FRM Part I · FRM Exam Part I

Interest Rates: formula sheet

Full chapter guide

Key formulas

Future value, m times per year: FV = A × (1 + Rm/m)^(m × n); Rm is the annual quoted rate, m the compounding periods per year, n the years.
Future value, continuous: FV = A × e^(Rc × n); Rc is the continuously compounded annual rate.
Effective annual rate: EAR = (1 + Rm/m)^m − 1; With continuous compounding, EAR = e^Rc − 1.
Periodic to continuous: Rc = m × ln(1 + Rm/m); Use the natural log, not log base 10.
Continuous to periodic: Rm = m × (e^(Rc/m) − 1); For annual compounding, set m = 1: R = e^Rc − 1.
Discount factor, continuous: PV = FV × e^(−Rc × n); Reverse of continuous growth.
Discount factor (annual compounding): d(T) = 1 ÷ (1 + z_T)^T; z_T is the T-year spot rate. For m compounding periods per year use (1 + z/m)^(mT).
Discount factor (continuous compounding): d(T) = e^(−z_T × T); Use when the question states continuous compounding.
Forward rate (annual compounding): (1 + z_2)^T2 = (1 + z_1)^T1 × (1 + f)^(T2 − T1), so f = [(1 + z_2)^T2 ÷ (1 + z_1)^T1]^(1 ÷ (T2 − T1)) − 1; f is the forward rate between T1 and T2. Equivalent to d(T1) ÷ d(T2) = (1 + f)^(T2 − T1).
Forward rate (continuous compounding): f = (z_2 × T2 − z_1 × T1) ÷ (T2 − T1); The forward rate is a time-weighted difference of the spot rates. Its compounding is also continuous.
Par yield: c = m × (1 − d(T)) ÷ Σ d(t_i); m is coupons per year. The sum runs over all coupon dates up to T. This gives the annual coupon rate per 1 of face value.
Bootstrapping a par bond (annual coupons): 100 = c × Σ d(t_i) for earlier dates + (100 + c) × d(T); Solve for d(T), then convert to z_T.
FRA value (receive fixed R_K, annual accrual, per unit notional): Value = N × (R_K − R_F) × τ × d(T2); R_F is the current forward rate for T1 to T2, τ is the accrual length and d(T2) discounts from the payment date. Use d(T1) instead if the FRA settles at T1 on a discounted basis. Reverse the sign for the pay-fixed side.
Price using zero rates (annual compounding): P = Σ C_t ÷ (1 + z_t)^t; Each cash flow C_t uses the zero rate z_t for its own maturity t. The final cash flow includes the principal.
Price using YTM (semiannual): P = Σ (c/2) ÷ (1 + y/2)^k + F ÷ (1 + y/2)^n; c is the annual coupon, F the face value, n the number of half-years, y the bond-equivalent yield. k runs from 1 to n.
Price with continuous compounding: P = Σ C_t × e^(−z_t × t); Use when zero rates are quoted with continuous compounding.
Yield to maturity definition: P = Σ CF_t ÷ (1 + y)^t, solve for y; No closed form for coupon bonds. Use a financial calculator (N, I/Y, PV, PMT, FV).
Convert semiannual yield to effective annual: EAY = (1 + y/2)^2 − 1; The bond-equivalent yield y is twice the semiannual rate.
Dirty price: Dirty price = Clean price + Accrued interest; Accrued interest = coupon × (days since last coupon ÷ days in coupon period).
Par, premium, discount: Price < par ⇔ YTM > coupon rate; Price > par ⇔ YTM < coupon rate; Price = par when YTM equals the coupon rate on a coupon date.
Macaulay duration: D_Mac = Σ [t × PV(CF_t)] ÷ P; t is in years. PV(CF_t) is the present value of each cash flow at the bond's yield. P is the bond price. For a zero-coupon bond, D_Mac = maturity.
Modified duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield, m the compounding periods per year. For annual compounding, divide by (1 + y).
Duration price approximation: ΔP/P ≈ −D_mod × Δy; Δy is in decimals (50 bp = 0.005). Works for small yield changes.
Dollar duration and DV01: Dollar duration = D_mod × P; DV01 = D_mod × P × 0.0001; DV01 is the dollar change for a 1 bp yield move. Scale by the face amount held.
Convexity (annual cash flows): C = (1/P) × Σ [t × (t + 1) × CF_t ÷ (1 + y)^(t + 2)]; With m payments per year and t counted in periods, divide the result by m² to express it in years squared.
Duration plus convexity approximation: ΔP/P ≈ −D_mod × Δy + ½ × C × (Δy)²; The convexity term is always positive when C is positive, so it adds to the price change.
Effective duration: D_eff = (P₋ − P₊) ÷ (2 × P₀ × Δy); P₋ and P₊ are prices after the yield falls and rises by Δy. Used for bonds with embedded options.
Effective convexity: C_eff = (P₋ + P₊ − 2 × P₀) ÷ (P₀ × (Δy)²); Same shifted prices as effective duration.
Portfolio duration: D_p = Σ w_i × D_i; w_i are market-value weights. DV01s of positions simply add.
Forward rate from spot rates (annual compounding): (1 + S₂)² = (1 + S₁) × (1 + f₁,₂); General form: (1 + Sₙ)ⁿ = (1 + Sₙ₋₁)ⁿ⁻¹ × (1 + fₙ₋₁,ₙ). Use the same compounding for all rates.
Pure expectations theory: forward rate = expected future spot rate; No risk premium. Long rate is a geometric average of expected short rates.
Liquidity preference theory: forward rate = expected future spot rate + liquidity premium; The premium is positive and rises with maturity, so forwards overstate expected spot rates.
Long rate as average of short rates (approximation): S₂ ≈ (S₁ + E[s₁,₂]) ÷ 2; Exact form is geometric. The average is a quick approximation for small rates.
Duration-based hedge ratio (DV01 form): N = − DV01 of portfolio ÷ DV01 of one futures contract; Negative means short futures. Use the CTD bond's DV01 for Treasury futures.
Duration-based hedge ratio (duration form): N = − (P × D_P) ÷ (F × D_F); P is portfolio value, F is the futures contract price (value per contract), D_P and D_F are the durations. Use consistent duration measures.
Futures DV01 from CTD: DV01 of futures ≈ DV01 of CTD bond ÷ conversion factor; Futures price ≈ CTD price ÷ conversion factor, so the futures DV01 per contract equals the CTD DV01 divided by the conversion factor (per unit of bond face value).
Cheapest to deliver: Cost = quoted bond price − (futures settlement price × conversion factor); The bond with the smallest cost is the CTD.
Cash price of a delivered bond: Cash received = (settlement price × conversion factor) + accrued interest; Used in Treasury futures delivery.
Eurodollar futures price: Quote = 100 − futures rate (%); A 1 basis point move equals 25 dollars on a 1 million dollar, three-month contract.
Convexity adjustment (approximation): Forward rate = futures rate − ½ σ² T₁ T₂; σ is the annual standard deviation of the short rate change, T₁ is the futures maturity and T₂ is the maturity of the underlying rate, in years.
FRA settlement: Payment = L × (R_floating − R_fixed) × τ ÷ (1 + R_floating × τ); L is notional, τ is the accrual fraction. Paid at the start of the period, so it is discounted. Positive means the fixed-rate payer receives.
Target duration hedge: N = (D_T − D_P) × P ÷ (D_F × F); Changes duration from D_P to a target D_T. Positive N means buy futures to lengthen duration.

Quick revision

Compounding conversion: (1 + Rm ÷ m)^m = e^Rc, so Rc = m × ln(1 + Rm ÷ m).
A higher compounding frequency gives a higher effective annual rate for the same quoted rate.
Bond price = Σ of coupons and principal, each discounted at the right rate.
Yield to maturity is the single discount rate that equates the present value of cash flows to the price.
A bond trades at a discount when its coupon rate is below its yield, and at a premium when above.
Par rate is the coupon rate that prices a bond at par.
Forward rate between T1 and T2 with continuous compounding: (R2 × T2 − R1 × T1) ÷ (T2 − T1).
Modified duration ≈ Macaulay duration ÷ (1 + y ÷ m).
Price change ≈ −D × Δy + ½ × C × (Δy)², using modified duration D and convexity C.
Convexity helps the holder: price gains from a yield fall exceed losses from an equal yield rise.
Expectations theory sees forward rates as expected future spot rates; liquidity preference adds a premium for longer maturities.
Hedge with futures using a ratio based on the portfolio and futures durations and values, and check the sign of the position.

Common mistakes

Dividing a continuous rate by m before using it, as in e^(Rc/m × m n) with a periodic formula. Fix: Continuous growth is only e^(Rc × n). Dividing Rc by m appears only when converting to a periodic rate.
Treating a semiannual 6% as 6% effective per year. Fix: Compute EAR = (1 + 0.06/2)^2 − 1 = 6.09%. Always attach the frequency to the rate.
Treating the 2-year par yield as the 2-year zero rate. Fix: The par yield reflects the coupon paid at year 1 too, so the 2-year zero rate differs from it (it is higher when the curve slopes up). Bootstrap to get the zero rate.
Taking the forward rate as the simple difference of spot rates, such as 4% − 3% = 1%. Fix: With annual compounding, divide the compounded growth factors: (1 + z_2)^2 ÷ (1 + z_1) − 1. With continuous compounding, use (z_2 T2 − z_1 T1) ÷ (T2 − T1). Both are time-weighted.
Using the annual coupon and annual yield with semiannual periods Fix: Halve the coupon and the yield, and double the years to get n.
Treating YTM as the spot rate for the bond's maturity Fix: A spot rate discounts one cash flow. YTM is one rate that fits all cash flows. They differ unless the curve is flat or the bond is a zero.
Using Macaulay duration directly in the price-change formula. Fix: Divide Macaulay duration by (1 + y/m) first. The price-change formulas use modified duration.
Entering Δy in percent or basis points instead of decimals. Fix: Convert to decimals first: 50 bp = 0.005. Check that a 100 bp move gives a price change near the duration number in percent.
Saying liquidity preference theory always predicts an upward curve. Fix: The curve is the expectations path plus a premium. Strongly expected rate cuts can still invert it.
Treating the forward rate as the expected spot rate under every theory. Fix: Only pure expectations sets forward = expected spot. With a liquidity premium, expected spot = forward − premium.

Exam tips

Look at the compounding frequency before reading anything else. Many wrong options come from using the wrong m.
Know the secured versus unsecured split cold: Treasury, repo and SOFR are secured or government-backed; LIBOR and fed funds are unsecured.
Remember LIBOR has been discontinued and SOFR is the US dollar replacement. Questions on benchmark reform test why a nearly risk-free rate differs from a bank-credit rate.
Use ln and e^x keys. Practise until the conversion takes under a minute.
Sanity-check direction. For the same growth, the quoted rate falls as m rises: continuous < periodic with high m < periodic with low m < annual, and the annual rate equals the EAR. You can often eliminate options without computing.
Check the compounding convention first. It is the most common trap, and each option usually corresponds to a different convention error.
Expect the curve-shape logic in conceptual questions: upward curve means forwards above zeros above par yields at long maturities, downward curve reverses it.
Bootstrapping needs only the formula for the last discount factor. Practise the calculator memory routine so a two- or three-period bootstrap takes under two minutes.