FIXED-INCOME SECURITIES
Chapter 7
Passive Fixed-Income Portfolio Management
Outline • • • • • • • • •
Passive Strategies Passive Funds Straightforward Replication Stratified Sampling Tracking Error Minimization Sample Covariance Estimate Exponentially-Weighted Covariance Estimate Factor-Based Covariance Estimate Out-Of-Sample Performance
Passive Strategies • A natural outcome of a belief in efficient markets is to employ some type of passive strategy • Passive strategies do not seek to outperform the market but simply to do as well as the market – The emphasis is on minimizing transaction costs – Any expected benefits from active trading or analysis are likely to be less than the costs
• Passive investors act as if the market is efficient – Take the consensus estimates of return and risk – Accepting current market price as the best estimate of a security's value
• If the market is totally efficient, no active strategy should be able to beat the market on a risk-adjusted basis
Passive Funds • In 1986, Vanguard started the first fixed-income passive fund: – Total Bond Market Index (VBMFX) – SEI Funds also started a bond index fund that year – In 1994, Vanguard created the first series of bond index funds of varying maturities, short, intermediate, and long – Today there are a large number of bond index funds
• Bond index fund managers now handle an estimated $21 billion • Bond index funds occupy a fairly small niche – Only 3% of all bond fund assets are in bond index funds – These assets are held disproportionately by institutional investors, who keep about 25% of their bond fund assets in bond index funds
Straightforward Replication • The most straightforward replication technique involves – Duplicating the target index precisely – Holding all its securities in their exact proportions
• Once replication is achieved, trading is necessary only when the make-up of the index changes • While this approach is often preferred for equities, it is neither practical nor necessary with bonds • For example, Lehman Brothers Aggregate Bond Index is a collection of 5,545 bonds (as of 12/31/99) – Many of the bonds in the indices are thinly traded – The composition of the index changes regularly, as the bonds mature
Stratified Sampling • One natural alternative is stratified sampling • To replicate an index, one has to represent its every important component with a few securities – First, divide the index into cells, each cell representing a different characteristic – Then buy one or several bonds to match those characteristics and represent the entire cell
• Examples of identifying characteristics are: – – – –
Duration (<5 years, > 5 years) Market sectors (Treasury, corporate, mortgage-backed) Credit rating (AAA, AA, A, BBB) Number of cells in this example: 2 x 3 x 4 = 24
Tracking Error Minimization • Risk models allow us to replicate indices by creating minimum tracking error portfolios • These models rely on historical volatilities and correlations between returns on different asset classes or different risk factors in the market • Typically, investment managers expect the correlation between the fund and the index to be at least 0.95 • The technique involves two separate steps: – Estimation of the bond return covariance matrix – Use of that covariance matrix for tracking error optimization
Optimization Procedure • The problem is to – Form a portfolio with N individual bonds (or derivatives) – Choose portfolio weights so as to replicate as closely as possible a bond index return
N
RP = ∑ wi Ri i =1
Min Var ( RP − RB ) =
w1 ,..., wN
N
N
i , j =1
i =1
2 w w σ − 2 w σ + σ ∑ i j ij ∑ i iB B
Bond Return Covariance Matrix Estimation • The key ingredient in this problem is the bond return variance-covariance matrix • Estimation problem: number of different inputs to estimate is N(N-1)/2 • Various methods can be used to improve the estimates of the variance-covariance matrix • Example: replicate JP Morgan T-Bond index using – – – – – – – –
6.25%, 31-Jan-2002 4.75%, 15-Feb-2004 5.875%, 15-Nov-2005 6.125%, 15-Aug-2007 6.5%, 15-Feb-2010 5%, 15-Aug-2011 6.25%, 15-May-2030 5.375%, 15-Feb-2031
Sample Covariance Estimate • First compute the correlation matrix Benchmark Benchmark Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8
1 0.035340992 0.570480252 0.762486545 0.80490507 0.873289816 0.987947611 0.932169847 0.912529511
Bond 1
Bond 2
Bond 3
1 0.037162337 0.03232004 0.030394112 0.023278035 0.03032363 0.023653633 0.022592811
1 0.539232667 0.928891982 0.865561277 0.573606745 0.439369201 0.586354587
1 0.675469702 0.698241657 0.771601295 0.782454722 0.608825075
Bond 4
Bond 5
Bond 6
Bond 7
Bond 8
1 0.982726525 1 0.810561264 0.880679105 1 0.684073945 0.774954075 0.89795721 1 0.788072503 0.858459264 0.866043218 0.932159141
– Note that medium maturity bonds exhibit highest correlation with the index – Not surprising: index average Maccaulay duration over the period is 6.73
• The simplest estimate is given by the sample covariance estimate T
(
)(
)
' 1 S= Rt − R Rt − R ∑ T − 1 t =1
1
Sample Covariance Estimate • Minimize portfolio tracking error
Min TE = Var ( RP − RB ) =
w1 ,..., w8
Sample Covariance Matrix with short-sales contraints without short-sales contraints
Bond 1 12.93% 1.99%
Bond 2 14.19% 39.92%
8
8
i , j =1
i =1
2 w w σ − 2 w σ + σ ∑ i j ij ∑ i iB B
Bond 3 0.00% -1.43%
Bond 4 0.00% 20.93%
Bond 5 0.00% -62.38%
Bond 6 62.41% 83.59%
Bond 7 8.33% 3.44%
• Compute the tracking error as a measure of quality of replication – – – –
Arbitrary equally-weighted portfolio of the 8 bonds: 0.14% daily Replicating portfolio deviates on average by 0.14% from the target Optimal replication in the presence short-sales constraints: 0.07% Optimal replication in the absence of short-sales constraints: 0.04%
Bond 8 2.13% 13.93%
Equally-Weighted Portfolio 110 108 106 104
Replicating Portfolio
102
Benchmark
100 98 96 3-Jan-02
3-Dec-01
3-Nov-01
3-Oct-01
3-Sep-01
3-Aug-01
94
Optimal Portfolio â&#x20AC;&#x201C; No Short Sales 110 108 106 104
Replicating Portfolio
102
Benchmark
100 98 96 3-Jan-02
3-Dec-01
3-Nov-01
3-Oct-01
3-Sep-01
3-Aug-01
94
Optimal Portfolio â&#x20AC;&#x201C; Short Sales Allowed 110 108 106 104
Replicating Portfolio
102
Benchmark
100 98 96 3-Jan-02
3-Dec-01
3-Nov-01
3-Oct-01
3-Sep-01
3-Aug-01
94
Exponentially-Weighted Covariance Estimate • One key problem is non stationarity of bond returns – More data is better because reduces estimation risk – Less data is better because uses more recent information
• A possible improvement is to allow for declining weights assigned to observations as they go further back in time (see Litterman and Winkelmann (1998)) T
(
)(
S = ∑ pt Rt − R Rt − R t =1
T −t +1 λ pt =
T
t λ ∑ t =1
)
'
Factor-Based Covariance Estimate • An alternative is reduce the number of estimates through the use of a factor model • We know that the term structure dynamics is driven by a very limited set of factors (2 or 3) Rit = µi +βi1F1t + ... + βikFkt + eit – Fjt is factor j at date t (j = 1,…,k) – eit is the asset specific return – βij measures the sensitivity of Ri to factor j, (j = 1,..., k)
• Use estimates of βij and σFj2 to obtain estimates of Cov(Ri,Rj) (2 factor case): only need kxN beta estimates + N + K volatility terms – Variance σi2 = βi12 σF12 + βi22 σF22 + 2βi1βi2Cov(F1,F2) + σei2 – Covariance σij = βi1βj1σF12 + βi2βj2σF22 + (βi1βj2 + βi2βj1)Cov(F1,F2 ) – These expressions simplify further when Cov(F1,F2 )=0 and σFi =1
Back to the Example • Using the same data is in the previous example, we regress the return on each of the 8 bonds and the benchmark on two factors – The first factor is the change in 3 month interest rates, regarded as a proxy for changes in the level of the term structure – The second factor is change in the spread between the 30 years rate and the 3 month rate, regarded as a proxy for changes in the slope of the term structure
• Regression results for the benchmark – R-squared > 90%
Intercept Factor 1 Factor 2
Coefficients Standard Error t Stat P-value 6.14942E-05 0.000180236 0.3411861 0.7335755 -27.0865211 1.264843741 -21.414915 2.737E-42 -22.2656083 0.996043688 -22.354048 4.799E-44
Back to the Example – Con’t • The following table displays the betas for each bond with respect to both risk factors Beta 1 Beta 2 Benchmark Bond 1 Bond 2 Bond 3 Bond 4 Bond 5 Bond 6 Bond 7 Bond 8
-27.0865211 -2.81376114 -11.1461975 -19.9855764 -23.1662469 -28.7309262 -32.4878358 -50.8107969 -52.2945817
-22.26560827 -1.084816042 -5.759743001 -13.46348007 -15.95768661 -22.18051504 -26.25424263 -48.54826524 -50.57663054
• One may want to impose further constraints in the optimization procedure 8 ∑ wi β i1 = −27.0865211 i =1 8 w β = −22.26560827 i i2 ∑ i =1
Out-Of-Sample Performance • • •
• • •
The relative performance of different estimators of the covariance matrix can be assessed on an out-of-sample basis Use the first 2/3 of the data for calibration of the competing estimates of the covariance matrix On the basis of those estimates, compute the best replicating portfolio in the presence and in the absence of short-sales constraints Record the performance of these optimal portfolios on the backtesting period, i.e., the last 1/3 of the original data set Compute the standard deviation of the excess return of these portfolio over the return on the benchmark This quantity is known as out-of-sample tracking error