Smart Beta In Fixed Income. Riccardo Rebonato EDEHC Business School EDHEC Research Institute

Transcription

1 Smart Beta In Fixed Income Riccardo Rebonato EDEHC Business School EDHEC Research Institute London, June 2017

2 EDHEC-Risk Institute traditional focus on Equity Smart Beta New focus on Fixed-Income What we understand by smart beta : cross-sectional vs time-series risk premia The landscape of SMFI Research avenues Finding rational, behavioural or institutional explanation The problem with proxies: eg, value in bonds Downside Risk CAPM in bonds TIPS vs nominals The data problem. Outline of the talk 2

3 Fixed Income (DM): A Very Complex Landscape 3

4 Risk Factors, Anomalies and Timing in Fixed Income A Bird s Eye View With cross-sectional strategies we answer the question: Given that we have to be invested today, is there a weighting scheme that performs better (in some sense) that market capitalization, by loading preferentially on well rewarded-factors (other than the market factor)? With timing strategies we ask whether at a given point in time the recognized factors are handsomely/poorly rewarded. The emphasis is on the time (state) dependence of the factors, not on the clever allocation. Factors may change sign! These neat distinctions can sometimes be blurred (eg, in the case of convexity), but they provide a useful map of the land. 4

5 Before we Begin: What is a True Risk Premium? The core of modern asset pricing can be summarized in one neat equation: S(t) = E t [m(t+1) X(t+1)] with m(t+1) (the stochastic discount factor) given by This means To understand what this equation is telling us, take a logarithmic utility function and see what we get. 5

6 Cross-Sectional Factors: Statement of the Problem Then, neglecting impatience, we get and therefore the risk premium is given by In sum: I get a positive (TRUE) risk premium when my security pays well in good states of the world; pays badly in bad states of the world. 6

7 Cross-Sectional Factors: Statement of the Problem If my consumption only comes from investment, then the only factor that matters to price any asset is the market return. This is the basis of all CAPM-like models: If for some securities the intercepts are significantly different from zero, then finding factors can be described as the identification of non-market-return variables, x i, such that with the new intercepts now either zero or such that and with w i the weights in the market portfolio. 7

8 Timing Strategies: Statement of the Problem Given a set of state variables, x i, that describe the yield curve (such as PCs), under no-arbitrage returns on a fixed-income portfolio, P t, are given by If the market prices of risk are assumed to depend on the state variables, the questions then arise of 1. identifying for which state variables the market price of risk is not zero; 2. for these rewarded state variables, identifying the dependence of the market price of risk on the state variables. For instance, for Treasuries, the rewarded variable is the (uncertainty) in the level of the yield curve (the 1 st PC), but the magnitude of the reward mainly depends on the slope (the 2 nd PC). The dependence of the market price of risk on the state variables introduces time dependence. 8

9 Excess Returns in Corporate Bonds A Bird s Eye (and Incomplete) View of the Factors 9

10 Excess Returns and Factors in Corporate Bonds Fundamental Challenges For corporate bonds, it is easy to explain yield changes, but difficult to explain spread changes. Both the theoretically-motivated variables and the ad-hoc factors have a limited explanatory power, with R² ranging from 19% to 25%. The first PC of the residuals explains a very large proportion of the observed variability. Therefore firm-specific factors are unlikely to account for the residuals: there is likely to be an important systematic factor that can account for the bulk of changes in credit spreads (as opposed to in yields), but we still don t really know what it is. Evidence of value and momentum factors has been found across asset classes (stocks, Treasuries, corporate bonds, currencies commodities). Ad hoc explanations unlikely to be valid: The strong correlation structure among value and momentum strategies across such diverse asset classes is difficult to reconcile under existing behavioural theories, while the high Sharpe Ratios of a global [ ] diversified portfolio presents an even more daunting hurdle for rational-risk-based models. (Assness et al, JoF, 2013). 10

11 The Problem with Proxies In a CAPM world consumption only derives from investment income. Logical extensions: risk compensation could also attach to uncertainty in labour income, unemployment, etc. Failure of direct applications of consumption-based CAPM Attempts to find consumption proxies (or to justify why found factors may proxy for consumption) In moving from equities to fixed-income, proxies of proxies eg, value in bonds, momentum, liquidity. 11

12 The Problem with Proxies [ctd] Asness et al, JoF, Value and Momentum Everywhere: [we] show that individual stock portfolios formed from the negative of past 5-year returns are highly correlated with those formed on BE/ME ratios in our sample. [ ] Hence, using past 5-year returns to measure value seems reasonable. Other measures: 5-year yield change (nominals) 10-year yield to 5-year inflation forecast (real bonds) 10-year yield to short rate Composite average of all three measures Fama s fishing expedition. At the latest count, 324 factors/proxies no wonder something is significant! 12

13 Rationalizing Proxies For value in bonds, use structural Gaussian model and look at the residuals. Akin to cheap/dear analysis, but with economic grounding for the residuals. Liquidity proxy: the dangers of throwing in too many proxies the Ludvigsson and Ng (2009) approach: parsimony and (hopefully) interpretability. Momentum: Fourier-based analysis? 13

14 Research Plan Part I A principled approach to proxy formation: Choose understandable and justifiable proxy first; test if significant then. Do value, momentum, (and liquidity) remain significant cross-sectional return-predicting factors when understandable, no data-mining quantities are used to proxy for them? We give a flavour of the research results by discussing three areas: Value in Treasury bonds; Momentum in Treasuriy bonds; Down-Side CAPM to explain cross-sectional differences 14

15 Preliminary Results Value in Treasuries Security-level (CUSIP-level) analysis Data from 1992 Fit to DMR Fit to covariance matrix first Fit to yield curve then Maturities above 6 years, exclude callable, puttable, STRIPs 15

16 Value in Treasuries An Economically Motivated TS Model The doubly-mean-reverting Vasicek (DMRV) model that we present in this section is a variation of a model introduced by Hull and White in the mid-1990s (Hull and White, 1994). These are its constitutive equations in the Q (pricing) measure: 16

17 Value in Treasuries An Economically Motivated TS Model The behaviour makes sense: the short rate still moves stochastically around a reversion level, but this level is not constant. A cursory look at yields in the 1970s and in the 2010s shows that this is not such a bad idea. Casual inspection also suggests that the level to which the short rate reverts does not wander aimlessly to infinity, but is in turn `anchored' to some long-term level. Apart from being generically plausible, this model can be seen as an incarnation of the macrofinancial and monetaryeconomics-inspired models. It is a particularly simple example of this class, in that there is no description of the reversion of the short rate to the target rate (a reversion which is assumed to instantaneous). Unless one is interested in the behaviour of very-short-maturity yields, this is a reasonable approximation. 17

18 Value in Treasuries An Economically Motivated TS Model The Strategy Focus on economic justifiability, not best fit. Separate fit to covariance matrix and yield curve. The fit to the covariance matrix ensures that the convexity contribution is (about) right, and avoids a tug of war' between reversion level and volatility Very short maturity bonds are excluded (expectations). No transaction costs NO PEEK AHEAD What do we get? 18

19 Preliminary Results CUSIP EA 19

20 Preliminary Results CUSIP EA 20

21 Preliminary Results Many CUSIPs 21

22 Proposed line of Investigation Value in Treasuries Results for the CUSIPs or the years shown typical of all Treasuries. As in Cieslak and Povala (2010), two timescales of mean reversion ( secular and cyclical ) Idea: form portfolios by sorting in quantiles cheap and dear bonds using rolling training period ; follow performance of sorted portfolios with no peek-ahead. How well does it work? 22

23 A typical pattern of P&Ls

24 The results depend on 3 parameters: Robustness Study for Value in Treasuries z : beyond how many standard deviations above the n-day moving average the long-short position is put on. cutoff : by how much the deviation from the n-day moving average must be reduced for the position to be taken-off by how much z must have reduced for the position to be taken off. n : the number of days in the moving average the deviations from which give the buy/sell signal. How robust are the results to different permutations of these parameters? 1. We look at moving averages with duration ranging from a week to a month 2. We look at a number of standard deviations between 0.4 and 2 3. We look at cutoffs from 0.5 to 2. NO PEEK AHEAD! 24

25 Robustness Study for Value in Treasuries Methodology and units: The value of a par bond is expressed as 1 (not 100). A profit of, say, 0.10, means that the strategy yielded 0.10 (for notional positions of 1) over a period of three years. The investment periods are blocks of three years. Every day the investor looks at the difference between richness/cheapness, establishes position accordingly as long as the signal is strong enough, and the positions are left on as long as the signal indicates trade. We only invest in bonds that are alive for all the three years NO PEEK AHEAD, BUT NO TRANSACTION COSTS 25

26 Robustness study: profitability for different combinations of the parameters (avg P&L per bond) 26

27 Robustness study: profitability for different combinations of the parameters 27

28 Robustness study: profitability for different combinations of the parameters 28

29 Robustness study: profitability for different combinations of the parameters 29

30 Robustness study: profitability for different combinations of the parameters Only 4 permutations out of 320 give a negative 3-year average (per bond) P&L. The highest 3-year average (per bond) P&L was , the lowest = The average P&Ls over the 80 combinations of parameters per 3- year block were [ ], [ ], [ ], [ ] 30

31 No duration immunization net short/long position out of 40 bonds

32 Typical pattern of long positions

33 All-portfolio profit ( ) for a typical parameter combination Sharpe Ratio = 1.23

34 Second Strand of Research Momentum in Treasuries Momentum has been documented first in equities, and then in a number of asset classes. The ubiquity of momentum poses problems to the original rational explanations. Little or no work has been done in Treasuries. Our investigation methodology: Use n-month past returns (cross-sectional) to sort bonds into winners and losers Create a self-financing portfolio long of the winners and short of the losers Invest for m months. Repeat every month with No Peak Ahead. 34

35 Profits from momentum strategy ( ) (9mx9m) Sharpe Ratio = 0.735

36 Sharpe Ratios InvHor/SigHor

37 Autocorrelation as a function of lag ( ) (9mx9m)

38 Frequency of top performers ( ) (9mx9m)

39 Frequency of bottom performers ( ) (9mx9m)

40 Serial a/c for the top three PCs

41 Further work The profitability of momentum strategies suggests positive autocorrelation in the first principal component and perhaps in other PCs. Idea: reconstruct the yield curve signal using only the first, second, third principal component which of these components is responsible for the profitability of the momentum strategy? Can this be picked up by Fourier Analysis? Wavelet decomposition? Why is the short end more frequently picked up as a winner and the long end as a loser? Is the success of momentum strategies correlated with ex-ante visible quantities (eg, shape of the yield curve which has business-cycle properties)? 41

42 Third Strand of Research DCAPM for Treasuries Lettau, Magiori and Weber present results using Downside CAPM for FX carry trades and to explain cross-sectional returns in many other asset classes. The intuition: While CAPM fails in explaining returns within asset classes (and performs poorly in ranking returns of different asset classes) it is enough to add as risk factor exposure to downside risk to achieve excellent cross-sectional explanation of returns, both within and across asset classes The intuition is powerful. Two questions: Can it be applied to fixed income? Should the measure of downside risk be adjusted? 42

43 43

44 44

45 45

46 The approach 46

47 The Lettau et al procedure 47

48 Full sample 48

49 First Half 49

50 Second Half 50

51 Change in sign of Beta_down 51

52 Is this the correct measure of downside risk? 52

53 Alternative market-return-recovering approaches 53

54 Alternative parsimonious explanations

55 Alternative parsimonious explanations

56 Results (full sample) 56

57 Explanation of returns (full sample) 57

58 Results (second half) 58

59 Explanation of returns (second half) 59

60 Results (first half) 60

61 Explanation of returns (second half) 61

62 Conclusions We have looked at cross-sectional return information in fixed income (Treasuries), by examining three very different lines of research value in bonds momentum compensation for downside risk (in various forms) There is an enormous amount of exciting work ahead old explanations do no work. A principled approach to proxy creation is indispensable. Another very important factor is liquidity (asset liquidity and funding liquidity) which may hold the key to the lowvolatility puzzle. 62