FRM Exam Part II · Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Convexity Effect and Jensen's Inequality in Interest Rates

Updated 11 October 2026 · Fact-checked

The convexity effect says bond prices are convex in yields, so by Jensen's inequality the expected price is higher than the price at the expected rate. Long-maturity yields therefore sit below expected short rates. In Model 1 the spot rate is r0 − σ²T²/6 and the forward rate is r0 − σ²T²/2.

Understand Convexity Effect and Jensen's Inequality in Rates

Start with a simple fact. A bond price is a convex function of its yield. Plot price against yield and the curve bends upward like a bowl. A zero-coupon bond paying 1 in T years has price 1 ÷ (1 + r)^T, and this curve is convex in r.

Jensen's inequality says that for a convex function f, E[f(X)] ≥ f(E[X]). Averaging after applying the function gives a higher number than applying the function to the average. For bonds: the expected discount factor is greater than the discount factor at the expected rate. In short, E[1 ÷ (1 + r)] ≥ 1 ÷ (1 + E[r]). The gap is bigger when rates are more uncertain.

Now read this in yield terms. If prices are on average higher than the expected-rate price, the yield that matches the expected price is lower than the expected rate. This is the convexity effect. It pulls long-term spot and forward rates below the path of expected short rates, even if there is no risk premium and no change in expected rates.

Volatility drives the size. In Model 1 (normal rates, zero drift, dr = σ dw), the spot rate is r0 − σ²T²/6 and the forward rate is r0 − σ²T²/2. Both effects grow with the square of volatility and the square of maturity. Double σ and the effect is four times larger. Double maturity and it is also four times larger. At short maturities the effect is tiny. At 20 or 30 years it can be large.

This is one of three forces on the curve: expected short rates, risk premium and convexity. The convexity term always pushes yields down, and it is why a curve can flatten or bend downward at long maturities when volatility is high. A related idea appears in markets: futures rates are higher than forward rates because futures are settled daily, and the difference is a convexity adjustment.

Key formulas to remember

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.

How to solve Convexity Effect and Jensen's Inequality in Rates questions

Use this method for any question on the convexity effect, whether it asks for a rate, a direction or a driver.

1Identify what is asked: spot rate, forward rate, futures-forward gap, or a qualitative effect of volatility or maturity.
2Check the model. For Model 1 use r0 − σ²T²/6 (spot) or r0 − σ²T²/2 (forward). For futures versus forward use ½σ²t1t2.
3Convert σ to decimals. A volatility of 100 bp a year is 0.01. Keep T in years.
4Compute the convexity term first: σ² × T², then divide by 6 for spot or by 2 for forward.
5Subtract the term from the starting rate r0 (or add it to the forward to get the futures rate).
6Convert the answer to basis points if the options are in bp, and sanity-check: spot effect must be smaller than forward effect, and both must be negative.
7If the question is qualitative, state the direction: higher volatility or longer maturity gives a larger convexity effect and lower long-term yields, all else equal.

Quickest way: Scale-and-compare shortcut

When to use it: Use when the question gives a base case and changes σ or T, or when you must rank options without full calculation.

Remember the effect scales with σ² and T². Double either one and the effect is four times larger.
Remember the ratio: forward effect = 3 × spot effect in Model 1.
Compute one number, for example σ²T²/6 in bp, then scale it rather than recomputing.
Eliminate any option where convexity raises a yield, or where the spot effect exceeds the forward effect.
For futures versus forward, the futures rate is always the higher one; the gap is ½σ²t1t2.

Common mistakes in Convexity Effect and Jensen's Inequality in Rates

Saying convexity raises long-term yields.
Students link convexity with higher bond prices and miss that higher price means lower yield.
Fix: Chain it: convex price, Jensen, higher expected price, lower implied yield. Convexity always lowers the yield.
Mixing up the divisors 6 and 2.
Both formulas look alike and both contain σ²T².
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.
Entering σ in percent instead of decimals.
Volatility is quoted as 1% or 100 bp and the number is dropped straight in.
Fix: Convert first: 100 bp = 0.01, so σ² = 0.0001. Then multiply by T² and divide.
Thinking the convexity effect depends on whether rates are expected to rise or fall.
Students mix it with the expectations component of the curve.
Fix: Convexity comes from volatility, not drift. It is present even with zero expected change in rates, and it is always negative.
Applying Jensen's inequality to the yield itself.
The inequality is remembered but not the function it applies to.
Fix: Expected yield is linear in r, so no gap there. The gap is in the convex function of the rate: the bond price or discount factor.
Getting the futures-forward sign wrong.
Daily margin makes the direction hard to recall, and it is easy to confuse a long position in the price with a long position in the rate.
Fix: Futures rate = forward rate + convexity adjustment, so the futures rate is higher than the forward rate. State the position first: take a futures contract whose price rises when rates fall (such as a Eurodollar-style contract) and go long the price. Gains arrive when rates are low, so they are reinvested at low rates. Losses arrive when rates are high, so they are financed at high rates. This makes the long futures less attractive than a forward, so the futures price is lower and the futures rate is higher.

Worked examples

Example 1

The one-year rate in a year's time is equally likely to be 3% or 7% (annual compounding). The expected rate is 5%. Compare the expected price of a 1-year zero (face value 1) bought at that time with the price at the expected rate, and find the implied yield from the expected price.

Show the solution

Expected rate = 0.5 × 3% + 0.5 × 7% = 5%.
Price at the expected rate = 1 ÷ 1.05 = 0.952381.
Price at 3% = 1 ÷ 1.03 = 0.970874. Price at 7% = 1 ÷ 1.07 = 0.934579.
Expected price = 0.5 × (0.970874 + 0.934579) = 0.952727.
Expected price − price at expected rate = 0.952727 − 0.952381 = 0.000346. This is about $346 per $1,000,000 face value.
Implied yield from the expected price = 1 ÷ 0.952727 − 1 = 4.96%.

Answer: The expected price (0.952727) exceeds the price at the expected rate (0.952381), as Jensen's inequality predicts. The implied yield is about 4.96%, below the 5% expected rate.

Example 2

In Model 1 (dr = σ dw, zero drift, continuous compounding), r0 = 4% and σ = 120 bp a year. Find the 10-year spot rate and 10-year forward rate, then find the 20-year spot rate.

Show the solution

σ = 0.012, so σ² = 0.000144.
10-year spot effect = σ²T²/6 = 0.000144 × 100 ÷ 6 = 0.0024 = 24 bp. Spot rate = 4% − 0.24% = 3.76%.
10-year forward effect = σ²T²/2 = 0.000144 × 100 ÷ 2 = 0.0072 = 72 bp. Forward rate = 4% − 0.72% = 3.28%.
20-year spot effect = 0.000144 × 400 ÷ 6 = 0.0096 = 96 bp. Spot rate = 4% − 0.96% = 3.04%.
Check: doubling maturity quadruples the effect (24 bp to 96 bp), and the forward effect is three times the spot effect (72 vs 24).

Answer: 10-year spot rate = 3.76%, 10-year forward rate = 3.28%, 20-year spot rate = 3.04%.

Exam tips

Expect a direction question: if volatility rises, do long yields rise or fall? With drift and risk premium unchanged, they fall.
Learn σ²T²/6 and σ²T²/2 exactly. Options are often built from the wrong divisor or from σ left in percent.
Use the square rule to scale answers: doubling σ or T gives four times the effect. It saves time.
In cases about curve shape, name the three drivers (expectations, risk premium, convexity) and say convexity matters most at long maturities and high volatility.
For futures versus forward rates, state that the futures rate is higher and the adjustment grows with σ² and with the product t1 × t2.

Practice questions from Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Convexity Effect and Jensen's Inequality in Rates: frequently asked questions

What is Jensen's inequality in bond pricing?

For a convex function, the average of the function is at least the function of the average. Bond prices are convex in yields, so the expected bond price is higher than the price at the expected yield. This is why long yields are pulled down when rates are uncertain.

How does volatility affect long-term yields through convexity?

Higher volatility increases the convexity term, which lowers long-term spot and forward rates. In Model 1 the effect is proportional to σ² and T², so doubling volatility makes the effect four times larger. The effect is small at short maturities and large at long ones.

Why is the forward rate's convexity effect three times the spot rate's?

In Model 1 the log of the bond price is −r0·T + σ²T³/6. The spot rate divides this by T, giving a σ²T²/6 term. The forward rate is the derivative with respect to T, giving σ²T²/2. The ratio is therefore 3 to 1.

What is convexity bias between futures and forward rates?

Futures are settled daily, so gains and losses are reinvested or financed at the prevailing rates. This makes the futures rate higher than the equivalent forward rate. The forward rate equals the futures rate minus a convexity adjustment, such as ½σ²t1t2 in a simple model.