CMA Final · Strategic Financial Management

Interest Rate Derivatives: formula sheet

Full chapter guide

Key formulas

Macaulay duration: D = Σ [t × PV(CFt)] ÷ Σ PV(CFt); Denominator equals the bond's current price. t is in years. For a zero-coupon bond, D equals its maturity.
Modified duration: MD = D ÷ (1 + y); y is the yield per period. For annual coupons, use the annual yield. For semi-annual coupons, divide D (in years) by (1 + y/2) and keep MD in years.
Approximate price change: ΔP ≈ −MD × Δy × P; Δy in decimal form (1% = 0.01). The negative sign shows price and yield move opposite ways. Valid for small yield changes.
Basis point value (PV01): BPV = MD × 0.0001 × P; The rupee change in price for a 1 basis point change in yield. For a portfolio, add the BPVs of the positions.
Basis point: 1 basis point = 0.01% = 0.0001; 100 basis points = 1%.
Contract period: Contract period = Second number − First number (in months); For a 3x6 FRA, the period is 3 months. Settlement is at month 3.
FRA rate (implied forward rate, simple interest): F = [ (1 + R_L × t_L) ÷ (1 + R_S × t_S) − 1 ] ÷ (t_L − t_S); R_L and R_S are annual rates for the long and short maturities. t_L and t_S are the times in years. Use this when the question asks you to price the FRA from spot rates.
Settlement amount (buyer's view): Settlement = N × (RR − FRA rate) × (d ÷ 360) ÷ [1 + RR × (d ÷ 360)]; N is notional, RR is the reference rate at settlement, d is the days in the contract period. Positive means the seller pays the buyer. Negative means the buyer pays the seller. Use 365 instead of 360 if the question says so.
Undiscounted interest difference: N × (RR − FRA rate) × (d ÷ 360); This is the amount that would be due at the end of the contract period. Discount it only if settlement is at the start.
Effective rate after hedging: Effective rate = Actual borrowing rate − (RR − FRA rate) for a buyer; If the loan is priced at the reference rate, the buyer's effective cost equals the FRA rate.
Cost of carry price (coupon bond): F = S × (1 + r × t) − FV of coupons; S is the full (dirty) spot price, r the financing rate p.a., t the time in years. Coupons are carried forward at the reinvestment rate to the futures expiry date. Adjust for accrued interest and conversion factor only if the question gives them.
Price and rate for rate futures: Futures price = 100 − futures rate (%); Used for T-bill or deposit-rate contracts. A fall in price means a rise in the implied rate.
Value of one basis point: Value of 1 bp = Notional × 0.0001 × (contract period in days ÷ day-count basis); For a 3-month contract on ₹1 crore, 1 bp = ₹1,00,00,000 × 0.0001 × 3/12 = ₹250. Use the day-count basis the question gives.
Number of contracts (rate hedge): N = (Exposure ÷ Contract size) × (Exposure period ÷ Contract period); Scale for both amount and time. A ₹10 crore exposure for 6 months needs twice as many 3-month contracts as one for 3 months.
Number of contracts (price-sensitivity hedge): N = (PVBP of portfolio) ÷ (PVBP of one futures contract); PVBP is the rupee change in value for a 1 bp change in yield. The duration form is N = (D of portfolio × value of portfolio) ÷ (D of futures × value of one futures contract).
Gain or loss on a futures position: Short: (Entry price − Exit price) × contract size; Long: (Exit price − Entry price) × contract size; For price-quoted rate contracts, convert the price move into bp and then into rupees using the value of 1 bp.
Effective hedged rate: Effective rate = Rate paid in the market − Futures gain (as a rate) or + Futures loss; If the hedge is perfect, the effective rate equals the rate locked in by the futures price.
Fixed leg payment: Fixed payment = Notional × Fixed rate × (Days ÷ 365), or × (1 ÷ payments per year); Use the day-count in the question. If none is given, use the period fraction such as 6/12 for half-yearly.
Floating leg payment: Floating payment = Notional × Floating rate set at start of period × period fraction; The rate is fixed at the beginning of the period and paid at its end.
Net settlement: Net = Fixed leg − Floating leg; If positive, the fixed payer pays the net. If negative, the fixed payer receives it.
Comparative advantage gain: Total gain = |Fixed spread difference − Floating spread difference|; Spread difference = rate of the weaker borrower minus rate of the stronger borrower in each market.
Net cost after swap: Net cost = Interest paid on own borrowing + Swap payment − Swap receipt; Compute it separately for each party and compare with the direct borrowing cost.
Swap value at a date: Value (to fixed payer) = Value of floating-rate bond − Value of fixed-rate bond = Par − PV of fixed-rate bond (immediately after a reset date); The floating-rate bond, including the notional, is worth par immediately after a reset date, once the coupon due on that date has been paid. Value the fixed-rate bond on its remaining coupons and notional, discounted at current market rates. Between reset dates, the floating leg is not exactly par.
Caplet payoff: Payoff = Notional × max(0, Reference rate − Cap strike) × (days ÷ 365); Use the day count given in the question. If none is given, use the period fraction such as 6/12 for half-yearly.
Floorlet payoff: Payoff = Notional × max(0, Floor strike − Reference rate) × (days ÷ 365); Paid to the buyer only when the reference rate is below the strike.
Effective rate with a cap (borrower): Effective rate = min(Reference rate, Cap strike) + Premium cost as a rate; Convert the premium to a per-annum rate over the loan life if the question asks for an all-in cost.
Effective rate with a collar (borrower): Floor strike ≤ Effective rate before premium ≤ Cap strike; Net premium = Cap premium paid − Floor premium received. It is zero for a zero-cost collar.
Net premium of a collar: Net premium = Premium on cap bought − Premium on floor sold; Reverse the signs for an investor who buys a floor and sells a cap.
Payer swaption payoff at expiry: Payoff = Notional × max(Market swap rate − Strike, 0) × Annuity factor; Annuity factor is the present value of ₹1 per year over the swap tenor. Payoff is before the premium.
Receiver swaption payoff at expiry: Payoff = Notional × max(Strike − Market swap rate, 0) × Annuity factor; Gains when swap rates fall below strike.
Net profit to buyer: Net profit = Payoff − Premium paid; Ignore time value of the premium unless the question asks for it.
Net settlement of a swap or OIS period: Net = Notional × (Floating rate − Fixed rate) × Days ÷ 365; Positive means the fixed-rate payer receives. Use the day-count the question gives.
Annuity factor: A = [1 − (1 + r)^−n] ÷ r; r is the discount rate per period and n is the number of periods.

Quick revision

A borrower on a floating rate pays more when rates rise; an investor holding fixed-rate bonds suffers a fall in the market value of the bonds, while a floating-rate investor earns higher interest income.
Bond prices and market interest rates move in opposite directions.
An FRA fixes a rate for a future period and settles in cash on the difference; the principal is notional and is not exchanged. The settlement is made at the start of the contract period, as the interest difference discounted at the market (settlement) rate, using the day-count given in the question.
The buyer of an FRA gains if the market rate settles above the agreed rate.
Interest rate futures are standardised, exchange traded and marked to market daily.
A swap exchanges interest payments, usually fixed for floating, on a notional principal that is not exchanged.
A fixed-rate payer in a swap gains when floating rates rise.
Net swap cost = interest on the underlying loan, adjusted by what you pay and receive under the swap.
A cap protects a borrower against rising rates; a floor protects an investor against falling rates.
A borrower's collar is a cap bought plus a floor sold; an investor's collar is a floor bought plus a cap sold. Either way, it lowers the net premium but limits the benefit.
A swaption gives the right, not the obligation, to enter a swap at a set rate.
Always compare the hedged outcome with the unhedged outcome before you recommend.

Common mistakes

Forgetting the negative sign and saying price rises when yield rises. Fix: Before writing the answer, ask: yield up, price down. Write the direction in words.
Using Macaulay duration directly as the price sensitivity. Fix: Divide by (1 + y) first to get modified duration. Use MD, not D, in ΔP and BPV.
Treating 3x6 as a 6-month contract, or as a contract that starts at month 6 Fix: The first number is the start and the second is the end. Length = second − first. A 3x6 FRA is a 3-month period starting in 3 months.
Forgetting to discount the settlement amount Fix: Settlement is paid at the start of the contract period, but interest accrues to its end. Always divide by 1 + RR × period fraction unless the question says settlement is at the end.
Buying futures when the company is a borrower fearing a rate rise. Fix: Think in prices. A borrower needs a gain when the price falls, so the borrower sells futures. Write 'rates up, price down, short gains' before you start.
Ignoring the time scaling when counting contracts. Fix: Multiply by (exposure period ÷ contract period). A 6-month exposure with 3-month contracts needs double the contracts.
Exchanging the full notional principal in a plain vanilla swap. Fix: State that notional is never exchanged in an interest rate swap. Only the net interest moves.
Using the annual rate for a half-yearly or quarterly period. Fix: Write the period fraction beside each leg before multiplying.
Treating a cap as protection for a lender. Fix: A cap limits the rate a borrower pays. A floor protects the lender or investor's minimum return.
Paying the cap payoff when the reference rate equals or is below the strike. Fix: Pay only when the rate is above the cap strike. Otherwise the payoff is zero.

Exam tips

Section A questions often give modified duration and ask for the price change or BPV. Learn the formulas so you can answer in under a minute.
In written answers, show the table of t, cash flow, PV and t × PV. Marks are given for method even if the final figure slips.
Always state the direction of change and the rupee amount, then add one line on what it means for the bank or company.
Keep decimals to at least three places in duration until the final step to avoid rounding drift.
Link the measure to a hedge: a high BPV points to the need for swaps, futures or FRAs.
Write the notation decoding as your first line. Examiners award marks for stating the settlement date and the contract period.
State the day-count basis you use (360 or 365) and follow the question if it specifies one. Mention it in one line.
Always end with a sentence saying who pays whom and how much. A correct number with no direction can lose marks.