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FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Drift

In the Model 1 framework, the model-implied term structure of spot rates is compared with the expected future short rate path. Since drift is zero, the expected short rate is flat at today's rate, yet model spot rates decline with maturity. Which explanation is correct?

Spot rates decline with maturity because of convexity (Jensen's inequality). Bond prices are convex in rates, so the average discount factor exceeds the one at the expected rate, lowering spot rates. This effect rises with maturity and volatility, and no risk premium or mean reversion is involved.

  1. AThe decline reflects a negative risk premium built into the drift
  2. BThe decline is due to a convexity effect from the Jensen's inequality, which lowers long-term spot rates relative to expected ratesCorrect
  3. CThe decline reflects mean reversion toward a lower long-run rate
  4. DThe decline reflects the fact that volatility falls with maturity

Explanation

Bond prices are convex in rates, so with rate volatility the expected discount factor exceeds the discount factor at the expected rate. This pushes spot rates below the expected short rate, and the effect grows with maturity and σ². Model 1 has neither a risk premium nor mean reversion, and volatility is constant.

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