Quick answer: a bank does not measure interest-rate risk with one magic number. It first maps when assets and liabilities reprice or pay cash, then estimates first-order sensitivity with duration, second-order curvature with convexity, and finally reprices the balance sheet under multiple yield-curve scenarios. The useful output is not merely “rates up = bad” or “rates down = good.” It is a structured picture of which cash flows move, by how much, under which assumptions, and where the model may fail.

Duration is a local approximation. A bank’s risk is a system of cash flows, options, behaviours and curves.

Why this belongs in Mathematics

Interest-rate risk is a good example of mathematics leaving the textbook and becoming an operating system. Present value uses exponents. Duration uses weighted averages and derivatives. Convexity uses second-order change. Scenario analysis uses functions evaluated under alternative inputs. Validation asks whether a model still works when assumptions are disturbed. The point is not to teach students to trade bonds. It is to show how algebra, calculus, approximation and model criticism work together in a real institution.

This is also why the Bukit Timah Tutor article A Formula Can Be Correct and Still Be the Wrong Model matters here. Duration can be calculated perfectly and still be an incomplete description of risk.

1. Start with the object being measured: discounted cash flow

For a fixed set of future cash flows, the basic pricing idea is present value. If cash flow CF_t is received at time t, a simplified price can be written as P = Σ CF_t / (1 + y/m)^mt, where y is a yield and m is the compounding frequency. As the discount rate rises, later cash flows are discounted more heavily, so the present value generally falls.

That inverse relation is the first clue. But a bank is not one bond. It holds loans, securities, deposits, derivatives and commitments with different repricing dates, currencies and behavioural features. The measurement problem therefore begins by converting a messy balance sheet into a time-structured set of cash flows and repricing exposures.

2. Duration: the first-order sensitivity

Modified duration can be interpreted as the approximate percentage price sensitivity to a small change in yield. In differential form, D_mod ≈ −(1/P)(dP/dy). For a small yield movement Δy, the first-order approximation is ΔP/P ≈ −D_mod × Δy.

Suppose a position has modified duration 4.2. A 1 percentage-point increase in yield means Δy = 0.01. The duration-only approximation is therefore about −4.2%. That is useful because it compresses many future cash flows into one sensitivity measure.

But the compression has a price: it throws information away. It works best for small rate changes, relatively stable cash flows and movements that resemble the yield shift assumed by the calculation.

3. Convexity: correcting the straight-line approximation

Bond prices are not normally a straight-line function of yield. The relationship curves. Convexity captures part of that curvature. A common second-order approximation is ΔP/P ≈ −D_modΔy + ½C(Δy)², where C is a convexity measure.

Using the same duration of 4.2, suppose convexity is 22 and yields rise by 1 percentage point. The duration term is −4.20%. The convexity adjustment is ½ × 22 × 0.01² = +0.11%. The second-order estimate is therefore about −4.09%. The difference looks small in this example, but it grows as the rate move becomes larger or the curvature becomes more important.

For ordinary option-free fixed-rate bonds, convexity is often positive. Instruments with embedded options can behave differently because the cash flows themselves may change when rates move. A mortgage borrower may prepay. A depositor may move funds. A callable security may be redeemed. At that point, the problem is no longer merely “discount the same cash flows at a new rate.”

4. A bank needs more than one yield movement

A single parallel shift is easy to teach but dangerous to mistake for the whole world. Yield curves can steepen, flatten, twist or move differently at different maturities. That is the same logical lesson as One Counterexample Can Be Enough: one non-parallel move can expose a weakness hidden by a parallel-shift model.

The Basel interest-rate-risk framework uses multiple prescribed shocks for economic-value measurement, including parallel-up, parallel-down, steepener, flattener and short-rate shocks. The purpose is not that these scenarios predict the future. Their purpose is to ask whether the balance sheet remains understandable under materially different curve shapes. See the Basel Framework on interest-rate risk in the banking book.

5. Two different questions: economic value and earnings

Lens	Question	Typical horizon
Economic value of equity (EVE)	How does the present value of asset cash flows minus liability cash flows change when rates move?	Longer horizon / full cash-flow profile
Net interest income (NII)	How might interest income minus interest expense change as positions reprice?	Shorter earnings horizon

These lenses can disagree without either being mathematically wrong. A structure may protect near-term earnings while exposing long-term economic value, or vice versa. This is an important modelling lesson: the answer depends partly on the target variable.

6. The algorithmic pipeline

Inventory the positions. Loans, securities, deposits, derivatives, off-balance-sheet commitments and hedges must be represented.
Map contractual timing. Record payment dates, maturities, reset dates, coupons, reference rates and currencies.
Add behavioural assumptions. Estimate prepayments, early withdrawals, deposit repricing, deposit decay and option exercise where contractual timing is not enough.
Build the baseline curves. Specify the discount and reference-rate curves used for valuation and repricing.
Compute local sensitivities. Duration, PV01/DV01, key-rate duration and convexity can show where the first derivatives and curvature are concentrated.
Generate scenarios. Parallel, steepening, flattening, short-rate and institution-specific stress scenarios are applied.
Reprice. Recalculate cash flows and present values when required; do not merely multiply every position by one duration number.
Aggregate by lens. Compute EVE, NII and other internal metrics by currency, portfolio and business line.
Diagnose concentration. Find maturities, assumptions, currencies or products that dominate the result.
Validate and challenge. Compare approximations with full repricing, test assumptions and track whether model outputs explain realised behaviour.

7. Inputs and outputs: what the machine actually needs

Input family	Examples	Why it matters
Contractual data	cash-flow dates, coupon, principal, reset frequency, maturity	Defines the mechanical cash-flow skeleton
Market data	yield curves, spreads, volatilities, basis relationships	Changes discounting and repricing
Behavioural data	deposit decay, beta, prepayment, withdrawal behaviour	Controls cash flows not fixed by contract
Optionality	caps, floors, calls, prepayment rights	Can make cash flows rate-dependent
Hedges	swaps, futures, options	May offset one risk while introducing basis or model risk
Output	EVE change, NII change, DV01, key-rate sensitivities, stress losses	Turns the balance sheet into measurable response

8. A worked sensitivity example — and why it is deliberately incomplete

Imagine a simplified asset portfolio worth S$100 million with modified duration 4.2 and convexity 22. Under a +100 basis-point parallel yield shock, the duration-convexity approximation gives a change of roughly −4.09%, or about −S$4.09 million.

That number is not yet a bank risk result. We have not modelled liabilities. We have not asked whether deposits reprice. We have not included hedges, currencies, basis movements or optionality. We have not tested a steepener. We have not calculated NII. The example is useful precisely because it shows the boundary between a mathematical component and a complete system.

9. Failure modes: where a neat number can mislead

Non-parallel curve moves. A single duration can hide concentration at particular maturities.
Cash flows change when rates change. Prepayments, calls and withdrawals can invalidate fixed-cash-flow assumptions.
Basis risk. Assets and liabilities may reference different rates that do not move together.
Deposit model risk. Non-maturity deposits have contractual immediacy but often exhibit behavioural persistence; assumptions about that persistence can dominate results.
Large shocks. First-order approximations deteriorate as changes become larger.
Hedge mismatch. A hedge may neutralise one sensitivity but leave another, such as curve, basis or optionality risk.
Model drift. Behaviour learned in one rate regime may not survive another.
Aggregation hides local weakness. A small total sensitivity can be the sum of large offsetting positions.

The 2023 failure of Silicon Valley Bank is a useful public case for studying the interaction of interest-rate, liquidity and governance weaknesses. The Federal Reserve review reports significant deficiencies in interest-rate and liquidity risk management; it is a reminder that a risk number is not a substitute for action. See the Federal Reserve review.

10. Diagnostics: questions that reveal weak modelling

Does the result change dramatically when deposit-life assumptions move slightly?
Do full-repricing results disagree materially with duration/convexity approximations?
Which key-rate bucket contributes most of the EVE change?
Does a steepener hurt far more than a parallel shock?
Are assets and liabilities linked to different reference rates?
Do option exercise or prepayment assumptions move in the economically expected direction?
Can the model explain the change from last month’s result?
Are there large positions excluded because data are missing or classified incorrectly?
Does the same hedge protect both EVE and NII, or only one?
What observation in the real world would show that a behavioural assumption is no longer credible?

11. Verification: make the model capable of disagreeing with itself

A strong checking system should be able to produce a contradiction. That is the idea behind A Good Mathematics Check Should Be Able to Disagree With the Working. For interest-rate risk, useful checks include finite-difference repricing against analytic duration, independent valuation, reconciliation of cash-flow totals to the general ledger, back-testing deposit and prepayment assumptions, and comparing predicted sensitivity with realised changes.

Supervisory guidance similarly emphasises data completeness, sensitivity testing, independent validation and scenarios beyond routine conventions. The OCC Interest Rate Risk handbook is a useful public example.

12. When should the model be updated?

the yield curve moves into a materially different regime;
deposit behaviour changes;
prepayment or refinancing patterns shift;
new products introduce options or reference rates;
hedges are added, removed or become ineffective;
model errors or unexplained variances persist;
regulatory definitions or prescribed shocks change;
realised outcomes repeatedly fall outside the model’s expected range.

13. What would falsify a confident conclusion?

Suppose someone says, “The bank is safe from rate rises because asset and liability durations match.” A falsifier would be a plausible scenario in which the matched-duration balance sheet still loses materially: for example, a non-parallel curve shift, deposit repricing faster than assumed, mortgage prepayments changing, or a basis relationship breaking. If such a scenario exists, the original statement was too strong.

This is the deeper mathematical habit: do not ask only whether the calculation is correct. Ask what had to be true for the calculation to deserve its conclusion.

Connections across the Bukit Timah Tutor finance-and-algorithms lane

Loan amortisation and recurrence relations — the cash-flow mechanics beneath many interest-bearing assets.
Credit-scoring algorithms — probability estimation and calibration rather than interest-rate sensitivity.
Payment-system routing — a different banking algorithmic problem where timing and queues matter.
Exponential and logarithmic functions — the school mathematics behind compounding and discounting.

Research anchors

The deeper lesson

Duration is useful because it simplifies. Convexity is useful because it repairs part of that simplification. Scenario analysis is useful because it asks whether the simplification survives a different world. The strongest model is therefore not the one with the most impressive formula. It is the one that makes its assumptions visible, can be challenged from several directions, and returns to observed reality for correction.

Educational note: This article explains mathematical and risk-management concepts. It is not investment advice, financial advice, a recommendation to buy or sell any security, or a substitute for institution-specific regulatory guidance.

Small Group Tutorials

How Banks Measure Interest-Rate Risk: Duration, Convexity and Scenario Shocks