UNDERSTANDING THE BUTTERFLY STRATEGY RI NOTES N°2002-01 L. MARTELLINI, P. PRIAULET and S. PRIAULET THE WORLD'S LOCAL BANK RESEARCH AND INNOVATION NOTES – N° 2002-01 UNDERSTANDING THE BUTTERFLY STRATEGY Lionel Martellini1, Philippe Priaulet2 et Stéphane Priaulet 3 Abstract A butterfly, which is a combination of a barbell and a bullet, is one of the most common active fixed-income strategies used by practitioners. While being neutral to small parallel shifts of the yield curve, a butterfly is purposely exposed to specific bets on particular changes of the yield curve. There exist four different types of butterflies, the cash-and $duration neutral weighting butterfly, the fifty-fifty weighting regression, the regression weighting butterfly and the maturity weighting butterfly. In this paper, we show that they generate a positive pay-off when the particular flattening or steepening move of the yield curve they were structured for capturing occurs. We also argue that one suitable way to detect the opportunity to enter a specific butterfly is to use spread indicators. Finally, we show that the curvature $duration obtained from the Nelson and Siegel (1987) model can be used to measure the curvature risk of this strategy. Keywords : fixed-income portfolio management, active strategy, butterfly, slope movement, spread indicators, curvature risk. Première version : March 2002 Dernière version : March 2002 Version : 1 1 Lionel Martellini is currently an Assistant Professor of Finance at the Marshall School of Business, University of Southern California. Email: [email protected] 2 Philippe Priaulet is Head of Fixed Income Research in the Research and Innovation Department of HSBC- CCF. Postal Address: CCF DRI – DEFE - 75419 Paris Cedex 08, France. Tél: (+33) 1 40 70 34 83. Fax: (+33) 1 40 70 30 31. E-mail: [email protected] 3 Stéphane Priaulet is Head of Quantitative Engineering in the Quantitative Research and Management team of the Structured Asset Management department at AXA Investment Managers. Email: [email protected] Research and Innovation Notes - N° 2002-01 CONTENTS INTRODUCTION................................................................................................................................................................................1 1. A CONVEX TRADE.......................................................................................................................................................................2 2. DIFFERENT KINDS OF BUTTERFLIES ..............................................................................................................................3 2.1 CASH AND $DURATION NEUTRAL WEIGHTING...................................................................................................................... 4 2.2 FIFTY-FIFTY WEIGHTING............................................................................................................................................................ 4 2.3 REGRESSION WEIGHTING........................................................................................................................................................... 5 2.4 MATURITY-WEIGHTING.............................................................................................................................................................. 6 3 HOW TO MEASURE THE PERFORMANCE AND THE RISK OF A BUTTERFLY ?...........................................7 3.1 TOTAL RETURN MEASURE ......................................................................................................................................................... 7 3.2 SPREAD MEASURES..................................................................................................................................................................... 9 4. LEVEL, SLOPE AND CURVATURE $DURATION RISK MEASURES ...................................................................10 CONCLUSION ...................................................................................................................................................................................13 REFERENCES ...................................................................................................................................................................................14 LIST OF RESEARCH AND INNOVATION NOTES................................................................................................................15 Research and Innovation Notes - N° 2002-01 UNDERSTANDING THE BUTTERFLY STRATEGY INTRODUCTION A butterfly is one of the most common fixed-income active strategies used by practitioners.4 It is the combination of a barbell5 (called the wings of the butterfly) and a bullet6 (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. On the other hand, as we let the yield curve be affected by more complex movements than parallel shifts, including slope and curvature movements, the performance of the strategy can be drastically impacted. It is in general fairly complex to know under which exact market conditions a given butterfly generates positive or negative pay-offs when all these possible movements are accounted for. There actually exist many different kinds of butterflies (some of which are not cash-neutral), which are structured so as to generate a positive pay-off in case a particular move of the yield curve occurs. Finally, we address the question of measuring the performance and the risk of a butterfly. 4 See Choudry (2001) for a description of this strategy including in particular an example of a butterfly analysis on Bloomberg. 5 A barbell portfolio is constructed by concentrating investments on the short-term and the long-term ends of the yield curve. 6 A bullet portfolio is constructed by concentrating investments on a particular maturity of the yield curve. - 1 - www.dri-ccf.com Research and Innovation Notes - N° 2002-01 1. A CONVEX TRADE 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. This point is illustrated in the following example. Example 1 We consider three bonds with short, medium and long maturities whose features are summarized in the following table. 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 These bonds are hypothetical bonds assumed to be default-risk free. The face value of bonds is normalized to be $100, YTM stands for yield-to-maturity, bond prices are dirty prices and we assume a flat yield-to-maturity curve in this example. We structure a butterfly in the following way: - we sell 1,000 5-year maturity bonds - we buy qs 2-year maturity bonds and ql 10-year maturity bonds The quantities qs and ql are determined so that the butterfly is cash and $duration neutral, i.e., we impose that they satisfy the following system: ì(q ´185.9) + (q ´ 772.2) =1,000 ´ 432.9 í s l î (qs ´100) + (ql ´100) = 1,000 ´100 whose solution is -1 æq s ö æ185.9 772.2ö æ432,900 ö æ578.65 ö ç ÷ = ç ÷ ´ ç ÷ = ç ÷ è ql ø è 100 100 ø è100,000 ø è421.35ø Of course, in a real market situation, we would buy 579 2-year maturity bonds and 421 10-year maturity bonds.7. We now draw the profile of the strategy gain depending on the value of the yield to maturity (see Figure 1). 7 On the market you actually buy bonds in terms of amount and not in terms of number of securities. You will or not round the amount you want to buy, depending on the minimum amount of the security that can be purchased. Here, for simplicity, the quantity of bonds is measured in terms of number. - 2 - www.dri-ccf.com Research and Innovation Notes - N° 2002-01 Figure 1. Profile of the P&L's butterfly strategy depending on the value of the yield to maturity 12 10 8 6 Strategy Gain 4 2 0 0% 2% 4% 6% 8% 10% 12% Yield to Maturity The butterfly has a positive convexity. Whatever the value of the yield to maturity, the strategy always generates a gain. This gain is all the more substantial as the yield to maturity reaches a level further away from 5%. 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%. We know, however, 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 movements8. It is in general fairly complex to know under what exact market conditions a given butterfly might generate a positive or a negative pay- off when all these possible movements are accounted for9. Some butterflies are structured so as to pay off if a particular move of the yield curve occurs. 2. DIFFERENT KINDS OF BUTTERFLIES While a feature common to all butterflies is that they always have a $duration equal to zero, they actually come in many very different shapes and forms that we now examine in details. In the case of a standard butterfly,