The Mathematics of Financial Modelingand Investment Management

21-Bond Portfolio Man Page 654 Wednesday, February 4, 2004 1:12 PM

654 The Mathematics of Financial Modeling and Investment Management

Determinants of Tracking Error

Using statistical techniques,^7 given the risk factors associated with a

benchmark index, forward-looking tracking error can be estimated for a

portfolio based on historical return data. The tracking error occurs

because the portfolio constructed deviates from the exposures for the

benchmark index. The tracking error for a portfolio relative to a bench-

mark index can be decomposed as follows:

I. Tracking error due to systematic risk factors:

A. Tracking error due to term structure risk factor

B. Tracking error due to nonterm structure risk factors

1. Tracking error due to sector


2. Tracking error due to quality


3. Tracking error due to optionality


4. Tracking error due to coupon


5. Tracking error due to MBS sector


6. Tracking error due to MBS volatility


7. Tracking error due to MBS prepayment
   II. Tracking error due to nonsystematic risk factors
   A. Tracking error due to issuer-specific risk
   B. Tracking error due to issue-specific risk

A manager provided with information about (forwarding-looking)

tracking error for the current portfolio can quickly assess if (1) the risk

exposure for the portfolio is one that is acceptable and (2) if the partic-

ular exposures are the ones being sought.

Illustration of the Multifactor Risk Model

We will now illustrate how a multifactor risk model is used to quantify

the risk profile of a portfolio relative to a benchmark and then explain

how optimization can be used to construct a portfolio. We will use the

Lehman Brothers multifactor model in the illustration. The bond market

index used as benchmark is the Lehman Brothers U.S. Aggregate Index.^8

(^7) Lev Dynkin of Lehman Brothers has described the statistical technique to the authors
as follows. The risk model uses decomposition of individual bond returns into carry,
yield curve, and spread components. The spread component is regressed on a certain
set of systematic (or common to all bonds in a peer group) risk factors using a prespec-
ified set of sensitivities. Residuals of this regression are used to estimate security-specific
risk. Factor realizations collected over many months form the covariance matrix of sys-
tematic risk factors. The current mismatch in risk sensitivities between the portfolio
and the benchmark is multiplied by this matrix to get the systematic tracking error.
(^8) The illustration in this section draws from Dynkin, Hyman, and Wu, “Multi-Factor
Risk Factors and Their Applications.”