HI

@Eustice_Langham Good question, and I do agree with you that "curve risk" is synonymous with non-parallel term structure shifts; e.g., as you already wrote, Tuckman writes " ... the resulting hedge does not protect against curve risk, that is, changes in the slope of the term structure" ... "The risk that rates along the term structure move by different amounts is known as curve risk."

To me, one distinction is between the

*term structure* and an

*interest rate factor* (I view spot/forward rates as special cases of interest rate factors). A

**term structure is a vector** of some interest rate (e.g., spot, forward, par) plotted against maturities (e.g., out to 30 years). The term structure shifts either in parallel or non-parallel (the concept of a parallel/non shift presumes a vector). While there can be an infinite variety of non-parallel shifts, I always liked how Fabozzi wrote that most of the observed non-parallel shifts are either

**twists **(i.e., Tuckman's flattening or steepening) or changes in the "

*humpedness*" (curvature). To me, which interest rate (spot, forward, par) isn't critical because we can translate back/forth.

We start by examining the single-factor risk of a parallel shift implied by shocking the yield (YTM). The YTM is not a vector, it is easy to illustrate that, regardless of the initial shape of the term structure vector, when we shock the yield (as a single risk factor) by X bps, it is (at least approximately) equivalent to a parallel shift in the term structure. Our duration is not the only duration, it is just the most common yield-based duration (ie, duration based on single-factor yield-to-maturity). Consequently, because the most common single-factor approach is yield-to-maturity, we tend to associate single-factor with parallel shift (although we can model non-parallel shifts with a single factor, hence the distinction between a generic factor and an interest rate. A factor can hold different things; e.g, PCA).

Then as a practical matter, the issue is "okay, yield-based duration and convexity are great for implicitly assuming a parallel shift in the term structure (a vector of whatever rates plotted out to 30 years), but how can we model non-parallel shifts?" The answer, by definition, is some multi-factor approach. Even if the key rate shift technique only assumes three (or even two) key rates, we've gone from parallel to non-parallel. I hope that's helpful perspective. Below is an early page from our Chapter 12 where I tried to introduce some terms that speak to some of these distinctions between term structure, factor, and interest rate (factor). I'll appreciate any feedback because I'm always trying to improve the clarity on this topic.

"Let’s use the simulation on the previous page to make several observations:

- An interest rate is a generic variable. Specific interest rates include
**spot rates** (aka, zero rates), forward rates, **par rates** (aka, par yields), and **yields **(aka, yield-to-maturity). The learning objective asks us to identify common examples of interest rate factors. The most common interest rate factor is a short-term spot rate. In the FRM’s Part 2, we will study Term Structure models (e.g., Ho-Lee, Vasicek). Most of those Term Structure models assume the interest rate factor is an instantaneous short-term spot rate. When the dynamic changes to a term structure, it can be described by a single factor (such as the instantaneous spot rate), then we have a single-factor model. There is a subtle distinction between a factor and an interest rate. A factor is a general variable or “container;” it is not necessarily an interest rate.
- The
**term structure **typically refers to the spot (aka, zero) rate term structure . The simulation above illustrates three different (and parallel) spot rate term structures. We did not illustrate a “yield curve” because the yield (YTM) is a single value that summarizes (as complex average of) the vector of spot rates. The yield must plot a flat (horizontal) line . There is not really a term structure of yields (YTMs): yield is a single interest rate that is variant to each bond’s cash flows and price. To describe a term structure, we require a vector (The obvious exception is the flat term structure famously used in exams. A flat term structure is super convenient because all of these interest rates—spot, forward, par, and yield, are the same when the term structure is flat!).
- When we shifted the spot rate term structure, above, in parallel by 100 basis points, the yields shifted by approximately, but not exactly, 100 basis points (0.9985%)! Because the yield is a single value (that is variant to the bond’s cash flows and therefore having no objective existence independent of the bond ), it does not make sense to simulate a “parallel shift in the yield curve.” Rather, we can shock the (single) yield value. Our simulation models a parallel shift in the spot rate term structure with a single factor model (e.g., plus 100 basis points added to all spot rates on the term structure), and this shift itself very closely approximates the resultant change in yield.
- In practice—by using yield-based duration, convexity, and DV01—we are shocking the yield and this shock implies (by way of necessary approximation) a parallel shift in the underlying term structure. The yield is a single-factor interest rate. Single-factor models do not necessarily imply a parallel shift , but the yield-to-maturity (YTM) does. Therefore, our yield-based DV01, duration, and convexity metrics also implicitly assume (by way of very close approximation) a parallel shift in the term structure.
- If we want to be sophisticated, we can model any non-parallel term structure shift by directly editing the vector of spot rates. This is very multi-factor! More likely, we’d shock a few selected key rates (e.g., key rate shift) which are also multi-factor. The simplest approach is to shock the yield and implicitly assume a parallel shift in the term structure." --- P1.T4. Valuation & Risk Models,

Chapter 12. Applying Duration, Convexity, and DV01, Bionic Turtle FRM Study Notes, **Page 5**

## Stay connected