Interest rates as an interest rate factor

[afterworkguinness]

Hello,
I don't understand how an interest rate (ie: spot rate or forward rate) can be a factor that causes a change in the term structure of rates . Is this because we are assuming in a single factor model that if one rate moves, all rates move in parallel ?

Also, related, how is the dollar value change of a 1bp decline in rates (DVO1) an interest rate factor ?

 

[ShaktiRathore]

Term structure of interest rates is the path the interest rate can take over a period of time i.e. level of interest rate over different periods in future. The interest rate is the expected interest rate. if there is 1 yr exp. interest rate i, we know from price of 2 yr bond the 2nd yr interest rate(bootstrapping) and so on so that entire term structure can be derived with help of 1 yr exp. interest rate and the price of bonds of different maturities. From this we get exp interest rate for different maturities we fill the gap with extrapolation. Hence any change in 1 yr exp interest rate will shift the entire term structure of interest rate logically.
You can also see this from expectations hypothesis theory that future rates are dependent on current interest rates so that any change in current rate will change the future rates and hence the entire term structure.
DV01 is the change in price of bond with 1 bps change in interest rate its nothing but duration in dollar terms. how much price change with change in interest rate that is DV01 is sensitive to interst rate and thus is an interest rate factor. DV01=Duration*(.0001)*P is evident from the formula that DV01 depends on duration other factors being constant and as duration in interst rate factor so is DV01.http://forum.bionicturtle.com/threads/difference-between-dv01-and-duration.1249/
thanks

 

[David Harper CFA FRM]

Your question speaks to the one of the more difficult ideas in Tuckman (frankly it took me 2+ years to feel like i got this part)

I want to quote Tuckman's 2nd edition: "An interest rate factor is a random variable that impacts interest rates in some way. The simplest formulations assume that there is only one factor driving all interest rates and that the factor is itself an interest rate."

The way I look at this is:

• A single-factor model merely means that our model employs only one interest rate factor as the random variable; e.g., yield (YTM), 10-year par rate. Single-factor often associates with parallel shift, but single-factor does not necessarily imply a parallel shift assumption.
• We can combine single-factor and parallel shift assumptions; e.g., single factor is the X-year spot (or forward | or par) rate and we will assume that all rates shift (in parallel) with the change in this X-year rate
• We can combine various combinations of X-factor and various non-parallel shift assumptions; e.g., we can use one-factor of 30-year spot rate but decide the other spot rates shift in non-linear manner, down to zero at year zero; we can use two spot rates (two factors) and interpolate the rates between them. So, the implication of "single-factor" is not parallel shift but rather than we've somehow approximated the entire (relevant) term structure, or at least all of the necessary pricing information, into one interest rate.
• Tuckman (and FRM) tends to use yield to maturity (YTM, aka, "yield") as the single factor because, by definition, it incorporates the (inherently multi-factor) term structure of spot/forward rates. It is more convenient to shock one yield (which indirectly contains the information of the term structure) than to shock all 30 or 60 or 120 spot rates.
• However, shocking the yield by one basis point is not exactly the same as shocking all the term structure's spot rates by one basis point: both are parallel shifts, but only the yield shock is also single-factor (we can alternatively designate, e.g., the 10-year spot rate as the single factor and assume the others shift in parallel ... then we've got a single-factor [10-year spot], parallel shift model, but it is not exactly the same as single-factor [yield], parallel shift).

So, that's how i look at it: First, we have an n-factor model (single? several key rates?). Second, the separate question of parallel (convenient) or non-parallel (in which case, we need a rule probably).

Among many choices, we tend to assume yields (YTM) that shift in parallel which is actually two assumptions (1. yield is the single-factor, 2. let's shift in parallel) out of convenience.

Re: how is the dollar value change of a 1bp decline in rates (DVO1) an interest rate factor ?
Great question. DV01 by itself does not specify the interest rate factor: technically, DV01 is the price change given a one basis drop in some interest rate factor. We happen to use the special (convenient) case where the interest rate factor is yield (YTM). Tuckman (2nd edition):

"Yield-based DV01 is a special case of DV01 introduced in Chapter 5. In particular, yield-based DV01 assumes that the yield of a security is the interest rate factor and that the pricing function P(y) is the price-yield relationship"

 

[afterworkguinness]

Thanks very much