Hi David,

am unable to comprehend both mathematically and intuitively :

1.why duration is a single factor model?
2.why duration assumes a parallel shift in term structure?

Plz explain as am very new to this field..

Thanks
Anil

Hi Anil,

Tuckman’s duration mathematically is:

Percentage price change = -Duration * 1% yield change. Or, multiplying by price
Change in price = -Duration * 1% yield change * Price, or
Change in price = -[Single factor called duration] * 1% yield change * Price

It is single factor because: it says the change in bond’s price is a function of only the single factor, duration.
(Its analog in equities could be the capital asset pricing model, which says, an equity’s return is a function of only the single factor, the equity risk premium).

I think your (1) and (2) are ultimately the same: the single factor *requires* a parallel shift. If you imagine a 10-year bond with 21 cash flows (coupons + par), and we compute, say, the yield (YTM) = 8%. That yield is a flat line across the entire term structure; i.e., all 21 cash flows into the future can be discounted back at the same 8% (this flat yield curve is consistent with, contains the information in, a sloping spot curve).

Now, try to use the duration without a parallel shift. You only have the one factor, so it must apply to the whole term structure. If i ask you to describe the curve shift with one number (e.g, the shift is +50 basis points), you can only be referring to the whole curve. (Even to say part of the curve does not change is to imply another factor!). You would need other factors to describe anything more creatively. That’s why the key rate is multi-factor: instead of one duration factor to “cover” the whole term, we have several key rates and we can say “okay this is more realistic, my bond will now change as a function of the 2-year rate shift, the 5-year rate shift, etc etc.”

This single factor is why there are so many attempts to improve on duration: a 30-year curve can shift in many non-parallel ways, so additional factors are needed to capture/model those changes. I hope that helps…

David

Hi David,

Change in price = -[Single factor called duration] * 1% yield change * Price

you mean to say that yield change is constant whether term structure is 2 years or 8 years.

like if we have duration for these two bonds with different maturities and different yields,then if we know the duration for these bonds.

now if interest rate rise then you will assume same shift in yields(is it YTM ?) to calculate the price change.

so the only term that is left is duration.that is why its single factor(duration) model.

Is my inference correct?

also if there is is a single bond with 30 year to maturity then to calculate the change is price what we are supposed to use is YTM not yield at any particular instant of time?

still confused a bit

Gaurav

Hi Gaurav,

I may have confused with YTM
(YTM is just a summary yield: the entire yield curve summarized in a single “flat” yield number, given the market price of the bond)

I apologize, I can’t quite follow the example. Start with: an upward sloping yield curve, any yield curve.
If, say, the duration = 7.0
we mean by this, if the yield shifts up by 1%, then bond price will go down by *approximately* 7%

Why is this one factor? Because we have only a single thing that is impacting the price change, duration. Regardless of the shape of the curve, because we have only this single factor we are implictly assuming a parallel shift in the term structure. Consider a desire to model a more realistic change in the curve, say a steepening of the curve. To model a steepening, we need more than one factor because short goes up less than long term rate. By saying: Price change is a function of = -duration * yield change, we are “stuck” with a single thing (i.e., a shift in the entire yield curve) impacting the price change. If we want to be more realistic, we need multi-factor; e.g.,

price change = “is a function of” = factor 1 (e.g., short term duration)*short term yield change + factor 2 (e.g., medium term duration)*short term yield change + etc etc

Also consider adding convexity makes it a two factor model:

Price change = function of duration + convexity adjustment

So now we are explaining price sensitivity with two factors, duration & convexity, and we are getting a bit more realistic. Let me know if this helps because, to see this, is to start to see the limitations of duration.

Thanks, David

Thanks David for such a wonderful reply....

one more doubt:

what exactly do we mean reversion.in the context of valuaing bonds through binomial trees?

(thanks!)

The Tuckman Chapter 9 is modeling a binomial were the short interest rate is without mean reversion; e.g., the short rate goes either up .5% or down .5% at each node. So, it can go up-up-up-up from 5% to 7% without mean reversion.

But generally interest rate is modeled with mean reversion, as in, unlike GBM for equities, we assume a rate will be pulled back to its long run average
(see this example, here is Vasicek, which is common mean reverting process for interest rates)

So, now we can insert some like a Vasicek into the binomial. But any number of processes can be inserted into the binomial...David

thanks again David for wonderful explanations..