Post on 16-Dec-2015
EXPLORING SOME ALTERNATIVE
FIXED-INCOME STRATEGIES
Philippe PRIAULET
HSBC-CCF and University of EVRY
2 AVRIL 2004
Fixed Income Strategy 22
CONTENTS
• Bond picking strategies
Results of a systematic trading strategy on the T-bond French market
• Swap barbells and butterflies
Results of a systematic trading strategy on the US, EUR and GBP
markets
• Revealing anomalies in forward and volatility curves
Anomalies in forward curves
Swaption and caplet break-evens
Fixed Income Strategy 33
Bond picking strategies
Fixed Income Strategy 44
• The bond relative value analysis
The goal of that analysis is to detect rich and cheap securities that historically present abnormal yields to maturity, taking as reference a theoretical zero-coupon yield curve fitted with bond prices.
The method can be developed both for Treasury and corporate bonds.
We take here the example of the French Treasury bond market.
We build a strategy that belongs to alternative fixed-income strategies, and back-test it from 1995 to 2001.
Fixed Income Strategy 55
• How it works ?
Bond rich-cheap analysis proceeds in five steps
1- We construct the adequate current zero-coupon yield curve with a spline model using data for assets with the same characteristics in terms of liquidity and risk.
2- Then compute a theoretical price for each asset to obtain the spread between the market yield to maturity and the theoretical yield to maturity.
3- For each asset, we implement a Z-score analysis so as to distinguish actual inefficiencies from abnormal yields. This statistical analysis provides signals of short or long positions to take in the market.
4- Short and long positions are unwound according to a criterion that is defined a priori.
Fixed Income Strategy 66
• Z-score analysis
At date t and for a given bond, we use the historical of the 60 last spreads.
1- We define the value Min such that x% of the spreads are below that value, and the value Max such that x% of the spreads are above that value. is the value of the spread at date t+1.
2- When converges to 1 or exceeds 1, the bond is considered cheap.On the other hand, when this ratio converges to zero or becomes negative, the bond is considered expensive.For other values of this ratio, we conclude that the bond is fairly priced.
1tS
MinMax
MinSt1
Fixed Income Strategy 77
• Example of Z-score analysis
Suppose that we obtain the following historical distribution for the spread of a given bond over the last 60 working days
For x = 5, Min = -0.0888% and Max = 0.0677%. One day later, the new spread is 0.0775% so that the ratio is equal to 1.063. The bond is cheap.
Historical Distribution
0
2
4
6
8
10
12
14
16
-0.1
0%
-0.0
8%
-0.0
6%
-0.0
4%
-0.0
2%
0.0
0%
0.0
2%
0.0
4%
0.0
6%
0.0
8%
Classes
Fré
qu
en
cy
Fixed Income Strategy 88
• When to unwind the position ?
The issue lies in the decision timing to reverse the position in the market.Many choices are possible. We expose here two of them:
- it can be the first time when the position generates a profit net of transaction costs
- another idea is to define new values Min (Max) such that y% of the spreads are below this value.
For example, if the signal is detected for x = 1, the position can be reversed in the market for y = 15, which means that the spread has now a more normal level.
Fixed Income Strategy 99
• Back-test of a systematic method on the French market
- We boost the performance of a monetary fund of Eur 50 million by benefiting of arbitrage opportunities detected by our model.
- Two different funds are created:one is defensive with a leverage coefficient of 2 as the other one is offensive with a leverage coefficient of 4.
- The Z-score analysis is performed over a 100-day period. The value x, which provides the signal to enter the position is equal to 3%. The fixed level, which is chosen to reverse the position is equal to 25%.
- Short and long positions are financed by means of the repo market. The repo rate raises by 50bp when the bond is cheap and decreases by 50bp when the bond is expensive.
Fixed Income Strategy 1010
• Back-test of a systematic method on the French market (2)
- An arbitrage opportunity is a pair of bonds which meets the three following rules:
* one bond cheap and one bond expensive* the difference of maturity between the two bonds is inferior to
1 year. * we buy a nominal of Eur 50 million of the cheap bond and sell the expensive bond for a nominal amount N such that the global position is $duration neutral.
- We applicate a stop-time of 30 calendar days on each position.
Fixed Income Strategy 1111
• Graph results
Evolution of the Net Asset Value from 31/05/95 to 31/12/01
50 000 000
55 000 000
60 000 000
65 000 000
70 000 000
75 000 000
80 000 000
85 000 000
90 000 000
31/05/95 26/03/96 20/01/97 16/11/97 12/09/98 09/07/99 04/05/00 28/02/01 25/12/01
Defensive Fund
Offensive Fund
Monetary Fund
Fixed Income Strategy 1212
• Regular performances
nb of months with positive performance for the defensive fund: 84 (100%)
mean of monthly total returns: 0.48%
higher total return: 3.47% (sept. 95) lower total return: 0.04% (oct. 95)
0,00%
0,50%
1,00%
1,50%
2,00%
2,50%
3,00%
3,50%
4,00%
Fixed Income Strategy 1313
• An uncorrelated strategy / An attractive Sharpe ratio
Money Market
French govt 10Y
MSCI Euro corporate
MSCI Euro Debt SP 500 CAC 40
DefensiveFund
Money Market 1,00 0,34 0,39 0,33 -0,06 -0,21 0,22French govt 10Y 1,00 0,87 0,94 0,00 0,03 -0,06MSCI Euro corporate 1,00 0,80 0,06 0,04 0,11MSCI Euro Debt 1,00 0,12 0,13 -0,01SP 500 1,00 0,68 0,08CAC 40 1,00 -0,12Defensive Fund 1,00
Money market
French govt 10Y
MSCI Euro corporate
MSCI Euro Debt SP 500 CAC 40
Def. Fund
risk 0,29% 2,96% 3,20% 3,66% 16,09% 20,31% 1,73%return 3,85% 6,54% 6,27% 7,93% 11,24% 13,33% 5,75%
Sharpe 0,912 0,758 1,115 0,460 0,467 1,097
Fixed Income Strategy 1414
• Risk measures
Skewness 3.84Kurtosis 17.58
Downside deviation 0.18%Upside deviation 0.46%
Maximum drawdown 0.97%Sortino ratio 3.08
Fixed Income Strategy 1515
• Leverage coefficients for the defensive fund
PON: Difference between bonds bought and bonds sold as a multiple of the initial value of the funds (Eur 50 million)
POA: Total of bonds bought as a multiple of the initial value of the funds (Eur 50 million)
POV: Total of bonds sold as a multiple of the initial value of the funds (Eur 50 million)
Leverage coefficients are multiplied by 2 for the offensive fund.
Max PON Min PON MoyennePON
Max POA MoyennePOA
Min POV MoyennePOV
1.96 -1.67 0.05 10.53 1.02 -11.25 -0.97
Fixed Income Strategy 1616
• Statistics on arbitrages
172 arbitrage opportunities from 31/05/95 to 31/12/01
average length of an arbitrage: 2 weeks
1- Total of transaction costs: Eur 7.5 million2- Total of repo costs: Eur -0.7 million
3- Total of gains: Eur 7.6 million
4- Total of gains for positive arbitrages: Eur 9 million5- Total of losses for negative arbitrages: Eur 1.4 million
6- Maximum gain for one arbitrage: Eur 3446167- Maximum loss for one arbitrage: Eur -138452
Fixed Income Strategy 1717
• Conclusion
At the moment, the number of arbitrage opportunities detected by the market is about 15 in a year.
To be really competitive, this method needs to be implemented on all the T-Bond markets of the Eurozone.
The model is also robust to consider arbitrage opportunities on investment grade markets.
See our Trade Ideas on HSBV (Bloomberg site of Fixed-Income Strategy) for such arbitrage opportunities.
Fixed Income Strategy 1818
Swap barbells and butterflies
Fixed Income Strategy 1919
Summary
Barbell/butterfly characteristics
Systematic positioning of numerous swap barbell/butterflies yields a high return
Trade-based rules revolve around Z-score measures that are adjusted to signal entry and exist of positions. Results are consistent for USD, EUR and GBP
Back-tests from 2000 to 2003 of 26 standard 50-50 and maturity-weighted swap barbells and butterflies identify more than 80% of profitable trades
Fixed Income Strategy 2020
P/L estimation of swap barbells and butterflies
For any $Duration-neutral butterfly, the approximate total return in $ is given by :
(1)
Where: Dm, Ds, Dl are the $Duration of the body, short- and long-wings, rm, rs and rl
the change in swap rates of the medium(body), short- and long-wings
and m, s and l are the weights which must satisfy the following constraint :
Rearranging (1) gives the following expression :
with
lllsssmmm rDrDrDLP &
lsmmm
lsmmm
lmm
lls
mm
ssmmm
rrrDLP
rrrDLP
rD
Dr
D
DrDLP
1&
1&
&
0 llssmm DDD
mm
ss
D
D
Fixed Income Strategy 2121
P/L estimation of swap barbells and butterflies
So the following spread measure is a good indicator of the performance of the butterfly :
In a barbell (a butterfly), the spread measure is expected to decrease (to increase)
Impact of the beta coefficient on the evolution of the spread measure
Relative value trades based on the assumption that this spread shows mean-reversion properties
A negative (positive) Z-score provides a signal to enter the butterfly (barbell)
lsm rrrSpread )1(
Fixed Income Strategy 2222
-70
-65
-60
-55
-50
-45
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.1
P/L estimation of swap barbells and butterflies
-55
-50
-45
-40
-35
-30
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.2
-35
-33
-31
-29
-27
-25
-23
-21
-19
-17
-15
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.3
-18
-16
-14
-12
-10
-8
-6
-4
-2
0
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.4
0
2
4
6
8
10
12
14
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.5
16
18
20
22
24
26
28
30
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.6
33
35
37
39
41
43
45
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.7
50
52
54
56
58
60
62
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.8
68
69
70
71
72
73
74
75
76
77
78
Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
Yie
ld S
pre
ad (
bp
)
2-5-10yr EUR Barbell @ β =0.9
Fixed Income Strategy 2323
P/L estimation of swap barbells and butterflies 50/50 swap buttefly
specific case with beta equals to 0.5
spread measure given by :
trade neutral to some small steepening and flattening movement as
Maturity-weighted butterfly
specific case with beta equals to
spread measure given by :
where Mm, Ms and Ml are the Maturities of the body, short- and long-wings
2ls
mrr
r
lsl
mls
sl
smm r
MM
MMr
MM
MMr
22
& mlsmmm
rrrrDLP
sl
sm
MM
MM
Fixed Income Strategy 2424
P/L estimation of swap barbells and butterflies Maturity-weighted butterfly
same weights as a 50/50 swap when Mm- Ms = Ml - Mm
designed to take into account the fact that short-term rates are much more volatile than long-term rates
neutral trade if the spread change between the long wing and the body is proportional to the spread change between the body and the short wing as shown by the following relationship :
Regression-weighted buttterfly
the coefficient beta is obtained by regressing the change in spread between the long wing and the body with the change in spread between the long wing and the short wing
this coefficient minimizes the variance of P&L of the position
smml
smsmml rr
MM
MMrrrr
1
Fixed Income Strategy 2525
P/L estimation of swap barbells and butterflies
Minimum Variance Butterfly
the idea is to minimize the variance of the spread measure as to increase the mean-reverting properties of the trades
the coefficient beta is the solution of the following minization program:
and is simply equal to the regression coefficient of the spread between the long wing and the body and the spread between the the long wing and the short wing
calculated over the last 100 working days
Combinations that are traditionally very directional when structured with the 50-50 weighting (such as 2-5-10 year, 2-5-30 year and 2-7-15 year) present stronger mean-reverting characteristics when a MV-weighting is used instead
lsm rrrVarMin )1(
Fixed Income Strategy 2626
P/L estimation of swap barbells and butterflies Minimum Variance Butterfly
8
10
12
14
16
18
20
22
24
26
28
Oct 03 Nov 03 Dec 03 Jan 04 Feb 04 Mar 04
50-5
0 Y
ield
Sp
read
(b
p)
92
94
96
98
100
102
104
106
108
110
112
MV
Yie
ld S
pre
ad (
bp
)
50-50 EUR 2-5-10 barbell (LHS)Minimum variance (MV) EUR 2-5-10 barbell (RHS)
Fixed Income Strategy 2727
Example: USD 2-5-10 50-50 barbell
30 July 03: Spread = 32bp Z-score = 2.7
8 August 03: Spread = 20bp Z-score = 0.9
Total return = 55bp
-5
0
5
10
15
20
25
30
35
Mar 03 Apr 03 May 03 Jun 03 Jul 03 Aug 03
Yie
ld S
pre
ad (
bp
)
2-5-10 yr USD barbell
Fixed Income Strategy 2828
Back-test results
Back-tests of 26 standard swap barbells/butterflies with different Z-scores from 2.5 to 5.0 (in absolute value) to enter the trade, and from 0.5 to 2.0 to exit the position
Additional constraints in terms of stop-time (between 20 and 60 working days) and number of trades (minimum of 150 trades)
Optimization with two criteria: cumulative total return and % of profitable trades
Best combinations (50-50 and maturity-weighted)
USD EUR GBPZ-score In 2.5 2.5 2.5Z-score Out 1.0 1.0 1.0Stop-time 40 working days 50 working days 60 working days
Fixed Income Strategy 2929
US statistics* for period 2000-2003
Source: HSBC *50-50 & maturity-weighted
-75
-50
-25
0
25
50
75
100
125
2000 2001 2002 2003 Total
To
tal
Ret
urn
s (b
p)
Max/Min Average Observations within +/-1sd
Fixed Income Strategy 3030
USD statistics* on different combinations
-5
0
5
10
15
20
25
30
35
40
45
45% 56% 67% 78% 89% 100%
Profitable Trades (%)
Ave
rag
e T
ota
l R
etu
rn (
bp
)
2-5-7
2-3-4
15-20-30
10-15-20
5-7-10
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3131
USD cumulative total returns*
0
10
20
30
40
50
60
70
80
2000 2001 2002 2003
Cu
mu
lati
ve T
ota
l R
etu
rns
(%)
USD cumulative total returns
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3232
USD annual cumulative returns*
0
10
20
30
40
50
60
70
80
90
Total 2000 2001 2002 2003
An
nu
al C
um
ula
tive
Ret
urn
s (%
)
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3333
USD - Statistics on trades
Number of trades = 454
These trades were initiated on 209 different dates with a maximum concentration of signals equal to 10 as of 11 Sep 01
Average carry = 17 working days
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3434
USD - Monthly distribution of trades
Source: HSBC
0
5
10
15
20
25
30
35
40
45
50
Mar 00 Dec 00 Sep 01 Jun 02 Mar 03 Dec 03
Nu
mb
er o
f T
rad
es
Maturity
50/50
Fixed Income Strategy 3535
EUR statistics for period 2000-2003*
-100
-80
-60
-40
-20
0
20
40
60
80
2000 2001 2002 2003 Total
To
tal
retu
rns
(bp
)
-100
-80
-60
-40
-20
0
20
40
60
80
Max/Min Average Observations within +/-1sd
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3636
EUR statistics on different combinations*
-2
0
2
4
6
8
10
12
14
16
60% 70% 80% 90% 100%
Profitable Trades (%)
Ave
rag
e T
ota
l R
etu
rn (
bp
)
2-5-10
2-7-15
3-5-7
15-20-307-10-15
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3737
GBP statistics for period 2000-2003*
-150
-100
-50
0
50
100
2000 2001 2002 2003 Total
To
tal
retu
rns
(bp
)
-150
-100
-50
0
50
100
Max/Min Average Observations within +/-1sd
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3838
GBP statistics on different combinations*
-10
-5
0
5
10
15
20
25
40% 60% 80% 100%
Profitable Trades (%)
Ave
rag
e T
ota
l R
etu
rn (
bp
)
7-15-20
3-5-7
7-10-15
15-20-30
Source: HSBC *50-50 & maturity-weighted
Fixed Income Strategy 3939
Revealing anomalies in forward and volatility curves
Fixed Income Strategy 4040
• Anomalies in forward curves
Forward rates are variables which are modelized for the pricing and hedging of fixed income derivatives
The pricing of the most simple products such as plain vanilla swaps or CMS swaps is obtained by discounting these forward rates
The detection of abnormal levels provide good opportunities to enter some trades
Example of a trade idea on the Euro Market on April 03:
EUR CMS curve steepener (see trade ideas on HSBV)
Fixed Income Strategy 4141
• 30 yr CMS forwards against 2 yr CMS forwards
Fixed Income Strategy 4242
• Forwards implying inversion of 30-2 yr curve
Fixed Income Strategy 4343
• EUR CMS curve steepener
The two previous figures show that the spread 30yr-2yr becomes negative after 2009, reaching a maximum of -57bp on 2019. Historical precedent suggest that this is very unlikely as since 1999, the flattest that the swap curve has been is in August 2000 when it reached +48bp.
There is an opportunity to enter a 10 year (or more) maturity swap to receive the 30 year CMS rate and pay the 2 year CMS rate. The value of the swap is zero at inception.
We implement a scenario analysis to judge the risk/return profile of that product.
Fixed Income Strategy 4444
• Results of the scenario analysis
The trade will be profitable as soon as the forward spread becomes positive.
The trade has a positive time value so as time passes it becomes more and more profitable.
Risks to this strategy centre on the forward spread becoming more negative over the next five years, making the value of the swap negative.
Also the curve could become inverted during the period 2009-2019.
Fixed Income Strategy 4545
• Swaption break-evens
We define the break-even of two swaptions on the same swap (for example a 10-year swap) with two different maturities t and T as the volatility which should be realized between t and T so that the two swaptions are correctly priced at the current date 0.
Denoting by and vol(T) the volatilities of the two
swaptions with maturities t and T, we have:
where is the break-even between t and T.
t
dsst
tvol0
2 )(1
)(
222 ).()(.)(. tTBEtTtvoltTvolT
tTBE
Fixed Income Strategy 4646
Finally we obtain
When the quantity is negative, we consider that the break-even is equal to zero.
• Detecting anomalies
Irregular break-evens can reveal good opportunities to enter trades.
tT
tvoltTvolTBEtT
22 )(.)(.
22 )(.)(. tvoltTvolT
Fixed Income Strategy 4747
Example: On 8 July 2003, EUR swaption break-evens for the 2-year maturity swap were:
The break-even is equal to zero which shows that the volatility of the 3-year maturity swaption is too low relatively to the volatility of the 2-year maturity swaption.
Between 8 July 2003 and 29 July 2003, the volatility of the 2-year and 3-year maturity swaption increased by 0.1% and 1% respectively, with the consequences that the break-even was on 29 July 2003 at a more adequate level of 11.8%.
mmBE63 yyBE21 yyBE32 yyBE75 yyBE107
29.6%12.9% 0% 9.8% 7%
Fixed Income Strategy 4848
• HSBV Bloomberg site
Fixed Income Strategy 4949
• References
L. Martellini, P. Priaulet and S. Priaulet, “Understanding the butterfly strategy”, Journal of Bond Trading and Management, 1(1), 9-19, 2002.
L. Martellini, P. Priaulet and S. Priaulet, “Fixed-Income Securities: Valuation, Risk Management and Portfolio Strategies”, Wiley, 2003.
F. Fabozzi, C. Dialynas, L. Martellini and P. Priaulet, “Indexing, Structured and Active Fixed-Income Portfolio Management”, Wiley, forthcoming 2005.
Fixed Income Strategy 5050
• Disclaimer
"Issued by CCF, a member of the HSBC Group. This material is for institutional and professional clients only and not for private customers.Courses and materials are for general information only and do not constitute recommendations or solicitation of any activity in relation to any investment. Accuracy or completeness of courses and materials cannot be guaranteed : any opinions therein are given in good faith but are subject to change without notice. Persons who attend a course or receive materials should make their own independent assessment of the merits or suitability of any investment referred to. No liability whatsoever is accepted by any member of HSBC Group for any direct or consequential loss arising from reliance upon information provided in a course or materials."