Interest Rate Risk Management: formula sheet

Key formulas

Repricing gap

Gap = Rate Sensitive Assets (RSA) − Rate Sensitive Liabilities (RSL)

Measured for each time bucket. Positive gap means income rises when rates rise.

Change in net interest income

ΔNII ≈ Gap × Δi

Δi is the change in interest rate in decimal. Valid for the bucket over the period considered, assuming the rate change applies for the full period.

Price effect of rate change

ΔP ≈ −Modified Duration × Δy × P

Approximation for small changes in yield. Rates up, price down.

Direction of the two bond risks

Rates ↑: price ↓, reinvestment income ↑. Rates ↓: price ↑, reinvestment income ↓

Use this to classify the risk named in a case.

Basis

Basis = Rate A − Rate B (two different benchmarks)

Basis risk is the risk that this difference changes.

Meaning of m x n

FRA starts after m months and ends after n months; contract period = (n − m) months

3x6 means a 3-month rate starting 3 months from now. 6x12 means a 6-month rate starting 6 months from now.

FRA settlement amount

Settlement = [(R_m − R_f) × Notional × (D ÷ 360 or 12-month fraction)] ÷ [1 + R_m × (D ÷ 360 or 12-month fraction)]

R_m = market (reference) rate on settlement date; R_f = FRA rate; use the same time fraction in both numerator and denominator. Use the day count given in the question (360 or 365).

Who pays whom

If R_m > R_f, seller pays buyer. If R_m < R_f, buyer pays seller.

Buyer gains when rates rise; seller gains when rates fall.

Effective result of hedge

Net borrowing cost = actual interest at R_m − FRA receipt (or + FRA payment) = interest at R_f

Compare on the same date. If the FRA is discounted, the receipt can be invested or used to reduce the borrowing at the start.

Implied rate from quote

Implied rate = 100 − futures price

Use for index-style quotes such as 94.50 meaning 5.50%.

Tick value (short-term rate contract)

Tick value = Contract size × Tick size × (Months of contract period ÷ 12)

Tick size as a rate, e.g. 0.01% = 0.0001. Use the period of the underlying instrument, such as 3 months.

Number of contracts (simple)

Contracts = (Exposure amount ÷ Contract size) × (Exposure period ÷ Contract period)

Work out the bracket (Exposure ÷ Contract size) first, then multiply by the period ratio. Apply the period ratio only when the exposure period differs from the contract period. If both are 3 months, the ratio is 1 and Contracts = Exposure ÷ Contract size.

Number of contracts (duration-based)

Contracts = (Portfolio value × Portfolio duration) ÷ (Futures value × Futures duration)

Used for bond portfolios. Use modified duration if the question gives it.

Profit or loss on futures

Short: (Sale price − Closing price) × ticks × tick value; Long: (Closing price − Purchase price) × ticks × tick value

Count the number of ticks in the price change, then multiply by tick value and number of contracts.

Hedge position rule

Fear rising rates (borrower, bondholder) → Sell futures; Fear falling rates (investor) → Buy futures

Check the direction with the inverse price–rate link.

Fixed-rate differential

Fixed differential = Fixed rate of weaker party − Fixed rate of stronger party

Use the rates each party would pay for direct borrowing.

Floating-rate differential

Floating differential = Floating spread of weaker party − Floating spread of stronger party

Compare spreads over the same benchmark, such as MIBOR.

Quality spread differential (QSD)

QSD = |Fixed differential − Floating differential|

This is the absolute difference between the two differentials. It is the total gain available to share. If it is zero, there is no scope for a swap gain.

Comparative advantage rule

Stronger party borrows where the differential is wider; weaker party borrows where the differential is narrower

The stronger party borrows in the market where the differential is wider. The weaker party borrows in the market where the differential is narrower.

Net gain per party

Gain per party = (QSD − intermediary fee) ÷ 2 if shared equally

Share equally only if the question says so. Otherwise follow the stated ratio.

Net swap settlement

Net payment = Notional × (Fixed rate − Floating rate) × Period fraction

Positive means the fixed payer pays. Negative means the fixed payer receives.

Effective cost after swap

Effective cost = Rate on own loan + Swap payment rate − Swap receipt rate

Add what you pay and subtract what you receive. Do this for both parties and check the total.

Value of a swap to fixed payer

Value to fixed payer = PV of floating leg − PV of fixed leg

Value to the fixed receiver is the opposite. Just after a floating reset, the floating leg is worth the notional.

Cap payoff per period (buyer)

Notional × max(0, Reference rate − Cap strike) × (days ÷ 360 or period fraction)

Paid at the end of each period. Use the day-count given in the question; if none, use the period as a fraction of the year.

Floor payoff per period (buyer)

Notional × max(0, Floor strike − Reference rate) × (days ÷ 360 or period fraction)

Paid when rates fall. Useful for lenders and investors.

Borrower's net interest with a cap

Interest on loan + Premium cost − Cap payoff

Effective rate is capped at (cap strike + loan spread) + premium cost, per annum.

Collar for a borrower

Buy cap at higher strike, sell floor at lower strike; net premium = Cap premium − Floor premium

The reference-rate component lies between floor strike and cap strike. Add the loan spread and the net premium to get the effective borrowing rate.

Collar payoff to the borrower

Cap payoff received − Floor payoff paid

If rate > cap strike, receive the excess. If rate < floor strike, pay the shortfall. Between the two, zero.

Premium as annual percentage

Premium ÷ Notional (as % of notional); annualise by dividing by years, or use annuity if asked

The simple method is common in exam answers. Follow the question's instruction on time value.

Payer swaption exercise rule

Exercise if market swap rate > strike rate

You pay the strike fixed rate and receive floating. Otherwise let it lapse and swap at the market rate.

Receiver swaption exercise rule

Exercise if market swap rate < strike rate

You receive the strike fixed rate. Otherwise let it lapse.

Annual saving on exercise

Saving per year = |Market swap rate − Strike rate| × Notional principal

This is the saving compared with entering a fresh swap at the market rate. Use the payment frequency if payments are not annual, e.g. half-yearly means × ½.

Net benefit

Net benefit = Total savings − Premium paid

Add the cost of financing the premium if the question asks for it. Present value of savings is better when discount rate is given.

Cap and floor payoff per period

Cap payoff = Max(0, Floating rate − Cap rate) × Notional × period; Floor payoff = Max(0, Floor rate − Floating rate) × Notional × period

Each period is settled separately; a cap or floor is a series of options.

Quick revision

• A floating-rate borrower fears rising rates; a floating-rate lender fears falling rates.
• An FRA fixes the rate for a future period; only the interest difference is settled in cash, not the principal.
• FRA settlement is usually discounted because it is paid at the start of the period, not the end.
• Interest rate futures are quoted as price = 100 − rate, so a rate rise means a price fall.
• A borrower hedges by selling futures; a lender hedges by buying them.
• Futures need margin and are marked to market; an FRA is over-the-counter and has no daily settlement.
• A plain vanilla swap exchanges fixed for floating on a notional that is not itself exchanged.
• Net swap cost = loan interest paid + swap payment − swap receipt; compare with the unhedged cost.
• A cap protects a borrower against rates above the strike; a floor protects a lender against rates below it.
• A collar buys a cap and sells a floor, so the premium received lowers the cost but limits the gain.
• A swaption gives the right, not the obligation, to enter a swap at a set rate.
• Always subtract the option premium (with its time value if asked) to get the true net result.

Common mistakes

• Treating basis risk and reinvestment risk as the same thing. Fix: Basis risk is about two different benchmarks moving unequally. Reinvestment risk is about reinvesting cash flows at a lower rate.
• Saying a rate rise is bad for every bond holder. Fix: A rise lowers price but raises reinvestment income. Which dominates depends on the holding period and duration.
• Using the full 6 months for a 3x6 FRA. Fix: The contract period is n − m = 3 months. The FRA starts at month 3 and ends at month 6.
• Forgetting to discount the settlement. Fix: Settlement is made at the start of the period, so divide by 1 + R_m × fraction. Always include this step unless the question says otherwise.
• Buying futures to hedge a future borrowing. Fix: Ask what happens to the futures price if rates rise. It falls, so you must be short (sell) to gain.
• Using annual tick value for a 3-month contract. Fix: Multiply by period ÷ 12. Contract size × 0.01% × 3/12 for a 3-month contract.
• Treating the QSD as the gain of each party instead of the total gain. Fix: Always write 'Total gain = QSD' and then divide it per the stated ratio after deducting any fee.
• Letting each party borrow where its absolute rate is lower. Fix: Compare the gaps, not the rates. The stronger party borrows where its advantage over the weaker party is larger.
• Treating the cap payoff as the full reference rate instead of the excess over the strike. Fix: Always write max(0, rate − strike) first, then multiply.
• Ignoring the period fraction and using the annual rate for a quarterly or half-yearly payoff. Fix: Multiply by months ÷ 12 or days ÷ 360 as given, every time.

Exam tips

• In case-scenario MCQs, name the risk from the direction test before reading the options.
• In written answers, define the risk in one line, apply it to the facts, then conclude with the hedge. This is the provision-facts-conclusion pattern.
• Always show the sign of the gap and what it means for income.
• Mention that hedges carry costs and may leave residual basis risk. Examiners reward this.
• Learn the instrument details in the separate FRA, futures, swap and option topics, because numerical questions are built on them.
• Write the notation decoding first, for example "3x6: starts in 3 months, period 3 months". It earns method marks and prevents period errors.
• Show the discounting step clearly. Examiners award a mark for it separately.
• Use the day count and time fraction given in the question, and keep it the same throughout.