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

A risk manager compares a model with a flat expected path of the short rate at 4% and positive interest rate volatility. In the reading's framework, with a risk premium of zero, how does the long-maturity forward rate compare with the expected short rate, and why?

The long-maturity forward rate is below 4%. Even with no risk premium, convexity from rate volatility raises bond prices relative to those computed at the expected rate, which lowers implied yields and forwards, increasingly so at longer maturities.

  1. AForward equals 4% because zero risk premium eliminates all differences
  2. BForward is above 4% because convexity raises bond prices and therefore raises forward rates
  3. CForward is below 4% because convexity increases bond prices, lowering yields and forwardsCorrect
  4. DForward is above 4% because volatility lowers bond prices

Explanation

Jensen's inequality makes expected discount factors exceed discount factors at the expected rate, so bond prices are higher, implying lower yields and forwards. The convexity effect grows with maturity and variance. Zero risk premium does not remove it.

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