The following article is reprinted from the Quarter 3, 2007 issue of

*On the Edge*, the Interactive Data Fixed Income Analytics quarterly newsletter.

**Back-to-Basics: Key Rate Durations***

*(*updated from a previous article on this topic)*

Teri Geske

Senior Vice President, Business & Product Development

**Introduction**
In previous Back-to-Basics articles we have reviewed the concept of effective duration (also known as option-adjusted duration) and why it is a meaningful measure of interest rate risk for all types of fixed income securities. While effective duration is a critical tool in portfolio analysis and risk management, it is important to recognize that this single duration number is an estimate of an asset’s (or liability’s) sensitivity to a parallel shift in interest rates. While effective duration is useful and convenient for many purposes, robust risk management requires us to recognize that although changes in interest rates are often highly correlated across the curve, in reality the slope and shape of the yield curve changes over time, sometimes dramatically so. The 10-year period from 1997 through 2006, illustrates this point: although weekly changes in the 10-year and 2-year Treasury yields were highly correlated over this period, with an R2 of 0.84, the slope of the curve, measured by the difference between the 10-year and 2-year yields, ranged from -51 basis points (the yield curve was inverted for almost all of the year 2000) to 271 basis points. Clearly, not all yield curve shifts are parallel.

It is well known that portfolios can achieve a particular effective duration with very different sensitivities to the short, intermediate or long end of the yield curve. Managing a portfolio’s interest rate risk requires tools to measure sensitivity to non-parallel shifts in the yield curve. This Back-to-Basics article discusses the concept of Key Rate Durations and how they can be used to measure exposure to this so-called yield curve risk.

**Computing Key Rate Durations**

Key Rate Durations (KRDs) measure an asset’s or portfolio's price sensitivity to independent shifts along the yield curve at selected “key” points. The relevant yield curve is the benchmark (risk-free) curve pertaining to the bond’s currency. KRDs are computed by decreasing and increasing each individual key spot rate by some number of basis points (e.g., ±50bps or ±100bps) and re-computing the asset’s (or liability’s) price given that shift, holding all other spot rates along the term structure constant. The average percentage change in the asset’s value resulting from a given pair of ± key rate shifts is its Key Rate Duration for that point on the curve. The sum of these “partial” durations is the asset’s overall effective duration – thus, KRDs “deconstruct” a security’s overall interest rate sensitivity into its component parts.

Each key rate shift is “anchored” (declines to zero) at the key rate points immediately preceding and following the point that is being measured – for example, assume the key points on the curve are defined as the 6-month, 1-year, 2-year, 3-year, 5-year and 10- and 30-year rates. To compute the 5-year Key Rate Duration, the shift at the 5-year point is fully absorbed by the time we reach either the 3-year or 10-year points on the curve:

**Key Rate Durations for Various Asset Types**

For bullet maturity assets and liabilities, the largest Key Rate Duration corresponds to a shift in the risk-free spot rate that is closest to the instrument’s maturity date, since a shift in this rate would have the greatest impact on the price (present value of the cash flows). The Key Rate Durations corresponding to shifts in spot rates earlier than the maturity date will be smaller, as they reflect the changes in discount rate used to compute the present value of the interest payments received relative to the overall present value.

KRDs are particularly informative for instruments with embedded options, including call or put options, embedded caps and floors, as well as prepayment options, as they show the impact on the overall duration of an asset or portfolio of the call, put or prepayment option’s sensitivity to a shift in a key interest rate. Consider the case of a bond maturing in ten years that is continuously callable at par after two years. Intuitively, we know that the price of this asset is sensitive to a change in both the 10-year and 2-year points on the yield curve, because a change in the 2-year rate would affect the likelihood of the call being exercised, while a change in the 10-year would affect the present value of the principal payment expected at the final maturity date if the bond is not called prior to maturity. We know that the bond’s effective duration will be somewhere between the effective duration of a 2-year bullet and a 10-year bullet security, but the bond cannot be viewed as either a “10-year” or a “2-year” instrument. Using Key Rate Durations we can quantify the relative sensitivity of the bond’s price to changes at both of these points on the curve.

For amortizing instruments and loans such as mortgage loans or pass-throughs, CMOs and asset-backed securities, Key Rate Durations indicate the relative importance of a shift at different parts of the curve given the pattern of principal repayments. Even without the complexity of changes in prepayment forecasts, the price of a mortgage-backed security sensitive to a shift in the Treasury rates that are used to discount the collateral cash flows (either passed through directly or distributed to the tranches in a CMO deal). If a security's cash flows are "front-loaded" relative to its average life, the earlier Key Rate Durations will be relatively large. If the cash flows are "back-end loaded" relative to its average life, the later Key Rate Durations will be larger. Of course, prepayable securities are also sensitive to a shift in the current mortgage rate that affects the borrower’s incentive to refinance, usually the 10-year point on the curve for fixed rate collateral. A well-protected PAC tranche will have a smaller 10-year key rate duration than a support tranche in the deal because a change in refinancings would impact the support tranche more than the PAC. KRDs help to explain why a CMO Interest Only (IO) tranche has a negative overall effective duration: a decline in mortgage rates, as driven by a change in the 10-year Treasury rate, causes prepayments to increase. The IO’s price falls when mortgage rates decline, as faster prepayments reduce the total interest payments received by the IO holder. The IO’s Key Rate Durations are positive except at the 10-year point, where the KRD is large and negative (a decline in price when rates fall produces a negative duration value), even if the IO is not expected to receive any cash flow at that time, because the 10-year Treasury rate drives the refinancing component of the prepayment forecast.

**Interpreting and Using Key Rate Durations**

There are a number of reasons one might wish to isolate a portfolio’s sensitivity to movements at specific points on the curve. Assume we wish to construct a portfolio so that its interest rate sensitivity matches that of a benchmark, or a liability the portfolio is intended to fund. Even if the option-adjusted durations of the portfolio and liability are perfectly matched, this says nothing about their relative sensitivities to non-parallel yield curve shifts. We know that we can structure a portfolio in a number of ways to achieve a duration target (e.g., barbelled, bulleted, laddered, etc.), and these various structures have different sensitivities to non-parallel yield curve moves. Portfolio managers often summarize term structure exposure using distributions, allocating each bond to a particular Maturity- or Duration-based “bucket”. While useful as a summary analysis, a significant shortcoming of this approach is that each asset can only be assigned to one bucket – thus, the 10-year bond callable in 2 years cannot be “counted” in both the 2-year and 10-year buckets. The limitation is even more striking for amortizing and prepayable bonds. Key Durations overcome these limitations, allowing us to do more rigorous risk management and portfolio vs. benchmark comparisons. Key Rate Durations allow us to quantify yield curve exposures between a portfolio and a benchmark, or a liability to be funded, by deconstructing the overall effective duration of each instrument in the portfolio and benchmark into its component parts. If one wishes to take a position on future movements in the yield curve (e.g., a steepening bet), Key Rate Durations can be used to quantify the extent to which that strategy has been implemented. In the Tracking Error analysis in BondEdge, the differences between a portfolio’s and benchmark’s key rate durations are used to measure the relative sensitivity to changes along the underlying yield curve. By multiplying the differences in KRDs by the volatilities associated with each key point on the yield curve, and incorporating the correlations across the potential yield curve shifts, the Tracking Error captures a full range of possible interest rate moves without relying on any single yield curve point to define “interest rate risk”.

One drawback of KRDs is that interpreting the individual key rate values themselves may not be particularly intuitive. Since it is extremely unlikely that a single point on the Treasury curve will exhibit an isolated “jump” upwards or downwards while all other points on the curve remain fixed, it is sometimes difficult to describe what the individual KRDs mean. However, when viewed in relative terms, we can easily make some useful observations. For example, if Portfolio #1 has a KRD at the 1-year point of say, 0.527 and a 5-year KRD of 0.844, and Portfolio #2 has a 1-year KRD of 1.19 and a 5-year KRD of 0.35, then Portfolio #2 is roughly twice as sensitive to shifts at the short end of the yield curve than Portfolio #1, and is less than half as sensitive to shifts at the intermediate part of the curve. We can also say that Portfolio #2 is much more sensitive to changes in short-term rates than to movements in the intermediate part of the curve, whereas Portfolio #1 is more sensitive to shifts in intermediate rates than to movements at the short end of the curve. A key rate duration analysis makes is easy to compare a portfolio to a benchmark or a liability in a way that may reveal structural mismatches not readily identified by other summary portfolio measures. At both the individual security and portfolio levels, key rate durations can provide valuable insights about term structure sensitivity and in a way that no single duration measure can^{1}.

^{1} For a more in-depth comparison of duration measures and yield curve risk, please refer to the white paper entitled, “Beyond Duration: Dissecting Yield Curve Risk”, available on the BondEdge Private Client Site.