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

In a model with zero risk premium and constant volatility, the term structure is driven by expectations and convexity. A risk manager notes that the 30-year spot rate is below the average of expected future short rates. Which effect best explains this, and how does its magnitude change with maturity?

Convexity lowers long-term spot rates below the average expected short rate because bond prices are convex in rates. The effect scales roughly with volatility squared and maturity squared, so it is negligible at short maturities but becomes large for long maturities such as 30 years.

  1. AConvexity lowers long-term rates, and its effect grows rapidly with maturity because it depends roughly on maturity squaredCorrect
  2. BConvexity raises long-term rates, and its effect is constant across maturities
  3. CThe risk premium lowers long rates, and it is largest at the short end
  4. DExpectations always dominate, so spot rates equal the average expected short rate

Explanation

Convexity (Jensen's inequality) makes bond prices higher than implied by expected rates, so yields are lower. The effect is proportional to sigma squared times roughly T squared, so it is small at short maturities and large at long ones. Option 4 ignores convexity.

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