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the second, and so on. Since these functions roughly correspond to our shift and twist motions, this approach attributes almost all of the curve change to these two modes, leaving a very small contribution from higher modes. Typical results attribute 90% of curve movements to shift changes, 8% to twist, and 2% to curvature (or butterfly) movements. However, the issue that these basis functions may be different from those in which the risk decisions were expressed is not widely appreciated.
125:
22:
2665:(1992). Ho defines a number of maturities on the yield curve as being the key rate durations, with typical values of 3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25 and 30 years. At each point, we define a duration that measures interest-rate sensitivity to a movement at that point only, with the effect of the duration at other maturities decreasing linearly to the neighboring points.
2433:. The modified duration of a bond assumes that cash flows do not change in response to movements in the term structure, which is not the case for an MBS. For instance, when rates fall, the rate of prepayments will probably rise and the duration of the MBS will also fall, which is entirely the opposite behavior to a vanilla bond. For this reason, effective duration
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movements are always assigned small weights in an attribution analysis. However, this is at the cost of a distortion of the other results. On the other hand, a naïve interpretation of the terms shift, twist, curvature when applied to yield curve movements may well give rise to higher-order movements that are much higher than investors would expect.
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and prices. There may also be inherent advantages in this approach with its ability to work with user-supplied risk numbers, since it allows the user to use sensitivity measures from in-house models, which is particularly useful where (for instance) the user has custom repayment models for mortgage-backed securities.
293:, then an attribution report will confirm this claim. Conversely, if the attribution report shows that this same manager is making non-zero returns from interest rate movements, then his exposure to interest rate risk is clearly not zero and his investment process clearly differs from his stated position.
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The simplest way to regard return on credit is to see it as return made by changes in a security's yield, after changes due to movements in the market's reference curve have been removed. This may be quite adequate for a simple portfolio, but for traders who are deliberately interest-rate neutral and
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To calculate the return arising from each effect, we can reprice the security from first principles by using a pricing formula, or some other algorithm, before and after each source of return is considered. For instance, in calculating yield return, we might calculate the price of the security at the
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An alternative way to regard the higher yields of credit instruments is to regard them as being priced off different yield curves, where these credit curves lie above the reference curve. The lower the credit rating, the higher the spread, thus reflecting the extra yield premium demanded for greater
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It is not always appropriate to use a single yield curve throughout a portfolio, even for instruments traded from a particular country. Inflation-linked securities use their own curve, whose movements may not show strong correlation with the yield curve of the broader market. Short-term money market
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use a large sample of historical yield curve data and construct a set of basis functions that can be linearly combined to represent these curve movements in the most economical way. The algorithm always attributes as much of the curve movement to the first basis function, then as much as possible to
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The data requirements for this approach to attribution are less onerous than for the first-principle approach. The perturbation equation does require externally calculated risk numbers, but this may not be a major obstacle, since these quantities are readily available from the same sources as yields
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This approach can easily be combined with the earlier decomposition into shift, twist and curvature components to give price changes due to these yield curve movement types. For instance, suppose we know the amount by which the yield curve has steepened at each key rate maturity. Then the return of
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To perform attribution on a portfolio, one must also run attribution on its associated benchmark, and this frequently presents substantial difficulties. To provide attribution information at the same level of detail for a benchmark, one needs extensive, detailed weights and returns, and these are
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The first source of return in a fixed-income portfolio is that due to interest. The majority of securities will pay a regular coupon, and this is paid irrespective of what happens in the marketplace (ignoring defaults and similar catastrophes). For instance, a bond paying a 10% annual coupon will
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To describe these movements in numerical terms, typically requires fitting a model to the observed yield curve with a limited number of parameters. These parameters can then be translated into shift, twist, and butterfly movements – or whatever other interpretation the trader chooses to use. This
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A more widely used approach to fixed-income attribution is to decompose the returns of individual securities by source of risk, and then to aggregate these risk-specific returns over an entire portfolio. Typical sources of risk include yield return, return due to yield curve movements, and credit
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While benchmarks may have much greater uniformity of instrument type than managed portfolios, the sheer number of securities – and the data maintenance issues required to reprice each one, and to ensure that the correct coupon amount and timing is used when a coupon is paid – means that detailed
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debt was downgraded to non-investment, or junk, status by the ratings agencies. As a result, the credit spread (or return demanded by investors for holding this riskier investment) rose by over 150 basis points, and the value of
General Motors bonds accordingly fell. The loss in performance this
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There are also problems in the exact definition of the terms shift and twist. Without fixing a twist point at the outset, there is no unique value for these terms in either a Nelson-Siegel or polynomial formulation. However, the location of this twist point may not match user expectations. For a
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The great advantage of a factor-based approach is that it ensures that as much curve movement as possible is attributed to shift movement, and that twist and curvature motion are given as small values as possible. This allows apparently straightforward reporting, because hard-to-understand curve
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Fixed-income attribution therefore provides a much deeper level of information than is available from a simple portfolio performance report. Typically, such a report only shows returns at an aggregated level, and provides no feedback as to where the investor's true skills lie. For these reasons,
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At the most basic level, we can break down yield changes in terms of treasury shift and credit shift. At any maturity, we can compare the change in the target security with the change in the corresponding government-backed security, which will have the highest credit rating and hence the lowest
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Firstly, there is no agreement as to what these fundamental modes actually are, since they depend on the historical dataset used in the calculation (unlike, say, a parallel curve shift – which may be defined in purely mathematical terms). Each market, over each analysis interval, will therefore
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Once a curve has been fitted, the user can then define various measures of shift, twist and butterfly, and calculate their values from the calculated parameters. For instance, the amount of shift in a curve modeled by a polynomial function can be modeled as the difference between the polynomial
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Of course, the yield curve is most unlikely to behave in this way. The idea is that the actual change in the yield curve can be modeled in terms of a sum of such saw-tooth functions. At each key-rate duration, we know the change in the curve's yield, and can combine this change with the KRD to
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While all these factors can be important in accounting for changes in MBS returns, in practice a particular user may only select a subset. The reason is that a perturbational analysis requires the provision of risk sensitivity numbers for each factor, and in some cases these may simply not be
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Roll return can occur when a yield curve is steeply sloped. In the absence of any changes in the curve, as a security is held over time its maturity will decrease and the yield (as read off the curve) will change. If the slope is positive, the yield will decrease and the security's price will
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in the portfolio. A portfolio manager may hold firm views on the ways in which these factors will change in the near future, so in three separate risk decisions he positions the assets in the portfolio to take advantage of these expected market movements. If all views subsequently prove to be
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Towards the end of the bond's life we often see a pull-to-parity effect. As maturity approaches, a bond's price converges to its nominal amount, irrespective of the level of interest rates, and this may cause a bond's price to move in a different way to what would normally be expected.
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Mortgage-backed securities (MBS) are substantially more complex to price than vanilla bonds, due to the uncertainties implied by the prepayment option included in the instrument's structure. Ideally, the returns generated by these other risks should be shown in the attribution report.
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Positioning a portfolio's assets to take advantage of a steeply sloping yield curve is sometimes called riding the yield curve. Strictly speaking, roll return belongs in a separate category, as it is neither a strict yield effect nor a return caused by a change in the yield curve.
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Conveniently, the perturbational approach may be extended to new asset types without requiring any new pricing code or types of data, and it also works for benchmark sectors as well as individual securities, which is useful if benchmark data is only available at sector level.
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Other ways to look at the return generated by credit spreads is to measure the yield of each security against an industry sector curve, or (in the case of
Eurobonds) to measure the spread between bonds of the same credit rating and currency but differing by country of issue.
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The shape of the modes may not match user expectations, and in practice it will be most unlikely that the portfolio will be managed and hedged with reference to these fundamental modes. A manager is more likely to view future curve movements in terms of a simple shift and
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The approach is also self-checking, in that the size of the residual returns should be very low. If this is not the case, there will be presumably be an error in the calculated return or the risk numbers, or some other source of risk will be distorting the returns.
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In other words, a key rate duration measures the effect of a change in the yield curve that is localized at a particular maturity, and restricted to the immediate vicinity of that maturity, usually by having the change drop linearly to zero at neighboring points.
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Since conventional risk analysis for fixed-income instruments usually assumes a parallel yield shift across all maturities, it would be most convenient if a parallel motion mode turned out to dominate the other modes, and in fact this is more or less what occurs.
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in the yield curve across all maturities. It does not take into effect other risk factors, such as non-parallel yield curve shifts, convexity, option-adjusted spreads, and others. However, effective duration may suffice for many managers as a basic risk measure.
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accurate inputs to these formulae, including market yields and other variable quantities such as the 90-day bank bill swap rate (BBSW) and consumer price index (CPI) factors for floating rate notes and inflation-linked securities, and regular updates for these
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spread shifts. These sub-returns can then be aggregated over time and sector to give the overall portfolio return, attributed by source of risk. For a description of the mechanics of combining these sub-returns in a self-consistent manner, see Bacon (2004).
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often hard to find. For instance, many widely used benchmarks contain thousands of bonds. Deriving the security-level returns of an industry benchmark so that the overall returns match the published figures remains a major challenge for most practitioners.
1056:, and many investment strategies are expressed in terms of changes in the curve. Any discussion of fixed-income attribution therefore requires an appreciation of how changes in the curve are described, and their effect on the performance of a portfolio.
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correct, then each decision will generate a profit. If one view is wrong, it will generate a loss, but the effect of the other bets may compensate. The overall performance will then be the sum of the performance contributions from each source of risk.
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The sheer variety of the fixed-income markets, and the pace of innovation in this area, means that provision of an attribution capability from scratch will continue to present significant challenges. In no particular order, issues to be faced include
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The situation is complicated by recent innovations in the credit markets and explosive growth of instruments that allow credit risk to be precisely targeted, such as credit-default swaps and the ability to split different tranches of instruments in
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For managers who need to account for changes in the shape of the yield curve in detail, a single risk measure for interest-rate sensitivity is insufficient and a more detailed way of measuring changes across the entire term structure is required.
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Attribution is therefore an extremely useful tool in verifying a fund manager's claims to possessing particular investment skills. If a fund is marketed as being interest-rate neutral while providing consistent returns from superior
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twist measures the degree to which the curve has steepened or flattened. For instance, one might measure the steepness of the
Australian yield curve as the difference between the 10-year bond future yield and the 3-year bond future
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parameters at successive dates. In practice, the Nelson-Siegel function has the advantages that it is well-behaved at long maturities, and that its parameters can be set to model virtually any yield curve (see Nelson and Siegel ).
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Since the yield of virtually any fixed-income instrument is affected by changes in the shape of the
Treasury curve, it is not surprising that traders examine future and past performance in the light of changes to this curve.
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For these reasons, a pricing model-based approach to attribution may not be the right one where data sourcing or reconciliation is an issue. An alternative solution is to perform a Taylor expansion on the price of a security
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Even pricing data can be difficult to come by in some cases. For some Asian benchmarks, illiquid markets can mean that accurate yield data is not published at all, which can make calculation of risks very difficult.
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curvature (or butterfly, or curve reshaping) measures the degree to which the term structure has become more or less curved. For instance, a yield curve that can be fitted to a straight line exhibits no curvature at
953:
634:{\displaystyle \delta P={\frac {\partial P}{\partial t}}\delta t+{\frac {\partial P}{\partial y}}\delta y+{\frac {1}{2}}{\frac {\partial ^{2}P}{\partial y^{2}}}\delta y^{2}+O\left({\delta t^{2},\delta y^{3}}\right)}
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start and end of the calculation interval, but using the yield at the beginning of the interval. Then the difference between the two prices may be used to calculate the security's return due to the passage of time.
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Changes in term structure form one of the most important sources of risk in a portfolio. Unlike an equity price, which just moves one-dimensionally, the price of a fixed-income security is calculated from sum of
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2173:, so that the curve movement on any given day is a linear combination of the basis eigenvectors. The eigenvalues of this matrix then give the relative weights, or importance, of these curve shifts. .
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is the decay factor: small values produce slow decay and can better fit the curve at long maturities, while large values produce fast decay and can better fit the curve at short maturities;
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produce a different set of fundamental modes and hence different attribution decompositions, and so it may be impossible to compare sets of attribution results over longer intervals.
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2358:, where the discount rate used depends on the interest rate at that maturity. The magnitude and shape of curve changes are therefore of major importance to fixed-income managers.
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risk. Using this model we can describe returns of, say, an A-rated security in terms of movements in the AAA curve, plus movements (tightening or widening) in the credit spread.
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benchmark modeling remains extremely difficult. There are also issues involving transparency of benchmark calculations, with many of the underlying actions remaining obscure.
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attribution. This is based on the standard
Brinson-Fachler attribution scheme, where the securities in the portfolio and benchmark are divided up into buckets based on their
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While there remain numerous challenges to solve, the state of fixed income attribution is much less murky than was the case even five years ago. The reasons include
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yield. All securities have yields equal or greater than their equivalent-maturity government securities, which act as a benchmark for movements in the marketplace.
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1564:{\displaystyle y\left(m\right)=\beta _{0}+\beta _{1}{\frac {\left}{m/\tau }}+\beta _{2}{\left({\frac {\left}{m/\tau }}-\exp \left({-m/\tau }\right)\right)}}
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While a factor-based decomposition of term structure changes is mathematically elegant, it does have some significant drawbacks for attribution purposes:
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2369:, with the size of this spread depending on current economic conditions and the credit rating of the individual security. For instance, in April 2005
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securities may be better modeled by a separate model for the bill curve, and other markets may use the swap curve rather than the treasury curve.
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If one is only interested in gross changes in the yield curve at a particular maturity, then one can read yields off the various datasets, using
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Another generalizing of Nelson-Siegel is the family of
Exponential Polynomial Model ("EPM(n)") where the number of linear coefficients is free.
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However, the effective yield on the bond may well be different, since the market price of the bond is usually different from the face value.
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is required. The most widely used nomenclature for describing yield curve changes uses the terms "shift", "twist" and "butterfly". Briefly:
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817:{\displaystyle \delta r=y\cdot \delta t-MD\cdot \delta y+{\frac {1}{2}}C\cdot \delta y^{2}+O\left({\delta t^{2},\delta y^{3}}\right)}
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By deciding to use such an approach, one is implicitly locked into a particular data history and (in practice) data/software vendor.
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Over a given interval, the return of each security will be made up of return from various sub-returns (see below for explanations)
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MBS have many more risk factors than are used for vanilla bonds, and an attribution scheme needs to model them all. They include
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2578:{\displaystyle D_{e}=-{\frac {P\left({y+\delta y}\right)-P\left({y-\delta y}\right)}{2\cdot P\left(y\right)\cdot \delta y}}}
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option-adjusted spread, or the extra yield demanded by the security holder to compensate for the mortgage repayment option;
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available. The return made by such uncomputed risks may be grouped into an ‘Other’ category in the attribution report.
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Svensson (1994) adds a "second hump" term; this is the Nelson–Siegel–Svensson (NSS) model. The additional term is:
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security-specific data, such as day-count conventions and whether a bond has a non-standard first and last coupons;
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is interpreted as the long run levels of interest rates (the loading is 1, it is a constant that does not decay);
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of this matrix. Each eigenvector corresponds to a fundamental model of the yield curve, and each eigenvector is
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146:
2947:{\displaystyle \delta r_{yield}^{steepening}=\sum \limits _{i=1}^{m}{KRD_{i}\cdot \delta y_{i}^{steepening}}}
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One of the most popular techniques to accomplish this is the use of key-rate durations (KRDs), introduced by
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shift measures the degree to which a curve has moved upwards or downwards, in parallel, across all maturities
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accurate pricing formulae including, where relevant, ex-coupon, settlement, and country-specific conventions;
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are supplied but there is none of the more detailed analysis supplied by a true fixed-income decomposition.
2784:. The sum may not be exact because modified duration assumes a flat yield curve, which is seldom the case.
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always pay 10% of its face value to the owner each year, even if there is no change in market conditions.
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3064:"Modeling the IR curve: the Exponential Polynomial Model ("EPM"), the true extension of Nielson-Siegel"
2056:{\displaystyle +\beta _{3}{\left({\frac {\left}{m/\tau _{2}}}-\exp \left({-m/\tau _{2}}\right)\right)}}
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measure first- and second-order interest rate sensitivity. These are conventionally referred to as the
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This scheme has the advantage that it is readily understandable, particularly by managers who have an
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a reconciliation function between existing performance measurement systems and the attribution system
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Historically, one of the most important drivers of return in fixed-income portfolios has been the
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Virtually no research has been published on the attribution of other sources of risk for MBS.
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are making all their returns from credit bets, something more detailed is probably necessary.
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If, on the other hand, one wants to describe curve movements in terms used by traders (or to
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This approach is simple in principle but can lead to operational difficulties. It requires
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A useful account of sector-based attribution, with worked examples, is provided in Dynkin
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fixed-income attribution is rapidly growing in importance in the investment industry; see
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2768:{\displaystyle \delta r_{yield}=\sum \limits _{i=1}^{m}{KRD_{i}\cdot \delta y_{i}}}
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return due to yield (equivalently coupon, or accrued interest, or running yield);
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is the medium-term component (it starts at 0, increases, then decays to zero);
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where the last term denotes higher-order corrections that may be ignored, and
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no standard approach to attribution – sector, yield-curve based, factor based
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The sum of an instrument's key rate durations is approximately equal to its
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A factor-based model of yield curve movements is calculated by deriving the
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calculate the overall change in value of the portfolio. In other words,
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Yield curve § Construction of the full yield curve from market data
948:{\displaystyle C={\frac {1}{P}}{\frac {\partial ^{2}P}{\partial y^{2}}}}
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Estimating and
Interpreting Foreward Interest Rates: Sweden 1992–1994
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is a better single-figure measure of interest-rate sensitivity, where
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where necessary, and there is no need to model any part of the curve.
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The simplest measure of interest-rate sensitivity for an MBS is its
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is the process of measuring returns generated by various sources of
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Active return § Active return in the context of
Brinson models
880:{\displaystyle MD=-{\frac {1}{P}}{\frac {\partial P}{\partial y}}}
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While compact, effective duration only measures the effect of a
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Many investment-grade securities are traded at a spread to the
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of yield shifts at predefined maturities, and calculating the
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The importance of benchmarks remains widely underestimated.
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Practical portfolio performance measurement and attribution
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Among the simplest fixed income attribution techniques is
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Attribution of portfolio performance relative to an index
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Forecasting the term structure of government bond yields
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is the short-term component (it starts at 1, and decays
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Principal component analysis § Quantitative finance
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the MBS due to a steepening
Treasury curve is given by
1186:{\displaystyle y\left(m\right)=a_{0}+a_{1}m+a_{2}m^{2}}
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caused was attributed entirely due to credit effects.
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Financial risk management § Investment management
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return due to movements in the reference yield curve;
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of the security, and are often called risk numbers.
2208:deeper discussion of this point, see Colin (2005).
149:. Unsourced material may be challenged and removed.
3141:Key rate durations: measures of interest rate risk
3044:better understanding of how to perform attribution
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383:First principles versus perturbational attribution
272:, as an example, include the overall level of the
2777:where the sum is across all key rate maturities.
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3015:many more risk factors than in the equity world
3136:, Lehman Brothers Fixed Income Research, March
1735:(see Diebold and Li ; Bolder and Stréliski ):
679:{\displaystyle \delta r={\frac {\delta P}{P}}}
3132:Dynkin, L., Hyman, J., Vankudre, P., (1998).
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3178:Institute for International Economic Studies
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3095:Yield Curve Modelling at the Bank of Canada
3041:cheaper and more powerful computing systems
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2152:
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365:return due to rolling down the yield curve;
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50:Learn how and when to remove these messages
3021:new types of instrument continually appear
378:(OAS), liquidity, inflation, paydown, etc.
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2416:Attribution on mortgage-backed securities
2272:{\displaystyle r_{yield}=y\cdot \delta t}
1091:model is often used for extrapolate CDS.
227:Learn how and when to remove this message
209:Learn how and when to remove this message
107:Learn how and when to remove this message
689:this leads to the perturbation equation
276:, the slope of the yield curve, and the
70:This article includes a list of general
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1094:Two of the most widely used models are
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1301:is the yield of the curve at maturity
643:Writing the return of the security as
3154:Parsimonious modeling of yield curves
3092:Bolder, D. and Stréliski, D. (1999).
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1326:Nelson-Siegel functions take the form
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1727:, are parameters to be fitted via a
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332:a parallel change in the yield curve
268:The risks affecting the return on a
147:adding citations to reliable sources
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3152:Nelson, C.R., Siegel, A.F. (1987).
3032:better third-party software systems
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442:{\displaystyle P\left({y,t}\right)}
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3018:much more complex instrument types
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76:it lacks sufficient corresponding
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3119:Diebold, F.X. and Li, C. (2006).
2618:is the price of the MBS at yield
2067:and the interpretation is as for
1270:are parameters to be fitted, and
1263:{\displaystyle a_{0},a_{1},a_{2}}
374:other sources of return, such as
31:This article has multiple issues.
3114:Mastering attribution in finance
2223:Yield return is calculated from
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61:
20:
3165:Advanced fixed income Analytics
2611:{\displaystyle P\left(y\right)}
2398:collateralized debt obligations
1599:{\displaystyle y\left(m\right)}
1294:{\displaystyle y\left(m\right)}
134:needs additional citations for
39:or discuss these issues on the
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1:
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260:are active at the same time.
256:, particularly when multiple
1102:(Nelson and Siegel (1987)).
371:return due to credit shifts;
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3062:Serge Moulin (March 2018).
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3167:, Frank Fabozzi Associates
2087:{\displaystyle \beta _{2}}
1888:{\displaystyle \beta _{2}}
1819:{\displaystyle \beta _{2}}
1786:{\displaystyle \beta _{1}}
1757:{\displaystyle \beta _{0}}
1700:{\displaystyle \beta _{2}}
1673:{\displaystyle \beta _{1}}
1646:{\displaystyle \beta _{0}}
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158:"Fixed-income attribution"
3160:, 60(4), pp. 473–489
3102:, Technical Report No. 84
3107:Fixed income attribution
2382:Appropriate yield curves
2321:{\displaystyle \delta t}
2153:Factor-based attribution
1028:Modeling the yield curve
305:Sector-based attribution
3146:Journal of Fixed Income
3129:, 130, pp. 337–364
3126:Journal of Econometrics
2349:Yield curve attribution
1100:Nelson-Siegel functions
1034:Bootstrapping (finance)
345:Yield curve attribution
91:more precise citations.
2948:
2877:
2769:
2730:
2632:
2612:
2579:
2454:
2322:
2295:
2273:
2142:
2108:
2088:
2057:
1889:
1862:
1842:
1820:
1787:
1758:
1721:
1701:
1674:
1647:
1620:
1600:
1565:
1315:
1295:
1264:
1211:
1187:
994:
974:
949:
881:
818:
680:
635:
443:
376:option-adjusted spread
3170:Svensson, L. (1994).
3038:easier access to data
2968:current-coupon spread
2949:
2857:
2770:
2710:
2633:
2613:
2580:
2455:
2453:{\displaystyle D_{e}}
2356:discounted cash flows
2328:is the elapsed time.
2323:
2296:
2274:
2143:
2141:{\displaystyle a_{0}}
2109:
2107:{\displaystyle \tau }
2089:
2058:
1895:achieves its maximum.
1890:
1863:
1861:{\displaystyle \tau }
1843:
1841:{\displaystyle \tau }
1821:
1788:
1759:
1722:
1720:{\displaystyle \tau }
1702:
1675:
1648:
1621:
1601:
1566:
1316:
1296:
1265:
1212:
1188:
1070:), then some form of
1042:Multi-curve framework
1032:Further information:
995:
975:
950:
882:
819:
681:
636:
444:
309:Further information:
3112:Colin, A.M. (2016).
3105:Colin, A.M. (2005).
3035:more demanding users
2794:
2679:
2622:
2591:
2466:
2437:
2425:Simple risk measures
2309:
2285:
2229:
2125:
2098:
2071:
1908:
1872:
1852:
1832:
1803:
1770:
1741:
1711:
1684:
1657:
1630:
1610:
1579:
1334:
1305:
1274:
1221:
1201:
1114:
1096:polynomial functions
984:
961:
892:
832:
695:
649:
459:
414:
143:improve this article
3158:Journal of Business
3149:, 2, pp. 29–44
3116:, Pearsons/FT Press
2942:
2853:
1868:also governs where
2957:Other risk factors
2944:
2901:
2800:
2765:
2653:Key rate durations
2628:
2608:
2575:
2450:
2431:effective duration
2391:Credit attribution
2318:
2301:is the security's
2291:
2269:
2138:
2104:
2084:
2053:
1885:
1858:
1838:
1816:
1797:and quickly to 0);
1783:
1754:
1717:
1697:
1670:
1643:
1626:are as above, and
1616:
1596:
1561:
1311:
1291:
1260:
1207:
1183:
990:
973:{\displaystyle MD}
970:
945:
877:
814:
676:
631:
451:higher-order terms
439:
3163:Phoa, W. (1998).
3079:Moulin, S. (2018)
3006:Future challenges
2782:modified duration
2631:{\displaystyle y}
2573:
2303:yield to maturity
2294:{\displaystyle y}
2159:covariance matrix
2005:
1619:{\displaystyle m}
1520:
1438:
1314:{\displaystyle m}
1210:{\displaystyle m}
1003:modified duration
993:{\displaystyle C}
943:
909:
875:
855:
748:
674:
571:
537:
518:
489:
354:Sources of return
328:equity background
321:modified duration
258:sources of return
237:
236:
229:
219:
218:
211:
193:
117:
116:
109:
54:
3201:
3072:
3071:
3059:
2953:
2951:
2950:
2945:
2943:
2941:
2909:
2894:
2893:
2876:
2871:
2852:
2820:
2774:
2772:
2771:
2766:
2764:
2763:
2762:
2747:
2746:
2729:
2724:
2706:
2705:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2607:
2584:
2582:
2581:
2576:
2574:
2572:
2562:
2541:
2540:
2536:
2512:
2508:
2486:
2478:
2477:
2459:
2457:
2456:
2451:
2449:
2448:
2327:
2325:
2324:
2319:
2300:
2298:
2297:
2292:
2278:
2276:
2275:
2270:
2253:
2252:
2212:Interest returns
2147:
2145:
2144:
2139:
2137:
2136:
2113:
2111:
2110:
2105:
2093:
2091:
2090:
2085:
2083:
2082:
2062:
2060:
2059:
2054:
2052:
2051:
2047:
2046:
2042:
2041:
2040:
2031:
2006:
2004:
2003:
2002:
1993:
1984:
1980:
1979:
1975:
1974:
1973:
1964:
1931:
1923:
1922:
1894:
1892:
1891:
1886:
1884:
1883:
1867:
1865:
1864:
1859:
1847:
1845:
1844:
1839:
1825:
1823:
1822:
1817:
1815:
1814:
1792:
1790:
1789:
1784:
1782:
1781:
1763:
1761:
1760:
1755:
1753:
1752:
1726:
1724:
1723:
1718:
1706:
1704:
1703:
1698:
1696:
1695:
1679:
1677:
1676:
1671:
1669:
1668:
1652:
1650:
1649:
1644:
1642:
1641:
1625:
1623:
1622:
1617:
1605:
1603:
1602:
1597:
1595:
1570:
1568:
1567:
1562:
1560:
1559:
1555:
1554:
1550:
1546:
1521:
1519:
1515:
1506:
1502:
1501:
1497:
1493:
1460:
1452:
1451:
1439:
1437:
1433:
1424:
1420:
1419:
1415:
1411:
1378:
1376:
1375:
1363:
1362:
1350:
1320:
1318:
1317:
1312:
1300:
1298:
1297:
1292:
1290:
1269:
1267:
1266:
1261:
1259:
1258:
1246:
1245:
1233:
1232:
1216:
1214:
1213:
1208:
1192:
1190:
1189:
1184:
1182:
1181:
1172:
1171:
1156:
1155:
1143:
1142:
1130:
1072:parameterization
999:
997:
996:
991:
979:
977:
976:
971:
954:
952:
951:
946:
944:
942:
941:
940:
927:
923:
922:
912:
910:
902:
886:
884:
883:
878:
876:
874:
866:
858:
856:
848:
823:
821:
820:
815:
813:
809:
808:
807:
792:
791:
768:
767:
749:
741:
685:
683:
682:
677:
675:
670:
662:
640:
638:
637:
632:
630:
626:
625:
624:
609:
608:
585:
584:
572:
570:
569:
568:
555:
551:
550:
540:
538:
530:
519:
517:
509:
501:
490:
488:
480:
472:
448:
446:
445:
440:
438:
434:
232:
225:
214:
207:
203:
200:
194:
192:
151:
127:
119:
112:
105:
101:
98:
92:
87:this article by
78:inline citations
65:
64:
57:
46:
24:
23:
16:
3209:
3208:
3204:
3203:
3202:
3200:
3199:
3198:
3184:
3183:
3176:, Papers 579 –
3139:Ho, T. (1992).
3076:
3075:
3060:
3056:
3051:
3008:
2988:
2959:
2910:
2905:
2889:
2885:
2878:
2872:
2861:
2821:
2804:
2795:
2792:
2791:
2758:
2754:
2742:
2738:
2731:
2725:
2714:
2689:
2685:
2680:
2677:
2676:
2655:
2623:
2620:
2619:
2597:
2592:
2589:
2588:
2552:
2542:
2523:
2519:
2495:
2491:
2487:
2485:
2473:
2469:
2467:
2464:
2463:
2444:
2440:
2438:
2435:
2434:
2427:
2418:
2393:
2384:
2351:
2338:
2310:
2307:
2306:
2286:
2283:
2282:
2236:
2232:
2230:
2227:
2226:
2214:
2155:
2132:
2128:
2126:
2123:
2122:
2099:
2096:
2095:
2078:
2074:
2072:
2069:
2068:
2036:
2032:
2027:
2020:
2016:
1998:
1994:
1989:
1985:
1969:
1965:
1960:
1953:
1949:
1936:
1932:
1930:
1929:
1925:
1924:
1918:
1914:
1909:
1906:
1905:
1879:
1875:
1873:
1870:
1869:
1853:
1850:
1849:
1833:
1830:
1829:
1810:
1806:
1804:
1801:
1800:
1777:
1773:
1771:
1768:
1767:
1748:
1744:
1742:
1739:
1738:
1712:
1709:
1708:
1691:
1687:
1685:
1682:
1681:
1664:
1660:
1658:
1655:
1654:
1637:
1633:
1631:
1628:
1627:
1611:
1608:
1607:
1585:
1580:
1577:
1576:
1542:
1535:
1531:
1511:
1507:
1489:
1482:
1478:
1465:
1461:
1459:
1458:
1454:
1453:
1447:
1443:
1429:
1425:
1407:
1400:
1396:
1383:
1379:
1377:
1371:
1367:
1358:
1354:
1340:
1335:
1332:
1331:
1306:
1303:
1302:
1280:
1275:
1272:
1271:
1254:
1250:
1241:
1237:
1228:
1224:
1222:
1219:
1218:
1202:
1199:
1198:
1177:
1173:
1167:
1163:
1151:
1147:
1138:
1134:
1120:
1115:
1112:
1111:
1050:
1044:
1030:
985:
982:
981:
962:
959:
958:
936:
932:
928:
918:
914:
913:
911:
901:
893:
890:
889:
867:
859:
857:
847:
833:
830:
829:
803:
799:
787:
783:
779:
775:
763:
759:
740:
696:
693:
692:
663:
661:
650:
647:
646:
620:
616:
604:
600:
596:
592:
580:
576:
564:
560:
556:
546:
542:
541:
539:
529:
510:
502:
500:
481:
473:
471:
460:
457:
456:
424:
420:
415:
412:
411:
385:
356:
347:
313:
307:
291:credit research
266:
233:
222:
221:
220:
215:
204:
198:
195:
152:
150:
140:
128:
113:
102:
96:
93:
83:Please help to
82:
66:
62:
25:
21:
12:
11:
5:
3207:
3197:
3196:
3182:
3181:
3168:
3161:
3150:
3137:
3130:
3117:
3110:
3103:
3100:Bank of Canada
3090:
3080:
3074:
3073:
3053:
3052:
3050:
3047:
3046:
3045:
3042:
3039:
3036:
3033:
3026:
3025:
3022:
3019:
3016:
3007:
3004:
2987:
2984:
2979:
2978:
2975:
2972:
2969:
2966:
2958:
2955:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2908:
2904:
2900:
2897:
2892:
2888:
2884:
2881:
2875:
2870:
2867:
2864:
2860:
2856:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2819:
2816:
2813:
2810:
2807:
2803:
2799:
2761:
2757:
2753:
2750:
2745:
2741:
2737:
2734:
2728:
2723:
2720:
2717:
2713:
2709:
2704:
2701:
2698:
2695:
2692:
2688:
2684:
2654:
2651:
2643:parallel shift
2627:
2606:
2603:
2600:
2596:
2571:
2568:
2565:
2561:
2558:
2555:
2551:
2548:
2545:
2539:
2535:
2532:
2529:
2526:
2522:
2518:
2515:
2511:
2507:
2504:
2501:
2498:
2494:
2490:
2484:
2481:
2476:
2472:
2447:
2443:
2426:
2423:
2417:
2414:
2392:
2389:
2383:
2380:
2371:General Motors
2367:Treasury curve
2350:
2347:
2337:
2334:
2317:
2314:
2290:
2268:
2265:
2262:
2259:
2256:
2251:
2248:
2245:
2242:
2239:
2235:
2213:
2210:
2201:
2200:
2196:
2193:
2154:
2151:
2135:
2131:
2103:
2081:
2077:
2065:
2064:
2050:
2045:
2039:
2035:
2030:
2026:
2023:
2019:
2015:
2012:
2009:
2001:
1997:
1992:
1988:
1983:
1978:
1972:
1968:
1963:
1959:
1956:
1952:
1948:
1945:
1942:
1939:
1935:
1928:
1921:
1917:
1913:
1899:
1898:
1897:
1896:
1882:
1878:
1857:
1837:
1827:
1813:
1809:
1798:
1780:
1776:
1765:
1751:
1747:
1716:
1694:
1690:
1667:
1663:
1640:
1636:
1615:
1594:
1591:
1588:
1584:
1572:
1571:
1558:
1553:
1549:
1545:
1541:
1538:
1534:
1530:
1527:
1524:
1518:
1514:
1510:
1505:
1500:
1496:
1492:
1488:
1485:
1481:
1477:
1474:
1471:
1468:
1464:
1457:
1450:
1446:
1442:
1436:
1432:
1428:
1423:
1418:
1414:
1410:
1406:
1403:
1399:
1395:
1392:
1389:
1386:
1382:
1374:
1370:
1366:
1361:
1357:
1353:
1349:
1346:
1343:
1339:
1328:
1327:
1323:
1322:
1310:
1289:
1286:
1283:
1279:
1257:
1253:
1249:
1244:
1240:
1236:
1231:
1227:
1206:
1194:
1193:
1180:
1176:
1170:
1166:
1162:
1159:
1154:
1150:
1146:
1141:
1137:
1133:
1129:
1126:
1123:
1119:
1108:
1107:
1088:
1087:
1083:
1079:
1029:
1026:
989:
969:
966:
939:
935:
931:
926:
921:
917:
908:
905:
900:
897:
873:
870:
865:
862:
854:
851:
846:
843:
840:
837:
812:
806:
802:
798:
795:
790:
786:
782:
778:
774:
771:
766:
762:
758:
755:
752:
747:
744:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
706:
703:
700:
673:
669:
666:
660:
657:
654:
629:
623:
619:
615:
612:
607:
603:
599:
595:
591:
588:
583:
579:
575:
567:
563:
559:
554:
549:
545:
536:
533:
528:
525:
522:
516:
513:
508:
505:
499:
496:
493:
487:
484:
479:
476:
470:
467:
464:
453:, which gives
437:
433:
430:
427:
423:
419:
407:
406:
403:
399:
396:
384:
381:
380:
379:
372:
369:
366:
363:
355:
352:
346:
343:
306:
303:
278:credit spreads
270:bond portfolio
265:
262:
235:
234:
217:
216:
199:September 2011
131:
129:
122:
115:
114:
69:
67:
60:
55:
29:
28:
26:
19:
9:
6:
4:
3:
2:
3206:
3195:
3192:
3191:
3189:
3179:
3175:
3174:
3169:
3166:
3162:
3159:
3155:
3151:
3148:
3147:
3142:
3138:
3135:
3131:
3128:
3127:
3122:
3118:
3115:
3111:
3108:
3104:
3101:
3097:
3096:
3091:
3088:
3084:
3081:
3078:
3077:
3069:
3065:
3058:
3054:
3043:
3040:
3037:
3034:
3031:
3030:
3029:
3023:
3020:
3017:
3014:
3013:
3012:
3003:
2999:
2995:
2991:
2983:
2977:cost of carry
2976:
2973:
2970:
2967:
2964:
2963:
2962:
2954:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2906:
2902:
2898:
2895:
2890:
2886:
2882:
2879:
2873:
2868:
2865:
2862:
2854:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2817:
2814:
2811:
2808:
2805:
2801:
2797:
2789:
2785:
2783:
2778:
2775:
2759:
2755:
2751:
2748:
2743:
2739:
2735:
2732:
2726:
2721:
2718:
2715:
2707:
2702:
2699:
2696:
2693:
2690:
2686:
2682:
2674:
2670:
2666:
2664:
2659:
2650:
2647:
2644:
2639:
2625:
2604:
2601:
2598:
2594:
2585:
2569:
2566:
2563:
2559:
2556:
2553:
2549:
2546:
2543:
2537:
2533:
2530:
2527:
2524:
2520:
2516:
2513:
2509:
2505:
2502:
2499:
2496:
2492:
2488:
2482:
2479:
2474:
2470:
2461:
2445:
2441:
2432:
2422:
2413:
2409:
2405:
2401:
2399:
2388:
2379:
2375:
2372:
2368:
2363:
2359:
2357:
2346:
2342:
2333:
2329:
2315:
2312:
2304:
2288:
2279:
2266:
2263:
2260:
2257:
2254:
2249:
2246:
2243:
2240:
2237:
2233:
2224:
2221:
2218:
2209:
2205:
2197:
2194:
2190:
2189:
2188:
2185:
2181:
2178:
2177:Factor models
2174:
2172:
2168:
2164:
2160:
2150:
2133:
2129:
2118:
2115:
2101:
2079:
2075:
2048:
2043:
2037:
2033:
2028:
2024:
2021:
2017:
2013:
2010:
2007:
1999:
1995:
1990:
1986:
1981:
1976:
1970:
1966:
1961:
1957:
1954:
1950:
1946:
1943:
1940:
1937:
1933:
1926:
1919:
1915:
1911:
1904:
1903:
1902:
1880:
1876:
1855:
1835:
1828:
1811:
1807:
1799:
1796:
1795:monotonically
1778:
1774:
1766:
1749:
1745:
1737:
1736:
1734:
1730:
1729:least-squares
1714:
1692:
1688:
1665:
1661:
1638:
1634:
1613:
1592:
1589:
1586:
1582:
1574:
1573:
1556:
1551:
1547:
1543:
1539:
1536:
1532:
1528:
1525:
1522:
1516:
1512:
1508:
1503:
1498:
1494:
1490:
1486:
1483:
1479:
1475:
1472:
1469:
1466:
1462:
1455:
1448:
1444:
1440:
1434:
1430:
1426:
1421:
1416:
1412:
1408:
1404:
1401:
1397:
1393:
1390:
1387:
1384:
1380:
1372:
1368:
1364:
1359:
1355:
1351:
1347:
1344:
1341:
1337:
1330:
1329:
1325:
1324:
1308:
1287:
1284:
1281:
1277:
1255:
1251:
1247:
1242:
1238:
1234:
1229:
1225:
1217:is maturity,
1204:
1196:
1195:
1178:
1174:
1168:
1164:
1160:
1157:
1152:
1148:
1144:
1139:
1135:
1131:
1127:
1124:
1121:
1117:
1110:
1109:
1105:
1104:
1103:
1101:
1097:
1092:
1084:
1080:
1077:
1076:
1075:
1073:
1069:
1064:
1062:
1061:interpolation
1057:
1055:
1049:
1043:
1039:
1035:
1025:
1021:
1017:
1013:
1011:
1010:
1005:
1004:
987:
967:
964:
955:
937:
933:
924:
919:
906:
903:
898:
895:
887:
871:
863:
852:
849:
844:
841:
838:
835:
827:
824:
810:
804:
800:
796:
793:
788:
784:
780:
776:
772:
769:
764:
760:
756:
753:
750:
745:
742:
737:
734:
731:
728:
725:
722:
719:
716:
713:
710:
707:
704:
701:
698:
690:
687:
671:
667:
664:
658:
655:
652:
644:
641:
627:
621:
617:
613:
610:
605:
601:
597:
593:
589:
586:
581:
577:
573:
565:
561:
552:
547:
534:
531:
526:
523:
520:
514:
506:
497:
494:
491:
485:
477:
468:
465:
462:
454:
452:
435:
431:
428:
425:
421:
417:
404:
400:
397:
394:
393:
392:
389:
377:
373:
370:
367:
364:
361:
360:
359:
351:
342:
340:
335:
333:
329:
324:
322:
318:
312:
302:
300:
294:
292:
286:
283:
279:
275:
271:
261:
259:
255:
252:
248:
244:
243:
240:Fixed-income
231:
228:
213:
210:
202:
191:
188:
184:
181:
177:
174:
170:
167:
163:
160: –
159:
155:
154:Find sources:
148:
144:
138:
137:
132:This article
130:
126:
121:
120:
111:
108:
100:
90:
86:
80:
79:
73:
68:
59:
58:
53:
51:
44:
43:
38:
37:
32:
27:
18:
17:
3194:Fixed income
3172:
3164:
3157:
3144:
3133:
3124:
3113:
3106:
3094:
3086:
3068:ResearchGate
3066:– via
3057:
3027:
3009:
3000:
2996:
2992:
2989:
2980:
2971:volatilities
2960:
2790:
2786:
2779:
2776:
2675:
2671:
2667:
2660:
2656:
2648:
2640:
2586:
2462:
2428:
2419:
2410:
2406:
2402:
2394:
2385:
2376:
2364:
2360:
2352:
2343:
2339:
2330:
2280:
2225:
2222:
2219:
2215:
2206:
2202:
2186:
2182:
2175:
2163:eigenvectors
2156:
2119:
2116:
2066:
1900:
1093:
1089:
1065:
1058:
1051:
1022:
1018:
1014:
1007:
1001:
956:
888:
828:
825:
691:
688:
645:
642:
455:
408:
390:
386:
357:
348:
338:
336:
325:
317:sector-based
316:
314:
295:
287:
267:
251:fixed income
239:
238:
223:
205:
196:
186:
179:
172:
165:
153:
141:Please help
136:verification
133:
103:
97:January 2017
94:
75:
47:
40:
34:
33:Please help
30:
2336:Roll return
2167:eigenvalues
1731:or similar
1068:extrapolate
1054:yield curve
449:and remove
402:quantities;
274:yield curve
242:attribution
89:introducing
3049:References
2986:Benchmarks
2341:increase.
2171:orthogonal
1046:See also:
957:The terms
264:Importance
169:newspapers
72:references
36:improve it
3083:Bacon, C.
2974:convexity
2899:δ
2896:⋅
2859:∑
2798:δ
2752:δ
2749:⋅
2712:∑
2683:δ
2663:Thomas Ho
2567:δ
2564:⋅
2547:⋅
2531:δ
2528:−
2514:−
2503:δ
2483:−
2313:δ
2264:δ
2261:⋅
2102:τ
2076:β
2034:τ
2022:−
2014:
2008:−
1996:τ
1967:τ
1955:−
1947:
1941:−
1916:β
1877:β
1856:τ
1836:τ
1808:β
1775:β
1746:β
1733:algorithm
1715:τ
1689:β
1662:β
1635:β
1548:τ
1537:−
1529:
1523:−
1517:τ
1495:τ
1484:−
1476:
1470:−
1445:β
1435:τ
1413:τ
1402:−
1394:
1388:−
1369:β
1356:β
1009:convexity
930:∂
916:∂
869:∂
861:∂
845:−
797:δ
781:δ
757:δ
754:⋅
732:δ
729:⋅
720:−
714:δ
711:⋅
699:δ
665:δ
653:δ
614:δ
598:δ
574:δ
558:∂
544:∂
521:δ
512:∂
504:∂
492:δ
483:∂
475:∂
463:δ
254:portfolio
42:talk page
3188:Category
3109:, Wileys
3089:, Wileys
3085:(2004).
341:(1998).
2400:(CDO).
2114:above.
280:of the
183:scholar
85:improve
2587:Here,
2305:, and
2281:where
2199:twist.
1575:where
1197:where
1082:yield.
1040:, and
339:et al.
185:
178:
171:
164:
156:
74:, but
282:bonds
249:in a
190:JSTOR
176:books
2165:and
2094:and
1707:and
1606:and
1098:and
1086:all.
1006:and
980:and
247:risk
162:news
2011:exp
1944:exp
1526:exp
1473:exp
1391:exp
145:by
3190::
3156:,
3143:,
3123:.
3098:.
1680:,
1653:,
1036:,
686:,
323:.
301:.
45:.
3180:.
3070:.
2939:g
2936:n
2933:i
2930:n
2927:e
2924:p
2921:e
2918:e
2915:t
2912:s
2907:i
2903:y
2891:i
2887:D
2883:R
2880:K
2874:m
2869:1
2866:=
2863:i
2855:=
2850:g
2847:n
2844:i
2841:n
2838:e
2835:p
2832:e
2829:e
2826:t
2823:s
2818:d
2815:l
2812:e
2809:i
2806:y
2802:r
2760:i
2756:y
2744:i
2740:D
2736:R
2733:K
2727:m
2722:1
2719:=
2716:i
2708:=
2703:d
2700:l
2697:e
2694:i
2691:y
2687:r
2626:y
2605:)
2602:y
2599:(
2595:P
2570:y
2560:)
2557:y
2554:(
2550:P
2544:2
2538:)
2534:y
2525:y
2521:(
2517:P
2510:)
2506:y
2500:+
2497:y
2493:(
2489:P
2480:=
2475:e
2471:D
2446:e
2442:D
2316:t
2289:y
2267:t
2258:y
2255:=
2250:d
2247:l
2244:e
2241:i
2238:y
2234:r
2134:0
2130:a
2080:2
2063:,
2049:)
2044:)
2038:2
2029:/
2025:m
2018:(
2000:2
1991:/
1987:m
1982:]
1977:)
1971:2
1962:/
1958:m
1951:(
1938:1
1934:[
1927:(
1920:3
1912:+
1881:2
1812:2
1779:1
1750:0
1693:2
1666:1
1639:0
1614:m
1593:)
1590:m
1587:(
1583:y
1557:)
1552:)
1544:/
1540:m
1533:(
1513:/
1509:m
1504:]
1499:)
1491:/
1487:m
1480:(
1467:1
1463:[
1456:(
1449:2
1441:+
1431:/
1427:m
1422:]
1417:)
1409:/
1405:m
1398:(
1385:1
1381:[
1373:1
1365:+
1360:0
1352:=
1348:)
1345:m
1342:(
1338:y
1321:.
1309:m
1288:)
1285:m
1282:(
1278:y
1256:2
1252:a
1248:,
1243:1
1239:a
1235:,
1230:0
1226:a
1205:m
1179:2
1175:m
1169:2
1165:a
1161:+
1158:m
1153:1
1149:a
1145:+
1140:0
1136:a
1132:=
1128:)
1125:m
1122:(
1118:y
988:C
968:D
965:M
938:2
934:y
925:P
920:2
907:P
904:1
899:=
896:C
872:y
864:P
853:P
850:1
842:=
839:D
836:M
811:)
805:3
801:y
794:,
789:2
785:t
777:(
773:O
770:+
765:2
761:y
751:C
746:2
743:1
738:+
735:y
726:D
723:M
717:t
708:y
705:=
702:r
672:P
668:P
659:=
656:r
628:)
622:3
618:y
611:,
606:2
602:t
594:(
590:O
587:+
582:2
578:y
566:2
562:y
553:P
548:2
535:2
532:1
527:+
524:y
515:y
507:P
498:+
495:t
486:t
478:P
469:=
466:P
436:)
432:t
429:,
426:y
422:(
418:P
230:)
224:(
212:)
206:(
201:)
197:(
187:·
180:·
173:·
166:·
139:.
110:)
104:(
99:)
95:(
81:.
52:)
48:(
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