Expectations, Risk Premium, Convexity and the Shape of the Term Structure: formula sheet

Key formulas

Forward rate from spot rates (annual compounding)

(1 + z_T)^T = (1 + z_t)^t × (1 + f(t,T))^(T − t)

z are spot rates, f(t,T) is the annualised forward rate from year t to year T.

One-year forward, one year ahead

f(1,2) = (1 + z₂)² ÷ (1 + z₁) − 1

The most common exam case.

Continuous compounding forward

f(t,T) = (z_T × T − z_t × t) ÷ (T − t)

Use only when the question gives continuously compounded rates.

Pure expectations hypothesis

f(t,T) = E[future spot rate for t to T]

Risk premium is zero. Forward is an unbiased predictor of the expected spot.

Forward with a term premium

f = E[future spot] + term premium

A positive premium means forwards sit above expected spots (liquidity preference view).

Yield decomposition

Long yield = average expected short rate over the bond's life + term premium (− convexity effect)

The convexity effect is small at short maturities and grows with maturity and volatility.

Term premium

Term premium = observed long yield − average expected short rate (ignoring convexity)

Not observable directly. It depends on the expected-rate forecast you use.

Expected excess return

Expected excess return = expected bond return − short rate ≈ λ × σ (per unit of duration-scaled volatility)

λ is the price of risk. Higher volatility or higher λ means a higher premium.

Real-world vs risk-neutral drift

Risk-neutral drift = real-world drift + λ × σ (usual setup)

With a positive λ, the risk-neutral drift is higher. Some texts define λ with the opposite sign, so the formula holds only under the convention stated. Pricing uses the risk-neutral drift.

One-period forward rate link

Forward rate = expected future short rate + forward (term) premium

If the premium is zero, the expectations hypothesis holds. A positive premium means forwards overstate expected rates.

Jensen's inequality (convex f)

E[f(X)] ≥ f(E[X])

Holds for convex f. Strict if f is strictly convex and X is not constant. Reverses for concave f.

Discount factor application

E[1 ÷ (1 + r)] ≥ 1 ÷ (1 + E[r])

Applies because 1 ÷ (1 + r) is convex in r for r > −1. Expected price is above the price at the expected rate.

Model 1 spot rate (continuous compounding)

r̂(T) = r0 − σ²T²/6

Zero drift, constant σ, risk-neutral. Comes from P(T) = exp(−r0·T + σ²T³/6).

Model 1 forward rate

f(T) = r0 − σ²T²/2

Convexity effect on the forward is three times the effect on the spot rate.

Futures-forward convexity adjustment (Hull, Ho-Lee style)

Forward rate = Futures rate − ½ σ² t1 t2

t1 is the time to futures maturity, t2 the time to maturity of the underlying rate, σ the annual short-rate volatility in decimals.

Spot rate decomposition

Spot rate(T) = Expectations(T) + Risk premium(T) − Convexity(T)

Convexity is entered as a positive amount that is subtracted. Check the sign convention in the question.

Expectations component

Expectations(T) = (1 ÷ T) × ∫ E[r(s)] ds, or simple average of expected short rates

With discrete periods, use the simple average of expected one-period rates as an approximation.

Convexity in the zero-drift normal model

Convexity(T) = σ² × T² ÷ 6

This is the spot-rate effect when short-rate volatility is σ (annual, in decimals) and drift is zero under the risk-neutral measure. It grows with T² and σ².

Spot rate in the normal model with constant risk-neutral drift λ

r(T) = r₀ + λ × T ÷ 2 − σ² × T² ÷ 6

Here λ is the constant risk-neutral drift of the short rate per year. It is not a risk premium, and no separate premium term is added in this formula. Use only if the question defines it.

Jensen's inequality for bond prices

E[P(r)] ≥ P(E[r]) for convex P

Because price is convex in yield, uncertainty raises expected price and lowers the yield.

Quick revision

• Forward rate: (1 + s₂)² = (1 + s₁)(1 + f₁,₂), with annual compounding.
• Pure expectations: forward rate equals the expected future spot rate.
• Upward-sloping curve under pure expectations means short rates are expected to rise.
• Term premium is extra expected return for holding long bonds over short ones.
• Realised forward rates are not unbiased predictors if a risk premium exists.
• Bond price is convex in yield, so E[P(r)] ≥ P(E[r]) (Jensen's inequality).
• The convexity effect lowers long-maturity yields relative to the expectations path.
• The convexity effect rises with maturity and with rate volatility.
• Forward rate ≈ expected spot + risk premium − convexity effect.
• Inverted curve can signal expected rate cuts, a negative premium, or a strong convexity effect.
• Name the driver first (expectations, premium or convexity), then compute.
• Check the compounding convention before using any forward formula.

Common mistakes

• Taking the forward rate as z₂ − z₁. Fix: Use the growth-factor ratio. The difference is only an approximation.
• Forgetting to raise to the power T − t for multi-year forwards. Fix: Take the (T − t)th root to annualise before subtracting 1.
• Treating the forward rate as the market's expected future short rate. Fix: Forward = expected rate + premium. Equate them only if the question assumes a zero premium.
• Calling any upward-sloping curve a forecast of rising rates. Fix: A slope can come from expected rises, a positive term premium, or both. Separate them before concluding.
• Saying convexity raises long-term yields. Fix: Chain it: convex price, Jensen, higher expected price, lower implied yield. Convexity always lowers the yield.
• Mixing up the divisors 6 and 2. Fix: Spot has /6 and forward has /2. The forward is the derivative of the total yield, so its effect is larger: three times the spot effect.
• Adding convexity instead of subtracting it Fix: Convexity raises bond prices, so it lowers yields. Subtract it from the spot rate.
• Assuming convexity is large at short maturities Fix: Compute it for each T. In the σ²T²÷6 formula, doubling maturity quadruples the effect.

Exam tips

• Check the compounding convention first. It decides which formula applies.
• Distinguish the pure expectations hypothesis (zero premium) from liquidity preference (positive term premium) in wording-based questions.
• Use f ≈ 2z₂ − z₁ to eliminate options fast, then compute exactly if the options are close.
• When asked about interpretation, mention both expected rates and risk premium.
• Link the topic to risk premium and convexity, which are the deviations from this benchmark.
• Always ask whether the question wants real-world or risk-neutral values. The two differ by the risk premium.
• Look for the phrase 'expectations hypothesis'. It means the term premium is zero and forwards equal expected rates.
• In decomposition questions, do the subtraction first, then check the sign against the story in the stem.