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FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models

In a Gauss+ framework, the short rate is the sum of two mean-reverting Gaussian factors, x1 with a slow mean-reversion speed and x2 with a fast mean-reversion speed. Both factors have the same volatility. After a shock to each factor, which description of the effect on long-maturity rates relative to short-maturity rates is most accurate?

The slowly mean-reverting factor has a lasting effect, so it moves long-maturity rates substantially, while the fast-reverting factor's shock decays quickly and mainly affects short maturities. Equal volatilities do not change this, because the maturity impact depends on the mean-reversion speed.

  1. AThe shock to the fast-reverting factor persists longest and dominates long-maturity rates
  2. BBoth factors affect long and short maturities identically because volatilities are equal
  3. CThe shock to the slow-reverting factor persists and has a larger effect on long-maturity rates, while the fast factor mainly affects short maturitiesCorrect
  4. DNeither factor affects long maturities because Gaussian factors are mean reverting

Explanation

Slow mean reversion means shocks decay slowly, so expected future short rates remain affected over long horizons, moving long-maturity yields. A fast factor's shock dies quickly, so it mainly moves the front end. Equal volatility does not equalize persistence.

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