Passive Fixed-Income Portfolio Management

1

FIXED-INCOME SECURITIES

• Chapter 7
• Passive Fixed-Income Portfolio Management

2
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

3
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

4
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

5
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

6
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 (lt5 years, gt 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

7
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

8
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

9
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

10
Sample Covariance Estimate

• First compute the correlation matrix

• 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

11
Sample Covariance Estimate

• Minimize portfolio tracking error

• 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

12
Equally-Weighted Portfolio
13
Optimal Portfolio No Short Sales
14
Optimal Portfolio Short Sales Allowed
15
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))

16
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