Bond Performance Attribution - SEB Group · PDF fileBond Performance Attribution SEB Asset Management. Editorial SEB Asset Management SEB-huset Bernstorffsgade 50 1577 Copenhagen V

2012 Vol. 1

Bond Performance AttributionSEB Asset Management

Editorial

SEB Asset ManagementSEB-husetBernstorffsgade 501577 Copenhagen VPhone: +45 33 28 14 00

Authors:Portfolio Manager, TAA: Peter Lorin RasmussenPhone: +45 33 28 14 22E-mail: [email protected]

Portfolio Manager, Fixed Income: Michael Denbæk Phone: +45 33 28 14 53 E-mail: [email protected]

Portfolio Manager, Fixed Income & TAA: Tore Davidsen Phone: +45 33 28 14 25 E-mail: [email protected]

This document produced by SEB contains general marketing information about its investment products. Although the content is based on sources jud-ged to be reliable, SEB will not be liable for any omissions or inaccuracies, or for any loss whatsoever which arises from reliance on it. If investment research is referred to, you should if possible read the full report and the disclosures contained within it. Information relating to taxes may become outdated and may not fit your individual circumstances. Investment products produce a return linked to risk. Their value may fall as well as rise, and historic returns are no guarantee of future returns; in some ca-ses, losses can exceed the initial amount invested. Where either funds or you invest in securities denominated in a foreign currency, changes in exchange rates can impact the return. You alone are responsible for your investment decisions and you should al-ways obtain detailed information before taking them. For more information, please see the relevant simplified prospectus for the funds, and the relevant information brochure for funds and for structured products. If necessary you should seek advice tailored to your individual circumstances from your SEB advisor.SEB is under supervision by Finanstilsynet in Denmark.Skandinaviska Enskilda Banken A/S, Bernstorffsgade 50, 1577 København V

Disclaimer

Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Price Source and Mortgage Model . . . . . . . . . . . . . . . . . . . . . . . 3

Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Time (Carry). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

OAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Return Attribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Performance Attribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Attribution Over Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

Appendix - Calculation of Return Contributions . . . . . . . . . . 12

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Introduction

Price Source And Mortgage Model

In this note we present our fixed income return attribution model. The pur-pose is to explain how to interpret the numbers and what you can and can-not do with them. With this knowledge the model becomes an invaluable tool in the communication between the portfolio manager and the client. It quantifies the contribution to the portfolio return from changes to the yield curve, volatility, option-adjusted spreads; as well as the contribution from time (carry). In short, it quantifies the effect of our active positioning. Furthermore, if a benchmark is associated with the portfolio, it allows us to quantify the contribution from the above-mentioned effects on the relative return (i.e. the difference between the return on the portfolio and the ben-chmark).

The purpose of this note is not to describe the underlying mathematics of our mortgage bond model; see the appendix for this. However, in order to clarify certain aspects we are forced to describe some tools that we use. We do hope that you, as the reader, is not intimidated by this, and the essence of the model should not be lost even though a couple of the mathematical expressions might seem exotic.

Not surprisingly, most clients will find that the primary return driver, besides carry, is changes to the yield curve. In other words the shifts, steepenings and bends of the curve. In addition most clients should, if all goes well, find a considerable pickup in the OAS component. This is the component that can most honestly be attributed our ability to select the right bonds in the Danish mortgage bond segment.

Before we dive into the actual model there are a couple of potential caveats which we shall mention.

First, the pricing source for the model is SEB Merchant Bank. This should not matter for the portfolio returns, as it is the same source that we use for standard reporting. However, it may matter for the benchmark return. So for example if we report a return on an EFFAS index, it might not be the same return that one would find on the official EFFAS pricing source (i.e. Bloom-berg). In practice however the difference should be minimal - especially over longer periods of time.

Second, to calculate the option-adjusted spread (OAS) and volatility effects for Danish mortgage bonds, an advanced statistical mortgage bond model must be used. Being a forward looking model the output (i.e. MOAD, OAS, OAC) is based on statistical estimates. In SEB, we have our own proprietary mortgage model, which has a long history of persistently performing better than our competitors.

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Factors

Time (Carry)

The model attributes the return of all relevant bonds to five main compo-nents. A time effect (carry), a curve effect, a volatility effect, an OAS effect, and finally a residual. We calculate the components such that they are ort-hogonal to each other and therefore becomes additive. In other words the return of a given bond is the sum of the percentage point-returns from the 5 main components.

In the following we go through all the effects separately. To illustrate them we show the accumulated effect for one of our mutual funds – SEBinvest Mellemlange Obligationer (DK00016015639). We focus on the period from 1 January 2011 to 30 November 2011.

The time effect is the isolated part of the return which we can attribute to time going by. In other words, what the return would be if the curve, volatility and option-adjusted spread all were completely unchanged from one day to the next. In practice, this effect grows almost linearly over time as one would expect.

This effect is illustrated in Figure 1 for our mutual fund and its benchmark. The difference between the carry effect of the portfolio and the benchmark is primarily due to our large exposure to mortgage bonds compared to the benchmark, which consists of Danish government bonds only. Historically mortgage bonds have a higher carry compared to government bonds.

Figure 1: The accumulated carry effect for SEBinvest Mellemlange Obligatio-ner, January to November 2011

Source: SEB internal calculations

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CurveThe curve effect is the part of the return which can be attributed to changes to the yield curve. This effect can be further split into four different subcom-ponents.

• A shift effect: The effect from parallel shifts in the yield curve.

• A steepness effect: The effect from changes in the steepness (i.e. the slope) of the yield curve.

• A curvature effect: The effect from changes in the curvature of the yield curve.

• And finally a residual, containing all the remaining changes to the yield curve. Because the first three effects explain the majority of the variance of the yield curve, the residual is usually very small. Note that the model is calibrated periodically to obtain the best fit of the movements of the yield curve.

The curve effects are all based on a singular value decomposition of chan-ges to the yield curve. This decomposition ensures that the effects are all orthogonal to each other. In other words, the effect from the curve is equal to the sum of the four subcomponents. Exactly like the return of a given bond is the sum of the time, the curve, the volatility, the option-adjusted spread and the residual effect.

The curve effect is usually the second-most important factor of the return in our bond portfolios. However, when the volatility of the yield curve in-creases the importance of this factor also increases, and it may therefore dominate the time (carry) effect. 2011 was a prime example of such a period.

The curve effect is illustrated in Figure 2. So what can be learned from the graph? First, we see that the return contribution from the curve is falling for both the portfolio and the benchmark from January to the middle of April. This is primarily due to the rising yields over the period. Then in April the return contribution starts to rise. This was when the European debt crisis began to escalate and yields started to drop significantly for both Danish and German government bonds. Second, we can see that our portfolio out-performed its benchmark in terms of the yield curve in the beginning of the year. The reason being that the portfolio had a relatively short duration exposure in the begining of the year. Conversely, when the flight to safety started we could not fully keep up with the benchmark, and some of our curve performance was lost.

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Figure 2: The accumulated curve effect for SEBinvest Mellemlange Obligationer


To most investors, the curve effect will be the most informative component of an attribution model. It quantifies our ability to add duration when yields are falling, and reduce when they are rising. Because the yield curve is such a macro sensitive parameter, it is our experience that going through the histo-rical development of the curve effect (such as in Figure 2) , forms a good ba-sis for a discussion of the economic events over the period. For example the flight to safety which followed the resurgence of the European debt crisis in April is clearly visible as a rising contribution to the return of the portfolio from falling yields.

The volatility effect is the part of the return which can be attributed to chan-ges in the volatility structure. This is a bit more technical than both the time and curve effects, because you need an estimate of the relevant volatility and the relation to the pricing of Danish (callable) mortgage bonds to cal-culate it. Fortunately, our proprietary mortgage bond model delivers these exact volatility numbers and thereby makes it is possible to isolate the de-composed effect from volatility.

The volatility component is illustrated in Figure 3, for our selected mutual fund and its benchmark.

First, note that the effect is zero for the benchmark. This is simply due to the fact that the benchmark consists purely of non-callable government bonds which are unaffected by changes in the volatility. Second, note that the ef-

Volatility

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fect goes from positive to negative over the year. The macro economic ex-planation hereof is the same as for the curve effects: As the European debt crisis started to escalate, the volatility increased, which in itself is negative for callable Danish mortgage bonds; hence the negative return contribu-tion. This can be interpreted in terms of the Black-Scholes model. When you buy a callable bond you are in effect selling a call option (the borrower’s op-tion to prepay at any given time in the future). When the volatility rises, the probability of a call rises (since the probability of the price of the bond being above 100 in the future increases), and the call option (which the investor has sold) becomes more valuable for the borrower (and less valuable for the investor).

Figure 3: The accumulated volatility effect for SEBinvest Mellemlange Obligationer


The OAS effect is the part of the return which can be attributed to changes in the option-adjusted spread. The OAS is the yield spread Danish mortgage bonds offer in excess of Danish government bonds. In a bit naïve interpreta-tion it is the same as the spread for a credit bond. If the firm/issuer is doing well, the option adjusted spread will tighten, and this will contribute positi-vely to the return of the bond. Like the volatility effect, the OAS effect is a result of our proprietary mortgage bond model and cannot be derived solely from public information. Being a result of a model, it is also an estimate and should be treated as such (i.e. with its uncertainty in mind).

The OAS effect is illustrated in Figure 4. The interpretation is quite straight-forward bearing in mind the macro economic story of the volatility and curve effects. The European debt crises resulted in a flight to safety. This implied very low government bond yields which resulted in an OAS widening. The

OAS

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Finally we have a residual in our return attribution model. The reason it exists is the following: The return components are all based on daily close prices. However, if we trade a given security on a given day, the trading price is probably different from the close price. Therefore in the portfolio we will probably get a different return on the bond than what the model gives rise to. In other words, we get a residual, which can be explained by intraday trades and bid-offer spreads. Note that this effect can be both positive and negative depending on whether the actual intraday traded price is higher or lower than the close price.

Residual

Return Attribution Combining the effects creates a full attribution of the return and perfor-mance of a portfolio. To illustrate, we first show the return attribution for our mutual fund, SEBinvest Mellemlange Obligationer, in Table 1.

To increase the readability of the table, it is only based on returns for No-vember 2011.

So what can we learn from this full return attribution of the portfolio? First,

reason for the non-zero OAS effect on the benchmark, which consists pu-rely of government bonds, lies in the interpolation of the government yield curve. In a perfect world this effect should be zero, so one should not put that much emphasis on it for government bonds.

Figure 4: The accumulated option-adjusted spread effect for SEBinvest Mellemlange Obligationer


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the portfolio returned 1.06% - see the South-East corner of the table. Se-cond, 23 basis points of the 1.06% can be attributed to the time (carry) ef-fect. This effect is almost evenly distributed between government bonds, non-callable and callable bonds. Third, a full 131 basis points can be attri-buted to the curve effect. This is primarily an effect of the declining yields in November, which can be seen by the large shift effect. Remember that the curve effect is the sum of the shift, steep, bend, and cres (curve residual) effects. Fourth, the increasing OAS contributes negatively by 42 basis points to the total portfolio return. Again, this was due to the uncertainty in the wake of the European debt crisis, which implied a flight to safety. A scena-rio in which mortgage bonds usually underperform government bonds. In other words, the option-adjusted spreads rise.

Table 1: Return attribution for SEBinvest Mellemlange Obligationer


Performance Attribution

Now instead of merely looking at the absolute returns we can also quantify the relative returns. This is also known as the performance of the portfo-lio. The benchmark for our mutual fund is the EFFAS Denmark 3-5 index. In November the benchmark consisted of two Danish government bonds. The absolute return contribution of the benchmark is illustrated in Table 2. Because the benchmark consist purely of government bonds the volatility effect is zero, and the option-adjusted spread effect is almost zero. Again we stress that the non-zero OAS effect is merely an artifact of the interpolation of the government bond curve. In theory it should be zero.

0.000.000.000.000.000.000.000.000.000.00PREPAYMENTS

-0.020.00-0.13-0.000.04-0.000.040.010.100.02CAPPED FLOATER

1.060.00-0.42-0.060.22-0.010.510.591.310.23TOTAL PORTFOLIO

-0.01-0.00-0.08-0.000.040.000.000.010.060.016 CF 2038

-0.010.00-0.06-0.000.00-0.000.040.000.040.015 CF 2018 IO

0.520.00-0.010.000.08-0.080.200.270.470.06DK GOV. BOND

0.000.000.000.000.00-0.000.000.000.00-0.006 DGB 2011

0.01-0.00-0.000.000.000.000.000.000.010.005 DGB 2013

0.000.00-0.000.000.000.00-0.000.000.000.004.5 DGB 2039

0.200.000.010.000.04-0.030.070.100.170.024 DGB 2019

0.160.00-0.010.00-0.00-0.000.070.080.140.024 DGB 2017

0.000.000.000.000.000.000.000.000.000.004 DGB 2015

0.01-0.00-0.000.000.010.000.000.000.010.004 DGB 2012

0.140.00-0.010.000.03-0.050.050.090.130.023 DGB 2021

0.000.000.000.000.000.000.000.000.000.002 DGB 2014

0.200.00-0.12-0.060.12-0.030.060.150.300.08CALLABLE

0.200.00-0.12-0.060.12-0.030.060.150.300.084 Callable 2041

0.36-0.00-0.160.00-0.020.100.210.160.450.07NON-CALLABLE

0.14-0.00-0.060.00-0.000.020.080.070.170.03Non-Call. 2016 Jan

0.15-0.00-0.060.00-0.030.050.090.070.180.03Non-Call. 2015 Jan

0.02-0.00-0.020.00-0.010.010.020.010.030.01Non-Call. 2014 Oct

0.05-0.00-0.030.000.020.020.020.010.070.01Non-Call. 2013 Jan

0.000.00-0.000.000.00-0.000.000.000.000.00Non-Call. 2012 Jan

TotalResOASVolCResBendSteepShiftCurveCarrySegment

0.000.000.000.000.000.000.000.000.000.00PREPAYMENTS

-0.020.00-0.13-0.000.04-0.000.040.010.100.02CAPPED FLOATER

1.060.00-0.42-0.060.22-0.010.510.591.310.23TOTAL PORTFOLIO

-0.01-0.00-0.08-0.000.040.000.000.010.060.016 CF 2038

-0.010.00-0.06-0.000.00-0.000.040.000.040.015 CF 2018 IO

0.520.00-0.010.000.08-0.080.200.270.470.06DK GOV. BOND

0.000.000.000.000.00-0.000.000.000.00-0.006 DGB 2011

0.01-0.00-0.000.000.000.000.000.000.010.005 DGB 2013

0.000.00-0.000.000.000.00-0.000.000.000.004.5 DGB 2039

0.200.000.010.000.04-0.030.070.100.170.024 DGB 2019

0.160.00-0.010.00-0.00-0.000.070.080.140.024 DGB 2017

0.000.000.000.000.000.000.000.000.000.004 DGB 2015

0.01-0.00-0.000.000.010.000.000.000.010.004 DGB 2012

0.140.00-0.010.000.03-0.050.050.090.130.023 DGB 2021

0.000.000.000.000.000.000.000.000.000.002 DGB 2014

0.200.00-0.12-0.060.12-0.030.060.150.300.08CALLABLE

0.200.00-0.12-0.060.12-0.030.060.150.300.084 Callable 2041

0.36-0.00-0.160.00-0.020.100.210.160.450.07NON-CALLABLE

0.14-0.00-0.060.00-0.000.020.080.070.170.03Non-Call. 2016 Jan

0.15-0.00-0.060.00-0.030.050.090.070.180.03Non-Call. 2015 Jan

0.02-0.00-0.020.00-0.010.010.020.010.030.01Non-Call. 2014 Oct

0.05-0.00-0.030.000.020.020.020.010.070.01Non-Call. 2013 Jan

0.000.00-0.000.000.00-0.000.000.000.000.00Non-Call. 2012 Jan


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The performance attribution is illustrated in Table 3. This table shows the at-tribution of the difference between the return of the portfolio and the ben-chmark. Again we can ask ourselves, what can be learned? First, the fund underperforms the benchmark by 40 basis points over the period. Second, this is primarily due to the rising OAS, which contributed with -40 basis points. Third, the fund gained 10 basis points from an excess carry effect.

Table 3: Performance attribution for SEBinvest Mellemlange Obligationer


Attribution Over Time Finally we can calculate the effects over time. In Table 4 the absolute re-turn contribution over time is shown. We can see that the fund gained most in July and August, where interest rates declined significantly. Furthermore, in October the fund lost 86 basis points due to rising yields.

The attribution over time makes is easy to see when and how a portfolio has achieved its return. The aggregated contributions are shown in the bot-tom row, while each effect can be isolated to specific periods of time (i.e. months). The portfolio manager is then able to explain to a client what hap-pened at that point in time, which in turn highlights the important return drivers and quantifies the effect on the specific portfolio.

Table 2: Return attribution for the EFFAS Denmark 3-5 index


1.460.00-0.000.00-0.100.270.660.511.330.13TOTAL BENCHMARK

1.460.00-0.000.00-0.100.270.660.511.330.13DK GOV. BOND

1.090.000.010.00-0.040.160.470.380.980.104 DGB 2015

0.370.00-0.010.00-0.060.110.180.130.350.032 DGB 2014


1.460.00-0.000.00-0.100.270.660.511.330.13TOTAL BENCHMARK

1.460.00-0.000.00-0.100.270.660.511.330.13DK GOV. BOND

1.090.000.010.00-0.040.160.470.380.980.104 DGB 2015

0.370.00-0.010.00-0.060.110.180.130.350.032 DGB 2014


0.000.000.000.000.000.000.000.000.000.00PREPAYMENTS

-0.400.00-0.42-0.060.31-0.27-0.140.08-0.020.10TOTAL RELATIVE

-0.020.00-0.13-0.000.04-0.000.040.010.100.02CAPPED FLOATER

-0.01-0.00-0.08-0.000.040.000.000.010.060.016 CF 2038

-0.010.00-0.06-0.000.00-0.000.040.000.040.015 CF 2018 IO

-0.940.00-0.000.000.17-0.34-0.46-0.24-0.87-0.07DK GOV. BOND

0.000.000.000.000.00-0.000.000.000.00-0.006 DGB 2011

0.01-0.00-0.000.000.000.000.000.000.010.005 DGB 2013

0.000.00-0.000.000.000.00-0.000.000.000.004.5 DGB 2039

0.200.000.010.000.04-0.030.070.100.170.024 DGB 2019

0.160.00-0.010.00-0.00-0.000.070.080.140.024 DGB 2017

-1.090.00-0.010.000.04-0.16-0.47-0.38-0.98-0.104 DGB 2015

0.01-0.00-0.000.000.010.000.000.000.010.004 DGB 2012

0.140.00-0.010.000.03-0.050.050.090.130.023 DGB 2021

-0.370.000.010.000.06-0.11-0.18-0.13-0.35-0.032 DGB 2014

0.200.00-0.12-0.060.12-0.030.060.150.300.08CALLABLE

0.200.00-0.12-0.060.12-0.030.060.150.300.084 Callable 2041

0.36-0.00-0.160.00-0.020.100.210.160.450.07NON-CALLABLE

0.14-0.00-0.060.00-0.000.020.080.070.170.03Non-Call. 2016 Jan

0.15-0.00-0.060.00-0.030.050.090.070.180.03Non-Call. 2015 Jan

0.02-0.00-0.020.00-0.010.010.020.010.030.01Non-Call. 2014 Oct

0.05-0.00-0.030.000.020.020.020.010.070.01Non-Call. 2013 Jan

0.000.00-0.000.000.00-0.000.000.000.000.00Non-Call. 2012 Jan


0.000.000.000.000.000.000.000.000.000.00PREPAYMENTS

-0.400.00-0.42-0.060.31-0.27-0.140.08-0.020.10TOTAL RELATIVE

-0.020.00-0.13-0.000.04-0.000.040.010.100.02CAPPED FLOATER

-0.01-0.00-0.08-0.000.040.000.000.010.060.016 CF 2038

-0.010.00-0.06-0.000.00-0.000.040.000.040.015 CF 2018 IO

-0.940.00-0.000.000.17-0.34-0.46-0.24-0.87-0.07DK GOV. BOND

0.000.000.000.000.00-0.000.000.000.00-0.006 DGB 2011

0.01-0.00-0.000.000.000.000.000.000.010.005 DGB 2013

0.000.00-0.000.000.000.00-0.000.000.000.004.5 DGB 2039

0.200.000.010.000.04-0.030.070.100.170.024 DGB 2019

0.160.00-0.010.00-0.00-0.000.070.080.140.024 DGB 2017

-1.090.00-0.010.000.04-0.16-0.47-0.38-0.98-0.104 DGB 2015

0.01-0.00-0.000.000.010.000.000.000.010.004 DGB 2012

0.140.00-0.010.000.03-0.050.050.090.130.023 DGB 2021

-0.370.000.010.000.06-0.11-0.18-0.13-0.35-0.032 DGB 2014

0.200.00-0.12-0.060.12-0.030.060.150.300.08CALLABLE

0.200.00-0.12-0.060.12-0.030.060.150.300.084 Callable 2041

0.36-0.00-0.160.00-0.020.100.210.160.450.07NON-CALLABLE

0.14-0.00-0.060.00-0.000.020.080.070.170.03Non-Call. 2016 Jan

0.15-0.00-0.060.00-0.030.050.090.070.180.03Non-Call. 2015 Jan

0.02-0.00-0.020.00-0.010.010.020.010.030.01Non-Call. 2014 Oct

0.05-0.00-0.030.000.020.020.020.010.070.01Non-Call. 2013 Jan

0.000.00-0.000.000.00-0.000.000.000.000.00Non-Call. 2012 Jan


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Table 4: Absolute return of SEBinvest Mellemlange Obligationer in 2011


There are a couple of do’s and dont’s in the interpretation of Table 4. First, the effects cannot be aggregated over time. This is because of compounded interest. So for example the combined carry effect is not the simple sum of all the monthly carry effects. However, the effects can be added across columns. For example the total return of January is still the sum of the carry, the curve (which in itself can be split), the volatility, the option-adjusted spread, and the residual effect.

SummaryIn this note we have presented our absolute return and relative performance attribution of Danish bond portfolios in SEB Asset Management. We have explained how the returns can be attributed to a set of different factors, and how you should interpret them. In addition we have shown how the factors add up, and when they do not.

We hope that after reading this short note you have a much better under-standing of how to interpret the effects. Hopefully this will lead to a better sense of understanding as to where the return of your portfolio comes from and how it was achieved.

5.43-0.09-0.79-0.260.23-0.060.292.913.383.19TOTAL PERIOD

1.060.00-0.42-0.060.22-0.010.510.591.310.23Nov

-0.63-0.01-0.020.00-0.22-0.01-0.29-0.34-0.860.27Oct

1.18-0.08-0.03-0.010.32-0.02-0.220.971.060.23Sep

1.880.020.13-0.08-0.09-0.000.361.291.550.27Aug

1.49-0.01-0.53-0.110.170.020.081.571.830.31Jul

0.120.01-0.150.090.09-0.030.39-0.58-0.120.30Jun

1.210.010.00-0.06-0.01-0.000.070.870.930.32May

0.48-0.00-0.060.00-0.02-0.00-0.000.240.210.33Apr

-0.59-0.010.210.040.050.01-0.06-1.15-1.160.33Mar

-0.13-0.02-0.160.01-0.03-0.01-0.09-0.12-0.250.29Feb

-0.73-0.020.26-0.06-0.27-0.00-0.44-0.49-1.200.30Jan

TotalResOASVolCResBendSteepShiftCurveCarryPeriod

5.43-0.09-0.79-0.260.23-0.060.292.913.383.19TOTAL PERIOD

1.060.00-0.42-0.060.22-0.010.510.591.310.23Nov

-0.63-0.01-0.020.00-0.22-0.01-0.29-0.34-0.860.27Oct

1.18-0.08-0.03-0.010.32-0.02-0.220.971.060.23Sep

1.880.020.13-0.08-0.09-0.000.361.291.550.27Aug

1.49-0.01-0.53-0.110.170.020.081.571.830.31Jul

0.120.01-0.150.090.09-0.030.39-0.58-0.120.30Jun

1.210.010.00-0.06-0.01-0.000.070.870.930.32May

0.48-0.00-0.060.00-0.02-0.00-0.000.240.210.33Apr

-0.59-0.010.210.040.050.01-0.06-1.15-1.160.33Mar

-0.13-0.02-0.160.01-0.03-0.01-0.09-0.12-0.250.29Feb

-0.73-0.020.26-0.06-0.27-0.00-0.44-0.49-1.200.30Jan

TotalResOASVolCResBendSteepShiftCurveCarryPeriod

Page 12

Appendix - Calculation of Return Contributions

In this appendix we illustrate how to calculate the different effects. The cal-culations are done on all calendar days since interest also accrue on non-business days. As a result, only the carry effect can be calculated on non-business days, since all other factors are kept constant.

First, let Ppri and acpri denote the price and accrued interest at the start of the day. Similarly, let Pult and acult denote the price and accrued interest at the end of the day.

Then the total return is given as:

Second, let Ptime, Ptime+curve, Ptime+curve+vol, Ptime+curve+vol+OAS denote the close prices calculated based on the parameters listed in table A.1.

Table A.1: Parameters that are used for the calculation of the different prices

Input Ptime Ptime+curve Ptime+curve+vol Ptime+curve+vol+OAS

Settlement day Ultimo Ultimo Ultimo Ultimo

Curve Primo Ultimo Ultimo Ultimo

Volatility Primo Primo Ultimo Ultimo

OAS Primo Primo Primo Ultimo

Based on these prices we can calculate the different effects.

1. The time (carry) effect: Let Rtime denote the contribution from time to the bond return. This is calculated as:

2. The curve effect: Let Rcurve denote the contribution from the curve to the bond return, and let Rtime+curve denote the return from the combined effect of the curve and time. Then Rtime+curve can be calculated as:

And Rcurve can be isolated as:

3. The volatility effect: Let Rvol denote the contribution from the volatility to the bond return and let Rtime+curve+vol denote the combined effect from the curve, time and volatility. Then Rtime+curve+vol can be calculated as:

Bond Performance Decomposition

9(10)

Appendix 1 – Calculation of decomposed return

In this appendix we illustrate how to calculate the different effects. The calculations are done on all calendar days since interest also accrue on non-business days. As a results, only the carry effect can be calculated on non-business days, since all other factors are kept constant. First, let priP and priac denote the price and accrued interest at the start of the day. Similarly, let

ultP and ultac denote the price and accrued interest at the end of the day.


1−++=

pripri

ultultTotal acP

acPR

Second, let timeP , curvetimeP + , volcurvetimeP ++ , oasvolcurvetimeP +++ denote the closing prices calculated based

on the parameters listed in table A.1. Table A.1: Parameters that are used for the calculation of the different prices

Input timeP curvetimeP + . volcurvetimeP ++ oasvolcurvetimeP +++





Based on these prices we can calculate the different effects. 1. The time effect: Let timeR denote the contribution from time to the bond return. This is

calculated as:

1−++=

pripri

ulttimetime acP

acPR

2. The curve effect: Let curveR denote the contribution from the curve to the bond return, and let

curvetimeR + denote the return from the combined effect of the curve and time. Then curvetimeR +

can be calculated as:

1−+

+= ++

pripri

ultcurvetimecurvetime acP

acPR

And curveR can be isolated as:

( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


10(10)

3. The volatility effect: Let volR denote the contribution from the volatility to the bond return and

let volcurvetimeR ++ denote the combined effect from the curve, time and volatility. Then

volcurvetimeR ++ can be calculated as:

1−+

+= ++++

pripri

ultvolcurvetimevolcurvetime acP

acPR

And volR can be isolated as:

( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol

volcurvetimevolcurvetime

RRR

RRR

4. The option-adjusted spread (OAS) effect: Let OASR denote the contribution from the option-

adjusted spread to the bond return and let OASvolcurvetimeR +++ denote the combined effect from

the curve, time, volatility, and OAS. Then OASvolcurvetimeR +++ can be calculated as:

1−+

+= ++++++

pripri

ultOASvolcurvetimeOASvolcurvetime acP

acPR

And OASR can be isolated as:

( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS

OASvolcurvetimeOASvolcurvetime

RRR

RRR

Thereby all the return components are isolated. Note that the total return is given as:

( ) ( )( )( )( )OASvolcurvetimeTotal RRRRR ++++=+ 11111

To make this additive we make a slight correction. In the following, effectRA denote the additive

return contribution from a given effect. All the additive effects can be found as:

( )( )( )

( )( )( )volcurvetimeOASOAS

curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,

Page 13

And Rvol can be isolated as:

4. The option-adjusted spread (OAS) effect: Let ROAS denote the contribution from the option-adjusted spread to the bond return and let Rtime+curve+vol+OAS denote the combined effect from the curve, time, volatility, and OAS. Then Rtime+curve+vol+OAS can be calculated as:

And ROAS can be isolated as:


To make this additive we make a slight correction. In the following, RAeffect denote the additive return contribution from a given effect. All the additive effects can be found as:


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


9(10)





1−++=

pripri

ultultTotal acP

acPR









calculated as:

1−++=

pripri

ulttimetime acP

acPR




1−+

+= ++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

+

time

curvetimecurve

curvetimecurvetime

RRR

RRR


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,


10(10)




1−+

+= ++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

+

++

+++

curvetime

volcurvetimevol


RRR

RRR




1−+

+= ++++++

pripri


acPR


( ) ( )( )

11

1111

−+

+=

⇔++=+

++

+++

+++++

volcurvetime

OASvolcurvettimeOAS


RRR

RRR





( )( )( )


curvetimevolvol

timecurvecurve

timetime

RRRRRARRRRA

RRRARRA

+++=++=

+==

111,11

,1,

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