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FIXED-INCOME SECURITIES
Chapter 7
Passive Fixed-Income
Portfolio Management
Outline
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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:
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–
–
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
N
N
i , j 1
i 1
Min Var RP  RB    wi w j ij  2 wi iB   B2
w1 ,..., wN
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
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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


2
w



 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
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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 – 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 – 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

pt 
T
t


t 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