FRM Part II · FRM Exam Part II

Expectations, Risk Premium, Convexity and the Shape of the Term Structure

This chapter explains why the yield curve slopes the way it does. Forward rates reflect expected future short rates, plus a risk premium, minus a convexity effect. To solve questions, split any yield or forward rate into these three parts, then match each part to its driver and sign.

What this chapter covers

This chapter builds one idea: a long-term yield is not just an average of expected short rates. It has three parts. The first is expectations of future short rates. The second is a risk premium (term premium) that investors demand for holding long bonds. The third is a convexity effect, which pulls long yields down because bond prices are convex in yield.

You start with the expectations hypothesis and forward rates. Under pure expectations, the forward rate equals the expected future spot rate. Then you add the risk premium, which explains why the average return on long bonds exceeds that of short bonds. Next comes Jensen's inequality: because price is a convex function of the rate, the expected price is higher than the price at the expected rate. That lowers yields at long maturities, and the effect grows with volatility and maturity.

The last topic ties it together: what makes the curve upward sloping, flat or inverted. This links to market risk (duration, convexity, key rate exposures), to investment management (carry and roll-down trades) and to Current Issues such as rising government debt, which can affect term premia. Expect applied questions that ask you to interpret a curve, not just compute a number.

Term structure logic sits behind many questions in market risk, liquidity and treasury risk, and investment management. The exam is 80 multiple-choice questions in 4 hours, so you need fast, accurate reasoning about forward rates, premia and convexity. Candidates who can decompose a forward rate into expectation, premium and convexity answer these items quickly and avoid traps built on sign errors and mixed-up concepts. The effort is modest compared with the reuse of the ideas across the paper.

Expectations, Risk Premium, Convexity and the Shape of the Term Structure: topics in the order to study them

1Expectations Hypothesis and Forward RatesIt is the base case. You need forward rate arithmetic and the no-premium benchmark before adding anything to it.
2Risk Premium and Term Premium in Bond ReturnsIt adds the first correction to the base case, explaining why expected long bond returns exceed short rates and why forwards can be biased predictors.
3Convexity Effect and Jensen's Inequality in RatesIt adds the second correction, a negative one, and needs the convexity of bond prices you already know from market risk.
4Drivers of the Shape of the Term StructureIt combines expectations, premium and convexity to interpret upward, flat and inverted curves, so it comes last.

How to prepare Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Work from formulas to interpretation. Most marks come from knowing which component drives a result and in which direction.

Practise forward rates until automatic: (1 + s₂)² = (1 + s₁)(1 + f₁,₂) for annual compounding. Solve for f, then check it against the curve shape.
Write the decomposition in your own words: forward rate = expected future spot rate + risk premium − convexity effect. Note the sign of each part.
Learn the two-sided meaning of premium: it is compensation for bearing interest rate risk, and it can change with uncertainty and supply of bonds.
Explain Jensen's inequality in one sentence: for a convex price function, E[P(r)] ≥ P(E[r]), so yields are lower than the expected-rate average implies.
Check how convexity grows with maturity and volatility. Longer bonds and higher rate volatility give a larger effect.
Do practice items that give a curve and ask for its cause. State the driver first, then the numbers.
Revise with a one-page sheet of formulas, signs and curve shapes, and re-test it two days before the exam.

Common mistakes in Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Treating the forward rate as the market's forecast of the future spot rate without any adjustment.
Fix: Always ask whether the question allows for a risk premium or convexity. If it does, adjust the forward before reading it as a forecast.
Adding the convexity effect instead of subtracting it from long yields.
Fix: Remember that higher convexity raises prices, which lowers yields. In the decomposition, convexity is a negative term.
Mixing up the term premium with the forward-spot gap in a way that ignores convexity.
Fix: Split the gap into premium and convexity, and note that convexity matters most at long maturities and high volatility.
Using the wrong compounding or period count when solving for forward rates.
Fix: Write the convention next to the formula before calculating, and keep it consistent through the item.
Reading an upward-sloping curve as proof that rates will rise.
Fix: List all three drivers. A positive slope can come from a term premium even if expected short rates are flat.
Stating Jensen's inequality without the convexity condition.
Fix: State it as: for a convex function, the expected value of the function is at least the function at the expected value.

Last-day revision: Expectations, Risk Premium, Convexity and the Shape of the Term Structure

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.

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

Expectations, Risk Premium, Convexity and the Shape of the Term Structure: frequently asked questions

What is the expectations hypothesis in simple terms?

It says long-term rates reflect the expected path of short-term rates. Under the pure form, the forward rate equals the expected future spot rate, with no risk premium. Real markets usually show some premium, so the pure form is a benchmark.

How does convexity affect the term structure?

Bond prices are convex in yield, so rate volatility raises expected bond prices. That pushes long-maturity yields lower than the expected rates alone would imply. The effect grows with maturity and volatility.

What is a term premium?

It is the extra expected return investors require for holding a long-term bond instead of rolling short-term ones. It compensates for interest rate risk and can change over time.

What causes an inverted yield curve?

Mainly expectations that short rates will fall. A negative or low term premium and a strong convexity effect at long maturities can also contribute. Read the question for which driver it points to.

How should I use the formulas in the exam?

Identify the driver first, then apply the relevant formula. Check the compounding convention and the sign of each component before you pick an answer.