# Measuring bond fair value (richness/cheapness) using basic regression models?

Background

Due to the nature of the curve (bond curve, swap curve etc), bond traders typically have some model that allows them to measure the "fair value" (FV) of a bond vs other bonds on the curve. This is where RV (relative value) trades come from: observing that a bond looks cheap relative to its FV against other bonds.

Regression-based model

Some models can be fairly involved, such as using splines to fit the curve. Lots of these complicated models break down when looking at things such as inflation swaps due to the seasonality in the index causing instability in the front-end, which propagates across the curve in most models.

Is there a way that a basic regression model can be used to observe fair value? We can use regression models to observe richening or cheapening. (Regress the change on day of a security against a bucket of of other securities, and accumulate the residuals of the model each day. A security deviating from the model over a number of days - i.e. richening - will see daily positive residuals that are not averaging to zero.) The issue here is that the deviation might have begun at a "cheap" point, so the residual deviations make the security look like it is "rich", when in actual fact it has just richened from cheap to FV.

The problem

So the problem is that we can observe richening or cheapening, but cannot see fair value. How can we develop a level for fair value using a basic model such as this? How can we avoid the problem outlined in the last sentence of the previous section?

Just on nomenclature. You cannot establish fair value but you can use a regression for RV.

OLS is perfectly legitimate when done in levels-as long as the series are cointegrated.

Now we know that the yield curve is driven by several factors (in the state-space modelling side of econometrics called “stochastic common trends” in levels), the first two of which are non-stationary and the third questionable. Technically this would mean that we need 3-4 bonds to establish cointegration. (Effectively cancelling our the factor exposures). This would be true if your US Treasury portfolio is a 2y note vs a 5y note vs a 10y bond vs a 30y bond. (I threw in the note and bond terminology just for the heck of it).

If we are focused on a sector you can easily establish fair value on bonds by looking at small flies or switches. Effectively the bonds in the same sector have close to the same factor loadings so you are only looking at changes in idiosyncratic prices. So yes people use regressions all the time when looking at switches and micro-flies. You should generally make sure the residuals remain stationary. If they get positive they don’t just keep drifting more and more positive but start to revert. Statistical changepoint tests can help to determine this but a well-trained eye knows the difference between a stationary (eg Ornstein-Uhlenbeck) and non-stationary (Brownian motion) process

You need to be careful to avoid bonds trading special on repo. The most recently issued bond can richen a lot to the curve. Is it rich? Not if you take the cheap financing into account. It may even make perfect sense to own that bond and repo it, as it typically gets “richer” for a little while before it cheapens as it turns into the old or old-old bond. Richness is also harder to establish for bonds which are deliverable into the futures contract. Again there are a lot of interesting possibilities in trading bonds as they move into and out of the basket. News matters and auctions and taps also have an impact.

So go right ahead as long as you limit the regression to the same sector. Outside of a sector it might be more common to use Nelson-Siegel (-Svennson) or 2+, and make sure you are factor neutral. These bigger models also give you an estimate of roll for free (not carry, you still need repo for that).

• What is a switch or flie? Feb 10, 2019 at 11:17
• Switch is the trade of “switching” one bond for another. It’s just a pairs trade, long one bond and short another. Usually it is done equal weighted or dv01 weighted. A fly or butterfly is a 3 legged trade such as long 1 unit of the 5y swap vs short 1/2 unit of the 2y and 1/2 unit of the 5y swap. You would always look at EUSA5 - 0.5* EUSA10 - 0.5*EUSA2 (using Bloomberg tickers
– NBF
Feb 10, 2019 at 11:31
• Finishing the comment. For flies you would track the spread and trade dv01 weighted versions of it. Sometimes people do regression wtd or PCA weighted flies.
– NBF
Feb 10, 2019 at 13:59
• Also in USTs sometimes one trades overall richness or cheapness of a bond by trading the PO vs the bond.
– NBF
Feb 10, 2019 at 14:00
• Finally I should comment that dv01 wtd switches are very different from cash for cash or par for par trades (which are essentially a type of forward trade in bond space)
– NBF
Feb 10, 2019 at 14:02