Term structure of interest rates: formula sheet

Key formulas

Discount factor from spot rate

v(t) = (1 + y_t)^(-t)

y_t is the annual effective spot rate for term t years. Present value of ₹1 due at time t.

Zero-coupon bond price

P = R × (1 + y_t)^(-t)

R is the redemption payment at time t. No coupons.

Spot rate from price

y_t = (R ÷ P)^(1/t) − 1

Use the same time unit for t as the rate (years for annual effective).

Spot rate from discount factor

y_t = v(t)^(-1/t) − 1

Same relation rearranged.

Value of cashflows using spot rates

PV = Σ C_t × (1 + y_t)^(-t)

Each cashflow C_t is discounted at the spot rate for its own term.

Continuous spot rate link

v(t) = e^(-δ_t × t), so δ_t = ln(1 + y_t)

δ_t is the force-of-interest equivalent of the spot rate for term t.

Implied forward rate (general)

(1 + i_{t+r})^(t+r) = (1 + i_t)^t × (1 + f_{t,r})^r

f_{t,r} is the annual effective rate from time t to t+r. Spot rates are annual effective rates.

Solving for f_{t,r}

f_{t,r} = [ (1 + i_{t+r})^(t+r) ÷ (1 + i_t)^t ]^(1/r) − 1

Take the r-th root of the ratio of accumulation factors.

One-year forward rate

1 + f_t = (1 + i_{t+1})^(t+1) ÷ (1 + i_t)^t

Here f_t is the rate for the year from t to t+1.

Spot rate from one-year forward rates

(1 + i_t)^t = (1 + f_0)(1 + f_1)…(1 + f_{t−1})

With f_0 = i_1. The spot rate is the geometric mean of the forward rates, less 1.

Discount factor from spot rate

v(t) = (1 + i_t)^(−t)

Equals the product of one-year forward discount factors (1+f_s)^(−1) for s = 0 to t−1.

Discount factor from spot rate

v_t = (1 + y_t)^(−t)

y_t is the annual effective spot rate for term t. Convert the answer back with y_t = v_t^(−1/t) − 1.

Price of a coupon bond using spot rates

P = Σ C_t × v_t, where C_t = Fn × coupon rate for t < n and C_n = Fn × coupon rate + redemption

Each payment is discounted at its own spot rate. Fn is the nominal (face) value.

Yield to maturity

P = Σ C_t ÷ (1 + i)^t

Solve for the single rate i. This usually needs a quadratic for 2 years, or trial and error or interpolation for longer terms.

Par yield (annual coupons, redeemed at par)

c_n = (1 − v_n) ÷ (v_1 + v_2 + ... + v_n)

Per unit of nominal. It comes from c × Σv_t + v_n = 1. It assumes coupons are paid annually.

Bootstrapping step

v_n = (P − C × (v_1 + ... + v_(n−1))) ÷ (C + R)

For a bond with annual coupon C and redemption R at time n. Use the same units for P, C and R.

Par bond check

If the price equals the redemption value and R = Fn, then YTM = coupon rate

For a bond redeemed at par with annual coupons, price at par means YTM equals the coupon rate.

Forward rate from spot rates (annual compounding)

(1 + s_t)^t = (1 + s_(t-1))^(t-1) × (1 + f_(t-1,t))

Here s_t is the t-year spot rate and f_(t-1,t) is the one-year forward rate from t-1 to t. Use it to test what the curve implies about expected rates.

Pure expectations theory

f_(t-1,t) = E[ future one-year spot rate at time t-1 ]

Forward rates are unbiased estimates of future spot rates. No risk premium.

Liquidity preference theory

f_(t-1,t) = E[ future spot rate ] + liquidity premium, with premium > 0 and increasing with term

Forward rates overstate expected future spot rates. This gives an upward bias to the curve.

Curve shape under expectations theory

Rising curve ⇒ expected rise in short rates; falling curve ⇒ expected fall

Holds only for the pure theory. Under liquidity preference, a rising curve does not by itself prove rates are expected to rise.

Present value at effective annual rate i

V(i) = Σ c_t · v^t, where v = 1 ÷ (1 + i)

c_t is the cash flow at time t. This is the value you differentiate.

Macaulay duration (discounted mean term)

D = Σ t · c_t · v^t ÷ Σ c_t · v^t

Measured in years. For a zero-coupon bond, D equals its term.

Volatility (modified duration)

ν = −V′(i) ÷ V(i) = D ÷ (1 + i)

Use (1 + i) for an effective annual rate. If the rate is a force of interest δ, then −V′(δ) ÷ V(δ) = D exactly.

Convexity

C = V″(i) ÷ V(i) = Σ t(t + 1) · c_t · v^(t+2) ÷ Σ c_t · v^t

With respect to i. With respect to δ, it is Σ t² · c_t · v^t ÷ V. State which version you use.

Change in value for a small change Δi

ΔV ÷ V ≈ −ν · Δi + ½ · C · Δi²

Using duration alone gives the first term only. Convexity corrects the estimate.

Redington condition (i)

V_A(i₀) = V_L(i₀)

Present value of assets equals present value of liabilities at the current rate i₀.

Redington condition (ii)

V_A′(i₀) = V_L′(i₀)

Given (i), this means the discounted mean terms of assets and liabilities are equal.

Redington condition (iii)

V_A″(i₀) > V_L″(i₀)

Asset convexity exceeds liability convexity. This is a strict inequality and guarantees surplus does not fall for small changes in i.

Quick revision

• Spot rate y_t is the yield on a zero-coupon bond paying 1 at time t: price = (1 + y_t)^(-t).
• Discount factor v(t) = (1 + y_t)^(-t).
• Forward rate from s to t: (1 + f_{s,t-s})^(t-s) = (1 + y_t)^t ÷ (1 + y_s)^s.
• For a one-year forward rate: 1 + f_{t,1} = (1 + y_{t+1})^(t+1) ÷ (1 + y_t)^t.
• Bootstrapping: solve spot rates in order of maturity, using the shorter spot rates already found.
• Yield to maturity is the single rate that makes the bond's price equal to the present value of its cash flows.
• A par yield is the coupon rate at which a bond prices at par.
• On a rising curve, the yield to maturity of a coupon bond is below the spot rate for its final maturity.
• Expectations theory: forward rates equal expected future spot rates; liquidity preference adds a premium for longer terms; market segmentation treats maturities as separate markets.
• Macaulay duration = Σ t × PV(cash flow at t) ÷ Σ PV(cash flow); modified duration = Macaulay duration ÷ (1 + i).
• Price change ≈ −(modified duration) × (change in yield) × price, and convexity improves the estimate for larger yield moves.
• Redington immunisation: PV assets = PV liabilities, equal duration, and asset convexity greater than liability convexity.

Common mistakes

• Using one rate for every cashflow. Fix: Read the term of each cashflow and pick the spot rate for exactly that term.
• Forgetting to subtract 1 when finding the spot rate. Fix: Always finish with y_t = (R ÷ P)^(1/t) − 1 and check that the answer looks like a percentage.
• Subtracting spot rates to get the forward rate, for example i_2 − i_1. Fix: Interest compounds. Use the ratio of accumulation factors, not the difference of rates.
• Forgetting to take the r-th root for multi-year forward rates. Fix: The ratio gives the r-year accumulation. Raise it to 1/r to get the annual rate.
• Treating the YTM of a coupon bond as the spot rate for that term. Fix: Remember a spot rate prices one payment. YTM is one rate for all payments. They agree only for a zero-coupon bond or a flat yield curve.
• Discounting every coupon at the YTM when asked to price with spot rates. Fix: If the question gives spot rates, use v_t = (1 + y_t)^(−t) for each payment. Use a single rate only when told to use YTM.
• Saying that under every theory an upward curve means rates will rise. Fix: Restrict that statement to pure expectations theory. Under liquidity preference, part of the slope is a premium.
• Treating the liquidity premium as decreasing with term. Fix: Longer bonds carry more price risk, so the premium increases with term.
• Forgetting to divide by (1 + i) when asked for volatility or modified duration. Fix: Read the question for the word volatility or modified. Then compute ν = D ÷ (1 + i) and state it separately.
• Weighting by cash flow instead of present value when computing duration. Fix: Always discount first. Duration is Σ t × PV ÷ Σ PV. The weights must sum to 1 after dividing by V.

Exam tips

• In written answers, define your notation first: y_t as the t-year spot rate and v(t) as the discount factor. Marks follow clear notation.
• Check the rate basis in the question. Many marks are lost by skipping a nominal-to-effective conversion.
• When asked to price a coupon bond from a spot curve, show a line for each cashflow. Partial marks are given for correct individual discounting.
• In computer-based questions, build a column of terms, spot rates, discount factors and present values, then sum. State the formula you used.
• Do a reasonableness check on every answer: discount factors below 1, falling with term when rates are positive.
• Write the no-arbitrage equation first. Examiners give method marks even if your arithmetic slips.
• State that spot rates are annual effective and that you assume no arbitrage. This often earns an assumption mark.
• Check the direction: a rising spot curve means forward rates above the spot rates. Use this to catch errors fast.