Economic value of equity sensitivity is arithmetic on durations. The hard part is the one duration nobody can observe: the deposits'.
Reprice every asset and liability for the shock with its duration and convexity, ΔV = −D × V × Δy + ½ × C × V × Δy², and take the change in assets less the change in liabilities. On an illustrative $8.0 billion bank, a 200 basis point rise cuts the economic value of equity by $85.6 million, 13.4 per cent of tier 1 capital, inside the Basel 15 per cent outlier threshold. Treat the non-maturity deposits as overnight money instead and the same shock costs 51.6 per cent.
Worked in full in Bank Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The bank is fictional and every figure is illustrative, in thousands of dollars. Market values are taken equal to book values at today's rates, so the starting economic value of equity (EVE) is 660,000; tier 1 capital is 640,000. Durations are modified durations; for non-maturity deposits they are behavioural assumptions, not contract terms.
| Item | Value | Duration | Convexity | Change at +200 bp |
|---|---|---|---|---|
| Cash and reserves | 400,000 | 0.00 | 0 | 0 |
| Securities, fixed rate | 2,800,000 | 4.20 | 24 | −221,760 |
| Loans, fixed rate | 2,900,000 | 2.80 | 11 | −156,020 |
| Loans, floating rate | 1,600,000 | 0.25 | 0 | −8,000 |
| Other assets | 300,000 | 0.00 | 0 | 0 |
| Assets | 8,000,000 | 2.54 | −385,780 | |
| Deposits, non-interest-bearing | 1,800,000 | 3.50 | 15.75 | −120,330 |
| Deposits, savings | 3,200,000 | 2.00 | 6 | −124,160 |
| Deposits, time | 1,500,000 | 0.90 | 1 | −26,700 |
| FHLB advances | 600,000 | 1.50 | 3 | −17,640 |
| Subordinated debt | 150,000 | 4.00 | 20 | −11,400 |
| Other liabilities | 90,000 | 0.00 | 0 | 0 |
| Liabilities | 7,340,000 | 2.12 | −300,230 | |
| Economic value of equity | 660,000 | 7.17 | −85,550 |
Duration gives the first-order price change and convexity the correction for curvature. For the securities book at +200 basis points:
ΔV = −D × V × Δy + ½ × C × V × Δy²
= −4.20 × 2,800,000 × 0.02 + 0.5 × 24 × 2,800,000 × 0.0004 = −235,200 + 13,440 = −221,760
Excel: =-D*V*dy+0.5*C*V*dy^2
Liabilities fall in value too when rates rise, which is a gain to the bank: a deposit that keeps paying a lagging rate becomes cheaper funding than the market offers, so the present value of what the bank owes falls. That is why the deposit duration matters as much as the securities duration.
ΔEVE = Δassets − Δliabilities = −385,780 − (−300,230) = −85,550
As a share of tier 1: 85,550 / 640,000 = 13.4 per cent; of starting EVE, 13.0 per cent
Cross-check with the duration gap: DA − DL × L/A = 2.54 − 2.12 × 0.9175 = 0.59 years, and 0.59 × 8,000,000 × 0.02 = 94,600 before convexity
The first-order estimate, 94,600, overstates the loss by 9,050 because both books are convex and the assets are more convex than the liabilities. A duration of equity of 7.17 years means EVE loses about 7.17 per cent for each 100 basis points before convexity, as a seven-year zero-coupon bond would.
The Basel standard on interest rate risk in the banking book prescribes six shock scenarios and treats a bank as an outlier when its EVE falls by more than 15 per cent of tier 1 capital in any of them; for dollar rates the parallel shock is 200 basis points. US supervisors do not apply that test mechanically, but expect banks to run shocks beyond 200 basis points. The parallel grid on this balance sheet:
| Shock | Δassets | Δliabilities | ΔEVE | % of tier 1 |
|---|---|---|---|---|
| −200 bp | 425,420 | 321,770 | 103,650 | 16.2% |
| −100 bp | 207,755 | 158,192 | 49,562 | 7.7% |
| +100 bp | −197,845 | −152,808 | −45,038 | −7.0% |
| +200 bp | −385,780 | −300,230 | −85,550 | −13.4% |
| +300 bp | −563,805 | −442,268 | −121,538 | −19.0% |
| +400 bp | −731,920 | −578,920 | −153,000 | −23.9% |
The bank passes at 200 basis points with 1.6 points to spare and would fail at 300. The full outlier test also needs the other four Basel scenarios (short rates up and down, steepener and flattener), which shock each tenor bucket by a different amount and cannot be read off a single duration; a bank this long in fixed-rate assets should run the short-rate-up shock before claiming a pass. Note the asymmetry from convexity: the 200 point fall adds 103,650 while the 200 point rise takes away 85,550. Quadrupling the 100 point result to estimate 400 points gives −180,150, 17.7 per cent worse than the convexity-adjusted −153,000.
Non-maturity deposits have no contractual maturity, so their duration is a modelling choice, and it moves the answer more than any other input. Holding the time deposits and everything else fixed, at +200 basis points:
| Non-interest-bearing | Savings | ΔEVE | % of tier 1 |
|---|---|---|---|
| 0.00 | 0.00 | −330,040 | −51.6% |
| 1.75 | 1.00 | −206,052 | −32.2% |
| 3.50 | 2.00 | −85,550 | −13.4% |
| 5.25 | 3.00 | 31,468 | 4.9% |
At contractual terms the bank is badly exposed; on generous behavioural terms it gains from rising rates. The Basel standard caps the average repricing maturity it will accept for core retail deposits (five years for transactional accounts), precisely because this assumption can manufacture a hedge. The deposit duration is only as good as the deposits' willingness to stay at a lagging rate, which is the deposit beta by another name; see how to calculate a deposit beta.
The expensive mistake is to report EVE sensitivity without the deposit assumption beside it. A committee that sees −13.4 per cent and not the 3.5 and 2.0 year durations behind it is approving a number that ranges from −51.6 to +4.9 per cent on the same balance sheet. The smaller mistakes: dropping convexity, which overstates the loss by 9,050 here, and scaling a 100 point result linearly to larger shocks.
EVE is a run-off value: it ignores new business and, in the Basel version, the bank's own equity. It answers what the existing balance sheet is worth after the shock, not what next year's earnings will be. Run earnings sensitivity beside it.
The weighted repricing gap, the duration and convexity marks and the EVE surface by deposit life and beta are live in the free workbook for this case.
Under the Basel standard on interest rate risk in the banking book, a bank whose economic value of equity falls by more than 15 per cent of tier 1 capital in any of six prescribed rate shock scenarios is an outlier and draws supervisory attention. On the illustrative bank here, the +200 basis point parallel shock costs 13.4 per cent of tier 1 and a +300 point shock would cost 19.0 per cent.
A behavioural one, estimated from how long balances stay and how slowly their rate follows the market, and capped under the Basel standard. It is the most powerful assumption in the model: on the bank here, moving the non-interest-bearing and savings durations from 3.5 and 2.0 years to zero turns a 13.4 per cent loss of tier 1 at +200 basis points into 51.6 per cent.
EVE sensitivity measures the change in the present value of the existing balance sheet after a shock, over its whole remaining life. Net interest income sensitivity measures the change in earnings over a short horizon, usually one year. A bank can pass one and fail the other; the illustrative bank here loses 13.4 per cent of tier 1 in value at +200 basis points whatever its next year's margin does.
This article is one calculation from Bank Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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