Properties of Interest Rates: formula sheet

Key formulas

Future value, compounding m times a year

FV = PV × (1 + R/m)^(m×T)

R is the quoted annual rate, m the number of compounding periods per year, T the years.

Future value, continuous compounding

FV = PV × e^(Rc×T)

Rc is the continuously compounded rate. PV = FV × e^(-Rc×T).

Effective annual rate

EAR = (1 + R/m)^m − 1

Use it to compare quotes with different frequencies. With continuous compounding, EAR = e^Rc − 1.

Periodic to continuous rate

Rc = m × ln(1 + Rm/m)

Rm is the rate compounded m times a year. ln is the natural logarithm.

Continuous to periodic rate

Rm = m × (e^(Rc/m) − 1)

Gives the rate quoted with m compounding periods per year.

Continuous rate from EAR

Rc = ln(1 + EAR)

Special case with m = 1.

Repo interest (simple, Actual/360)

Repurchase price = Sale price × (1 + r × days ÷ 360)

USD repo is quoted on an Actual/360 basis. Interest = Sale price × r × days ÷ 360.

Compounded overnight rate (daily compounding in arrears)

Compounded rate = [Π (1 + rᵢ × dᵢ ÷ 360) − 1] × 360 ÷ D

rᵢ is the overnight rate for day i, dᵢ the number of calendar days it applies, D the total days in the period. This is how SOFR-based loans and OIS accrue.

Spread over the risk-free rate

Rate on a risky instrument = Risk-free rate + Spread

The spread compensates for credit, liquidity and term risk.

Discounting with the risk-free rate (continuous)

PV = FV × e^(−r × T)

r is the continuously compounded risk-free rate and T is time in years.

Discount factor from a zero rate (annual compounding)

d(T) = 1 ÷ (1 + z)^T

z is the T-year zero rate. With continuous compounding, d(T) = e^(−zT).

Bond price using zero rates

P = Σ C_t × d(t) + F × d(T)

Each cash flow is discounted at its own zero rate. C is the coupon, F the face value.

Bond yield (YTM)

P = Σ C_t ÷ (1 + y)^t + F ÷ (1 + y)^T

One rate y for all cash flows. Solve with a financial calculator (N, PV, PMT, FV, then CPT I/Y).

Par yield (annual coupons)

c = (1 − d(T)) ÷ Σ d(t)

c is the annual coupon rate as a fraction of face value, with the sum over all coupon dates t = 1 to T. Price then equals face value.

Bootstrapping the final zero rate

d(T) = (P − Σ C × d(t) for t < T) ÷ (F + C)

Use for a bond whose last cash flow is F + C. Then z = (1 ÷ d(T))^(1/T) − 1.

Convert between compounding conventions

(1 + z_annual) = e^(z_cc)

z_cc = ln(1 + z_annual). For m compounding periods per year, (1 + z_m ÷ m)^m = e^(z_cc).

Forward rate, continuous compounding

F = (R2 × T2 − R1 × T1) ÷ (T2 − T1)

R1 and R2 are zero rates to T1 and T2. Use decimals and years.

Forward rate, annual compounding

F = [(1 + R2)^T2 ÷ (1 + R1)^T1]^(1 ÷ (T2 − T1)) − 1

Use when the question gives annually compounded zero rates.

Forward rate from discount factors (simple rate)

F = (DF1 ÷ DF2 − 1) ÷ (T2 − T1)

Gives the simple, non-compounded forward rate for the period. Common for FRAs.

FRA payoff at T2 (receive fixed, pay floating)

Payoff = L × (R_K − R) × (T2 − T1)

L is notional, R is the actual rate for the period. Flip the sign for the floating receiver.

FRA value (receive fixed R_K)

V = L × (R_K − R_F) × (T2 − T1) × DF(T2)

R_F is the current forward rate and DF(T2) is the discount factor to T2. Both rates must use the same compounding.

FRA settlement at T1

Settlement = L × (R_K − R) × τ ÷ (1 + R × τ)

τ is the accrual fraction. This is the market practice of paying at the start of the period.

Macaulay duration

D_Mac = Σ [t × PV(CF_t)] ÷ P

t in years; PV(CF_t) is the discounted cash flow at time t; P is the bond price.

Modified duration

D_mod = D_Mac ÷ (1 + y/m)

y is the annual yield, m is the number of compounding periods per year. With continuous compounding, D_mod = D_Mac.

Price change from duration

ΔP/P ≈ −D_mod × Δy

Δy in decimals: 25 bp = 0.0025.

Convexity

C = (1/P) × d²P/dy² = Σ [t(t + 1/m) × PV(CF_t)] ÷ [P × (1 + y/m)²]

Positive for option-free bonds. Units are 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.

Dollar duration and DV01

Dollar duration = D_mod × P; DV01 ≈ D_mod × P × 0.0001

DV01 is the price change for a 1 bp rise in yield, quoted as a positive number.

Portfolio duration

D_port = Σ w_i × D_i

w_i is the market value weight. Dollar durations and DV01s add directly.

Zero-coupon bond duration

D_Mac = T

Modified duration is T ÷ (1 + y/m).

Hedge with DV01

N = −DV01_position ÷ DV01_hedge

Choose N units of the hedge instrument so net DV01 is zero. A long bond is hedged by a short position.

Expectations theory (two periods, annual compounding)

(1 + y2)² = (1 + y1) × (1 + E[f])

y1 is the 1-year spot rate, y2 the 2-year spot rate, E[f] the expected 1-year rate one year ahead. Under pure expectations, E[f] equals the forward rate.

Forward rate from spot rates

f(1,2) = (1 + y2)² ÷ (1 + y1) − 1

Annual compounding. This is the rate locked in today for year 2.

Liquidity preference theory

Forward rate = Expected future spot rate + Liquidity premium

The premium is positive and tends to increase with maturity.

Shape summary

Expectations: slope reflects expected rate path; Segmentation: supply and demand by bucket; Liquidity preference: usually upward

Learn the one-line prediction of each theory.

Quick revision

• Continuous rate: Rc = m × ln(1 + Rm ÷ m), and Rm = m × (e^(Rc ÷ m) − 1).
• Value with continuous compounding: A × e^(R × T).
• A bond price is the sum of its cash flows discounted at the matching zero rates.
• Par yield is the coupon rate that prices the bond at par.
• Bootstrapping solves for each new zero rate using the earlier ones, working from short to long maturities.
• Forward rate with continuous compounding: F = (R2 × T2 − R1 × T1) ÷ (T2 − T1).
• With an upward-sloping zero curve, forward rates sit above zero rates for the same maturity.
• Modified duration ≈ Macaulay duration ÷ (1 + y ÷ m).
• ΔP ÷ P ≈ −D × Δy + ½ × C × (Δy)².
• Convexity adds to price gains when yields fall and reduces losses when yields rise.
• DV01 is the price change for a one basis point move in yield.
• Expectations theory ties the long rate to expected future short rates; liquidity preference and market segmentation add other explanations.

Common mistakes

• Using the nominal rate as the effective rate Fix: Whenever m > 1, compute EAR = (1 + R/m)^m − 1. It is higher than R.
• Forgetting to divide the rate by m or multiply the periods by m Fix: Always use (R/m) per period and m×T periods. Write both before calculating.
• Saying LIBOR was based on actual transactions. Fix: LIBOR relied on bank submissions of estimated borrowing costs. SOFR is based on actual repo transactions.
• Treating SOFR as unsecured. Fix: SOFR is secured by Treasury collateral, so it has almost no bank credit risk.
• Discounting every cash flow at the bond yield when the question gives zero rates. Fix: If zero rates are given, use a separate rate for each cash flow date.
• Forgetting that the last cash flow includes the principal when bootstrapping. Fix: Divide by face value plus the final coupon.
• Averaging zero rates instead of using the growth identity Fix: Always take R2 × T2 minus R1 × T1, then divide by T2 − T1. Never average.
• Mixing compounding conventions Fix: Convert everything to one basis before comparing. Note the compounding stated in the question.
• Using Macaulay duration directly in the price change formula. Fix: Divide Macaulay duration by (1 + y/m) first. Use that modified duration in ΔP/P ≈ −D_mod × Δy.
• Forgetting the minus sign or mixing up the direction. Fix: Write the sign before computing. Yield up means price down for an option-free bond.

Exam tips

• Most questions are one conversion. Do it in one calculator pass with ln or e^x, then check direction.
• Use the ordering rule to cut options: for the same quote, more frequent compounding means a higher effective yield, and Rc is lower than the annual rate.
• Read the compounding label in every option. Distractors often give the right number with the wrong frequency.
• Keep at least five decimals for ln and e values to avoid rounding errors.
• Practise converting both ways, since either can be asked in a forward rate or bond pricing question.
• Know the three contrasts: secured versus unsecured, actual trades versus submissions, overnight versus term.
• Expect a conceptual question on why LIBOR was discontinued and what replaced it. Name SOFR for USD.
• Remember the risk-free rate used for discounting derivatives has moved from LIBOR and swap rates to OIS and overnight rates.