No Slide Title
Download
Report
Transcript No Slide Title
EXPLORING SOME ALTERNATIVE
FIXED-INCOME STRATEGIES
Philippe PRIAULET
HSBC-CCF and University of EVRY
2 AVRIL 2004
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
2
Bond picking strategies
Fixed Income Strategy
3
• 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
4
• 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
5
• 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.
S t 1 is the value of the spread at date t+1.
S
Min
2- When t 1
converges to 1 or exceeds 1, the bond is considered
Max Min
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.
Fixed Income Strategy
6
• 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
Historical Distribution
16
Fréquency
14
12
10
8
6
4
2
0.08%
0.06%
0.04%
0.02%
0.00%
-0.02%
-0.04%
-0.06%
-0.08%
-0.10%
0
Classes
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.
Fixed Income Strategy
7
• 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
8
• 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
9
• 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
10
• Graph results
Evolution of the Net Asset Value from 31/05/95 to 31/12/01
90 000 000
85 000 000
80 000 000
Defensive Fund
Offensive Fund
75 000 000
Monetary Fund
70 000 000
65 000 000
60 000 000
55 000 000
50 000 000
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
Fixed Income Strategy
11
• 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)
4,00%
3,50%
3,00%
2,50%
2,00%
1,50%
1,00%
0,50%
m
ai
-9
se 5
pt
-9
ja 5
nv
-9
m 6
ai
-9
se 6
pt
-9
ja 6
nv
-9
m 7
ai
-9
se 7
pt
-9
ja 7
nv
-9
m 8
ai
-9
se 8
pt
-9
ja 8
nv
-9
m 9
ai
-9
se 9
pt
-9
ja 9
nv
-0
m 0
ai
-0
se 0
pt
-0
ja 0
nv
-0
m 1
ai
-0
se 1
pt
-0
ja 1
nv
-0
2
0,00%
Fixed Income Strategy
12
• An uncorrelated strategy / An attractive Sharpe ratio
Money Market
French govt 10Y
MSCI Euro corporate
MSCI Euro Debt
SP 500
CAC 40
Defensive Fund
risk
return
Sharpe
Money
market
0,29%
3,85%
Money
Market
1,00
French govt
10Y
0,34
1,00
French govt
10Y
2,96%
6,54%
0,912
MSCI Euro
corporate
0,39
0,87
1,00
MSCI Euro
corporate
3,20%
6,27%
0,758
MSCI Euro
Debt
0,33
0,94
0,80
1,00
MSCI Euro
Debt
3,66%
7,93%
1,115
SP 500
-0,06
0,00
0,06
0,12
1,00
SP 500
16,09%
11,24%
0,460
CAC 40
-0,21
0,03
0,04
0,13
0,68
1,00
CAC 40
20,31%
13,33%
0,467
Defensive
Fund
0,22
-0,06
0,11
-0,01
0,08
-0,12
1,00
Def. Fund
1,73%
5,75%
1,097
Fixed Income Strategy
13
• Risk measures
Skewness
Kurtosis
3.84
17.58
Downside deviation
Upside deviation
0.18%
0.46%
Maximum drawdown
Sortino ratio
0.97%
3.08
Fixed Income Strategy
14
• Leverage coefficients for the defensive fund
Max PON Min PON Moyenne Max POA Moyenne Min POV Moyenne
PON
POA
POV
1.96
-1.67
0.05
10.53
1.02
-11.25
-0.97
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.
Fixed Income Strategy
15
• 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 million
2- 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 million
5- Total of losses for negative arbitrages: Eur 1.4 million
6- Maximum gain for one arbitrage: Eur 344616
7- Maximum loss for one arbitrage: Eur -138452
Fixed Income Strategy
16
• 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 TBond 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
17
Swap barbells and butterflies
Fixed Income Strategy
18
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
19
P/L estimation of swap barbells and butterflies
For any $Duration-neutral butterfly, the approximate total return in $ is given by :
P & L m Dmrm s Ds rs l Dl rl
(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 :
m Dm s Ds l Dl 0
Rearranging (1) gives the following expression :
D
D
P & L m Dm rm s s rs l l rl
m Dm
m Dm
P & L m Dm rm rs 1 rl
P & L m Dm rm rs 1 rl
with
s Ds
m Dm
Fixed Income Strategy
20
P/L estimation of swap barbells and butterflies
So the following spread measure is a good indicator of the performance of the butterfly :
Spread rm rs (1 )rl
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)
Fixed Income Strategy
21
P/L estimation of swap barbells and butterflies
Yield
(bp)
Yield Spread
Spread (bp)
-45
-30
-15
14
30
45
62
78
0
-17
77
-2
12
28
43
60
-50
-35
-19
76
-4
10
26
-21
75
41
58
-6
-55
-40
-23
74
24
8
-8
-25
39
56
73
-10
22
6
-60
-45
-27
72
-12
37
54
-29
71
20
4
-14
-65
-50
-31
70
35
52
18
2
-16
-33
69
-70
-55
-35
-18
16
33
50
68
0
Nov 03
Dec 03
Jan 04
Feb 04
Mar 04
2-5-10yr
2-5-10yr EUR
EUR Barbell
Barbell @
@ ββ =0.8
=0.5
=0.6
=0.7
=0.9
=0.1
=0.2
=0.3
=0.4
Fixed Income Strategy
22
P/L estimation of swap barbells and butterflies
50/50 swap buttefly
specific case with beta equals to 0.5
spread measure given by :
r r
rm s l
2
trade neutral to some small steepening and flattening movement as
r r r r
P & L m Dm m s l m
2
2
Maturity-weighted butterfly
specific case with beta equals to
spread measure given by :
Mm Ms
Ml M s
Mm Ms
Ml M m
rl
rm
rs
Ml M s
Ml M s
where Mm, Ms and Ml are the Maturities of the body, short- and long-wings
Fixed Income Strategy
23
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 :
rl rm
M Ms
rm rs m
rm rs
1
Ml M m
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
Fixed Income Strategy
24
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 meanreverting properties of the trades
the coefficient beta is the solution of the following minization program:
Min Var rm rs (1 )rl
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 meanreverting characteristics when a MV-weighting is used instead
Fixed Income Strategy
25
P/L estimation of swap barbells and butterflies
28
112
26
110
24
108
22
106
20
104
18
102
16
100
14
98
12
96
10
94
8
92
Oct 03
MV Yield Spread (bp)
50-50 Yield Spread (bp)
Minimum Variance Butterfly
Nov 03
Dec 03
Jan 04
Feb 04
Mar 04
50-50 EUR 2-5-10 barbell (LHS)
Minimum variance (MV) EUR 2-5-10 barbell (RHS)
Fixed Income Strategy
26
Example: USD 2-5-10 50-50 barbell
35
30 July 03:
Spread = 32bp
Z-score = 2.7
Yield Spread (bp)
30
25
8 August 03:
Spread = 20bp
Z-score = 0.9
20
15
Total return = 55bp
10
5
0
-5
Mar 03
Apr 03
May 03
Jun 03
Jul 03
Aug 03
2-5-10 yr USD barbell
Fixed Income Strategy
27
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
GBP
Z-score In
2.5
2.5
2.5
Z-score Out
1.0
1.0
1.0
Stop-time 40 working days 50 working days 60 working days
Fixed Income Strategy
28
US statistics* for period 2000-2003
125
Total Returns (bp)
100
75
50
25
0
-25
-50
-75
2000
Max/Min
Source: HSBC
2001
Average
2002
2003
Total
Observations within +/-1sd
*50-50 & maturity-weighted
Fixed Income Strategy
29
USD statistics* on different combinations
Average Total Return (bp)
45
15-20-30
40
35
30
25
20
10-15-20
5-7-10
15
10
5
2-3-4
0
-5
45%
2-5-7
56%
67%
78%
89%
100%
Profitable Trades (%)
Source: HSBC
*50-50 & maturity-weighted
Fixed Income Strategy
30
USD cumulative total returns*
Cumulative Total Returns (%)
80
70
60
50
40
30
20
10
0
2000
2001
2002
2003
USD cumulative total returns
Source: HSBC
*50-50 & maturity-weighted
Fixed Income Strategy
31
Annual Cumulative Returns (%)
USD annual cumulative returns*
90
80
70
60
50
40
30
20
10
0
Total
Source: HSBC
2000
2001
2002
2003
*50-50 & maturity-weighted
Fixed Income Strategy
32
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
33
USD - Monthly distribution of trades
50
Number of Trades
45
Maturity
40
50/50
35
30
25
20
15
10
5
0
Mar 00
Dec 00
Sep 01
Jun 02
Mar 03
Dec 03
Source: HSBC
Fixed Income Strategy
34
Total returns (bp)
EUR statistics for period 2000-2003*
80
80
60
60
40
40
20
20
0
0
-20
-20
-40
-40
-60
-60
-80
-80
-100
-100
2000
Max/Min
Source: HSBC
2001
Average
2002
2003
Total
Observations within +/-1sd
*50-50 & maturity-weighted
Fixed Income Strategy
35
EUR statistics on different combinations*
Average Total Return (bp)
16
14
12
10
15-20-30
8
7-10-15
6
3-5-7
4
2
0
2-5-10
-2
60%
2-7-15
70%
80%
90%
100%
Profitable Trades (%)
Source: HSBC
*50-50 & maturity-weighted
Fixed Income Strategy
36
Total returns (bp)
GBP statistics for period 2000-2003*
100
100
50
50
0
0
-50
-50
-100
-100
-150
-150
2000
Max/Min
Source: HSBC
2001
Average
2002
2003
Total
Observations within +/-1sd
*50-50 & maturity-weighted
Fixed Income Strategy
37
GBP statistics on different combinations*
Average Total Return (bp)
25
20
15-20-30
15
7-10-15
10
5
3-5-7
0
-5
-10
40%
7-15-20
60%
80%
100%
Profitable Trades (%)
Source: HSBC
*50-50 & maturity-weighted
Fixed Income Strategy
38
Revealing anomalies in forward and volatility curves
Fixed Income Strategy
39
• 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
40
• 30 yr CMS forwards against 2 yr CMS forwards
Fixed Income Strategy
41
• Forwards implying inversion of 30-2 yr curve
Fixed Income Strategy
42
• 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
43
• 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
44
• 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 vol(t )
1 t 2
( s )ds and vol(T) the volatilities of the two
t 0
swaptions with maturities t and T, we have:
2
T .vol(T ) 2 t.vol(t ) 2 (T t ).BEtT
where BEtT is the break-even between t and T.
Fixed Income Strategy
45
Finally we obtain
BEtT
T .vol(T ) 2 t.vol(t ) 2
T t
When the quantity T .vol(T ) 2 t.vol(t ) 2
break-even is equal to zero.
is negative, we consider that the
• Detecting anomalies
Irregular break-evens can reveal good opportunities to enter trades.
Fixed Income Strategy
46
Example: On 8 July 2003, EUR swaption break-evens for the 2-year maturity
swap were:
BE3m6m
29.6%
BE1 y 2 y
12.9%
BE2 y 3 y
0%
BE5 y 7 y
9.8%
BE7 y10 y
7%
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%.
Fixed Income Strategy
47
• HSBV Bloomberg site
Fixed Income Strategy
48
• 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
49
• 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."
Fixed Income Strategy
50