for a 200bp parallel rise, in £m (negative for a fall)?

£m

Question 2 · objective 2

Assets worth £100bn have a duration of 2 years; liabilities worth £95bn have a duration of 1.5 years. What is the duration of equity (£5bn), in years?

years

Question 3 · objective 3

In the parallel up shock, RiskAtoms Bank's EVE falls by £257.2m. Why does most of that loss come from the structural hedge?

AEquity is excluded from EVE, so the part of the receive-fixed ladder that invests equity has no matching liability and counts as an open positionBThe structural hedge swaps were written at the wrong ratesCThe structural hedge pays fixed, so it loses when rates riseDCore deposits are slotted overnight, so they gain nothing when rates rise

Question 4 · objective 4

A rise of 275bp adds £254.7m to the book's year-1 NII but takes £257.2m off its EVE. Which reading is right?

ABoth: NII looks one year ahead on a constant balance sheet, EVE values today's positions to maturity, so they can move in opposite directionsBThe NII figure is wrong, because EVE is the true measureCThe EVE figure is wrong, because NII is what shareholders receiveDThey should always agree in sign; the book must contain an error

Question 5 · objective 2

The book's dollar convexity Σ t²·CF·DF is £52,402m·years². What does the second-order term ½ × C × Δ² add to the duration estimate for a 275bp shift, in £m?

£m

Glossary

Economic value of equity (EVE): Economic value of equity: the present value of a bank's rate-sensitive assets less its rate-sensitive liabilities, plus or minus its hedges, with equity itself excluded.
ΔEVE: The change in EVE when the curve is shocked: the value of every cash flow on the shocked curve less its value on today's curve.
Run-off balance sheet: A run-off balance sheet: only the positions held today, with their cash flows to maturity or repricing. No new business is assumed.
Commercial margin: The part of a customer rate above the risk-free rate. Banks may include it in the cash flows they discount for EVE, or leave it out.
Duration of equity: Duration of equity: minus the derivative of EVE with respect to a parallel shift, divided by EVE. The approximate percentage fall in EVE for a one percentage point rise in rates.
Dollar duration: The sum of each cash flow's time × present value, Σ t·CF·DF. It is duration × value, in pounds × years.
DV01: The fall in value for a one basis point rise in rates: dollar duration × 0.0001.
Structural hedge: A rolling ladder of receive-fixed swaps that turns the floating return on core deposits and equity into a steadier fixed return.
Earnings-adjusted economic value: An economic value measure that gives equity the rate profile of the assets that hedge its earnings, instead of leaving equity out.

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, Interest rate risk in the banking book, BIS, April 2016: the principles, the disclosure rules for ΔEVE and the standardised framework. Intermediate.
Basel Committee on Banking Supervision, Basel Framework, SRP98 Application guidance on interest rate risk in the banking book (economic value and earnings measures, the treatment of equity), BIS. Advanced.

Origins

Basel Committee on Banking Supervision, Basel Framework, SRP31 Interest rate risk in the banking book, version in force from 1 January 2026, BIS.
Basel Committee on Banking Supervision, Basel Framework, SRP98 Application guidance on interest rate risk in the banking book, paragraphs 98.45 and 98.46, version in force from 1 January 2026, BIS.
Basel Committee on Banking Supervision, Standards: Interest rate risk in the banking book, BIS, April 2016, Principle 8 and paragraph 70 (ΔEVE computed with own equity excluded; discounting with or without commercial margins).

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