F
IXED
I
NCOME
P
ERFORMANCE
A
TTRIBUTION
A
NALYSIS OF AM
ULTI-C
URRENCYB
ONDP
ORTFOLIODiploma thesis submitted to
Swiss Federal Institute of Technology, Zürich
University of Zürich, Swiss Banking Institute
for the degree of
Master of Advanced Studies in Finance
presented by
B
LAISER
ODUITlic. sciences éco.
Supervisors
Dr. Nils Tuchschmid
Dr. Anna Holzgang
Abstract
The two key asset classes available to investment managers are equities and bonds. Equity attribution has been around for a while and well-established methods of attribution have been developed. It is therefore tempting to generalize these methods to fixed income attribution. However, in doing this the performance analyst ignores essential characteristics of fixed income investments. In many points, risk factors in fixed income investments are fundamentally different from those in equity. Some of them do not even have an equivalent in the equity attribution universe – these include yield curves and credit spreads. Furthermore, the effect of yield curve moves and spread changes on bond value is non-trivial. This paper proposes in the first part to review the different factor decompositions and methodologies used in the fixed income industry. A special emphasis is put on the yield curve shift effects (parallel, twist, butterfly, reshape) which play a central role in performance attribution. In the second part we discuss the practical problems of data quality that usually occur when implementing a fixed income performance attribution. Then we will run a Fixed Income Performance Attribution analysis (FIPA) on a real portfolio and interpret the results obtained. We finish by checking which FIPA factors are the main driver of excess returns and if excess returns identified are still present under a risk-adjusted basis.
C
ONTENTS1. Introduction ... 1
1.1. Performance attribution... 1
1.2. Fixed income performance attribution ... 1
2. Theoretical framework ... 2
2.1. Fixed income return decomposition ... 2
2.1.1. Carry return – Coupon income... 2
2.1.2. Carry return - Roll-down... 3
2.1.3. Market return – Yield curve ... 4
2.1.4. Market return – Spread... 6
2.1.5. Market return – Volatility ... 6
2.1.6. Market return – FX rate... 7
2.1.7. Timing return... 7
2.2. Yield curve construction ... 8
2.2.1. Yield to maturity (YTM) curve ... 8
2.2.2. Zero coupon yield curve... 8
2.3. Yield curve decomposition... 10
2.3.1. Principal component analysis method... 10
2.3.2. Empirical method ... 13
2.3.3. Polynomial method ... 16
2.3.4. Duration based method... 17
2.4. Linking return effects to multiple periods... 18
2.4.1. The arithmetic model ... 18
2.4.2. The geometric model... 19
3. Issues in practice... 20
3.1. Data quality ... 21
3.1.1. Assets without price or with an incorrect price... 21
3.1.2. Corporate actions... 21
3.2. Cash flows and management fees ... 22
3.2.1. Management fees... 22
3.2.2. Accounting of reclaimable withholding taxes... 22
3.2.3. Reinvestment of coupons ... 22
3.3. Gross / Net basis... 22
3.4. Replicating the benchmark in general ... 23
4. Characteristics of the portfolio analyzed ... 24
4.1. Constraints on the portfolio... 24
4.2. Style of the portfolio manager... 24
4.3. Set up of the fixed income performance analysis ... 25
4.3.1. The yield curve... 25
4.3.2. The YC decomposition factors... 26
4.3.3. Linking method ... 26
5. The results... 27
5.1. The FIPA attribution for the global portfolio... 27
5.1.1. Global return (TWR) ... 28
5.1.2. Direct return ... 28
5.1.4. YC shift 1 (parallel shift) ... 29
5.1.5. YC reshape ... 29
5.1.6. Sector spread return (credit spread) ... 30
5.1.7. YC spread return (issue spread) ... 31
5.1.8. Fixed income timing... 31
5.1.9. Fixed income currency return ... 32
5.2. The key ratios ... 32
5.2.1. The Alpha ... 32
5.2.2. The Beta ... 33
6. Interpretation ... 34
6.1. Excess returns and FIPA factors ... 34
6.1.1. Distribution of excess returns... 34
6.1.2. Interaction between FIPA factors and excess returns ... 34
6.1.3. Multivariate analysis ... 37
6.2. Performance on a risk-adjusted basis ... 39
6.2.1. Alpha and FIPA factors... 39
6.2.2. Excess returns on a risk-adjusted basis ... 40
7. Conclusion... 44
8. Acknowledgments... 46
9. References ... 47
10. Appendix ... 48
Appendix 1: US government yield curve principal component analysis ... 48
1.
I
NTRODUCTION1.1. Performance attribution
A manager has a return of 8% for the year 2005 while the benchmark only performs 6%. How did he get it? What could have caused an excess return of 2%? Hopefully it has something to do with the manager’s conscious decisions. That is, with something the manager meant to do. But, in reality, a whole lot of the return might have to do with things the manager didn’t do, right? Like, the effects of the market at large. The economy. The overall movement of industries relative to actions of the Federal Reserve or other bodies. Even some unintended consequences of the manager’s actions! Performance attribution tries to answer these questions.
The purpose of performance attribution is to understand realized excess returns and to relate this information to the active decisions made in the investment organization, in order to understand the sources of out-performance and identify the active decisions that have generated the excess returns. Attribution models are designed to identify the relevant factors that impact performance and to asses the contribution of each factor to the final result. This information can then be communicated to clients, management and (not least) the portfolio managers that conducted the active bets. In doing so the performance analysis can over time add value by assisting in the identification of the investment management particular skills and of the areas where skills appear to be lagging.
1.2. Fixed income performance attribution
The slump in equity markets during the last couple of years has changed many investors attitude towards fixed income. From being a low returning low volatile asset class bond investments are now considered more than just a safe-haven. Measured on a risk-adjusted basis the long-term returns from bond investments compare favorably with equity returns. In order to understand the active decisions made during the investment process it is essential to understand the characteristics of the underlying asset classes and relevant risk factors that drive the investments, since it is these assets classes and risk factors that the portfolio manager analyzes when designing portfolios.
Two key asset classes available to investment managers are equity and bonds. Equity attribution has been around for a while and well-established methods of attribution have been developed. It is therefore tempting to generalize these methods to fixed income attribution. However, in doing this the performance analyst ignores essential characteristics of fixed income investments.
In many points, risk factors in fixed income investments are fundamentally different from those in equity. Some of them do not even have an equivalent in the equity attribution universe – these include yield curves and credit spreads. Furthermore, the effect of yield curve moves and spread changes on bond value is non-trivial.
For all these reasons, Fixed Income Attribution has been one of the key challenges in the portfolio management industry; though there is now an extensive set of research into differing methodologies, there is still no agreed industry standard.
This paper proposes in its first part to review the different factor decompositions and methodologies used in the fixed income industry. A special emphasis is put on the yield curve shift effects (parallel, twist, butterfly, reshape) which play a central role in performance attribution. In the second part we will discuss briefly the different problems that usually occur in practice when implementing the attribution. Then we will run a Fixed Income Performance Attribution analysis (FIPA) on a real portfolio and interpret the results obtained. We finish by checking which FIPA factors are the main driver of excess returns and if the excess returns identified are still present under a risk-adjusted basis.
2.
T
HEORETICAL FRAMEWORK2.1. Fixed income return decomposition
It is generally admitted that the value generated by holding bonds is composed of three different components. Unlike the case for equities, the return generated from periodic cash flows is significant. In addition to the periodic return, bond returns are sensitive to changes in the fundamental market variables or fixed income risk factors. Finally the return is affected by timing of trades. These three different sources of return are usually denoted by carry, market and timing return:
Timing Market
Carry
Total r r r
r = + +
Fig. 1. Fixed income return components
2.1.1. Carry return – Coupon income
The carry return is composed of two components. The central component is the (typically annual) coupon being paid out to the investor – we denote this component direct return. This component is always positive. This direct return is theoretically defined as:
t y P t C rDirect = ⋅∆ = Current⋅∆
where C is the annual coupon, t is time passed, P is the initial price and y denotes yield.
More generally a direct return is computed as follows within a “end of the day cash-flow / geometric model”:
(
)
(
)
(
)
∑
∑
− − − − − − − − ⋅ + ⋅ ⋅ + + ⋅ = 1 1 1 1 1 Coupon 1 1 Direct , 1 t t t t t t t t t t t X AI P N X C AI P N rwhere t: time, N: nominal amount, AI: accrued interest, P: price, C: coupon, X: FX rate.
Coupons 1 days interest earned
Redemptio
n
+ Co
upon
Coupons 1 days interest earned
Redemptio
n
+ Co
upon
Fig. 2. Direct return
2.1.2. Carry return - Roll-down
A less pronounced component of carry is the passage of time. Bonds usually do not trade at par, but they are eventually redeemed at par, therefore at maturity the market price must converge towards par. For longer-dated bonds this effect is minor, whereas it can be significant for shorter-dated bonds trading away from par. This return component is called
roll-down return. The effect is positive for discount bonds (the roll effect will pull the price
up towards par) and negative for premium bonds (the roll effect will pull the price down towards par).
The roll-down return can be interpreted as:
(
)
(
)
(
)
(
)
(
)
∑
∑
− − − − − − − − − − ⋅ + + ⋅ ⋅ + + ⋅ = 1 Coupon 1 1 1 1 Coupon 1 1 1 1 RollDown , 1 , t t t t t t t t t t t t t t X C AI P N X C AI YCS YC P N rwhere t: time, N: nominal amount, P: price, X: FX rate, YC: yield curve, YCS: yield curve spread.
Remark: The artificial price “Pt(.)” is calculated by a function of different factors like yield
curve, yield curve spread, volatility for example (the number of factors depends on the model complexity). Artificial prices are needed to sequentially calculate and decompose return effects (see Fig. 1.).
YCt-1 Time Ze ro ra te 1 Day YCt-1 Time Ze ro ra te YCt-1 Time Ze ro ra te 1 Day YCt-1 Time Ze ro ra te 1 Day YCt-1 Time Ze ro ra te YCt-1 Time Ze ro ra te 1 Day
Fig. 3. Roll-down return when the bond is overvalued
2.1.3. Market return – Yield curve
In contrast to the carry return components the market return is less predictable. The market return is driven by the market variables on which bond value depends. In fixed income the yield curve is the central market variable. Traditionally the yield curve is based on bonds issued by government entities. The rationale has been that this provides a default free yield curve per country. Therefore the market value of government bonds are normally driven entirely by movements in this curve.
The basic approach to modeling yield curve movements is to calculate the difference between the final and the initial yield curve for the period for which performance is measured.
Fig. 4. Yield curve movements
Often portfolio managers decompose yield curve shifts further into basic movements. Typically the number of basic movements vary between 2 and 5. This number is arbitrarily chosen by the portfolio manager who constructed the portfolio and who did bets on yield curve moves. The number of basic movements is consequently a trade-off between the explanation power of the model and the complexity of the interpretation.
Recent studies suggest that most of the yield curve shift can be explained pretty well by essentially three factors: parallel shift, slope (or twist) and curvature (or butterfly). The unexplained shift left is normally statistically small and put in a residual factor called reshape.
Of course in market crisis situations these three first factors might be insufficient to leave the
reshape small and to do a good performance attribution. a) Parallel shift
A parallel shift appears when the rates at standard maturities move uniformly. Note that parallel shifts in yields are captured directly by the bond duration as rParallel ≅−D⋅∆ycParallel where r denotes return, D is modified duration and YC is the yield curve.
b) Twist (steeping / flattening)
We can see a twist effect when short term and long term rates move in opposite direction but proportionately in relation to the distance from some “pivot point” maturity (usually defined at 5 years).
c) Curvature
The curvature or butterfly effect occurs when short term and long term rates move in same direction while medium term rates move in an opposite direction, still proportionately.
By decomposing the yield curve movements into contributions from these shifts the bond fixed income portfolio return that is due to the yield curve moves can be decomposed into:
Reshape Curvature Twist Parallel YieldCurve r r r r r = + + +
Fig. 5. Example of parallel, slope (steepness) and curvature shifts
The financial literature identifies several methods to extract these factors and quantify them. Four, at least, can be mentioned:
• Statistical Principal Component Analysis (PCA) • Empirically constructed user-defined factors
• Polynomial fit mimicking a Taylor decomposition of the yield curve function • Factor model based on duration analysis.
As the returns that are generated by the yield curve shifts are the heart of a fixed income attribution analysis, we are going to review these four methods in detail a bit later in the paper.
2.1.4. Market return – Spread
In addition to the general yield levels, non-sovereign debt is also sensitive to credit risk. The market measure of credit risk is the spread. This is the additional yield that an investor will require in order to invest in such bonds. The Implied Yield Curve Spread (YCS), which is the discounting spread necessary to add to the yield curve in order to match the market price of the given bond. So the YCS of a bond is the solution to the equation:
(
)
∑
+ + = t t t YCS y C V ) 1where Ct denotes cash flow at time t, V the value of the bond and yt is the zero yield for time t.
The return resulting from the yield curve spread is fully defined by the following equation:
(
)
(
)
(
)
(
)
∑
∑
− − − − − − ⋅ + ⋅ ⋅ + ⋅ = + 1 1 1 1 1 YCS , 1 , , , , 1 t t t t t t t t t t t t t t t t X AI Vol YCS YC P N X AI Vol YCS YC P N rwhere t: time, N: nominal amount, P: price, X: FX rate, YC: yield curve, YCS: yield curve spread, Vol: volatility.
The magnitude of the spread reflects the credit quality of the issue. The spread is typically decomposed into two subcategories – the sector spread (industry specifics and rating specifics) and the issue spread (issuer specifics). The first subcategory reflects aspects common across bonds issued by corporations with similar ratings and in similar industries; the second category reflects issue/issuer specific considerations. Therefore the spread can be decomposed as:
Issue Sector
Spread r r
r = +
The spread return component is a market return. Typically sector spreads vary substantially over time and the spread return can be sizeable - also in comparison to curve returns. As an example the contagious spread of the Russian credit crisis that occurred in the autumn of 1998 meant that increases in spreads were of the same magnitude as falls in government yields.
2.1.5. Market return – Volatility
For standard domestic bonds the previous factors are the main drivers. For more complex instruments other market variables can add value. An important category of bonds is bonds
with embedded options. Often asset-backed, mortgage-backed and corporate bonds have built-in options in the form of put, call or prepayment options. For such bonds changes to
implied volatility is an important factor behind market value, since the volatility drives the
option value. For most vanilla bond portfolios the volatility return is small compared to the direct, curve and spread return components. However for portfolios with large options positions or many mortgage bonds the volatility effect can be significant.
Volatility is computed as follows:
(
)
(
)
(
)
(
)
(
)
(
)
∑
∑
− − − − − − − − − ⋅ + + ⋅ ⋅ + + ⋅ = + 1 Coupon 1 1 1 1 Coupon 1 1 1 Vol , 1 , , , 1 t t t t t t t t t t t t t t t t t X C AI YCS YC P N X C AI Vol YCS YC P N rwhere t: time, N: nominal amount, P: price, C: coupon, X: FX rate, YC: yield curve, YCS: yield curve spread, Vol: volatility.
2.1.6. Market return – FX rate
For foreign investments the FX rate development is another key risk factor that impacts the performance. The currency effect is generic (not specific to fixed income) and it is treated exactly as for equity portfolios.
The FX effect is calculated with:
(
)
(
)
(
)
(
)
∑
∑
− − ⋅ + + ⋅ ⋅ + + ⋅ = + 1 Coupon Coupon Currency , 1 1 t t t t t t t t t t t t X C AI P N X C AI P N rwhere t: time, N: nominal amount, P: price, C: coupon, X: FX rate.
2.1.7. Timing return
Timing return component arises due to the trading activities in a portfolio. Performance
measurement is typically done based on end of day prices. Usually trading is conducted during the trading hours and therefore some discrepancy will occur. The effect of this trading is compounded into the timing return component. In case the trader has executed on attractive levels relative to end of day pricing the effect will show up as a positive return component.
Timing return can be defined as:
(
)
(
)
(
)
(
)
(
)
∑
−∑
− − − − ⋅ + + ⋅ ⋅ + + ⋅ = + 1 Coupon 1 , 1 1 Coupon Timing , 1 , 1 t t t t t t t t t t t t t t t X C AI Vol YCS YC P N X C AI P N rwhere t: time, N: nominal amount, P: price, C: coupon, X: FX rate, YC: yield curve, YCS: yield curve spread, Vol: volatility.
Note that in this section 2.1. all formulas come from an “end of the day cash-flow / geometric model”. The formulas change a bit if we are in a “beginning of the day and/or arithmetic”
setting. However the logic behind remains the same. This concludes the study of fixed income return decomposition as described in Fig. 1. The following section is dedicated to the most important component for fixed income attribution – the yield curve effect.
2.2. Yield curve construction
The yield curves are extracted from bonds available on the markets. Basically there are two main types of yield curves - the yield to maturity curve and the zero coupon yield curve. The
zero coupon yield curve is easier to model than the YTM curve. Therefore a consequent
amount of academic research has been done on the zero coupon yield curve where maybe the most well known model is the stochastic model of Vasicek.
2.2.1. Yield to maturity (YTM) curve
The yield to maturity (YTM) curve is computed with the yield to maturity, which is a security’s internal rate of return, or the anticipated yield of the bond if held to maturity. The YTM is the rate used when calculating the present value of all cash flows, so that they add up to the current market price. In other words, it is the compounded rate of return that investors receive if the bond is held to maturity and all cash flows are reinvested at the same rate of interest. If r is the current yield to maturity, then a bond price is given by:
(
r)
C(
r)
C(
r)
B(
r)
P C + − + + − + + + −n+ + −n = 1 1 ... 1 1 1 2where C is an annual coupon, n is the number of years to maturity, B is the par value of the bond, P is the current market price of the bond.
2.2.2. Zero coupon yield curve
The zero coupon yield curve is computed with zero coupon yield, which is the return it would show if all coupons were stripped out. Note that for securities that do not pay coupons, such as zero-coupon bonds or bills, there is only one repayment cash flow at maturity. In this case, the yield to maturity is identical to the zero coupon yield.
Thanks to its simplicity a lot of evaluation methods have been developed for the zero coupon
yield curve. The following list is not exhaustive:
• Bootstrapping • Cubic spline • Nelson Siegel • Cox Ingersoll Ross
• Cox Ingersoll Ross (inflation) • Vasicek
• Longstaff Schwartz • Maximum smoothness • Natural spline
The evaluation principles may be divided into three groups: bootstrapping methods, mathematical methods and term structure models.
For all models, except the bootstrapping method, the underlying functional form is estimated using ordinary least squares. Therefore, theoretical and observed prices on the bonds, which have provided data for the yield curve will usually deviate. On the other hand, in the bootstrapping method theoretical and observed prices on the bonds that have provided data for the yield curve are always equal due to the calculation principle.
In the bootstrapping method, the zero coupon yield curve is approximated using a continuous, piece-wise linear, function. The number of pieces are equal to the number of bonds (or money market, FRA/IRF and/or swap quotes) within the segment to provide data for the yield curve. The break points are defined by the time to maturity of the bonds. Therefore, if the segment includes 18 bonds, the yield curve is defined by 18 parameters, i.e. the slopes of the 18 linear pieces.
With the mathematical methods (cubic spline, natural spline, Nelson spline), estimation techniques are used to create yield curves. We invite the reader to refer to a statistical book for further details1.
Finally an alternative is to use term structure models (Cox Ingersoll Ross, Cox Ingersoll Ross
[inflation], Vasicek, Longstaff Schwartz). They are descriptions of changes to interest rates
over time. Some of these models are characterized by having closed form solutions to the price of zero bonds, which may be used in the yield curve estimation. Basically the parameters in the interest rate process are used as variables in the estimation. By varying these parameters, it is possible to find the process that fits the prices of the instruments used in the estimation best. With theses approaches the interest rate models have good asymptotic behavior (such as converging, as term to maturity is large) and sometimes the parameters may have a financial interpretation. However, the approach is rather pragmatic. The models are simply used to produce zero curves with ideal features and no further interpretation is attempted.
Vasicek and Longstaff-Schwartz are somewhat more complex than the rest of the models. In
these models the zero coupon yield curve is approximated by an equation that is derived as a solution to a stochastic differential equation. The change in interest rates is decomposed into a drift term and a stochastic term. The Longstaff-Schwartz model even includes two stochastic differential equations. In these models the underlying stochastic differential equations relate to so called factors, which are presumed to describe the pricing in the financial market.
As an example we present here a brief model specification of the Vasicek model.
Vasicek model2
The change in interest rates is modeled with a stochastic differential equation:
(
)
( )
0 r0 r dW dt r r a dr t = + − =σ
1See for example [11] Knott G.D., “Interpolating cubic splines”, Birkhäuser, 2000
2
See for a complete specification [15] Vasicek O., “An equilibrium characterisation of the term structure”, Journal of Financial Economics, 5: 177-187, 1977
where r is the interest rate. The other parameters are defined as follows: • a > 0 : speed of mean-reversion
• r > 0 : level of mean-reversion (the average value where the interest rate converges) •
σ
> 0 : absolute volatility• Wt : a standard Brownian motion at time t
Note that negative interest rates are possible with positive probability with these settings, which is a weakness of the model.
The solution of the stochastic differential equation given above is for 0≤s<t:
( )
( )(
( )
)
( )( )
∫
− − − − − + + = t s u t a s t a r s r e dW u e r t rσ
Given Fs, the filtration at time s, r(t) is normally distributed with mean and variance:
( )
[
]
( )(
( )
)
( )
[
]
(
a( )t s)
s s t a s e a t r Var r s r e r t r E − − − − − = − + = 2 2 1 2σ
F FThe zero coupon yield curve can then be modeled with r(t).
For a more detailed discussion concerning the characteristics of these term structure models, please refer to relevant financial literature covering these models3.
2.3. Yield curve decomposition
As we saw in the sections above, the yield curve can be decomposed in 3 main factors in order to explain the global shift – the factors being the parallel shift, twist, butterfly, plus a residual. Now we are going to review the four methods that people usually use in the industry to decompose the yield shift. These methods are generally applied on zero coupon yield curves.
2.3.1. Principal component analysis method
To explain all the possible distortions by using n maturity points to define the curve, n scenarios on each yield curve are required. PCA is a coordinate transformation that reduces the redundancy contained within the data by creating a new series of components in which the axes of the new coordinate systems point in the direction of decreasing variance. The resulting components are often more interpretable than the original images. The mean of the original data is the origin of the transformed system with the transformed axes of each component mutually orthogonal.
3
See for example [9] Hughston L., “Vasicek and Beyond: Approaches to Building and Applying Interest Rate Models”, Risk Books, 1997
The methodology is as follows:
1. Import the interest rate series. For example daily yield curves of selected maturities up to 30 years.
2. Compute the stationary series of differences.
3. Compute the eigenvalues and eigenvectors of the series. The eigenvectors represent the factor loadings, while the eigenvalues represent the significance of the factors. Eigenvalues are reported in descending order.
4. The relative weight of the eigenvalues gives the explanatory power of the various factors.
5. The matrix of components is created. By construction because of orthogonality, the components are mutually uncorrelated.
6. Every element of the original series can be reconstructed using the components and the loading matrix.
7. As reported in the interest rate literature4, the three factors represent different aspects of interest rate movements. Typically, the first factor is responsible for parallel shifts, the second one for twist changes and the third one for butterfly adjustments.
Below you see a PCA analysis of the US Government yield curve. The time period chosen goes from January 1997 to August 2005. We took monthly data. The Fig. 6. shows the different indexes that compose the US Government yield curve (1 month, 3 months, 6 months, 9 months, 1 year, 2 years, 3 years, 5 years, 10 years and 30 years).
USD Govt Yield Curve Indexes
0 1 2 3 4 5 6 7 8 01 /1 9 9 7 05 /1 9 9 7 09 /1 9 9 7 01 /1 9 9 8 05 /1 9 9 8 09 /1998 01 /1 9 9 9 05 /1999 09 /1 9 9 9 01 /2 0 0 0 05 /2 0 0 0 09 /2 0 0 0 01 /2 0 0 1 05 /2 0 0 1 09 /2001 01 /2 0 0 2 05 /2002 09 /2 0 0 2 01 /2 0 0 3 05 /2 0 0 3 09 /2 0 0 3 01 /2004 05 /2 0 0 4 09 /2004 01 /2 0 0 5 05 /2 0 0 5 % USD.INDEX.1M USD.INDEX.3M USD.INDEX.6M USD.INDEX.9M USD.INDEX.1Y USD.INDEX.2Y USD.INDEX.3Y USD.INDEX.5Y USD.INDEX.10Y USD.INDEX.30Y
Fig. 6. Indexes that compose the USD Government yield curve from Jan. 1997 to August 2005
This period is particularly interesting to analyze because it encloses a yield curve reversion (shorter rates higher than longer rates) from May 2000 to January 2001, then a steeping of the curve for 2001 and finally a flattening from May 2004.
4
By applying a standard PCA5 analysis, we obtain the following factor loadings and the cumulative explained variance:
USD.INDEX.1M USD.INDEX.3M USD.INDEX.6M USD.INDEX.9M USD.INDEX.1Y USD.INDEX.2Y USD.INDEX.3Y USD.INDEX.5Y USD.INDEX.10Y USD.INDEX.30Y 0.0 0.1 0.2 0.3 0.4 Factor 1 loadings USD.INDEX.1M USD.INDEX.3M USD.INDEX.6M USD.INDEX.9M USD.INDEX.1Y USD.INDEX.2Y USD.INDEX.3Y USD.INDEX.5Y USD.INDEX.10Y USD.INDEX.30Y -0.2 0.0 0.2 0.4 Factor 2 loadings USD.INDEX.1M USD.INDEX.3M USD.INDEX.6M USD.INDEX.9M USD.INDEX.1Y USD.INDEX.2Y USD.INDEX.3Y USD.INDEX.5Y USD.INDEX.10Y USD.INDEX.30Y -0.2 0.0 0.2 0.4 Factor 3 loadings
Fig. 7. Factor loadings of the first three principal components
0 5 10 15 2 0 2 5 F.1 F.2 F.3
Cumulative variance explained
Va ri a n c e s 0.971 0.995 0.999
Fig. 8. Cumulative variance explained by the first three factors
The factor loadings of the first principal component are as expected typically large and similar for all variables. An upward shift in the first principal component therefore induces a roughly parallel shift in all variables. For this reason the first principal component is called parallel
shift. With PCA method, the parallel component is not strictly speaking a translation of yield
returns but rather a level change impacting the short and long term slightly differently. Here for example the shorter rates are proportionally less affected than the longer rates. The first component explains here 97% of the variation during the data period in consideration.
5
In this example, an upward movement in the second principal component induces a change in slope of the yield curve, where short maturities move up but long maturities move down with an unchanged point at approximately 2 years. This second component is called twist and explains about 2.5% of the variation.
The third principal component influences the convexity of the yield curve. The factor weights are positive for the short rates, but decreasing and becoming negative for the medium term rates and then increasing and becoming positive again for the longer maturities. This is the
butterfly effect. This effect explains 0.4% of the variation.
The unexplained variation (less than 0.1%) is sometimes called reshape and considered as the residual of the PCA decomposition.
Note that the PCA method is a pure statistical decomposition and does not involve making strong assumptions on the magnitude and direction of yield changes occurring on a given period. The principal components are perfectly uncorrelated, making the performance numbers attached to each curve effect additive and clearly definable and explain most of the yield changes variance.
PCA does not require that functional form of the parallel, twist and butterfly are defined a priori. We generally observe that the first component identified as the parallel shift is not even over the term structure of the yield curve and shows more movement at the short end than the long end. However a method exists to force the first component to be strictly parallel by reprocessing the components to orthogonalize them.
Furthermore, we still have to make assumptions on the horizon length. There is a fine line between statistical data relevance and explanatory relevance. Statistically speaking, the longer the horizon the better, however the changes in yield curve shape from a past period may be less relevant than recent events. For a performance attribution, a time window of 3 months seems to be appropriate.
2.3.2. Empirical method
The empirical method decomposes returns of the portfolio in a very similar manner to the PCA method. The difference is that instead of using a statistical analysis to define the parallel, twist and butterfly components, changes in zero-coupon yield at the beginning and end periods are measured empirically. It consists in a decomposition of the yield curve changes into a combination of three basic components: parallel, twist and butterfly.
Unlike the PCA, the components are not statistically determined through a set of axis rotations in the spot rates, but rather by an empirical analysis of the yield curve. A method developed by Lehman brothers6 uses a piecewise function with 5 maturity points on the yield curve (2, 3, 5, 10, 30 years), the pivot point being at 5 years.
6
See: [5] Dynkin L., Hyman J. & Konstantinovsky V., “A return Attribution Model for Fixed Income Securities”, Handbook of Portfolio Management, 1998
The method used by Lehman brothers is:
1. First define the beginning and end of the period and compute the changes in zero-coupon yields between those two dates for the five reference maturities.
2. The three piecewise functions for parallel, twist and butterfly are defined:
• Parallel shift called p is intuitively set to equal the average yield changes over the five references maturities:
(
2 3 5 10 30)
5 1 y y y y y p= ∆ +∆ +∆ +∆ +∆• Twist returns are defined with a pivot point set at the year maturity point. The 5-year point is consequently not affected by the twist change. The twist magnitude is defined as:
2
30 y
y t=∆ −∆
t is applied such as the 30-year moves up by t/2 and the 2-year point moves down
by t/2 too.
• Butterfly returns is defined as:
(
2 30)
5 2 1 y y y b= ∆ +∆ −∆b is applied such as the 2-year and 30-year move up by b/3 and the 5-year point
moves down by 2b/3.
The empirical approach uses full revaluation. The results are very consistent across all securities and the whole portfolio. It is also interesting to notice that the shape effect (residual) is small, which means that the empirically defined curve distortions explain a very important part of the returns. The flexibility of the empirical method allows the portfolio manager to customize the attribution relatively to his bets. For example he can move the pivot point to better match his investment positions. Such refinements are an open discussion that can lead to more accurate measures of performance attribution.
One way to exploit at best this flexibility would be to calibrate the pivot point by using a PCA. Hence we will keep the flexibility of the empirical method whilst leveraging the PCA to describe the yield curve environment in a pertinent manner.
When the shift factors (parallel, twist, butterfly,...) are well defined, a factor loading is computed: ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + + ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ + ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ = ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⋅ ⋅ − − N N N N t t N t t R R F F I F F I YC YC YC YC 1 5 1 5 5 1 1 1 1 1 1 1 1 ...
Yield curve, t Yield Curve, t-1 Shift 1 Shift 5 Reshape (residual)
where N represents the number of maturity points, F the factors and I the factor loadings. With these factor loadings we can therefore quantify the yield curve shift explained by each factor. Two approaches are broadly used to define these loadings - the first is the sequential
OLS, the second one the standard OLS. a) Sequential OLS
In the Sequential Ordinary Least Square procedure, the loadings are calculated for each factor using simple algebra factor by factor. For example loading 1 is calculated by maximizing:
1 1 1 , 1 1 F F YC F I T tt T ⋅ ∆ = −
where ∆YCt,t-1 is the total shift in the yield curve between time t-1 and time t, I the factor
loading and F the factor.
Once the first loading is calculated the next loading (I2) is computed by maximizing:
(
)
2 2 1 1 1 , 2 2 F F F I YC F I Ttt T ⋅ ⋅ − ∆ = −This process continues until all loadings are calculated (sequential OLS), or the process stops at the desired number of factors and the remaining unexplained yield curve shift is the residual (reshape) change. The basic idea behind this method is that the first factor explains the most yield curve variance as possible and leaves a residual. Then the second factor only explains the residual as best as possible and gives another smaller residual and so on.
By analyzing factors loadings computed with a sequential OLS, we have to be careful because a factor effect can offset another one. For example a yield curve shift can be partially offset by a twist effect. However in practice, portfolio managers first think in term of duration (i.e. a parallel shift) and then with a twist for example. Therefore, even though this method seems to be mathematically less correct, it fits better the methodology of portfolio managers.
b) Standard OLS
Alternatively all loadings can be calculated directly using an Ordinary Least Square procedure. Here the vector of loadings is the maximized solution to the following problem:
[
1] [
1]
Reshape1
, F,...,F I ,...,I YC
YCtt = n ⋅ n T +∆
∆ −
where ∆YC is the total shift in the yield curve between time t-1 and time t, I the factor loading and F the factor.
Here all factors maximize the variance explained in one step. Mathematically this result is optimal but less interpretable than the sequential OLS.
2.3.3. Polynomial method
The polynomial method does not involve a multi-dimensional statistical analysis, but rather uses a more intuitive of extracting the zero, first and second order changes from polynomials that fit the yield curve at the beginning and end of period and model the yield changes as the difference between each polynomial of the same degree.
The polynomial approach relies on defining coefficients from fitting polynomials at the beginning and end of the period and using them to estimate magnitude changes in the portfolio yield.
The methodology:
1. Fit three polynomials respectively of degree zero, one and two to the yield curve at the beginning and end of period and extract respectively three sets of polynomials:
(
)
(
) (
)
(
) (
) (
End)
2 Begin 2 End 1 Begin 1 End 0 Begin 0 End 1 Begin 1 End 0 Begin 0 End 0 Begin 0 , , , , , , , , ,γ
γ
γ
γ
γ
γ
β
β
β
β
α
α
2. Compute for each yield curve component the magnitude of change due to each effect:
a. The parallel magnitude is defined as:
Begin 0 End 0
α
α
− = pb. The twist magnitude is defined as:
( )
t(
) (
)
t s = − + − Begin ⋅ 1 End 1 Begin 0 End 0β
β
β
β
c. The butterfly magnitude is defined as:
( )
(
) (
) (
Begin)
2 2 End 2 Begin 1 End 1 Begin 0 End 0 t t t b =γ
−γ
+γ
−γ
⋅ +γ
−γ
⋅where t is the time period on the yield curve term structure.
These parameters are proxies for the parallel (p), twist (s) and butterfly (b) effects at each maturity point of the zero-coupon yield curve.
In practice, relying on zero-degree polynomials to measure the parallel shift effects leads to a poor outcome and a minuscule attribution of the return to the parallel component. This method attributes returns mainly to the twist and more predominantly to the residual, emphasizing a redundancy or double-counting.
To explain these deficiencies of the polynomial decomposition we have to understand that every one of the polynomial fits is independent from the others and explains as much of the variance as possible in a non-orthogonal space. We understand then that without a correlation effect correction an over-estimation of the total return and a disproportionate residual is obtained.
The only applicable way to use this method is to measure the parallel shift with the empirical method and then apply the first order polynomial for the twist and the second-order polynomial for the butterfly. This sequential attribution will ensure that parallel shift explains most of the return.
2.3.4. Duration based method
The duration approach decomposes returns of the portfolio based on its yield, duration and convexity. The calculation can be applied at every level of the portfolio and is very intuitive and easy to implement.
Following the method detailed in Fong7, the duration method breaks down the yield curve movement using the duration and convexity measures. The duration component explains the parallel effect and the convexity component captures the twist.
The parallel shift can be simply calculated as follows: R
D⋅∆ − = ∆Parallel
where ∆R is the change in zero-coupon yield from the beginning of period to the end of period and D is the modified duration.
Similarly, the twist effect is measured as the second order term of a Taylor expansion:
7
See [8] Fong G., Yoo D., & Zelaya Z.M., “Global Performance Attribution, Perspectives on International Fixed Income Investing”, 1998
2 Twist 2 1 R Ci ⋅∆ = ∆
where C is the effective convexity.
The advantage of the duration approach is that it does not require the definition of terms and conditions of securities. Secondly portfolio managers and traders have the intuition for YTM, duration and convexity values, as these measures are widely accepted and used in fixed income analytics.
The key assumption is that the distributed cash flows of a fixed income instrument are approximated by a concentrated cash flow at the duration of the security. Consequently this method may not work well with a bond featuring big distributed cash flows scattered about the full term structure.
2.4. Linking return effects to multiple periods
Two broadly used methods are available to link returns over multiple time period: the geometric model and the arithmetic model.8
2.4.1. The arithmetic model
In arithmetic attributions the daily excess return contribution is simply obtained by addition of the different factors:
∑
−− t = ti t
t r
r 1, 1,
where i is used as indicator for the factors, i={Direct, Roll-down, YC shift 1,…, Currency} We can compound this return into multiple periods with:
(
)
∏ ∑
(
)
∏
+ − = + − = + i t t t t r r r 1 1, 1 1, 1Compounding will however result in cross products of the different return effects as it is not possible to swap between sums and products. These cross products create residuals difficult to attribute and interpret. Up to now, some methods have been developed to handle this problem. For a complete description please refer for example to David Spaulding’s book. We can shortly mention the most used methods:
• Arithmetic linking + a residual • Geometric linking + a residual
• Logarithmic linking + a residual distributed along each effect with a repartition key • Optimized approach (similar to the logarithmic one)
8
From the unpublished white papers we read to write this thesis, it appears that many attribution vendors typically use the geometric method to link the sub-period effects. Geometric method has the merit to be simple and easy to comprehend.
2.4.2. The geometric model
Geometric attribution is not as “linking challenged” as arithmetic.
In geometric attributions the daily return contribution is obtained by multiplication and the return can be decomposed as:
(
)
∏
− − = + + i t t t t r r 1, 1 1, 1where i is used as indicator for the factors, i={Direct, Roll-down, YC shift 1,…, Currency} Compounding into multiple periods is really straight forward:
(
)
∏∏
(
)
∏
+ − = + − = + i t t i t t r r r 1 1, 1 1, 1The big advantage of the geometric method is that it uses multiplication properties to link return effects in multiple periods. This calculation is consequently very easy and without any residuals. Other benefits of this method are the convertibility and proportionality properties:
a) Proportionality
One advantage to geometric excess return (relative to a benchmark) is that it takes into consideration the magnitude of the individual returns. That is, it provides some dimension to what is going on.
For example, let’s say our portfolio had a return of 11% versus a benchmark of 10%. Arithmetically, we would have an excess return of 1%. Likewise, if our portfolio was 25% versus 24% benchmark, we would show an excess return of 1%.
Geometrically, we get different numbers:
% 991 . 0 1 1 10 . 0 1 11 . 0 − = + + and 1 0.81% 1 24 . 0 1 25 . 0 − = + +
The differences occur because the 1% addition earned relative to 10% counts a whole lot more than it does relative to 24%. Make sense?
b) Convertibility
Another benefit of the geometric approach is that it reports the same excess return, regardless of the currency.
For example, let’s say on January 1, our portfolio starts out at $100. On that date, the conversion rate to Euro was 1.13305 (i.e. for $1, we get €1.13305). Our conversion to Pounds Sterling is 0.696676 (i.e. for $1 we get roughly 69 pence).
Twelve months go by and our US portfolio has gone up 10%, to $110. The benchmark (in US dollars) has gone up 8% during this time. The new FX rates are for Euro 1.18970 (i.e. for $1 we get €1.18970) and for Pounds Sterling 0.710610 (i.e. for 1$ we get £0.710610).
The following table shows the starting and ending values in the three currencies for thee portfolio and benchmark. We also show the returns and excess returns.
Portfolio Index Portfolio Index Portfolio Index Arithmetic Geometric
US $ 100.00 100.00 110.00 108.00 10.00% 8.00% 2.00% 1.85%
Euro 113.31 113.31 130.87 128.49 15.50% 13.40% 2.10% 1.85%
Pounds 69.67 69.67 78.17 76.75 12.20% 10.16% 2.04% 1.85%
Starting values Ending values Return Excess return
Fig. 9. Comparison of the arithmetic and geometric returns
For example, the geometric and arithmetic excess return for Euro is computed as:
% 10 . 2 1340 . 0 1550 . 0 % 85 . 1 1 1340 . 0 1 1550 . 0 1 ,€ ,€ = − = = − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = A G ER ER
As the table shows, the arithmetic excess return varies from country to country because of the exchange rate differences. The fact that the geometric excess return shows the same value regardless of the exchange is considered an advantage, especially for firms that market internationally.
With these words we are ending the theoretical parts of this master thesis. After having reviewed the theoretical framework underlying a fixed income performance attribution we would like to continue this thesis by describing shortly the different problems usually encountered in practice.
3.
I
SSUES IN PRACTICETo perform a fixed income performance attribution the entire portfolio has to be recalculated each day in order to extract the different return effects. Furthermore, a decomposition is typically done with subcategories like for example currency and maturity. Consequently index benchmarks provided by the market are not sufficient, internal benchmarks on security level have to be constructed as well to match each subcategory. These internal benchmarks have to replicate exactly the index benchmark to which they depend.
We then understand that the IT system, price sources, price quality, cash out- and inflows must be handled in a very rigorous way.
A good performance attribution without an excellent performance measurement is nothing! Database maintenance is therefore the first obligatory step prior to any performance
attribution. And we would highly recommend people not to underestimate this data quality issue! The next paragraph gives a quick view of the principal issues that will usually arise when implementing a fixed income performance analysis.
3.1. Data quality
Performance attribution requires a very high data quality, which is certainly the most sensible part of this kind of analysis because it is costly and time consuming to monitor and maintain a high-quality database. To give a time indication, it is not unusual that firms invest more of a year for cleaning the data history. For example daily prices coming in the system must be closely monitored. Here you find a non-exhaustive list of inputs that require a close monitoring:
• The different price sources that feed the FIPA analysis • The booking of non-standard corporate actions
• The booking of management fees • The treatment of withholding taxes • The reinvestment of coupon
• The dynamic changes in ratings
• The dynamic changes of maturity buckets (e.g. for multi-step bonds or callable bonds) • The dynamic changes of business classes
You find next a more complete description of some issues one will get for sure by implementing a FIPA analysis:
3.1.1. Assets without price or with an incorrect price
Generally databases get prices from different sources like Morgan Stanley, Merrill Lynch, Pictet, Lehman, JP Morgan,… Priority lists are set up to prioritize the prices. A problem comes when delivered prices with the highest priority are false. And this will happen for sure. To remedy this problem a daily process with the back office should be put in place to correct the wrong prices.
Another source of incorrect prices can be caused by banking holidays abroad and not in the home country of the portfolio. This causes an important number of assets to have no price although it was a working day in the home country. Here again a close monitoring has to be put in place.
3.1.2. Corporate actions
Other minor price errors can be caused by special corporate actions (principally for stocks) like splits, new issue rights, dividends in stocks, bond convertible issues. A timing error of the corporate action booking is often responsible for the error. In fact, in most cases the database gets 3 dates: the ex-date, the recording date and the payable date which are not always standardized and may cause errors.
3.2. Cash flows and management fees
Another issue is the booking of the different cash in- and outflows of the portfolio, which must be performance neutral. Here are listed the main sources of cash flows:
3.2.1. Management fees
Whether you want to compute a performance attribution on a gross or net basis, management fees have to be taken into account or not. If you choose a net performance, management fees have to be added and your relative performance to the benchmark will be a bit less. This reflects the point of view of your client. If, in the contrary, you choose a gross attribution without management fees, your portfolio will be directly comparable with the benchmark portfolio, which is what a portfolio manager wants. But by removing management fees, on a cumulative basis, a linear trend will appear between the portfolio value calculated for your performance attribution and the real accounting value.
3.2.2. Accounting of reclaimable withholding taxes
Returns should be calculated net of non-reclaimable withholding taxes. Reclaimable withholding taxes should consequently be accrued. The main problem here is that taxes policy may differ with the country where the bond was issued, but also with the owner of the bond.
3.2.3. Reinvestment of coupons
The methodology for the reinvestment of coupon concern principally benchmark portfolios. In fact if one decides to create benchmark portfolios on a security level, portfolios have to replicate exactly their benchmark index. Unfortunately different practices are used by the main benchmark index providers, for example,
• Lehman Brothers records coupons on an account without interest rate and reinvests them every month.
• Merrill Lynch records coupons on an account with interest rate and reinvest every month.
• JP Morgan and Morgan Stanley aggregate the coupon with the daily returns of the according security.
3.3. Gross / Net basis
What is treated as a cash flow should be performance neutral. Programs can generally calculate performance with or without taxes and fees, which means gross or net.
SPPS, which stands for Swiss Performance Presentation Standards is the Swiss version of the international recognized Global Investment Performance Standards (GIPS). The aim of these standards is to provide fair performance presentations for clients, which allows an objective
comparison between investment fund companies. When a company is a member of SPPS, the company is asked to follow these SPPS standards.
The differences between gross and net performance is summarized in the Fig. 7.
Federal Direct Tax & anticipatory Tax
Federal Stamp Courtage Fee
Mgmt. Fee (inkl. MwSt) Depot Fee
Net (SPPS)
Index Gross Div. reinvested
Index Net Div. reinvested Reclaimable
Gross (SPPS)
Fig. 10. Gross / Net performance
3.4. Replicating the benchmark in general
The main problem with security-level benchmark data supplied by most vendors is that rates of return are not available on security-level. To calculate returns of individual securities, one needs to know their prices and all their cash flows. This in turn requires knowledge of the pricing formula used, the ex-coupon conventions, treatment of cash coming from a coupon payment and non-standard features such as non-uniform first coupon payment, multi-coupons, step-up coupons, callable and so on.
Perhaps surprisingly, the main problem in replication of fixed income benchmark returns lies in the calculation of coupon timing and amounts. The timing of coupons depends on the bond issuer, the ex-period convention used and the frequency of coupon. In addition, the amount of coupon paid can depend on whether the bond has a non-standard first coupon period, in which case the first coupon may be more or less than the standard payment. The same considerations may apply to other coupon payments.
In principle, these coupons may be recalculated from first principles if we know the inception date, first and last coupon dates, maturity date, annual coupon payment and coupon frequency and ex-date convention for each bond. In practice, this imposes a substantial burden on the index calculator, who has to obtain and verify large amounts of bond data. In addition, the ex-day conventions for many bonds are obscure. There is no easy answer to these problems and the person who wants to implement a FIPA analysis is strongly advised to consult and expert in this field.
In our case, it took us more than one year to replicate almost perfectly every benchmark used in the company. But we will spare the reader the explanations of this tedious work...
Finally, after having solved all the technical issues presented in this section, we are finally ready to implement a FIPA analysis of quality on a real portfolio.
4.
C
HARACTERISTICS OF THE PORTFOLIO ANALYZED4.1. Constraints on the portfolio
The portfolio we are going to analyze is a multi-currency bond portfolio with reporting currency in CHF. Its size exceeds 1 billon CHF. The portfolio has a customized credit benchmark with fixed weights which are rebalanced every month. The constituents are indices from Morgan Stanley. The portfolio invests in investment grades credits (from AAA to BBB) and government bonds. There are no derivatives in the portfolio as well as in the benchmark.
4.2. Style of the portfolio manager
The portfolio is constructed to take proactive bets on credit spread while being neutral on the interest rate and currency risk. The portfolio is then claimed to generate returns and alpha via the credit analysis ability of its manager.
Fig. 11. The red color represents the interest rate effect, the blue represents the credit spread and the green the currency effect. The portfolio is mainly active in credit spread.
Fig. 12. Graphs representing the style of the portfolio manager. Note the active bets on credit (rating). Currency Distribution Relative to
Benchmark -0.2% -0.1% -0.1% 0.0% 0.1% 0.1% 0.2% CAD AUD USD EUR GBP SEK DKK
Duration Distribution Relative to Benchmark -0.06 -0.04 -0.02 0.00 0.02 0.04 CAD AUD USD EUR GBP SEK DKK
Rating Distribution Relative to Benchmark
-20.0% -10.0% 0.0% 10.0% 20.0% AA A AA A BB B NR
AAA AA A BBB NR 0-1 1-3 3-5 5-7 7-10 10+ 16.03% 3.36% 0.82% 0.00% 0.00% 9.55% 2.59% 1.11%1.95% 0.09% 8.48% 2.15% 1.67% 1.84% 0.00% 12.35% 1.72% 2.53% 2.10% 0.00% 8.30% 1.12%2.16% 1.13% 0.00% 2.17% 0.50% 1.37% 0.80% 0.00% 0.00% 2.00% 4.00% 6.00% 8.00% 10.00% 12.00% 14.00% 16.00% 18.00%
Fig. 13. Rating buckets per maturity in the portfolio
But is this statement really true? Is the portfolio manager’s credit analysis ability really the main driver of the over-performance? Given that the exposures toward interest rate and currency risk are neutral in any point in time, over-performances must come from the credit part. An interesting question to analyze is: does the performance come from an overweight in A and BBB (asset allocation and taking more risk) or is it due to selection skills?
4.3. Set up of the fixed income performance analysis
To answer this question we propose here to run a FIPA analysis in order to decompose the return of the portfolio. We hope that this decomposition will put light on the main performance drivers of the portfolio and help us understand better where the performance come from.
The followings settings have been used to analyze the portfolio:
4.3.1. The yield curve
Yield curves play a major role in fixed income attribution analysis, because movements in yield curves have a large effect on the pricing and hence the return of fixed income assets. A large and complex literature exists on yield curves, reflecting the central part they play in fixed income market pricing. Many hundreds of research papers and several textbooks have been written on their construction and modeling and yield curve experts continue to devise ever-improved software systems incorporating bootstrap techniques for constructing curves, the dynamics of stochastic interest rate modeling and sophisticated techniques to match bill strips to bond curves.
So, the construction of yield curves and forecasting how they behave, are deep and complex areas. Furthermore for a good FIPA analysis some constraints have to be put on the yield curve behavior:
• The curve should be smooth, with no glitches or discontinuities. Otherwise, arbitrage opportunities will arise.
• Curves of different credit ratings should not cross or intersect. Otherwise, identical bonds with different repayment risks may show the same yield.
• The curve should intersect the cash rate at zero maturity
First we tried to generate yield curves from different term structure models9 with all bonds available in our universe (sometimes more than 300 bonds per yield curve), but the outliers produced biases in the computation. If you go this way, you will be inevitably forced to reduce the number of bonds (maybe between 10 to 30 bonds) to remove the outliers and make the model works. But then occurs the sensible question of which bond should be selected. Even if you do a meaningful selection, it is likely that your modeled curves will not always satisfy all the constraints mentioned above. This yield curve generation is therefore non-trivial.
However, note that what we are doing here is a Fixed Income Performance Attribution (FIPA) and not a bond pricing. The differences are:
• In Pricing the absolute value of the curve plays a central role, while in FIPA only the daily relative moves are important.
• In Pricing you try to forecast the curve, while in FIPA you only need historical curves • In Pricing the degree of precision has to be much higher than in FIPA.
• In Pricing the curves are not always computable in illiquid markets. In FIPA if you use credit curves, you must be able to generate a curve per day, per currency and per rating even if the market is not very liquid. You cannot afford to miss a curve for one day because the attribution is done on a daily basis.
To respect the constraints and to come up with reasonable curves each day for each currency and each rating, we finally used a hybrid method to construct the yield curve. We imported fair market indices that play the role of bonds. Then we computed the yield curve by cubic spline method. This method is simple, easy to implement and we think that the curve reflects at best the maturity term structure of the market.
4.3.2. The YC decomposition factors
The yield curve decomposition was done with the empirical method (see 2.2.2.). A PCA has been used to calibrate the factors. Under the period into consideration, as the spread are pretty tied, the butterfly effect is statistically non-significant. So we decided to remove this effect. We are left with:
• A parallel shift (more than 95% of the global shift)
• A reshape that includes the twist effect (4%), the butterfly effect (0.5%) and a residual
4.3.3. Linking method
A geometric approach has been chosen for diverse reasons. First, the convertibility property (see 2.4.2. b) allows a consistency in returns through different currencies. For the portfolio
9
analyzed, the reporting currency is in CHF but the portfolio is invested in USD, EUR, GBP, CAD, AUD, SEK and DKK. Second, by reading different white papers on this topic, it appears that geometric linking becomes the standard in the industry.
5.
T
HE RESULTS5.1. The FIPA attribution for the global portfolio
To compute time weighted returns (TWR), transaction costs have been taken into account but not management fees and returns generated by cash accounts. In fact only the “pure” bond bucket of the portfolio is analyzed, which allows a direct and clean comparison with the benchmark. The bond bucket is consolidated in CHF. Returns are given in basis points.
Date TWR RC (Bps.) Direct return (Bps.) Roll down (Bps.) YC shift 1 (Bps.) YC reshape (Bps.) Sector spread return (Bps.) YC spread return (Bps.) Fixed income timing (Bps.) Fixed income currency return (Bps.) 01/2005 336.4 39.5 -4.8 10.9 17.5 13.4 12.3 0.1 245.0 02/2005 -203.7 40.2 -7.4 -102.9 -1.2 7.6 22.9 -0.1 -162.8 03/2005 175.7 45.6 -6.8 -19.0 1.1 -16.8 -17.1 0.1 188.9 04/2005 119.2 40.6 -5.9 108.1 21.2 -32.3 29.1 -0.3 -41.1 05/2005 345.6 40.1 -5.3 76.7 12.4 1.0 -14.1 -0.3 232.2 06/2005 295.9 43.7 -8.3 46.6 15.5 -5.9 7.3 1.7 193.1 07/2005 -53.3 37.7 -5.7 -130.0 -7.9 30.0 -7.1 0.0 30.6 08/2005 6.2 44.5 -6.5 83.6 19.6 -1.3 -2.1 0.0 -130.2 Attribution
Fig. 14. Fixed income attribution for the portfolio in bps per month
Date TWR benchmark RC (Bps.) Direct return benchmark (Bps.) Roll down benchmark (Bps.) YC shift 1 benchmark (Bps.) YC reshape benchmark (Bps.) Sector spread return benchmark (Bps.) YC spread return benchmark (Bps.) Fixed income timing benchmark (Bps.) Fixed income currency return benchmark (Bps.) 01/2005 325.4 36.3 -2.1 10.9 14.7 11.9 5.6 0.0 246.1 02/2005 -207.4 36.6 -3.8 -102.2 -3.2 9.7 17.7 0.0 -162.4 03/2005 171.8 42.2 -3.2 -18.6 0.4 -19.0 -18.1 0.0 188.6 04/2005 112.3 37.3 -2.5 107.9 22.7 -31.9 20.4 0.3 -41.6 05/2005 343.3 38.0 -2.6 74.5 10.6 5.4 -14.9 0.0 229.7 06/2005 289.1 40.8 -6.1 46.8 13.6 -0.7 1.6 0.4 190.5 07/2005 -51.6 35.2 -5.6 -125.6 -10.0 22.8 2.7 -1.9 31.5 08/2005 0.4 41.5 -4.2 81.1 17.1 -1.7 -2.8 0.0 -129.3 Attribution
Fig. 15. Fixed income attribution for the benchmark in bps per month
Date Excess return RC (Bps.) Direct return excess (Bps.) Roll down excess (Bps.) YC shift 1 excess (Bps.) YC reshape excess (Bps.) Sector spread return excess (Bps.) YC spread return excess (Bps.) Fixed income timing excess (Bps.) Fixed income currency return excess (Bps.) 01/2005 11.0 3.1 -2.7 0.0 2.9 1.4 6.7 0.1 -1.0 02/2005 3.7 3.6 -3.6 -0.7 2.1 -2.1 5.2 -0.1 -0.5 03/2005 3.9 3.4 -3.6 -0.4 0.8 2.2 1.0 0.2 0.3 04/2005 6.9 3.3 -3.4 0.2 -1.5 -0.4 8.7 -0.7 0.5 05/2005 2.3 2.1 -2.6 2.2 1.8 -4.4 0.9 -0.2 2.5 06/2005 6.8 2.9 -2.1 -0.2 1.9 -5.2 5.6 1.2 2.5 07/2005 -1.7 2.5 -0.1 -4.4 2.1 7.1 -9.8 1.9 -0.9 08/2005 5.8 3.0 -2.4 2.5 2.4 0.4 0.7 0.0 -0.9 Attribution