1. Chapter 8 Active Fixed-Income Portfolio Management
FIXED-INCOME SECURITIES
Chapter 8
Active Fixed-Income Portfolio Management

2. 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

3. 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)

4. 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)

5. Riding the Yield Curve
Riding the yield curve is a technique that fixed-income 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

6. Example Example Price at t=0 Maturity Zero-coupon rate Price at t=1
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
Zero-coupon
rate
Price at t=0
Price at t=1
1 y
3.90% $
2 y
4.50% $
3 y
4.90% $
4 y
5.25% $
5 y
5.60% $

7. 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 $
Hold until maturity
Total return

8. Example – Cont’ Option 2.1 Option 2.2 Option 2.3 Option 2.4
Buys at $ at t=0
Receives a $6 coupon payment at t=1
Sells the bond with 1-year residual maturity at $
Total return
Option 2.2
Option 2.3
Option 2.4

9. 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%

10. 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

11. 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 years, $ duration is -$670
In case of a 1% drop in YTM
Absolute Profit = - $670x(-1%) = $6.7
Relative Profit = x(-1%) = 6.296%

12. 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

13. Example – Expect Decrease in Rates
Consider at date t a flat 5% curve and five bonds delivering annual coupon rates with the following features
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
Bond
Maturity
Coupon
rate
YTM
Price 1
2 y 5%
$100 2
10 y 3
30 y 4
7.5%
$138.43 5
10%
$176.86

14. Example – Cont’
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%
Bond
Modified
Duration
$Duration
Relative
Gain
Absolute 1
1.859
185.9
0.936%
$0.936 2
7.722
772.2
3.956%
$3.956 3
15.372
1537.2
8.144%
$8.144 4
14.269
1975.3
7.538%
$10.436 5
13.646
2413.4
7.196%
$12.727

15. 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
Buy in one year a 4-year maturity bond
Suppose interest rates stay at 6% from then on

16. Roll-Over Example – Cont’
Option 1
Cash-flows
where
Annual rate of return
Dates
T = 0
1 year later
2 years later
Cash-Flows
-$100 $5
3 years later
4 years later
5 years later
$105

17. Roll-Over Example – Cont’
where
Option 2
Cash-flows
Annual rate of return
Dates
T = 0
1 year later
2 years later
Cash-Flows
-$100 $5 $6
3 years later
4 years later
5 years later
$106

18. 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

19. Barbells, Bullets and Ladders
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 2-year 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

20. 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

21. 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 pay-off if the yield curve is affected by a positive or a negative parallel shift
Example:

22. 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
Solution: qs = and ql =

23. Example - P&L Profile
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%

24. 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

25. Cash and Dollar Duration Neutral
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
Solution: qs = 5,553.5 and ql = 4,513.1

26. 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'‘

27. 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

28. 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

29. 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

30. Fixed-Income Active Asset Allocation Strategic Versus Tactical Asset Allocation
Unconditional
Conditional
Slow evolving
weights
Dynamic
Constant

31. 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

32. Fixed-Income Active Asset Allocation Performance of a Perfect Timer

33. 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 %), on average the Lehman high yield index under performs the Lehman Investment grade index by 7.64% more than the unconditional mean

34. Fixed-Income Active Asset Allocation Contemporaneous Factor Analysis - Example
Performance of Bond Indices under Different Contemporaneous Economic Conditions – the Example of Changes in Implicit Volatility (monthly data over the period )

35. 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

36. Fixed-Income Active Asset Allocation Lagged Factor Analysis - Example
Performance of Bond Indices under Different Lagged Economic Conditions – The Example of the Term Spread (monthly data over the period )

37. 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)

38. 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

39. 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)

40. 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)).

41. 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
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
Detect rich and cheap securities that historically present abnormal yield-to-maturity
Taking as reference a theoretical zero-coupon yield curve fitted with bond prices

42. 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

43. Trading on Market Inefficiencies Example
The price of strips with maturity 02/15/02, 02/15/03 and 02/15/04 are respectively 99.07, 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 x x =
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 ( )% = $19,300

44. 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

45. 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

46. 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 ) =
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 =
This bond is expensive and can be shorted
Confidence level is high: Proba ( -1 < U < 1 ) =

47. 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= %, 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 % and (S-Min)/(Max-Min)=1.063
We conclude that this bond is cheap and can be bought

48. 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 b which is chosen to reverse the position is equal to 15%
Taking b >a ensures Min(b )>Min(a) and Max(b )<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 The ratio (S-Min/(Max-Min) is equal to 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)

49. Trading on Market Inefficiencies Historical Distribution of the Spread on 10/04/96

50. Trading on Market Inefficiencies Position and P&L
TC stand for transaction costs, and FC for financing cost

51. Trading on Market Inefficiencies Rich-Cheap Analysis – Performance
EONIA stands for Euro Overnight Index Average : average European Interbank Offered Rate

52. 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 bid-ask spread)
Operational Risk
Price errors may exist in the database
Bonds with embedded options can be mistaken for straight bonds

53. 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