550.447
Quantitative Portfolio Theory
& Performance Analysis
1.1
Fixed Income PFs
 Quantitative methods for managing bond portfolios are a more recent development than for managing equities
 Managing bond portfolios quantitatively is much more prevalent than in equities and growing faster
 Growth is both in style and scope
 In large part because of innovations in the derivatives markets
 Also, because of advances in modeling the key sources of risk faced by the bond PF manager
 We look at the particular characteristics of bond investments and the impact on the modeling approaches for performance analysis
1.3
Assignment




For April 22 (This Week)
 Read: A&L, Chapter 8
 Read: E&G Chapter 22
 Problems E&G: 8.5; E&G: 16.3, 16.4 (Due April 22)
 Project Definition & Discussion (Continued)
For April 29 (Next Week)
 Read: A&L Chapter 7
 Read: E&G Chapter 25&27
Last Day of Class: Wednesday, May 1 st
Final Project Presentation: Monday, May 13 th ;
9:00am – Noon (Whitehead 203)
1.2
Fixed Income PFs

Bond Portfolios – Sources of Risk/Return
 Interest Rates – the Term Structure
 Coupon income
 Gains or losses from early liquidation – not held to maturity
 Prevailing levels of (comparable) rates
 Rolling Down the Yield Curve
 Credit/Sector
 Quality of individual issuer & the likelihood of the payment of interest and the return of principal
 Correlation of Default
 Sector vs. Individual issues
 Structure
 Liquidity 1.4
1

Fixed Income PFs

Bond Portfolios – Sources of Risk/Return
 Interest Rates
 Term Structure Concepts and Models for interest rates
 Main reason for the advance of quantitative techniques for bond portfolio management
 Concepts – YTM/IRR, spot rates, discount function, term structure, term premium, forward rates, expected future spot rates, duration, convexity
 Models – static, dynamic, stochastic; spot rate & forward rates
1.5
Fixed Income PFs

Managing a Bond Portfolio
 Defining the Risks & Sources of Return
 Default/Credit Risk
 The Risk of Default will drive the price the market will set for the potential that interest and principal may not be paid in full
 This price is expressed as a yield or price spread over otherwise equivalent, risk-free issues
 The spread, s , may be approximated through the recovery,
R , in the event of default and a measure of the likelihood of default (the default intensity per year, λ ) s = (1-R) x λ
 Rating agencies provide discrete measures to express credit risk - investment grade (AAA, AA, A, BBB; Aaa, Aa, A, Baa); with sub-investment grades (BB, B, CCC, CC, C)
1.7
Fixed Income PFs

Managing a Bond Portfolio
 Defining the Risks & Sources of Return
 Market Risk
 Interest Rate (IR) Risk – duration, convexity, etc.
 Prepayment (see structure risk)
 Liquidity Risk
 Modeling techniques for the impact of IR risk on the PF
 Factor Models for explaining YC shifts
 PCA analysis – parallel, steepening/flattening, barbell
 Key rate sensitivities (correlations) for DVBP analysis (JP
Morgan)
 Forecasting Returns w/ horizon – total return & (stochastic) horizon/scenario analysis 1.6
Fixed Income PFs

Managing a Bond Portfolio
 Defining the Risks & Sources of Return
 Default/Credit Risk
 Methods of Approaching Default Risk
 Firm-based (or structural) models – where the likelihood of default is defined by the relationship between the value of a firms assets to relevant threshold
 Reduced Form models – where the likelihood of default is drawn from historical or implied (from the market) measures of a population of “like companies”
 Systematic credit risk or the nature of contagion – credit risk correlation (industry, geography, quality, etc.)
1.8
2

Fixed Income PFs

Managing a Bond Portfolio
 Defining the Risks & Sources of Return
 Structure Risk
 Can be an interplay between credit risk and market risk
 Prepayment permission to a borrower – usually paid for at time of issue, but the time of the event is uncertain so the charge may or may not have been appropriate
 Senior/subordinate and other forms of credit enhancement
 Modeled through cash flow models and Monte Carlo scenario analysis – provides scenario/horizon return measures
 Volatility effects call structure/value
1.9
Fixed Income PFs

Managing a Bond Portfolio
 Bond Investing Strategies
 The construction of a bond portfolio relies more on the choice of, and allocation to, (a) category(ies) of bonds presenting the required characteristics (of duration, credit, maturity, etc.) than on specific choice of individual bonds
 Strategies
 Active – prudent assumption of risk to achieve additional return
 Associated with a rationale (forecast) for taking on the risk
 Passive – replicating a benchmark
 Passive with an active overlay
1.11
Fixed Income PFs

Managing a Bond Portfolio
 Analysis Framework
 Portfolio Objective and Constraints
 Asset Allocation
 Dividing the PF into different classes: sectors, duration, activepassive, diversification, liquidity
 Benchmark
 Choosing a “Standard” Benchmark or Defining the Bespoke
Benchmark
 Once the benchmark is specified, return objectives and risk constraints may be quantified in terms of deviations from the benchmark
1.10
Fixed Income PFs
 Management of Bond Portfolios
 Major Source of Risk & Return in Bond PFs
 Foremost, the yield curve
 To a large extent, dominates any thoughts of MPT
 Never-the-less, to manage the bond PF one needs estimates of future (next period) returns and their attendant risk
 Then manage the PF to take the risk or insulate from the risk
 Bond returns have two sources
 Interest Income and P/L from price changes
 Price changes from the passage of time & the yield curve
 Traditional measures of exposure to the yield curve include
 Duration & Convexity
 Convexity – positive & negative 1.12
3

Fixed Income PFs
 Management of Bond Portfolios
1.13
Fixed Income PFs
 Management of Bond Portfolios
 Two techniques for insulating an indexed bond PF from shifts in the yield curve – passive management
 Immunization
 Strategies
 Barbell vs. Focused
 Barbell allows bonds of durations quite different from liability requirement
 Focus requires bonds with durations close to liability need
 Focus works much better as non-parallel shifts are better immunized
1.15
Fixed Income PFs
 Management of Bond Portfolios
 Two techniques for insulating an indexed bond PF from shifts in the yield curve – passive management
 Dedication – the exact matching of asset cash flows to a liability requirement
 Usually, the lowest cost PF that does the job
 Surplus cash – cash carry forward
 Risks of dedication
 Assets default
 Reinvestment risk for cash carry forward
1.14
Fixed Income PFs
 Management of Bond Portfolios
 Two techniques for insulating an indexed bond PF from shifts in the yield curve – passive management
 Immunization – matching duration (and other characteristics) of the assets PF to the liability requirement
 Risks
 Rates rise – loss on bonds with remaining term; reinvest cash at a higher rate
 Rates fall – bond value gains, but reinvest at a lower rate
 Simple examples show a net cancelation – but not always/exactly
 Many managers also match convexity – but at a cost
1.16
4

Fixed Income PFs
 Management of Bond Portfolios
 Two techniques for insulating an indexed bond PF from shifts in the yield curve – passive management
 Immunization
 Dedicated PFs are Immunized
 Dedicated PFs are more expensive
 Many managers will use a dedication strategy on a portion of the PF and immunize the remainder
1.17
Fixed Income PFs
 Bond PF management of yearly returns
 Cell Matching
 Delineate the important characteristics of the index, usually by duration cell
 Duration, coupon, rating, sector
 Replicate as best as possible through available bonds
 Active Bond Management Strategies
 Interest Rate forecasting
 Sector Selection and Rotation
 Individual bond selection
1.19
Fixed Income PFs
 Bond PF management of yearly returns
 Many bond managers are not concerned with managing against a particular liability, but are measured by their yearly return
 Against an Index
 For Total Return
 When a manager is measured against an index, often employ some variation of an index replication strategy
 The index often has thousands of bonds
 To replicate, the manager need only duplicate the bond proportions in the index
 But cannot because they are likely not available
 Alternative is to do cell matching 1.18
Fixed Income PFs
 Bond PF management of yearly returns
 Interest Rate forecasting
 Shorten or lengthen duration to gain exposure or limit exposure according to rates moving down or up
 Manager has a rationale for predicting and anticipating these moves
 If right 60% of time, bond PF can exhibit superior performance vs. other alternatives
 Will take a number of periods before luck vs. skill becomes apparent
1.20
5

Fixed Income PFs
 Bond PF management of yearly returns
 Sector Selection
 A manager might have reason to believe some sector will have superior performance over others – for example, high yield and the pay-up for added default risk might be in excess of what the manager believes is necessary
 Sector Rotation
 Overweight a sector where performance is expected to be superior in the next period
 If wide spreads are expected to tighten – as was the case immediately after the credit crisis, when widening was
“over done”
 Volatility is mispriced for option laden debt
1.21
Fixed Income PFs
 Active Bond Selection using MPT
 Estimating Expected Return
 Coupon Income
 P/L from term structure
 Expectations theory – current 1-period forward rates forecast expected forward rate in 1-period
1.23
Fixed Income PFs
 Bond PF management of yearly returns
 Individual Selection of mispriced bonds
 Usually as it relates to credit rating (too good or too bad)
 Best done with relative value models for price (“yield” or
“spread”) – models should include optionality, maybe even term or other risk premiums
1.22
Fixed Income PFs
 Active Bond Selection using MPT
 Estimating Expected Return
 Example: 5-year bond, $8 interest per period, $100 principal, market price is $82
 P
0Eqilibrium
=86.16
 P
1Eqilibrium
=86.77
1.24
6

Fixed Income PFs
 Active Bond Selection using MPT
 Estimating Expected Return
 Price in Equilibrium = $86.16
 Equilibrium price in 1-period is $86.77
 If market price was equilibrium, 1-period expected return is
$8 interest, $.61 capital gains, total return 8.61/86.16=10%
 If bond could have been bought at $82 and it returned to equilibrium, 1-period return would have been estimated at
(8.61+4.77)/82=15.57%
1.25
Fixed Income PFs
 Active Bond Selection using MPT
 Estimating Expected Return
 Today’s Equilibrium Price is the same = $86.16
 1-period Equilibrium price is now = $87.05 (vs. $86.77)
 1-period equilibrium return is (8+.61+.28)/86.16 = 10.32%
 The extra 32 bps is due to the liquidity premium and pays the investor for taking term risk
1.27
Fixed Income PFs
 Active Bond Selection using MPT
 Estimating Expected Return
 Liquidity Premium Theory for calculating expected return
 Assumed Forward rates (%)
 FR w/LP removed, stay the same, remove 1 st
 LP is slid forward to get new FR w/LP to value
1.26
Fixed Income PFs
 Active Bond Selection using MPT
 Single-Index Models
 Total Return = Exp Return + Return Due to
Unanticipated + e
R i
 R i
    i
 Where we use a bonds duration w/ unexpected % IR change
 Extending the simple 1-factor model from stocks w/ index,
R m
  i i i
  i i i
  i i i
   i
X i m  i
 R m

 R m

 i
 i
X D i
X D i

X i m  i
 
R m
 D m i
where D m
 i
X D i
1.28
7

Fixed Income PFs
 Active Bond Selection using MPT
 Single-Index Models
 The return on bond i is, R i
 R i
 D i
 
 Solving for ∆ and substituting into single index model gives
 

R m
 R m

/ D m

R i
 R i
   e i
 R i

D
D i m

R m
 R m

  i
 Where we can make the association, assumed independent of bond index
 i

D i
D m
 Where the beta has the traditional meaning as
, where the ε i
 i
 cov
 Except no need to estimate an otherwise deterministic relationship, a ratio of durations are


R R m
2 m

1.29
Fixed Income PFs
 Active Bond Selection using MPT
 Multi-index models
 Reasons for multi-index models
 More accurately measure effect of IR
 Change in yield spreads to Treasury
 Sector spreads
 Change in value of a call – volatility
1.30
8