The use of financial derivatives to hedge against interest rate risk: formula sheet

Key formulas

Interest cost change from a rate move

Change in interest = Exposed amount × Change in rate × Time in years

Use only the floating-rate, unhedged amount. For part-years use months ÷ 12.

Net floating exposure

Net exposure = Floating-rate liabilities − Floating-rate assets

A positive figure means you lose when rates rise. A negative figure means you lose when rates fall.

Expectations theory (annual compounding)

(1 + s₂)² = (1 + s₁) × (1 + f₁,₂)

s₁ and s₂ are spot rates for 1 and 2 years. f₁,₂ is the implied forward rate for year 2.

Implied forward rate

f₁,₂ = (1 + s₂)² ÷ (1 + s₁) − 1

Works for annual rates. Rearranged form of the expectations formula.

Settlement amount (paid at start of period)

Settlement = N × (Market rate − FRA rate) × (m ÷ 12) ÷ [1 + Market rate × (m ÷ 12)]

N is the notional amount, m is the length of the FRA period in months. Positive means the bank pays a borrower; negative means the borrower pays the bank.

Undiscounted interest difference

N × (Market rate − FRA rate) × (m ÷ 12)

This is the amount you would pay at the end of the period. Discount it to get the actual settlement.

Reading mXn

Start = m months from now; End = n months from now; Period = n − m months

3v9 gives a 6-month period starting in 3 months.

Quote rule

Borrower pays the higher quoted rate; depositor receives the lower quoted rate

Use the rate that is worse for you as the customer.

Effective rate with FRA

Effective cost = Actual interest paid on loan ± FRA settlement (compounded to end of period)

If hedged properly it equals the FRA rate, plus any margin over the reference rate on the loan.

Futures price

Futures price = 100 − implied annual interest rate (%)

A price of 94.80 implies a rate of 5.20%. Price and rate move in opposite directions.

Number of contracts

Contracts = (Amount to hedge ÷ Contract size) × (Hedge period in months ÷ Contract period in months)

Round to a whole number. A 6-month loan hedged with 3-month contracts needs twice as many contracts.

Tick size and tick value

Tick value = Contract size × 0.01% × (Contract months ÷ 12)

For a $1,000,000 3-month contract, one tick (0.01) is worth $25. Use the contract details given in the question.

Futures gain or loss

Profit or loss = Price movement in ticks × Tick value × Number of contracts

Seller gains when the price falls. Buyer gains when the price rises.

Basis

Basis = (100 − current spot rate) − current futures price

Cash price minus futures price. It converges to zero at expiry.

Unexpired basis

Basis at a date = Opening basis × (Months remaining to expiry ÷ Months at start)

Linear convergence is an assumption. State it if you use it.

Hedge direction rule

Borrower: sell futures now, buy back later. Depositor: buy futures now, sell later.

Check the direction by asking which way the futures price must move for you to gain.

Effective rate and hedge efficiency

Effective rate for a borrower = (Cash interest − futures gain) ÷ Amount ÷ Years. Effective rate for a depositor = (Deposit interest + futures gain) ÷ Amount ÷ Years. Hedge efficiency = Futures gain ÷ Loss on the underlying position

Compare the futures gain with the extra cost caused by the rate move. A gain larger than the loss means the hedge over-compensated. If the futures position makes a loss, the sign of the futures adjustment reverses.

Futures price

Futures price = 100 − interest rate (%)

A rate of 4.6% is a futures price of 95.40. Use this to convert strike prices to rates.

Tick value

Tick value = contract size × 0.01% × (months ÷ 12)

For a $1,000,000 three-month contract: 1,000,000 × 0.0001 × 3/12 = $25.

Number of contracts

Contracts = loan ÷ contract size × (loan months ÷ contract months)

Round to a whole number. The exam usually states the contract size and period.

Exchange-traded borrower hedge

Buy put options. Exercise if futures price at expiry < strike price. Gain = (strike − futures price) in ticks × tick value

A lender or depositor buys calls and exercises if the futures price is above the strike.

Effective rate (OTC cap)

If market rate > strike: strike + annualised premium. If market rate ≤ strike: market rate + annualised premium

Annualised premium = flat premium % × 12 ÷ number of months in the period. This ignores the time value of the premium.

Effective rate (exchange-traded put)

Effective rate = (100 − strike price) + premium, if exercised

Premium is in annual percentage points. For a 3-month loan, strike of 95.50 and premium 0.30 gives 4.50% + 0.30% = 4.80%.

Collar

Net premium = cap premium paid − floor premium received. Rate paid is capped at cap rate and floored at floor rate, then add net premium

A borrower buys the cap and sells the floor. Between the two strikes you pay the market rate.

Net saving from comparative advantage

Net saving = |Fixed rate differential − Floating rate differential|

Differential = the difference between the two companies' rates in that market. Compute both, then take the gap between them.

Gain after bank fee

Gain to share = Net saving − Bank fee

If the bank fee is quoted as a total, deduct it before sharing. Share equally unless the question says otherwise.

Net cost with a swap (payer of fixed)

Net cost = Loan interest + Fixed paid on swap − Floating received on swap

If the loan is floating and the floating received matches it, the floating parts cancel and the cost becomes fixed.

Net cost with a swap (payer of floating)

Net cost = Loan interest + Floating paid on swap − Fixed received on swap

Use this for the party that wants floating exposure.

Annual swap cash flow

Cash flow = Notional principal × rate difference × (months ÷ 12)

Use the rate difference for the net settlement. Adjust for the period length if payments are not annual.

Swaption outcome

Exercise only if the market swap rate is worse than the swaption's strike rate

The premium is paid either way. Compare the outcome with and without the swaption.

FRA settlement idea

Borrower pays fixed FRA rate effectively; bank pays (market rate − FRA rate) × amount × period if market rate is higher

For a borrower, FRA gives an effective rate equal to the FRA rate, whichever way rates move. The settlement is normally discounted at the market rate.

Interest rate future price

Futures price = 100 − expected interest rate (%)

A borrower sells futures to hedge; a depositor buys. A rate rise lowers the price, so the seller gains.

Futures hedge number of contracts

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

Round to a whole number of contracts. Hedge outcome includes basis difference at closing.

Futures gain or loss

Gain/loss = Ticks moved × tick value × number of contracts

For a 3-month contract, a tick of 0.01% on a standard contract is worth the contract size × 0.01% × 3/12.

Option outcome for a borrower

Effective rate = lower of (market rate, option strike rate) + premium cost

The option is exercised only if the market rate is above the strike. Include premium, ideally with its financing cost.

Swap net rate for a borrower

Net cost = rate paid on loan + swap rate paid − swap receipt

Check what is received and what is paid. Add any bank fee. The swap changes the rate basis, not the loan itself.

Collar

Borrower buys cap at higher strike and sells floor at lower strike

The floor premium reduces the cap cost. The effective rate lies between the floor and cap strikes, plus any net premium.

Quick revision

• A borrower loses when rates rise; a depositor loses when rates fall.
• An FRA fixes the rate for a future period; you receive or pay the difference to the reference rate at settlement.
• FRA rates are quoted as, for example, 3v9, meaning starting in 3 months and ending in 9 months.
• Interest rate futures prices move opposite to interest rates: price = 100 − rate.
• A borrower hedges with futures by selling; a depositor hedges by buying.
• With options, a borrower buys put options on interest rate futures (equivalent to a cap on rates); a depositor buys call options on futures (equivalent to a floor on rates).
• Futures hedges can be imperfect because of basis risk and contract size, which means rounding the number of contracts.
• Options give a right, not an obligation, so a company keeps the benefit of favourable moves but pays a premium.
• For a borrower, a collar is a bought cap and a sold floor; for a depositor, it is a bought floor and a sold cap. The premium received on the sold option offsets the premium paid on the bought option, which reduces the net cost, but the company gives up the benefit of favourable moves beyond the sold strike.
• A swap exchanges interest payment streams, and the principal is not exchanged.
• Use comparative advantage to see whether a swap saves cost for both parties.
• Compare hedges on certainty, cost, flexibility, and counterparty or margin risk.
• Always end with a recommendation that links to the company's size, view on rates and risk appetite.

Common mistakes

• Treating all debt as exposed to rate rises. Fix: Tag each item fixed or floating first. Fixed debt has no cash flow exposure to rate moves, though it carries an opportunity cost.
• Confusing basis risk with gap risk. Fix: Gap risk is about timing: items reprice on different dates. Basis risk is about the rate: items follow different benchmarks or a hedge does not track the exposure.
• Misreading 3v9 as a 9-month period or a 3-month period. Fix: Period = second number minus first number. 3v9 starts in 3 months and lasts 6 months.
• Using the wrong side of the bank quote. Fix: A borrower pays the higher rate. A depositor receives the lower rate.
• Buying futures to hedge a loan, or selling them to hedge a deposit. Fix: Remember that price = 100 − rate. A rise in rates is a fall in price. A borrower needs a gain when prices fall, so the borrower sells.
• Forgetting to scale the number of contracts for the loan period. Fix: Always apply the period ratio: months of exposure ÷ months of the contract. Write the formula out before calculating.
• Forgetting the premium in the effective rate, or forgetting it when the option lapses. Fix: Always add the premium to the result, whether or not you exercise. It is a sunk cost paid on day one.
• Using the wrong option type on futures: buying calls when the borrower needs puts. Fix: Futures price = 100 − rate. Rates up means price down. A borrower profits from a price fall, so buys puts.
• Subtracting the wrong rates, such as comparing one company's fixed rate with the other's floating rate. Fix: Always compute differentials within the same market: fixed minus fixed, floating minus floating. Then compare those two gaps.
• Forgetting to deduct the bank fee before sharing the gain. Fix: Write 'Saving − fee = gain to share' as a line in your answer. Then share it.

Exam tips

• Start every answer by finding the net floating exposure. Examiners reward this first step.
• Name the risk type precisely. Use gap, basis, yield curve or refinancing risk, rather than saying only interest rate risk.
• When discussing the curve, give at least two theories. Say which fits the scenario and why.
• Link the exposure to a hedging recommendation and its limits. Do not stop at identification.
• Show your workings with months ÷ 12 clearly. Marks follow method even if arithmetic slips.
• Write the mXn decode first. Examiners often test whether you know the start date and the period length.
• Show the borrower or depositor decision and the bank quote you chose. These are easy marks and the scenario usually hints at them.
• Prove the hedge by showing the effective rate equals the FRA rate. It earns marks and checks your arithmetic.