# Duration/Yield Curve Risk in Non-Parallel shifts

I’m looking for a clarification on why Duration is “a poor approximation of the sensitivity of the value of a bond portfolio to non-parallel shifts in the yield curve” (Schweser). I understand that yields can change by different amounts based on the different time to maturities within the portfolio. Can’t duration be weighted to capture these different effects? Or are we running into a similar situation to where IRR loses its usefulness when there are multiple IRRs?

They are just talking about plain vanilla Macauley duration. You can tailor a duration measure to macth whatever kind of yield curve shift you are thinging about but it would be beyond the scope of at least L1.

For a simple example, take a zero-coupon; duration equals maturity, call it 10 years. Now rotate/twist the yield curve around the 5y point (say short end up, long end down): price increases (the rate drops for the sole cash flow, at maturity), but duration would predict 0 change – if you average the change in yield to 0.0 over the curve. I dont’ even know how “change in yield” is defined for non-parallel shifts; I wouldn’t say that the duration approach is a poor approximation; I’d say it’s not even defined.

Fixed Income asset managers will use what is called “Key Rate Duration” which is the instrument or portfolio’s sensitivity to shifts at particular points in the yield curve. In actual practice yield curve movement is almost never parallel but rather described as “steppening” (long end rises more or declines less than short end) or “flattening” (long end falls more or increases less than short end).

Thanks for the clarification guys. Much appreciated.

DarienHacker Wrote: ------------------------------------------------------- > For a simple example, take a zero-coupon; duration > equals maturity, call it 10 years. Now > rotate/twist the yield curve around the 5y point > (say short end up, long end down): price > increases (the rate drops for the sole cash flow, > at maturity), but duration would predict 0 change > – if you average the change in yield to 0.0 over > the curve. > > I dont’ even know how “change in yield” is defined > for non-parallel shifts; I wouldn’t say that the > duration approach is a poor approximation; I’d say > it’s not even defined. There’s some interesting stuff out there that addresses these issues. For example, there are all kinds of useful models of term structures and evolution of forward rates. I personally like Heath, Jarrow, Morton (HJM) type models which provide arbitrage free evolution of the entire forward rates and lots of flexibility in modelling volatility changes. The important thing about the model is that it is arbitrage free, it corresponds to currently observed interest rates, and you can work hard at getting a good volatility process. Of course, if you have such a model you also have a risk-free bond price and you can say a lot about the evolution and volatility of the bond price using Ito’s lemma (that everyone thinks is so complex and is about as difficult as the chain rule in Freshman calculus). But now you have a bond price model and a corresponding model of forward rate evolution. That means that you can take the derivative of the bond price wrt any forward rate and get something that looks at the change in bond price due to the evolution of the term structure under your model (i.e., “the change in yield”). While I don’t think I have ever seen a paper on this, if you did a half-ass job of specifying the HJM model I’ll bet the HJM duration would predict bond price changes much better than any key rate duration or Macauley yield-to-maturity duration methodology. (BTW - anybody want to do a master’s paper and look for a topic? I could practically dictate that paper and you would just have to get your favorite bond data (risk-free). I’ll help.)

What about PhD thesis ideas, Joey?

Are you doing one? Sure I have some ideas but it’s best to have an advisor with ideas. Picking the right advisor is a more important choice.

I can’t go back to school full time and as far as I know all PhD programs are full time. I am planning on publishing several papers and then try to get a PhD based on them. What are your thoughts on that?

You can’t do that unless you want some online type degree or the papers are so good that someone is going to give you an honorary degree which isn’t a real Ph.D. anyway. To get a Ph.D. you probably need at least a couple of years of classes which you do well in, pass your qualifiers (by far the hardest part in my program), write a dissertation.

I guess I need to start looking for an advisor then.