Outline how the Gauss+ model is structured and discuss what that structure means for its capacity to reproduce interest rate behaviour seen in practice.
The idea
The Gauss+ model describes the term structure with three rate factors: a short rate r, a medium-term rate m and a long-term rate l. They are linked as a cascade of mean reversion. The short rate is pulled toward the medium rate, the medium rate is pulled toward the long rate, and the long rate is pulled toward a constant long-run level θ. The short rate follows the equation below, with its own speed kr and volatility σr. dr = kr(m − r) dt + σr dW. The medium rate is pulled toward the long rate, as the next equation shows. dm = km(l − m) dt + σm dW. The long rate is pulled toward the constant θ, as the last equation shows. dl = kl(θ − l) dt + σl dW. The speeds satisfy kr > km > kl, so the short factor snaps back and the long factor is persistent. Because the factors are distinct, the curve can change in level, slope and curvature, which a single factor cannot do. The usual mistake is to treat them as independent rates reverting to constants. Exam point: the cascade.
A practice question
Which description best matches the structure of the Gauss+ model?
- Three independent rates, each reverting directly to its own constant, with no link between the short, medium and long factors.Why not: Treats the factors as independent constants-reverting rates; in Gauss+ each factor's target is the next factor in the chain.
- Three rate factors in a cascade: r reverts to m, m reverts to l, and l reverts to a constant level. - correct
- A single short-rate factor that reverts to a constant level, with the medium and long rates obtained only by bond-pricing formulas from it.Why not: Describes a one-factor model such as Vasicek; Gauss+ has three genuine rate factors.
- Three factors in which the long rate reverts to the short rate and the medium rate reverts to the long rate.Why not: Reverses the direction of the cascade; the short rate is pulled toward the medium rate, not the other way round.
Gauss+ chains mean reversion: r is pulled to m, m is pulled to l, and l is pulled to a constant. This is what gives the three factors different speeds and lets the curve change shape.
A harder one
In the Gauss+ cascade, which ordering of mean-reversion speeds is consistent with the interpretation of the factors?
- k_r > k_m > k_l: the short rate reverts fastest and the long rate very slowly. - correct
- k_l > k_m > k_r: the long rate reverts fastest because it sits furthest from the short rate.Why not: Reverses the ordering; the long factor reflects slow-moving macro expectations and reverts slowest.
- k_r = k_m = k_l: equal speeds, so that each factor has the same half-life.Why not: Equal speeds would make the three factors behave alike and remove the distinct curve movements the model is meant to capture.
- k_m > k_r > k_l: the medium rate reverts fastest because it sits between the other two factors.Why not: Gives the middle factor the highest speed; the short rate is the one that is pulled back quickly.
The short rate is tied to policy and snaps back toward m quickly. The medium factor moves more slowly toward l, and the long factor reverts very slowly. That gives k_r > k_m > k_l.
The hardest tier
A Gauss+ model has k_r = 1.386, k_m = 0.347 and k_l = 0.0693 per year. Which statement about the long factor's persistence is closest to correct?
- Its half-life is about 14.4 years, which is 1 divided by k_l, the mean life of a shock.Why not: Uses the mean life 1/k instead of the half-life ln(2)/k.
- About 10 years, so a shock to the long factor is nearly permanent over one year. - correct
- Its half-life is about 7.2 years, which is 0.5 divided by k_l, a linear shortcut.Why not: Uses 0.5 in place of ln(2) for the half-life, which is not the half-life of exponential decay.
- Its half-life is about 0.05 years, which is k_l multiplied by ln(2), a very fast decay.Why not: Multiplies by ln(2) instead of dividing; a smaller speed means slower, not faster, decay.
Half-life = ln(2)/k = 0.693/0.0693 = 10.0 years. After one year only about 1 - e^(-0.0693) = 6.7% of a shock has decayed, so the long factor behaves like a near-permanent level shift.
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