# 3 Factor HJM model, do these factors have an economic meaning?

In the HJM model, in case we have 3 factors, do these factors have an economic meaning at all ?

One of the motivations for multifactor models such as Two-Factor-HW, HJM and LMM (Lobor-Market-Model) is derived from the properties of the yield curve. One can run a Principal-Component-Analysis on yield-curve data in order to analyse the number of independant factors contributing to yield curve movements.

It has been shown that there are generally three types of movements a yield curve can undergo (and thus also three factors driving it's movements):

1. rougly flat (parallel shift → average rate)
2. upward or downward sloping (tilt → slope )
3. hump shaped (flex →curvature)

One factor models such as Hull-White can only capture parallel shifts of the yield curve. Two factor models can either reproduce the tilt or the hump. With a three factor model all three common yield curve patterns can be modelled.

Fixed Income Markets and Their Derivatives (See p. 136 ) mentions/explains that the first two componetns of a PCA explain up to 95-98% of the variations of the yield curve. The remaining variation is mostly picked up by the fird factor. (I will also refer you to Interest Rate Risk Modeling: The Fixed Income Valuation Course as an alternative source)

• Thanks for sharing this info. This is however too abstract for me, I understand PCA decerns 3 indep modes of changes of the yield curve (info contribution should be linked to variance I suppose) but I cannot link the shift/tilt/curvature of a yeild curve to the actual HJM model formulation ( my understanding is that 3 factos <=> 3 volatility term structures ), a simple example would help ( in a layman's point of view ) if you have any. – DKK Mar 7 '14 at 13:48
• okey so you would like to see how the HJM is actually able to produce those shapes with it's three factors correct? Thus how the mechanics translate into the actual yield curve form ? – Probilitator Mar 7 '14 at 13:51
• Well, I would like to understand the logic behind the choice of these factors, and if this translates into capturing information yield curve then yes I would be happy to have a "mecanical" example or something that could help me understand , thanks :) – DKK Mar 7 '14 at 14:04
• First you would like to know/understand why you would use three factors instead of two or one - what are the indicators for such a descicion ? Second you would like to see how this factors are actually chosen - thus what correlation do we assume and which forward rates do we chose to propage correct ? – Probilitator Mar 7 '14 at 14:08
• Yes that'd be great. – DKK Mar 7 '14 at 14:40