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The following article is reprinted from the Quarter 3, 2004 issue
of On the Edge, the Interactive Data Fixed Income Analytics Quarterly newsletter. Effective Duration: Subtleties and Considerations
Teri Geske It is generally accepted that the effective duration of a bond is calculated by revaluing the security under higher and lower rate environments, where "higher" and "lower" are assumed to be parallel movements in interest rates of a certain magnitude, e.g. 25 bps, 50 bps or 100 bps. The average percentage change in the security's value under the higher and lower rate environments relative to the starting value (adjusted for the magnitude of the shift) is the security's effective duration. However, the specific implementation of this general definition can vary with respect to what is meant by "interest rates", and the duration values that result will differ. Beginning with version 5.21, BondEdge offers clients a choice of two methodologies in computing Effective Duration and Convexity, referred to as the "Par Curve" and "Spot Curve" methods. We have made this enhancement to offer total return managers a more consistent comparison with the values published by the purveyors of the major bond indices, who use the "Par Curve method, while still retaining the "Spot Curve" method which we believe has a number of advantages as described later in the article. In previous newsletter articles we described the mechanics of these two methodologies (click here to access these articles), while in here we provide some comparisons and analyses to help portfolio managers understand these two approaches. To summarize, the Par Curve method imposes parallel shifts on the initial Par bond yield curve and derives two new spot curves from the shifted par curves1 . These two resulting spot curves, each one derived from the shifted par bond yield curve, are used to value the security in the higher and lower rate environments, holding the security's OAS constant. The average percentage change in the price of the security versus its current price is its Par Curve effective duration2. In this case, while the yield curve shifts are parallel, the spot curves that are derived from the shifted yield curve are not parallel to the starting spot curve, except in the limiting case where the starting yield curve is flat. The Spot Curve method, which has been the standard in BondEdge, first derives the initial spot curve from the current par curve, then imposes parallel shifts on the initial spot curve. Two new prices for the security are computed based on the shifted initial spot curve, holding constant the OAS derived from the starting price. The average percentage change in the price of the security computed this way is its Spot Curve duration. In this case, by definition, the shifted spot curves are parallel to the initial spot curve. In both methods, the OAS for the security is derived from the initial spot curve, since OAS is the spread that must be layered onto a spot curve (not a yield curve) in order to equate the present value of the expected future cash flows to the market price of the security. A few observations: (1) The shorter the maturity (or average life) of the security, the smaller the difference between the Par Curve and Spot Curve durations. Thus, for most MBS pass-throughs, ABS and short to intermediate corporate bonds this distinction is somewhat irrelevant. However, for long maturity bullet bonds and some CMOs, the difference can be substantial (a difference of 0.70 or more, in some cases). (2) A steep yield curve increases the difference between the two methods - a flat yield curve minimizes the difference. (3) In some cases, such as for zero-coupon bonds, the Par Curve method can produce non-intuitive results. (4) The Spot Curve method may be considered more internally consistent, because the bond's OAS is derived from the same curve that is shifted to derive its Effective Duration. Consider the following examples (all based on market conditions as of month-end June 2004):
As noted above, the difference between the par curve and spot curve durations increases with maturity. Due to the mathematics of deriving spot rates from par coupon rates, when the yield curve is positively sloped (as is typically the case), for a given change in par bond yields, the associated change in spot rates is more pronounced, i.e. the spot curve increases by more than the par curve when interest rates rise, and decreases by more when rates fall, except when the par curve is flat (in this case, changes in the par curve and spot curve are the same). In other words, a positively sloped par curve that is shifted upwards (downwards) by X bps produces a spot curve that is more than X bps higher (lower) than the initial spot curve (the spot curve implied by the starting par bond curve). This is understandable because a par coupon rate of a given maturity may be thought of as a cash flow-weighted average of the spot rates up to that maturity.
Thus, in a normal interest rate environment we can see why par curve effective durations are greater (longer) than spot curve durations for most securities: effective duration is the percentage change in price due to a change in interest rates and prices are computed from spot curves; since spot curves move more when the initial par curve is shifted compared to when the initial spot curve itself is shifted, the prices derived from shifting the par curve change by more (relative to the starting price), than prices derived from shifting the spot curve. Since a duration derived from shifting the par curve is longer than the duration based on shifting the spot curve, this can result in a duration that is longer than the maturity of the security, as with the 10 year 0% coupon bond shown above. While our focus here is on effective (option-adjusted) durations, it is also interesting to compare a bond's par curve and spot curve durations to its Macaulay's Modified Duration (this comparison must be restricted to option-free securities, as Modified duration is not meaningful for securities with any embedded options, such as call or put features, prepayments, reset or lifetime caps, etc.). For option-free securities, Modified Duration is a valid "point measure" of price sensitivity - i.e., the "first order approximation" of the percentage change in price for a very small change in its yield-to-maturity (YTM). We observe that the Modified Duration and spot curve effective durations of option-free securities are almost identical, but this does not hold true for Par Curve durations. Upon closer examination, this result makes intuitive sense: The price of a non-callable bond is equal to the sum of the present values of its remaining coupon and principal payments, where each payment is discounted at the spot rate corresponding to the time the payment is received, (plus, for non-risk free bonds, a spread which is assumed to be a compensation for credit risk, liquidity risk and other factors). So, price (present value) is clearly a direct function of spot rates, not par coupon yields. We also know that YTM is the internal rate of return that equates the present value of the remaining coupon and principal payments to the price of the bond, so we can see that the YTM must be essentially a weighted average of the spot rates that are used to determine price. So, since price is a function of spot rates and YTM is a function of price, a change in price given a change in YTM must be closely related to the change in price derived from a change in spot rates. Given that Macaulay's duration is an approximation of the percentage change in a bond's price for a small change in its YTM, it makes sense that the Modified Duration for non-callable bonds more closely resembles the duration obtained by shifting the spot curve than a duration obtained by shifting the par bond yield curve. Note that BondEdge previously set the Modified and spot curve-based Effective Durations of option-free securities to be equal to each other. While we now separately compute and display the Modified and spot curve Effective Durations, for many option-free securities the values will continue to be virtually identical, with differences showing for longer maturity bonds. Since most fixed income benchmark purveyors (including Lehman Brothers and Citigroup) use the Par Curve method in computing the duration of their indices, effective with version 5.21, the Par Curve method is used to compute the duration and convexity of the benchmark Indices in BondEdge to achieve the most precise replication of the "official" benchmark characteristics. To ensure consistency in portfolio versus benchmark comparisons, clients can elect to use the Par Curve method for Portfolio versus Benchmark comparisons. However, since we believe there are certain drawbacks to this method and some advantages to the Spot curve method, and we are committed to providing independent, unbiased analytics to the fixed income community, BondEdge continues to use the Spot Curve method in "Aged" portfolio and individual security simulations, as well as in Performance Attribution. Conclusion __________________________________ 1A "par curve" is constructed from a set of hypothetical securities, all priced at par with coupon rates set equal to the YTMs of existing, maturity-matched instruments. 2For convenience, we use the term "price" throughout the article, although the actual calculation of effective duration is based on price plus accrued interest. | ||||||||||||||||||||||||||||||||||
 
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