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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 normally distributed rate changes and volatility sigma, a term structure analyst separates the long-maturity forward rate into expectations, risk premium and convexity components. Holding expectations and risk premium constant, what is the effect of an increase in volatility on long-maturity spot rates, and why?

Higher volatility lowers long-maturity rates. Bond prices are convex in yields, so greater volatility raises expected discount factors through Jensen's inequality, which means lower yields. The convexity effect grows with maturity, so it is most pronounced at long maturities.

  1. ARates rise, because bond prices are concave in yield and volatility lowers expected prices
  2. BRates fall, because convexity raises the expected bond price, which lowers the yieldCorrect
  3. CRates are unchanged, because convexity only affects bond durations
  4. DRates fall only for short maturities, because convexity is irrelevant beyond one year

Explanation

Bond prices are convex in yields, so by Jensen's inequality higher volatility raises expected bond prices (discount factors). A higher price means a lower yield, and the effect grows with maturity (roughly proportional to sigma squared times maturity squared). The effect is largest at long maturities, not short.

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