CMA Final · Risk Management in Banking and Insurance

Interest Rate Risk Management: formula sheet

Full chapter guide

Key formulas

Net interest income: NII = Interest income − Interest expense; The earnings view measures the change in this amount when rates change.
Repricing gap: Gap = Rate sensitive assets (RSA) − Rate sensitive liabilities (RSL); Calculated for each time bucket. A positive gap means the bank is asset sensitive.
Change in NII from a rate shift: ΔNII ≈ Gap × Δi; Δi is the change in rate. Use the gap for the period within which items reprice, and adjust for the part of the year remaining if asked.
Economic value of equity: EVE = PV of assets − PV of liabilities (including off-balance sheet items); The economic value view looks at the change in EVE when rates shift.
Repricing gap: Gap = RSA − RSL; Calculate for each time bucket. Positive gap = asset sensitive; negative gap = liability sensitive.
Change in NII: ΔNII = Gap × Δi; Δi is the change in rate as a decimal. Valid for the bucket's horizon when the rate change applies to the whole period. If the rate change applies only for part of the year, multiply by that fraction.
Cumulative gap: Cumulative gap = sum of gaps of all buckets up to the chosen horizon; Use the cumulative gap for a one-year horizon NII impact.
Gap ratio: Gap ratio = RSA ÷ RSL; Above 1 means positive gap, below 1 means negative gap.
Duration gap: DGAP = DA − (L ÷ A) × DL; DA and DL are durations of assets and liabilities; L and A are market values of liabilities and assets.
Change in equity value: ΔE ≈ −DGAP × A × Δi ÷ (1 + i); Positive DGAP means equity value falls when rates rise. i is the current yield level.
Macaulay duration: D = Σ [t × PV(CFt)] ÷ P, where P = Σ PV(CFt); t is time in years. PV is discounted at the yield to maturity. Result is in years.
Modified duration: D_mod = D ÷ (1 + y/m); m is the number of coupon payments per year. For annual coupons, divide by (1 + y).
Price change using duration: ΔP ÷ P ≈ − D_mod × Δy; Use Δy as a decimal (1% = 0.01). The sign is negative: yield up, price down.
Convexity (annual cash flows): C = [Σ t(t + 1) × PV(CFt)] ÷ [P × (1 + y)²]; Measured in years squared. Use the version given in the question if it differs for non-annual periods.
Price change with convexity: ΔP ÷ P ≈ − D_mod × Δy + ½ × C × (Δy)²; The convexity term is positive for both rises and falls in yield.
Duration gap: DGAP = D_A − (L ÷ A) × D_L; A is the market value of assets, L of liabilities. D_A and D_L are the weighted durations.
Change in equity value: ΔE ≈ − DGAP × A × Δy ÷ (1 + y); Positive DGAP: equity falls when rates rise. Negative DGAP: equity rises when rates rise.
Net interest income: NII = Interest income − Interest expense; Earnings perspective works on this figure, usually over a 12-month horizon.
Change in NII (gap approximation): ΔNII ≈ Periodic gap × Δi × (fraction of year remaining after repricing); Periodic gap = RSA − RSL for the bucket. Use the time left in the horizon after the midpoint of the bucket. Valid for small, parallel rate changes.
Economic value of equity: EVE = PV of assets − PV of liabilities (± off-balance sheet items); Cash flows are discounted at market rates for the whole remaining life.
Change in EVE: ΔEVE = EVE after shock − EVE before shock; A negative value is a loss of economic value. It is compared with capital.
Duration approximation for EVE: ΔEVE ≈ −(D_A × A − D_L × L) × Δi ÷ (1 + i); Uses modified-duration style approximation with D_A and D_L as duration of assets and liabilities. Good only for small parallel shifts.
Unit spread (lending unit): Spread = Customer lending rate − Transfer rate charged; This is the margin earned by the business unit for credit and customer service.
Unit spread (deposit unit): Spread = Transfer rate credited − Deposit rate paid; This rewards the unit for raising funds cheaply.
Treasury (ALM) spread: Treasury spread = Transfer rates charged to lenders − Transfer rates credited to depositors; Weight by amount. This is the result of the maturity mismatch, kept with the treasury.
Total net interest margin check: Sum of all unit spreads + Treasury spread = Bank's net interest income; Use it to confirm your allocation is complete.
Forward rate from spot rates: (1 + s₂)² = (1 + s₁) × (1 + f); f is the one-year rate one year ahead. s₁ and s₂ are annual spot rates for 1 and 2 years.
Expectations theory: Long rate ≈ average of expected short rates over the period; Liquidity preference adds a positive premium to this.
FRA settlement amount (paid at the start of the period): Settlement = Notional × (Reference rate − FRA rate) × (days ÷ 360) ÷ [1 + Reference rate × (days ÷ 360)]; Positive means the FRA buyer receives; negative means the buyer pays. Use 365 days if the question says so. Discounting is because settlement is made at the start of the period.
FRA settlement (undiscounted): Notional × (Reference rate − FRA rate) × (days ÷ year basis); Use only if the question says settlement is at the end of the period.
Swap net payment (fixed payer): Net = Notional × (Floating rate − Fixed rate) × (period ÷ year); Positive means the fixed payer receives net. Negative means the fixed payer pays net.
Effective cost after swap: Effective rate = Loan rate paid + Fixed rate paid on swap − Floating rate received; If the loan floating and swap floating are the same, the floating terms cancel.
Cap payoff per period: Notional × max(0, Reference rate − Cap strike) × (period ÷ year); Net cost = payoff received − premium paid, shown against the loan interest.
Floor payoff per period: Notional × max(0, Floor strike − Reference rate) × (period ÷ year); Protects a lender or investor in floating assets.
Collar: Borrower collar = Long cap + Short floor; Effective rate stays between the floor and cap strikes, adjusted for net premium.
Repricing gap: Gap = Rate Sensitive Assets (RSA) − Rate Sensitive Liabilities (RSL); Computed for each time bucket. Positive gap means assets reprice faster than liabilities.
Change in NII (simple gap approach): ΔNII ≈ Gap × Δi; Δi is the change in rate over the bucket period. Use the gap of the relevant bucket and a time fraction if the bucket is shorter than one year.
Change in economic value: ΔEVE = EVE(shocked) − EVE(base); Present value of assets minus liabilities (plus off-balance items) under the shocked curve versus the base curve.
Basel IRRBB outlier test: ΔEVE ÷ Tier 1 capital > 15% → supervisory outlier; Applied under the standardised shock scenarios. It triggers supervisory scrutiny, not an automatic capital charge.
Ratio of RSA to RSL: RSA ÷ RSL; Above 1 means asset-sensitive; below 1 means liability-sensitive.

Quick revision

Interest rate risk comes from repricing mismatch, basis risk, yield curve risk and optionality.
Gap = rate-sensitive assets − rate-sensitive liabilities for a time bucket.
Change in net interest income ≈ Gap × change in rate, for the bucket period.
A positive gap gains when rates rise; a negative gap gains when rates fall.
Gap analysis ignores value effects and the timing of cash flows within a bucket.
Price change ≈ −Modified duration × change in yield × price.
Modified duration = Macaulay duration ÷ (1 + y ÷ n), for n payments a year.
Convexity improves the duration estimate, especially for large rate moves.
The earnings view looks at near-term income; the economic value view looks at present value of all cash flows.
Funds transfer pricing credits and charges business units so rate risk sits with the treasury.
A pay-fixed swap hedges a rising-rate liability exposure; an FRA fixes a future borrowing or lending rate.
IRRBB covers rate risk in the banking book, assessed through both earnings and economic value.

Common mistakes

Confusing repricing risk with basis risk Fix: Repricing risk is about when items reprice. Basis risk is about which benchmark they follow. Items can reprice at the same time and still have basis risk.
Treating yield curve risk as just a rise or fall in rates Fix: Yield curve risk is about a change in shape or slope. A parallel shift is the simplest case, and it is not the whole risk.
Including all assets and liabilities in RSA and RSL. Fix: Include only items that reprice or mature within the stated bucket. Exclude equity, fixed assets and longer fixed-rate items.
Reversing the NII effect of a negative gap. Fix: Always compute Gap × Δi with the sign of Δi. A negative gap with a negative Δi gives a positive result.
Using modified duration where Macaulay duration is needed, or the reverse. Fix: Remember that Macaulay is in years and measures time. Modified is the price sensitivity and equals Macaulay ÷ (1 + y/m). Use modified for price change.
Putting Δy as 1 instead of 0.01. Fix: Convert every yield change to a decimal before substituting. Check that the answer is a sensible percentage.
Treating NII change and EVE change as the same measure Fix: Remember NII is a short-term earnings measure, EVE is a long-term value measure. They can have opposite signs.
Applying the full-year effect to every bucket's gap Fix: Multiply each bucket's gap by the fraction of the horizon remaining after repricing.
Using one pooled average cost of funds for every loan. Fix: Use a transfer rate matched to the loan's tenor or repricing date from the yield curve. A pooled rate hides maturity mismatch.
Reversing the spread direction for depositors. Fix: Remember: lenders earn customer rate minus FTP. Depositors earn FTP minus deposit rate.

Exam tips

MCQs often give a short scenario and ask for the source. Use the four-source tagging to answer quickly.
In descriptive answers, always give a one-line example for each source. It shows application.
Do not skip the sign of the gap in numerical questions. State whether the bank is asset or liability sensitive.
Link the sources to the earnings and economic value views in your conclusion.
Start every numerical answer by writing Gap = RSA − RSL, then classify each item. Marks are given for correct classification.
State the sign of the gap and its meaning in words. Examiners look for the positive-versus-negative interpretation.
In case-based MCQs, read the bucket and time horizon carefully. A rate change that applies for part of the year needs a fraction.
Close descriptive answers with a recommendation, such as matching repricing dates or using swaps, and mention that gap analysis assumes a parallel rate shift.