Expectations, Convexity, and Risk Premium in the Term Structure

Why does the yield curve usually slope upward? Three forces shape the level and slope of the term structure:

1. Expectations — the market's forecast of future short rates. If rates are expected to rise, long-term bonds must yield more than short ones to compete with the strategy of rolling over short positions.
2. Convexity — a purely mathematical effect. Zero-coupon bond prices are convex functions of the short rate. Jensen's inequality implies that, under uncertainty, the discount factor is higher than its deterministic-world counterpart — which always lowers long-term spot rates relative to expected future short rates.
3. Risk premium — compensation for bearing interest rate risk. Risk-averse investors require an expected return above the short rate for holding long-duration bonds.

• state the pure expectations hypothesis and derive its implication that forward rates are expected future spot rates;
• use Jensen's inequality to explain why convexity always lowers long spot rates relative to the expectations benchmark;
• define the risk premium and explain why risk-averse investors require it for holding duration;
• state the relationship: forward rate = expected future short rate + convexity adjustment + risk premium;
• explain why the yield curve can be upward-sloping even when rate expectations are flat;
• describe the empirical evidence against the pure expectations hypothesis.

1. 1. Expectations in a deterministic world
2. 1. Convexity: Jensen's inequality lowers spot rates
3. 1. Risk premium: what risk-averse investors require
4. 1. The term-structure decomposition
5. 1. The pure expectations hypothesis and its failure
6. 1. Summary: what determines the shape