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

In a one-factor model with no risk premium, a risk manager compares the price of a zero-coupon bond computed using the expected future short rates with the price obtained in an arbitrage-free model with volatile rates. Which statement best describes the relationship, and why?

The arbitrage-free price is higher than the price using expected rates, because the bond price is convex in the rate. By Jensen's inequality the expected discount factor exceeds the discount factor at the expected rate, so volatility lowers the implied yield even with no risk premium.

  1. AThe arbitrage-free price is higher, because the convexity of the bond price in rates makes the expected discount factor exceed the discount factor at the expected rateCorrect
  2. BThe arbitrage-free price is lower, because volatility raises the expected rate and thereby the discount rate
  3. CThe two prices are equal, because with no risk premium expectations alone determine the price
  4. DThe arbitrage-free price is higher, because investors demand a premium for bearing volatility

Explanation

Bond price is a convex function of the rate (e^{-r}). By Jensen's inequality E[e^{-r}] > e^{-E[r]}. So the price under volatility is higher and the implied yield is lower than the expectations-only yield. The 'equal' option ignores convexity, and the premium option confuses convexity with a risk premium.

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