FIXED-INCOME SECURITIES
Chapter 8
Active Fixed-Income Portfolio Management
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
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)
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)
Riding the Yield Curve • Riding the yield curve is a technique that fixedincome 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
Example • Example – 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 1y 2y 3y 4y 5y
Zero-coupon rate 3.90% 4.50% 4.90% 5.25% 5.60%
Price at t=0
Price at t=1
$102.021 $102.842 $103.098 $102.848 $102.077
$102.021 $102.842 $103.098 $102.848
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 $102.021 – Hold until maturity – Total return
106 − 102.021 = 3.9% 102.021
Example – Cont’ • Option 2.1 – – – –
Buys at $102.842 at t=0 Receives a $6 coupon payment at t=1 Sells the bond with 1-year residual maturity at $102.021 Total return 6 + 102.021 − 102.842
• Option 2.2 • Option 2.3 • Option 2.4
102.842
= 5.036%
6 + 102.842 − 103.098 = 5.571% 103.098 6 + 103.098 − 102.848 = 6.077% 102.848 6 + 102.848 − 102.077 = 6.633% 102.077
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%
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
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 6.296 years, $ duration is -$670 – In case of a 1% drop in YTM • Absolute Profit = - $670x(-1%) = $6.7 • Relative Profit = - 6.296x(-1%) = 6.296%
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
Example – Expect Decrease in Rates • Example – Consider at date t a flat 5% curve and five bonds delivering annual coupon rates with the following features
Bond 1 2 3 4 5
Maturity 2y 10 y 30 y 30 y 30 y
Coupon rate 5% 5% 5% 7.5% 10%
YTM
Price
5% 5% 5% 5% 5%
$100 $100 $100 $138.43 $176.86
– 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
Example – Cont’ Bond 1 2 3 4 5
Modified Duration 1.859 7.722 15.372 14.269 13.646
$Duration 185.9 772.2 1537.2 1975.3 2413.4
Relative
Gain 0.936% 3.956% 8.144% 7.538% 7.196%
Absolute
Gain $0.936 $3.956 $8.144 $10.436 $12.727
• 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%
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 – Hold it until maturity – Buy in one year a 4-year maturity bond
• Suppose interest rates stay at 6% from then on
Roll-Over Example – Cont’ • Option 1 – Cash-flows Dates Cash-Flows Dates Cash-Flows
1 year later
T=0 -$100 3 years later $5
$5 4 years later $5
2 years later $5 5 years later $105
– Annual rate of return 1/ 5
128.85 − 100 100
− 1 = 5.092%
where
128.85 = 5 ×1.06 4 + 5 ×1.063 + 5 × 1.06 2 + 5 × 1.06 + 105
Roll-Over Example – Cont’ • Option 2 – Cash-flows Dates Cash-Flows Dates Cash-Flows
T=0 -$100 3 years later $6
1 year later $5 4 years later $6
2 years later $6 5 years later $106
– Annual rate of return 1/ 5
132.56 − 100 100
− 1 = 5.8%
where
132.56 = 5 × 1.06 4 + 6 × 1.063 + 6 ×1.06 2 + 6 ×1.06 + 106
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
Barbells, Bullets and Ladders • Bullets – 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 2year 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
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
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 payoff if the yield curve is affected by a positive or a negative parallel shift • Example: 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
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 ( qs ×185.9) + ( ql × 772.2 ) = (1000 × 432.9) (qs ×100) + (ql × 100) = (1000 ×100) • Solution: qs = 578.65 and ql = 421.35
Example - P&L Profile 1200
1000
Total Return in $
800
600
400
200
0 0%
2%
4%
6%
8%
10%
12%
Yield to Maturity
• •
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%
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
Cash and Dollar Duration Neutral Maturity YTM Bond Price Quantity $Duration 2 years
4. 5%
100 . 936
qs
188 . 6
5 years
5. 5%
97. 865
10, 000
421 . 17
10 years
6%
92. 64
ql
701 . 14
• 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
( qs × 188.6) + ( ql × 701.14 ) = (10,000 × 421.17 ) (qs × 100.936) + (ql × 92.64) = (10,000 × 97.865)
• Solution: qs = 5,553.5 and ql = 4,513.1
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'‘
( qs ×188.6 ) + ( ql × 701.14 ) = (10,000 × 421.17 ) qs = 11,165.7 ⇒ (qs × 188.6) − (ql × 701.14) = 0 ql = 3,003.5
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
( qs × 188.6 ) + ( ql × 701.14 ) = (10,000 × 421.17 ) q = 7,443.8 s ⇒ 1 (qs ×188.6) − ( × ql × 701.14) = 0 ql = 4,004.7 2
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 Mm − Ms β=
Ml − Mm
( qs ×188.6 ) + ( ql × 701.14 ) = (10,000 × 421.17 ) Mm − Ms qs = 8,374.3 (qs × 188.6) − ( M − M × ql × 701.14) = 0 ⇒ ql = 3,754.4 l m =3 5
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
Fixed-Income Active Asset Allocation Strategic Versus Tactical Asset Allocation Constant weights
Strategic
Unconditional
Slow evolving weights
Dynamic weights
Tactical
Conditional
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
Fixed-Income Active Asset Allocation Performance of a Perfect Timer
T-Bond Investment Grade High Yield Mean Standard Deviation Year 1994 1995 1996 1997 1998 1999 2000 2001 Average St. Dev.
T-Bond -10.22% 10.90% -4.61% 3.72% 2.31% -8.07% 8.32% -1.34% 0.12% 7.07%
T-Bond
Investment Grade
High Yield
1.00 0.77 0.12
1.00 0.24
1.00
0.37% 4.92%
0.40% 5.60%
0.22% 6.64%
Corporate -12.42% 13.08% -5.14% 4.51% -1.41% -8.68% 5.19% 2.07% -0.35% 7.74%
High Yield -11.39% 9.45% 0.20% 2.63% -9.00% -6.80% -7.46% 5.89% -2.06% 7.15%
LGBI -10.86% 10.24% -4.03% 3.49% 0.37% -7.70% 5.95% 0.09% -0.31% 6.57%
Perfect Timer -10.22% 13.08% 0.20% 4.51% 2.31% -6.80% 8.32% 5.89% 2.16% 7.19%
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 +59.22%), on average the Lehman high yield index under performs the Lehman Investment grade index by 7.64% more than the unconditional mean
Fixed-Income Active Asset Allocation Contemporaneous Factor Analysis - Example
Change in Implicit Volatility Difference between Conditionnal Values and Unconditionnal Values
Low
Medium
High
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum
-33.15%
-4.39%
-3.80%
4.45%
4.46%
59.22%
Unconditionnal Values
Mean
Stdev
Mean
Stdev
Mean
Stdev
Mean
Stdev
Correlation
CREDIT BOND INDEX - LEHMAN AGGREGATE
0.87%
0.26%
0.45%
-0.31%
-1.32%
-0.02%
0.25%
1.73%
-0.33
HIGH YIELD BOND INDEX - LEHMAN AGGREGATE
7.40%
-0.42%
1.56%
-0.31%
-8.96%
0.10%
0.07%
7.40%
-0.44
TREASURY BOND INDEX - LEHMAN AGGREGATE
0.14%
0.01%
2.89%
-0.20%
-3.03%
-0.01%
0.21%
2.89%
0.00
HIGH YIELD BOND INDEX - CREDIT BOND INDEX
6.53%
-0.17%
1.11%
-0.21%
-7.64%
-0.02%
-0.18%
7.59%
-0.36
Performance of Bond Indices under Different Contemporaneous Economic Conditions – the Example of Changes in Implicit Volatility (monthly data over the period 1991-2001 )
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
Fixed-Income Active Asset Allocation Lagged Factor Analysis - Example
Term spread Difference between Conditionnal Values and Unconditionnal Values
Low
Medium
High
Minimum
Maximum
Minimum
Maximum
Minimum
Maximum
-0.61%
0.99%
1.03%
2.35%
2.48%
3.91%
Unconditionnal Values
Mean
Stdev
Mean
Stdev
Mean
Stdev
Mean
Stdev
Correlation
CREDIT BOND INDEX - LEHMAN AGGREGATE
-0.81%
0.54%
0.52%
-0.49%
0.29%
-0.20%
0.24%
1.75%
0.09
HIGH YIELD BOND INDEX - LEHMAN AGGREGATE
-3.17%
1.52%
-0.73%
-0.18%
3.90%
-1.64%
0.05%
7.44%
0.12
TREASURY BOND INDEX - LEHMAN AGGREGATE
0.33%
-1.42%
-0.04%
0.41%
-0.29%
0.63%
0.26%
2.92%
-0.03
HIGH YIELD BOND INDEX - CREDIT BOND INDEX
-2.37%
1.28%
-1.25%
-0.26%
3.61%
-1.14%
-0.19%
7.62%
0.10
Performance of Bond Indices under Different Lagged Economic Conditions – The Example of the Term Spread (monthly data over the period 1991-2001 )
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)
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
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)
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)). Risk Return Analysis
TSA Fund
Lehman Brothers Global
Cumulative Return
36.31%
-2.65%
Annualised Return
10.45%
-0.84%
Annualised Std Deviation
4.16%
3.46%
Downside Deviation (3.0%)
2.56%
3.46%
Sortino (3.0%)
2.91
-1.11
Sharpe (Risk Free Rate = 3.0%)
1.79
-1.11
1st Centile
-1.54%
-2.30%
% Negative Returns
11.11%
52.78%
Up Months in Up Market
88.24%
Down Months in Down Market
10.53%
Up Market Outperformance
52.94%
Down Market Outperformance
89.47%
Worst Monthly Drawdown
-2.08%
-2.48%
Maximum Drawdown
-2.08%
-6.75%
1
7
in progress
26
Months in Max Drawdown Months to recover
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
• Pure 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
• Speculative arbitrage opportunities – Detect rich and cheap securities that historically present abnormal yield-tomaturity – Taking as reference a theoretical zero-coupon yield curve fitted with bond prices
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
Trading on Market Inefficiencies Example • On 09/14/01 – The price of strips with maturity 02/15/02, 02/15/03 and 02/15/04 are respectively 99.07, 96.06 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 0.9907 + 10 x 0.9606 +110 x 0.9254 = 121.307 – 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 (121.5-121.307)% = $19,300
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
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
Trading on Market Inefficiencies Z-Score Analysis • Criterion 1, based on a normality assumption – Compute the average value m and standard deviation σ of, say, the last 60 spreads – Assume the spread S is normally distributed and define the normalized spread U = (S-m)/2σ – In particular, we have that Proba ( -1 < U < 1 ) = 0.9544 – 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)/kσ
• Example – Mean value m and standard deviation σ of the 60 last spreads are: m=0.03% and σ =0.04% – One day later, the new spread is -0.11% – Considering a confidence level k=3, we obtain U = (S-m)/3σ = -1.166 – This bond is expensive and can be shorted – Confidence level is high: Proba ( -1 < U < 1 ) = 0.9973
Trading on Market Inefficiencies Z-Score Analysis – Con’t • Criterion 2, more general – Define the value Min such that α% of the spreads are below that value, and the value Max such that α% of the spreads are above that value – Take α% 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=0.0888%, 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 0.0775% and (S-Min)/(Max-Min)=1.063 – We conclude that this bond is cheap and can be bought
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 α which provides signals for short and long positions is 1% The level β which is chosen to reverse the position is equal to 15% Taking β >α ensures Min(β )>Min(α) and Max(β )<Max(α), 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 3.17. The ratio (S-Min/(Max-Min) is equal to 1.359. 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)
Trading on Market Inefficiencies Historical Distribution of the Spread on 10/04/96
20 18 16
Frequency in %
14 12 10 8 6 4 2 0 -0.7
-0.5
-0.3
-0.1
0.1
0.3
0.5
0.7
0.9
Classes
1.1
1.3
1.5
1.7
1.9
2.1
2.3
Trading on Market Inefficiencies Position and P&L Bond OAT 02/27/04
Price MD YTM Position Investment 5.39
5.64
Long
-9,999,977
OAT 10/25/04 112.81 5.82
5.74
Short
9,276,006
Date
Bond
10/07/96 OAT 02/27/04
120.4
Price Position Investment
TC
120.4
Long
-9,999,977
2,500
10/07/96 OAT 10/25/04 112.81
Short
9,276,006
2,319
10/17/96 OAT 02/27/04 120.92
Short
10,042,836
2,511
10/17/96 OAT 10/25/04 112.97
Long
-9,289,568
2,322
Total
29,297
FC Profit and Loss
9,652 -804
TC stand for transaction costs, and FC for financing cost
18,840
Trading on Market Inefficiencies Rich-Cheap Analysis â&#x20AC;&#x201C; Performance 46 000 000
45 000 000
44 000 000
43 000 000
42 000 000
41 000 000
40 000 000
Eonia + Bond Rich-Cheap Eonia
39 000 000 01/01/1995 15/04/1995 28/07/1995 09/11/1995 21/02/1996 04/06/1996 16/09/1996 29/12/1996
EONIA stands for Euro Overnight Index Average : average European Interbank Offered Rate
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 bidask spread)
• Operational Risk – Price errors may exist in the database – Bonds with embedded options can be mistaken for straight bonds
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