FIXED-INCOME SECURITIES
Chapter 8
Active Fixed-Income
Portfolio Management
Outline
• Market Timing: Trading on Interest Rate Predictions
–
–
–
–
–
Naive Strategies
Roll-Over Strategies
Riding the Yield Curve
Barbells, Bullets, Ladders and Butterflies
Active Fixed-Income Style Allocation Decisions
• Bond Picking: Trading on Market Inefficiencies
– Trading within a Given Market: Bond Relative Value Analysis
– Trading across Markets: Spread and Convergence Trades
Active Strategies
• Investors who do not accept the EMH pursue active
investment strategies
• Pursuit of an active strategy assumes that investors
possess some advantage relative to other market
participants
– Superior information
– Superior analytical or judgment skills
– Ability or willingness to do what other investors are unable to do
• Two types of active strategies
– Market timing (trading on interest rates predictions)
– Bond picking (trading on market inefficiencies)
Market Timing
• Portfolio managers are making bets on changes in
the Treasury yield curve
– Bets based on no changes in the yield curve (riding the yield curve)
– Bets on changes in interest rate level (naive or roll-over strategies)
– Bets based both on level, slope and curvature moves of the yield curve
(butterflies)
• Managers need scenario analysis tools to estimate
the return and the risk of implemented strategies
– Evaluation of break-even point from which the strategy will start making or
losing money
– Assessment of the risk that the expectations are not realized
• One can also implement timing on various asset
classes or fixed-income styles (e.g., Treasury versus
Corporate, or Bonds versus Stocks)
Riding the Yield Curve
• Riding the yield curve is a technique that fixedincome portfolio managers traditionally use in order
to enhance returns
– When the yield curve is upward sloping
– and is supposed to remain unchanged
• Enables an investor to earn a higher rate of return by
– Purchasing fixed-income securities with maturities longer than the desired
holding periods and
– Selling them to profit from falling bond yields as maturities decrease with
the passage of time
Example
• Example
– Consider at date the following zero-coupon curve and five bonds with the
same $100 nominal value and 6% annual coupon rate
– Prices of these bonds are given at date t=0 and one year later at date t=1
assuming that the zero-coupon yield curve has remained unchanged
Maturity
1y
2y
3y
4y
5y
Zero-coupon
rate
3.90%
4.50%
4.90%
5.25%
5.60%
Price at t=0
Price at t=1
$102.021
$102.842
$103.098
$102.848
$102.077
$102.021
$102.842
$103.098
$102.848
Example
• A portfolio manager who has money to invest with a
1 year horizon
– Option 1: Invest in 1-y maturity bond
– Option 2: ride the yield curve
• Option 2.1: buy the 2 year bond and sell it back in 1 year
• Option 2.2: buy the 3 year bond and sell it back in 1 year
• Option 2.3: buy the 4 year bond and sell it back in 1 year
• Option 2.4: buy the 5 year bond and sell it back in 1 year
• Total return TR for option 1
– Buy at $102.021
– Hold until maturity
– Total return
 106  102.021 

  3.9%
 102.021 
Example – Cont’
• Option 2.1
–
–
–
–
Buys at $102.842 at t=0
Receives a $6 coupon payment at t=1
Sells the bond with 1-year residual maturity at $102.021
Total return
 6  102.021  102.842 
• Option 2.2
• Option 2.3
• Option 2.4

  5.036%
102.842


 6  102 .842  103 .098 

  5.571 %
103 .098


 6  103 .098  102 .848 

  6.077 %
102 .848


 6  102.848  102.077 

  6.633%
102.077


Example – Cont’
• Options riding the curve are better than option 1
• Best option is 2.4
• The longer the maturity of the bond bought at the
outset, the higher the return to the strategy
• Now, if zero-coupon rates had increased
– Total return would have been less than 6.633%
– Could have been less than 3.90%
Timing Bets on Interest Rate Level
• Strategies based on changes in the level of interest
rates are very naive
• Assume that only one factor drives the yield curve
– Focus on YTMs
– Assume only translation moves (parallel shifts)
• They are only two possible moves, an up move and
a down move
– If you think that interest rates will decrease in level, you want to buy bonds
or future contracts with longest possible duration
– On the other hand, if you think that interest rates will increase in level, you
want to shorten the $ duration or modified duration of your portfolio by
selling bonds or futures contracts,
– Alternatively you will hold short-term instruments until maturity and roll over
at higher rates
Naive Strategies – Expect Decrease in Rates
• Focus on modified duration if you care for relative
capital gains
– Relative P&L = - Modified Duration x Change in Rate
• Focus on $ duration if you care for absolute capital
gains
– Absolute P&L = $ Duration x Change in Rate
• Example
– Bond with nominal amount $100, 10% coupon rate, 9% YTM
– Price = $106.42, modified duration is 6.296 years, $ duration is -$670
– In case of a 1% drop in YTM
• Absolute Profit = - $670x(-1%) = $6.7
• Relative Profit = - 6.296x(-1%) = 6.296%
Naive Strategies – Expect Decrease in Rates
• Recall
– The longer the maturity T, the higher the coupon rate c, the higher
the $ duration of a bond
– The longer the maturity T, the lower the coupon rate c, the higher
the modified duration of a bond
• Strategy
– Invest in bonds with long T and high c to optimize absolute gain
– Invest in bonds with long T and low c to optimize relative gain
Example – Expect Decrease in Rates
• Example
– Consider at date t a flat 5% curve and five bonds delivering annual
coupon rates with the following features
Bond
1
2
3
4
5
Maturity
Coupon
rate
2y
10 y
30 y
30 y
30 y
5%
5%
5%
7.5%
10%
YTM
Price
5%
5%
5%
5%
5%
$100
$100
$100
$138.43
$176.86
– A portfolio manager think that the YTM curve will decrease to 4.5%
– Which bond would he pick if he cares for
• Absolute capital gains
• Relative capital gains
Example – Cont’
Bond
1
2
3
4
5
Modified
Duration
1.859
7.722
15.372
14.269
13.646
$Duration
185.9
772.2
1537.2
1975.3
2413.4
Relative
Gain
0.936%
3.956%
8.144%
7.538%
7.196%
Absolute
Gain
$0.936
$3.956
$8.144
$10.436
$12.727
• If focuses on absolute gain, choose the 30-year maturity
bond with 10% coupon rate (bond # 5)
• If focuses on relative gain, invest in the 30-year maturity
bond with 5% coupon rate (bond # 3)
• Difference in terms of relative gain between bond 1 and
bond 3 is 7.208%
Roll-Over Example – Expect Increase in Rates
• Consider a flat 5% yield-to-maturity curve
– Investment horizon : 5 years
– Expects an interest rate increase by 1% in one year
• Option 1
– Buy a 5-year maturity bond
– Hold it until maturity
• Option 2
– Buy a 1-year maturity T-Bill
– Hold it until maturity
– Buy in one year a 4-year maturity bond
• Suppose interest rates stay at 6% from then on
Roll-Over Example – Cont’
• Option 1
– Cash-flows
Dates
Cash-Flows
Dates
Cash-Flows
1 year later
T=0
-$100
3 years later
$5
$5
4 years later
$5
2 years later
$5
5 years later
$105
– Annual rate of return
 128.85  100 


100


1/ 5
 1  5.092%
where
128.85  5 1.06 4  5 1.063  5 1.06 2  5 1.06  105
Roll-Over Example – Cont’
• Option 2
– Cash-flows
Dates
Cash-Flows
Dates
Cash-Flows
T=0
-$100
3 years later
$6
1 year later
$5
4 years later
$6
2 years later
$6
5 years later
$106
– Annual rate of return
 132.56  100 


100


1/ 5
 1  5.8%
where
132.56  5 1.06 4  6 1.063  6 1.06 2  6 1.06  106
Bets on Specific Moves of the Yield Curve
• We know that the yield curve is potentially affected
by many other movements than parallel shifts
• These include in particular pure slope and curvature
movements, as well as combinations of level, slope
and curvature movements
• It is in general fairly complex to know under what
exact market conditions a given strategy might
generate a positive or a negative pay-off when all
these possible movements are accounted for
• We study
– Simple bullet and barbell strategies
– More complex butterfly strategies and other semi-hedged strategies
Barbells, Bullets and Ladders
• Bullets
– Definition: a bullet portfolio is constructed by concentrating investments on
a particular maturity of the yield curve
– Example: a portfolio invested 100% in a 5-year maturity T-Bond
• Barbells
– Definition: a barbell portfolio is constructed by concentrating investments at
the short-term and the long-term ends of the yield curve
– Example: a portfolio invested half in a 6-month maturity T-Bill and half in a
30-year maturity T-Bond
• Ladders
– Definition: a ladder portfolio is constructed by investing equal amounts in
bonds with different maturity.
– Example: a portfolio invested 20% in a 1-year maturity T-Bond, 20% in a 2year maturity T-Bond, 20% in a 3-year maturity T-Bond, 20% in a 4-year
maturity T-Bond and finally 20% in a 5-year maturity T-Bond
Butterflies
• Barbells, bullets and ladders are building blocks
used in the construction of complex strategies
• A butterfly is one of the most common fixed-income
active strategies used by practitioners
– It is the combination of a barbell (called the wings of the butterfly) and a
bullet (called the body of the butterfly)
– The purpose of the trade is to adjust the weights of these components so
that the transaction is cash-neutral and has a $duration equal to zero
– The latter property guarantees a quasi-perfect interest-rate neutrality when
only small parallel shifts affect the yield curve
– Besides, the butterfly, which is usually structured so as to display a positive
convexity, generates a positive gain if large parallel shifts occur
– There actually exist many different kinds of butterflies which are structured
so as to generate a positive pay-off in case a particular move of the yield
curve occurs
Butterflies – Example
• When only parallel shifts affect the yield curve, the
strategy is structured so as to have a positive
convexity
• The investor is then certain to enjoy a positive payoff if the yield curve is affected by a positive or a
negative parallel shift
• Example:
Maturity Coupon Rate YTM Bond Price $Duration Quantity
2 Years
5%
5%
100
185 . 9
qs
5 Years
5%
5%
100
432 . 9
1, 000
10 Years
5%
5%
100
772 . 2
ql
Example – Cont’
• The face value of bonds is normalized to be $100,
and we assume a flat yield-to-maturity curve in this
example
• We structure a butterfly in the following way:
– we sell $1,000 worth of 5-year maturity bonds
– we buy qs 2-year maturity bonds and ql 10-year maturity bonds
• The quantities qs and ql are determined so that the
butterfly is cash and $duration neutral, i.e., they are
solutions to the following system
qs 185.9  ql  772.2  1000  432.9

 (qs 100)  (ql 100)  (1000 100)
• Solution: qs = 578.65 and ql = 421.35
Example - P&L Profile
1200
1000
Total Return in $
800
600
400
200
0
0%
2%
4%
6%
8%
10%
12%
Yield to Maturity
•
•
The butterfly has a positive convexity; whatever the YTM, the strategy always
generates a gain
The gain has a convex profile with a perfect symmetry around the 5% X-axis;
for example, the total return reaches $57 when the yield-to-maturity is 4%
Different Kinds of Butterflies
• Seems too good to be true!
• We know, however, that the yield curve is affected by
many other movements than parallel shifts
– Pure slope and curvature movements
– Combinations of level, slope and curvature movements
• It is in general fairly complex to know under what
market conditions a given butterfly generates a
positive or a negative pay-off
• Some butterflies are structured so as to pay off if a
particular move of the yield curve occurs
–
–
–
–
Cash and $Duration Neutral Weighting
Fifty-Fifty Weighting
Regression Weighting
Maturity-Weighting
Cash and Dollar Duration Neutral
Maturity YTM Bond Price Quantity $Duration
2 years 4. 5%
100 . 936
qs
188 . 6
5 years 5. 5%
97. 865
10, 000
421 . 17
10 years
92. 64
ql
701 . 14
6%
• We structure a butterfly in the following way:
– we sell $10,000 5-year maturity bonds
– we buy qs 2-year maturity bonds and ql 10-year maturity bonds
• The quantities qs and ql are determined
 qs 188.6  ql  701.14  10,000  421.17 

(qs 100.936)  (ql  92.64)  (10,000  97.865)
• Solution: qs = 5,553.5 and ql = 4,513.1
Fifty-Fifty Weighted
• Adjust the weights so that the portfolio has a zero
$duration and the same $duration on each wing
• Aim is make the trade neutral to small steepening
and flattening moves
• A fifty-fifty weighted butterfly is neutral to curve
movements such that the spread change between
the body and the short wing is equal to the spread
change between the long wing and the body
– Steepening scenario: “-30/0/30'‘
– Flattening scenario: “30/0/-30'‘
qs 188.6  ql  701.14  10,000  421.17 qs  11,165.7


(qs 188.6)  (ql  701.14)  0

ql  3,003.5
Regression Weighting
• As short-term rates are much more volatile than
long-term rates, we normally expect that the short
wing moves more from the body than the long wing
– Regress changes in the spread between the long wing and the body on
changes in the spread between the body and the short wing
– Assuming that we obtain a value of, say, beta = 0.5 for the regression
coefficient, it means that for a 20 bps spread change between the body
and the short wing, we obtain on average a 10 bps spread change
between the long wing and the body
– Trade is quasi curve neutral to a ”-30/0/15'' steepening scenario, or a
“30/0/-15'' flattening scenario
– Fifty-fifty weighting butterfly = regression weighting with beta=1
qs 188.6  ql  701.14  10,000  421.17  q  7,443.8

s

1

(qs 188.6)  (  ql  701.14)  0
ql  4,004.7


2

Maturity-Weighting
• Maturity-weighting butterflies are structured similarly
to the regression butterflies
• Instead of searching for a regression coefficient beta,
the idea is to weight each wing of the butterfly
depending on the maturities of the three bonds
• Similar to regression weighting, with beta equal to
maturity weighting coefficient
Mm  Ms

Ml  Mm
qs 188.6  ql  701.14  10,000  421.17 

Mm  Ms
qs  8,374.3

 (qs 188.6)  ( M  M  ql  701.14)  0 
l
m
ql  3,754.4




3

5
Other Semi-Hedged Strategies
Example: Ladder Hedged against a Slope Movement
• Make a particular bet on a movement of the yield
curve while being hedged against all other
movements
• This can be done using the level, slope and
curvature $durations of the Nelson and Siegel model
• An example is a ladder hedged against a slope
movement
• Of course many different products may be structured
in a similar way
• Example is a cash and sensitivity neutral butterfly
hedged against a curvature movement
Fixed-Income Active Asset Allocation
Strategic Versus Tactical Asset Allocation
Constant
weights
Strategic
Unconditional
Slow evolving
weights
Dynamic
weights
Tactical
Conditional
Fixed-Income Active Asset Allocation
TAA and TSA
• A tactical asset allocation (TAA) strategy is a “strategy
that allows active departures from the normal asset mix
according to specified objective measures of value. (…) It
involves forecasting asset returns, volatilities and
correlations. The forecasted variables may be functions of
fundamental variables, economic variables, or even
technical variables.” Campbell Harvey (online finance
glossary)
• Kao and Shumaker (1999) and Amenc, Martellini and Sfeir
(2002) have formalized the concept of style timing or tactical
style allocation
– Involves dynamic trading in various investment styles:
• Growth, value, large cap, small cap for equity investing
• T-Bond, investment grade and high yield for fixed-income investing
(also maturities)
– The industry has started to look into TSA strategies
Fixed-Income Active Asset Allocation
Performance of a Perfect Timer
T-Bond
Investment Grade
High Yield
Mean
Standard Deviation
Year
1994
1995
1996
1997
1998
1999
2000
2001
Average
St. Dev.
T-Bond
-10.22%
10.90%
-4.61%
3.72%
2.31%
-8.07%
8.32%
-1.34%
0.12%
7.07%
T-Bond
Investment Grade
High Yield
1.00
0.77
0.12
1.00
0.24
1.00
0.37%
4.92%
0.40%
5.60%
0.22%
6.64%
Corporate
-12.42%
13.08%
-5.14%
4.51%
-1.41%
-8.68%
5.19%
2.07%
-0.35%
7.74%
High Yield
-11.39%
9.45%
0.20%
2.63%
-9.00%
-6.80%
-7.46%
5.89%
-2.06%
7.15%
LGBI
-10.86%
10.24%
-4.03%
3.49%
0.37%
-7.70%
5.95%
0.09%
-0.31%
6.57%
Perfect Timer
-10.22%
13.08%
0.20%
4.51%
2.31%
-6.80%
8.32%
5.89%
2.16%
7.19%
Fixed-Income Active Asset Allocation
Contemporaneous Factor Analysis - Example
• Intuition
– High yield bonds are a good investment in periods of low uncertainty
– On the other hand, they are dominated by higher quality bonds in periods
of higher uncertainty
– Flight-to-quality
• Confirmation
– 1/3 largest decreases in implicit volatility on equity corresponds to drops
ranging from –33.15% to –4.39%
– Under these conditions, the Lehman high yield index performs rather well,
since on an annual basis it over performs the Lehman Investment grade
index by 6.53% more than the unconditional annualized mean, a small
negative –0.18%
– On the other hand, when implicit volatility on equity is decreasing
significantly (from +4.46% to +59.22%), on average the Lehman high yield
index under performs the Lehman Investment grade index by 7.64% more
than the unconditional mean
Fixed-Income Active Asset Allocation
Contemporaneous Factor Analysis - Example
Change in Implicit Volatility
Difference between Conditionnal Values and
Unconditionnal Values
Low
Medium
High
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum
-33.15%
-4.39%
-3.80%
4.45%
4.46%
59.22%
Unconditionnal Values
Mean
Stdev
Mean
Stdev
Mean
Stdev
Mean
Stdev
Correlation
CREDIT BOND INDEX - LEHMAN AGGREGATE
0.87%
0.26%
0.45%
-0.31%
-1.32%
-0.02%
0.25%
1.73%
-0.33
HIGH YIELD BOND INDEX - LEHMAN AGGREGATE
7.40%
-0.42%
1.56%
-0.31%
-8.96%
0.10%
0.07%
7.40%
-0.44
TREASURY BOND INDEX - LEHMAN AGGREGATE
0.14%
0.01%
2.89%
-0.20%
-3.03%
-0.01%
0.21%
2.89%
0.00
HIGH YIELD BOND INDEX - CREDIT BOND INDEX
6.53%
-0.17%
1.11%
-0.21%
-7.64%
-0.02%
-0.18%
7.59%
-0.36
Performance of Bond Indices under Different Contemporaneous Economic Conditions – the
Example of Changes in Implicit Volatility (monthly data over the period 1991-2001 )
Fixed-Income Active Asset Allocation
Lagged Factor Analysis - Example
• Intuition
– An upward slopping yield curve signals expectations of increasing short
term rates
– This typically associated with scenarios of economic recovery
– These are conditions under which high yield bond tend to over-perform
safer bonds
• Confirmation
– The 1/3 lowest values for the term spread range from –0.61% to 0.99%
– Under these conditions, Lehman high yield index performs rather poorly,
since on an annual basis it under performs the Lehman Aggregate bond
index –3.17% below the unconditional annualized mean, a small positive
0.05%
– On the other hand, when the yield curve is very upward slopping implicit
(term spread ranging from 2.48% to 3.01%), on average the Lehman high
yield index over performs the Lehman Aggregate bond index by 3.90%
more than the unconditional mean
Fixed-Income Active Asset Allocation
Lagged Factor Analysis - Example
Term spread
Difference between Conditionnal Values and
Unconditionnal Values
Low
Medium
High
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum
-0.61%
0.99%
1.03%
2.35%
2.48%
3.91%
Unconditionnal Values
Mean
Stdev
Mean
Stdev
Mean
Stdev
Mean
Stdev
Correlation
CREDIT BOND INDEX - LEHMAN AGGREGATE
-0.81%
0.54%
0.52%
-0.49%
0.29%
-0.20%
0.24%
1.75%
0.09
HIGH YIELD BOND INDEX - LEHMAN AGGREGATE
-3.17%
1.52%
-0.73%
-0.18%
3.90%
-1.64%
0.05%
7.44%
0.12
TREASURY BOND INDEX - LEHMAN AGGREGATE
0.33%
-1.42%
-0.04%
0.41%
-0.29%
0.63%
0.26%
2.92%
-0.03
HIGH YIELD BOND INDEX - CREDIT BOND INDEX
-2.37%
1.28%
-1.25%
-0.26%
3.61%
-1.14%
-0.19%
7.62%
0.10
Performance of Bond Indices under Different Lagged Economic Conditions – The Example
of the Term Spread (monthly data over the period 1991-2001 )
Fixed-Income Active Asset Allocation
Lagged versus Contemporaneous Factor Analysis
• We have just seen a couple of examples illustrating that both
contemporaneous and lagged economic and financial variables
had an impact on fixed-income style differentials
• Forecasting economic variables is a difficult art, with the failures
often leading to all systematic TAA being abandoned
• Two ways of considering tactical style allocation
– Forecasting returns is based on forecasting the values of economic variables
(scenarios on the contemporaneous variables)
– Forecasting returns is based on anticipating market reactions to known
economic variables (econometric model with lagged variables)
• The anticipation of market reactions to known variables is easier
– It leads one to think that the performance does not result from privileged
information but an analysis of the reactions of the market to its publication
– The market is guided by the information (informational efficiency) but some
players can hope to manage the consequences better than others
(inefficiency or reactional asymmetry)
Fixed-Income Active Asset Allocation
Forecasting Returns
• We are interested in forecasting Rt as a function of
lagged variables Zt-1
• Start with a linear regression model (can be
generalized)
• The linear regression model is:
Rt = d0 + d1Z1,t-1 + …+ dkZk,t-1 + residualt
• Use state-of-the-art econometric techniques
– Check for the presence of autocorrelation, multi-colinearity,
heteroskedasticity,
– Check for robustness of model through time
– Check for robustness of linear specifications
– Check for robustness of distributional assumption
Fixed-Income Active Asset Allocation
Both Art and Science
• Principle 1: Parsimony Principle
–
–
Other things equal, simple models are preferable to complex models
KISS principle (“Keep It Sophisticatedly Simple”): simple model dos not mean naïve
model
• Principle 2: Data Mining versus Economic Analysis?
–
–
Preferable to select a short list of meaningful variables on the basis on previous
evidence of their ability to predict asset returns, as well as their natural influence on
asset returns
Trying to screen thousands of variables through stepwise regression techniques
usually leads to high in-sample R-squared but low out-of-sample R-squared
(robustness problem)
• Principle 3: Dynamic versus Static Modeling
–
–
The relationship between styles differentials and forecasting factors is not constant
through time
Because some factors are relevant in some periods, but not in others, use a dynamic
recursive modeling framework (calibration, training and trading periods)
Fixed-Income Active Asset Allocation
Performance of TSA Portfolios
Risk/Return Analysis of TSA Portfolio. This table features the monthly return
from January 1999 to December 2001 for a TSA portfolio based on the
econometric approach presented above, and dollar neutrality with a level of
leverage equal to 2 (from Amenc, Martellini and Sfeir (2002)).
Risk Return Analysis
TSA Fund
Lehman Brothers Global
Cumulative Return
36.31%
-2.65%
Annualised Return
10.45%
-0.84%
Annualised Std Deviation
4.16%
3.46%
Downside Deviation (3.0%)
2.56%
3.46%
Sortino (3.0%)
2.91
-1.11
Sharpe (Risk Free Rate = 3.0%)
1.79
-1.11
1st Centile
-1.54%
-2.30%
% Negative Returns
11.11%
52.78%
Up Months in Up Market
88.24%
Down Months in Down Market
10.53%
Up Market Outperformance
52.94%
Down Market Outperformance
89.47%
Worst Monthly Drawdown
-2.08%
-2.48%
Maximum Drawdown
-2.08%
-6.75%
1
7
in progress
26
Months in Max Drawdown
Months to recover
Trading on Market Inefficiencies
Trading within a Given Market
• Bond relative value is a technique which consists in
detecting underpriced and overpriced bonds
• Two types of investment opportunities exist
– Pure arbitrage opportunities
– Speculative arbitrage opportunities
• Pure arbitrage opportunities
– Compare the price of two products with the same cash-flows
– Typically a bond and the sum of strips that reconstitute exactly the bond
– If a difference in prices exists, it is riskfree arbitrage opportunity
• Speculative arbitrage opportunities
– Detect rich and cheap securities that historically present abnormal yield-tomaturity
– Taking as reference a theoretical zero-coupon yield curve fitted with bond
prices
Trading on Market Inefficiencies
Comparing a Bond to a Portfolio of Strips
• Strips (Seperate Trading of Registered Interest and
Principal) are zero-coupon securities
–
–
–
–
Mainly issued by government bonds of the G7 countries
The strips program was created in 1985 by the US Treasury
In the US where more than 150 Treasury strips on September 2001
In France, Treasury strips are about 80 on September 2001
• We know that, in the absence of arbitrage
opportunities, the price of a bond must be equal to
the weighted sum of zero-coupon bonds or strips
• If the bond price is different from the prices of the
sum of strips, there is an arbitrage opportunity
Trading on Market Inefficiencies
Example
• On 09/14/01
– The price of strips with maturity 02/15/02, 02/15/03 and 02/15/04 are
respectively 99.07, 96.06 and 92.54
– The principal amount of strips is $1,000
– The price of the bond with maturity 02/15/04, coupon rate 10% and
principal amount $10,000 is 121.5
– Is there any arbitrage opportunity?
• First compute the price of the reconstructed bond
using strips
– Price: 10 x 0.9907 + 10 x 0.9606 +110 x 0.9254 = 121.307
– Sell 1,000 bonds, buys a quantity of 1,000 strips with maturity 02/15/02,
1,000 strips with maturity 02/15/03 and 11,000 strips with maturity
02/15/04.
– Profit from the trade: 10,000 x 1,000 x (121.5-121.307)% = $19,300
Trading on Market Inefficiencies
Speculative Arbitrage
• Jordan, Jorgensen and Kuipers (2000) find
– Such arbitrage opportunities appear to be rare
– When price differences occur, they are usually small and too short-lived to
be exploited
• Because pure arbitrage opportunities are extremely
hard to find, more speculative trades are often
performed
• Bond rich and cheap analysis is a common market
practice
• The idea is to obtain a relative value for bonds which
is based upon a comparison to a homogeneous
reference
Trading on Market Inefficiencies
Rich-Cheap Analysis
• Rich-cheap analysis is a several steps process
– Step 1: Construct an adequate current zero-coupon yield curve using data
for assets with homogenous characteristics in terms of liquidity and risk
– Step 2: Compute a theoretical price for each asset as the sum of the
discounted cash-flows
– Step 3: Compute the market YTM maturity and compare it to the
theoretical YTM
– Step 4: The spread (market yield - theoretical yield) allows for the
identification of an expensive (spread<0) or a cheap asset (spread>0)
– Step 5: Use statistical analysis (Z-Score analysis) of historical spreads for
each asset in an attempt to distinguish actual inefficiencies from abnormal
yields related to specific features of a given asset (liquidity effect,
benchmark effect, coupon effect, etc...).
– Step 6: Combine short and long positions to create a portfolio that is quasi
insensitive to interest rate changes (market neutral HF)
– Step 7: Reverse short and long positions according to a criterion which is
defined a priori
• At the first time when the position generates a profit net of costs
• When the spread has come back to a more “normal” level
Trading on Market Inefficiencies
Z-Score Analysis
• Criterion 1, based on a normality assumption
– Compute the average value m and standard deviation s of, say, the last 60
spreads
– Assume the spread S is normally distributed and define the normalized
spread U = (S-m)/2s
– In particular, we have that Proba ( -1 < U < 1 ) = 0.9544
– If actual normalized spread is larger (smaller) than 1, the bond can be
regarded as relatively cheap (expensive)
– Can make the confidence level stronger by taking U = (S-m)/ks
• Example
– Mean value m and standard deviation s of the 60 last spreads are:
m=0.03% and s =0.04%
– One day later, the new spread is -0.11%
– Considering a confidence level k=3, we obtain U = (S-m)/3s = -1.166
– This bond is expensive and can be shorted
– Confidence level is high: Proba ( -1 < U < 1 ) = 0.9973
Trading on Market Inefficiencies
Z-Score Analysis – Con’t
• Criterion 2, more general
– Define the value Min such that a% of the spreads are below that value,
and the value Max such that a% of the spreads are above that value
– Take a% equal to 10, 5, 1%, or any other value, depending upon the
confidence level the investor requires for that decision rule
– When (S-Min)/(Max-Min) gets close to or above 1, the bond is regarded as
cheap, when (S-Min)/(Max-Min) gets close to or below 0, the bond is
regarded as expensive
• Example
– The value Min such that 5% of the spreads are below that value is Min=0.0888%, and the value Max such that 5% of the spreads are above that
value is Max=0.0677%.
– One day later, the new spread is 0.0775% and (S-Min)/(Max-Min)=1.063
– We conclude that this bond is cheap and can be bought
Trading on Market Inefficiencies
Rich-Cheap Analysis – An Illustration
• Experiment
–
–
–
–
–
–
Apply Z-score analysis on the French T-Bond market from 1995 to 1996
The value a which provides signals for short and long positions is 1%
The level  which is chosen to reverse the position is equal to 15%
Taking  >a ensures Min( )>Min(a) and Max( )<Max(a), as it should be
Short and long positions are financed with the repo market
For each transaction, we buy or sell for Eur 10 millions and seek a
$duration neutral portfolio
• On 10/07/96, we detect an opportunity to buy the 02/27/04 OAT
– On 10/04/96, the value Min and Max at the 1% level are, respectively, 0.65% and 2.16%
– The spread value on 10/07/96 is 3.17. The ratio (S-Min/(Max-Min) is equal
to 1.359. Buy the OAT 02/27/04 for Eur 9,999,977 and sell the OAT
10/25/04 for Eur 9,276,006 so that the position is $duration neutral
– The value Max such that 15% of the spreads are below that value, is 1
– The first time (10/17/04) when the spread goes below 1 we close the
position (the spread value is then equal to 0.67)
Trading on Market Inefficiencies
Historical Distribution of the Spread on 10/04/96
20
18
16
Frequency in %
14
12
10
8
6
4
2
0
-0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
Classes
1.1
1.3
1.5
1.7
1.9
2.1
2.3
Trading on Market Inefficiencies
Position and P&L
Bond
OAT 02/27/04
Price MD YTM Position Investment
5.39
5.64
Long
-9,999,977
OAT 10/25/04 112.81 5.82
5.74
Short
9,276,006
Date
Bond
10/07/96 OAT 02/27/04
120.4
Price Position Investment
TC
120.4
Long
-9,999,977
2,500
10/07/96 OAT 10/25/04 112.81
Short
9,276,006
2,319
10/17/96 OAT 02/27/04 120.92
Short
10,042,836
2,511
10/17/96 OAT 10/25/04 112.97
Long
-9,289,568
2,322
Total
29,297
FC Profit and Loss
9,652 -804
TC stand for transaction costs, and FC for financing cost
18,840
Trading on Market Inefficiencies
Rich-Cheap Analysis – Performance
46 000 000
45 000 000
44 000 000
43 000 000
42 000 000
41 000 000
Eonia + Bond
Rich-Cheap
40 000 000
Eonia
39 000 000
01/01/1995 15/04/1995 28/07/1995 09/11/1995 21/02/1996 04/06/1996 16/09/1996 29/12/1996
EONIA stands for Euro Overnight Index Average : average European Interbank Offered Rate
Trading on Market Inefficiencies
Rich-Cheap Analysis – The Risks
• Interest Rate Risk
– $duration neutrality only guarantees a perfect immunization against small
parallel changes
– We are still affected by large shifts, and changes in slope and curvature
– See chapter 5 for possible remedies
• Liquidity Risk
– For example a bond may seem to be cheap at a given date, but if it is not
liquid it won't probably mean-revert to the “normal” level
– It may be useful to include a proxy for liquidity in the analysis (e.g., the bidask spread)
• Operational Risk
– Price errors may exist in the database
– Bonds with embedded options can be mistaken for straight bonds
Trading Across Markets
Spread Trades
• Swap-Treasury or Corporate-Swap Spread Trades
– Consist in detecting the time when swap spreads are relatively high or low
– The statistical technique used for judging the cheapness or richness of
swap spreads is basically the same as the one used to perform bond
relative value analysis
• Example (Swap-Treasury)
– Suppose the current Euro 10-year swap spread amounts to 40 BP
– Assume that its mean and standard deviation over the last 60 working days
amount to respectively 25 BP and 5 BP
– Corresponding Z-score is equal to (40-25)/5 = 3 => swap spread is cheap
– Go long a AA 10-year corporate bond and short a 10-year Treasury bond in
order to lock in a high spread.
– Suppose that three months later the swap spread is 20 BP. Its 60-day
mean and standard deviation are equal to respectively 25 BP and 2 BP.
– The corresponding Z-score is equal to (20-25)/2 = -2.5.
– So the swap spread can be considered rich: time to unwind the position