EXPLORING SOME ALTERNATIVE FIXED-INCOME STRATEGIES Philippe PRIAULET HSBC-CCF and University of EVRY 2 AVRIL 2004

1. EXPLORING SOME ALTERNATIVE FIXED-INCOME STRATEGIES Philippe PRIAULET HSBC-CCF and University of EVRY 2 AVRIL 2004

2. 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

3. Bond picking strategies

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. 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.

11. Graph results

12. 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)

13. An uncorrelated strategy / An attractive Sharpe ratio Money French govt MSCI Euro MSCI Euro Defensive Market 10Y corporate Debt SP 500 CAC 40 Fund 0,22 Money Market 1,00 0,34 0,39 0,33 -0,06 -0,21 -0,06 French govt 10Y 1,00 0,87 0,94 0,00 0,03 0,11 MSCI Euro corporate 1,00 0,80 0,06 0,04 -0,01 MSCI Euro Debt 1,00 0,12 0,13 0,08 SP 500 1,00 0,68 -0,12 CAC 40 1,00 1,00 Defensive Fund Money French govt MSCI Euro MSCI Euro Def. Fund market 10Y corporate Debt SP 500 CAC 40 risk 0,29% 2,96% 3,20% 3,66% 16,09% 20,31% 1,73% 3,85% 6,54% 6,27% 7,93% 11,24% 13,33% 5,75% return Sharpe 0,912 0,758 1,115 0,460 0,467 1,097

14. Risk measures Skewness 3.84 Kurtosis 17.58 Downside deviation 0.18% Upside deviation 0.46% Maximum drawdown 0.97% Sortino ratio 3.08

15. 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.

16. 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

17. 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.

18. Swap barbells and butterflies

19. 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

20. 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

21. 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)

22. P/L estimation of swap barbells and butterflies

23. 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

24. 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

25. 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

26. P/L estimation of swap barbells and butterflies • Minimum Variance Butterfly

27. 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

28. 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)

29. US statistics* for period 2000-2003 Source: HSBC *50-50 & maturity-weighted

30. USD statistics* on different combinations Source: HSBC *50-50 & maturity-weighted

31. USD cumulative total returns* Source: HSBC *50-50 & maturity-weighted

32. USD annual cumulative returns* Source: HSBC *50-50 & maturity-weighted

33. 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

34. USD - Monthly distribution of trades Source: HSBC

35. EUR statistics for period 2000-2003* Source: HSBC *50-50 & maturity-weighted

36. EUR statistics on different combinations* Source: HSBC *50-50 & maturity-weighted

37. GBP statistics for period 2000-2003* Source: HSBC *50-50 & maturity-weighted

38. GBP statistics on different combinations* Source: HSBC *50-50 & maturity-weighted

39. Revealing anomalies in forward and volatility curves

40. 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)

41. 30 yr CMS forwards against 2 yr CMS forwards

42. Forwards implying inversion of 30-2 yr curve

43. 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.

44. 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.

45. 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.

46. 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.

47. 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%.

48. HSBV Bloomberg site

49. 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.

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