Smart Beta Strategies in Fixed Income

Riccardo Rebonato, Professor of Finance at EDHEC Business School, Member of EDHEC-Risk Institute### 1 Introduction

In the last decade, the search for priced non-market equity risk factors, and the
implementation of smart beta strategies for equities have been a major focus of
applied and theoretical research. It is now generally acknowledged that, *in the
equity space*, these strategies permit the construction of more desirable portfolios
than naive passive allocations (such as equal or market-capitalization weighting
schemes).

Recently, this focus has been shifted to other asset classes (see, eg, Asness,
Moskowitz and Pedersen, 2013) and to fixed income in particular. Given the
huge size of the fixed-income market^{1}, the natural question is whether smart beta strategies will prove effective for this asset class.

In this article:

- we put the search for factors and beta strategies in the context of asset pricing, and we show that compensation for non-market factors is not just allowed, but actually required, by financial theory;
- we explain the different, and complementary, questions answered by time-series and cross-sectional analyses of risk premia;
- we then focus on fixed-income instruments, and present the time-series and cross-sectional formulations for the search of priced risk factors;
- we explain the unique challenges encountered in identifying priced risk factors in fixed-income products;
- we present the main findings obtained to date;
- we suggest avenues for fresh research.

*some*cogent explanation for the putative factors.

### 2 Excess Returns – Background

In the first incarnation of the "modern" approach to asset pricing (the body of work that, starting from Markowitz (1952), led to the CAPM (Treynor (1961), Sharpe (1964), Lintner (1965)), the excess return earned by any security was derived to be proportional (via the famous "market beta") to a single factor – the market excess return over the riskless rate.

As a corollary to this result, it followed that (barring leverage) increasing exposure to the market factor was the only way for an investor to increase excess return.

The CAPM has fared better theoretically, and indeed among practitioners^{2},
than empirically. Indeed, statistical tests have robustly and convincingly rejected
the validity of the CAPM model. This rejection did not imply, however,
that the market factor played *no* role in explaining excess returns. Rather,
the empirical studies revealed the untenable claim that the market factor was
the *only* factor, and suggested that additional, non-market, factors, could have
significant explanatory power: the market risk factor had to be complemented
by other explanatory variables, these empirical studies said, not *tout court* jettisoned. These empirical studies were silent, however, as to the nature of the
additional factors.

Is it reasonable to accept the existence of non-market factors? It certainly is, both normatively and descriptively. A positive risk premium reflects the compensation for the fact that a security is expected to pay well in states of the world when investors are doing well (high-consumption states), and to have poor payouts when investors feel poor (low-consumption states). Now, the CAPM implicitly assumes, among other things, that investors only draw their income (and hence derive their consumption) from their investment portfolios. If this were true, high and low consumption would indeed only be linked to the performance of the market portfolio.

In reality, investors face a number of macroeconomic risks to their consumption
stream: unemployment, for instance, would affect their labor income;
inflation would erode the nominal value of their nominal assets; productivity
shocks are known to be related to stock returns^{3}; etc.

In principle, every source of consumption risk can therefore command a compensation for bearing that risk, and hence a risk premium.

This line of thought led to extensions of the CAPM model in which several
consumption-affecting factors were allowed to influence the expected returns of
stocks.^{4} This, in turn, motivated, or at least provided the theoretical justification for, the empirical search of non-market factors.

In parallel, studies in behavioural finance and in the institutional workings
of financial markets pointed, on the one hand to the bounded rationality of
investors^{5}, and on the other to the "frictions" that taxes, laws, and regulations impose on the functioning of the financial system. For the present discussion
the important point is that both these sources of "imperfection" ("irrationalities"
and "frictions") could in principle introduce new explanatory variables (which
may, but need not, be proper "factors") to account for excess returns.

### 3 An Expression for the Factors

These qualitative considerations can be made more precise as follows.

Consider first the statistical regression of the excess return, *r _{i}*, over the riskless rate,

*r*, from security

_{f}*i*, on the market excess return,

*r*–

_{m}*r*:

_{f}If we take Equation (1) purely as a statistical regression, there are no constraints on the intercepts. As we discussed, the CAPM makes the strong statement that all the intercepts, *α _{i}*, should be statistically indistinguishable from zero, (and that the residual should be uncorrelated with the left-hand variables).

If one empirically finds, as one does, that some intercepts *are* statistically different from zero, then finding "factors" can be described as the identification of *n* non-market-return variables, *x _{i}*, such that

with the new intercepts of now either zero or at least such that

(In the equation above, the quantities *w _{i}* are the weights in the market portfolio.) The identification of new factors turns at least a part of the "undigested" intercepts of the CAPM-inspired regression (the

*α*in Equation (1)) into new interpretable "betas" (the in Equation (2)).

_{i}In the equity space, where most of the theoretical and empirical work has
been carried out, Fama and French (1993) pioneered the search for the factors
*x _{k}*. In their early work they identified, in addition to the market portfolio, two additional factors: the small-minus-big factors, and high-minus-low factor
(where "small" and "big" refer to the size of a firm, and "high" and "low" to the
ratio of the book to market value).

^{6}

In the wake of these findings, an immense literature blossomed on the search
for additional explanatory variables of excess returns. Regression studies which
directly used macroeconomic variables as factors were met with limited success.
Given the difficulty to quantify macroeconomic variables (think, for instance,
of creating a time series of productivity shocks), the practices therefore became
common first to use well-identifiable traded proxies^{7}, and then to use an array of market-observable variables that were posited to have some link to a consumption risk story.

The degree of theoretical rigour and statistical robustness of these studies
varied greatly^{8}. So, alongside the factors that traditional asset pricing theory would readily understand, a richly populated menagerie of more opaque "anomalies" was born.^{9} Admittedly, it did not always prove easy – albeit not beyond the ken of an ingenious financial economist – to "map" these empirically determined factors to the sources of consumption risk that would justify calling them "factors".

After the initial research dust settled, the academic and practitioner consensus in equities finally coalesced around the proposition that a small number of robust factors (from which the small-minus-big was often dropped and to which the momentum frequently added) could be identified.

When a statistically sound and economically principled approach to factor identification has been employed, the implications of these findings for asset management have been profound. As new, robust (and sometimes economically interpretable) factors were identified, portfolio weighting schemes other than the market capitalisation were soon created in the equities arena that would tilt the portfolio composition towards the non-market rewarded factors.

The degree and nature of the weight tilt would be determined in such a way as to exploit diversification in order to obtain what the CAPM had claimed to be unattainable: a higher-than-CAPM return for the same risk; or a lower-than-CAPM risk for the same return. Since in the old CAPM world the only way to gain extra unleveraged return was to increase the exposure to the market beta, the new, CAPM-beating portfolio weighting schemes became known as "smart beta" strategies. Their success in the equity space has been widely documented, and it is now an established, text-book "fact" of asset pricing. See, eg, Ang (2014).

### 4 Smart Beta: From Time-Series to Cross-Sectional Analysis for Fixed Income

Until very recently, the search for risk premia and excess returns had a very
different complexion in the fixed-income arena. Most of the studies were focussed on (mainly US-issued) Treasury bonds, for which good quality data has
been available for decades. However, the high degree of correlation amongst
Treasuries (it is well known that two or three Principal Components explain
over 95% over the observed price variations) makes the identification of *cross-sectional* differences less promising than for equities. Time series analysis of
excess returns has therefore been prevalent for Government bonds, and the associated research programme that until very recently was their staple diet of
risk-premium research in fixed income can be summarised as follows.

Given a set of state variables, *x _{i}*, that describe the (typically Treasury) yield curve (such as Principal Components), under no-arbitrage the time-

*t*returns on a fixed-income bond of maturity

*T*,

*P*, are given by

_{t}^{T}where is volatility of the *i*th factor and its associated market price of risk.

If the "market prices of risk" are assumed to depend on the state variables,

the search for time- (state-) dependent risk premia boils down to

- identifying for which state variables the market price of risk is not zero;
- for these "rewarded" state variables, identifying the dependence of the
market price of risk on the state variables – in the last decade there has
been the vibrant research programme associated with the search for the
return-predicting factors, ie, with linear combinations of state variables
which have
*ex ante*(predictive) power about the sign and magnitude of the excess returns.

Until the mid 2000s cutting-edge research in Treasury risk premia was (and still is) focused on the identification of return-predicting factors more efficient than the slope. See, eg, Cochrane and Piazzesi (2005), Cieslak and Povala (2010a, 2010b), and the references therein.

Time series and cross-sectional studies are both valuable, but answer different questions. When the state and time dependence of the risk premium for a
given asset class is investigated via a time-series analysis and the identification
of a return-predicting factor, the question being answered is whether "today" is a good or bad time to invest (be "overweight") in the asset class *as a whole*. When the cross-sectional differences within a given asset class are explored, the question being answered is to which securities within the asset class one should give more weight, given that an investment in that asset class "has to" be made.

In the fixed-income area, time-series analysis has typically resulted in the decision of whether to construct a portfolio with longer or shorter duration than the benchmark. A cross-sectional analysis has typically been approached via cheap/dear analysis using empirical (Nelson-Siegel, 1987) or structural (see, eg, Kim-Wright (2005), or Adrian, Crump and Moench (2014, 2015)) models. In the fixed-income area, this type of analysis has usually been "tactical" in nature, and has typically given rise to the construction of duration-neutral relative-value portfolios.

This state of affairs is rapidly changing. In the last few years practitioners
and academics have begun to look at fixed-income products from a *smart-beta (cross-sectional) perspective*. Given the size of the international government and corporate debt outstanding, the lateness of this development is at first blush
surprising. This lateness can be partly accounted for by the relative poverty
of the data quality for large sections of the fixed-income universe. Another,
and arguably more compelling, explanation is the sheer complexity of the fixed-income lay of the land, some salient aspects of which are shown in Fig (1) (which
only looks at Developed Market, DM).

As the picture shows, under the capacious tent of the "fixed-income" denomination one gathers

- truly riskless government debt,
- "somewhat"-to-extremely credit-risky government debt,
- corporate debt that ranges in creditworthiness from better than most government instruments to junk,
- real and nominal bonds (which come in government and corporate flavour),
- funded and unfunded (ie, cash versus swap) instruments,
- corporates for which public data are available (and for which accountancy-related characteristics can be extracted) and corporate for which this is not possible.

Not surprisingly, empirical studies so far have focused on (often rather limited) subsections of this investment universe. We briefly review in the next section some of the more salient findings.

### 5 Empirical Findings To Date

Looking at the results with a broad brush, one can say the following.

For corporate bonds, it is easy to explain yield changes, but difficult to
explain spread changes. When the attempt has been made to find explanatory
variables to account for spread changes (see, eg, Collins-Dufresne, Goldstein
and Martin (2001)), both the theoretically-motivated variables^{10} and the *ad-hoc* factors have been shown to have a limited explanatory power, with *R ^{2}* ranging from 19% to 25%.

It was also found that the first Principal Component of the residuals could explain 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.

One could, of course, take the first Principal Component of the residuals as the "factor", but this would not allow any meaningful economic interpretation, and there would be no guarantee of the stability of this factor.

Howling and van Zundert (2014) find empirical evidence that "the Size, Low-Risk and Momentum factors have economically meaningful and statistically significant risk-adjusted returns in the corporate bond market". They find that their factors can be combined to form a more attractive (better Sharpe-ratio) overall portfolio, and that the results are robust when transaction costs are included, when the factor proxies are defined somewhat differently, and when the portfolios are built in different but reasonable ways.

The low-risk factor is echoed in the work by de Carvalho et al (2014), who find that low-volatility bonds have better Sharpe Ratios than high-volatility bonds. However, the Sharpe Ratio associated with some of these low-volatility portfolios may well be high, but the leverage required to make the expected returns comparable to, say, expected returns from equities can be as high as 50 or 60. (This, by the way, may well be an explanation of why the "anomaly" is there in the first place.)

It has been claimed that more efficient portfolios can be built by reducing
exposure to corporates or sectors with large issuance size. For individual corporates, of course, the variable of interest is leverage, not debt size *per se*, but this
quantity is only computable for companies with public data. As for "excessive" issuance in particular sectors, the "explanation" of why size may be negatively correlated with performance points to debt issuance "bubbles" (such as the volume of issuances for Telecoms or tech companies in 2000, or for financials in 2005-2006).

Liquidity affects different issuers to very different extents, and is poorly correlated with creditworthiness: Italy, for instance, has a similar credit spread (to Bunds) as Spain, but the issuance size, and hence the normal-times liquidity, is much larger in BTPs than in Bonos. Much work needs to be done in this area, which is one of the least explored (probably because of the difficulty in constructing "non-tautological" proxies).

Momentum has been observed in fixed income as well, but the choice of the trailing window is delicate and the optimal choice for the length of the momentum "run" is not universal. Short-term mean-reversals have been observed to compete with momentum, complicating the analysis.

Value has been found difficult to define in the case of bonds. For issuers for which reliable yield curves can be built (mainly government bonds, bonds issued by semi-government agencies, and a handful of corporates) cheap/dear analysis has been successfully undertaken by market practitioners for a long time, but few, if any, systematic studies have appeared in the literature. Asness, Moskowitz and Pedersen (2013) provide a (not obviously intuitive) proxy for value, and find that high "value" bonds tend to perform better than low "value" ones.

It must be stressed that evidence of value and momentum factors has been
found across a number of asset classes (stocks, Treasuries, corporate bonds,
currencies commodities). This suggests that *ad hoc* explanations are 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."
(Asness, Moskowitz and Pedersen, 2013).

Finally, the "fallen angels" effect (which is a classic example of a "friction" generated by a regulatory-like constraint) seems to still be present, although downgrade-tolerant strategies are becoming increasingly widespread.

### 6 Conclusions

In this note, we have put in context the recent cross-sectional studies of excess returns in the fixed-income space. We have highlighted both the promises and the difficulties associated with the identification of these fixed-income factors. Many seem to be variants of the factors that have already been identified for equities. As the value factor shows, however, the "transliteration" from one asset class to another often requires careful handling.

A convincing economic interpretation of the factors still remains elusive: if anything, having found similar factors at play in the fixed-income market makes their economic justification more, not less, challenging.

Overall, it seems fair to say that "fixed-income smart beta" is an exciting new area of research, where a lot of empirical and theoretical work still needs to be carried out to build a convincing, and practically exploitable, understanding of which factors are "really there", of why they exist in the first place, of how they can be best captured, and of how desirable portfolios can be built.

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**Footnotes:**

^{1}According to the BIS, the size of the global debt market is approximately $22tn (as reported in the *Financial Times*, 10 November 2016, page 18, Lex).
^{2}"Even though the CAPM is firmly rejected by data, it remains the workhorse of finance: 75% of finance professors advocate using it, *and 75% of CFOs employ it in actual capital budgeting decisions*", Ang (2016), page 197, emphasis added. See also Welch (2008) and Graham and Harvey (2001), quoted therein.
^{3}Ang (2016) reports a 48% correlation between a five-year moving average of productivity shocks and stock returns. In real business cycle models, productivity shocks affect not only stock returns, but also growth, investments and savings, and therefore indirectly affect non-investment consumption – for instance, through wage growth.
^{4}The first to include macro factors in equities in the cross-sectional search for systematic source of risk were Chen, Roll and Ross (1986).
^{5}and to the difficulty to arbitrage these irrationalities away (See, eg, Schleifer and Vishny, 1997).
^{6}More precisely, the Fama and French factors were factor-mimicking portfolios, ie, long-short portfolios of stocks that would mimick the factors of interest.
^{7}for instance, the VIX index is an obvious proxy for volatility risk.
^{8}As Fama famously said, abandoning the requirement to link a factor to a cogent consumption story was equivalent to issuing a "fishing licence". The dangers of data mining are particularly salient in this context, given the large amount of data required to create a training and a back-testing set. See, eg, Abu-Mostafa, Magdon-Ismail and Lin (2012) in this respect.
^{9}The distinction between "true" risk factors and "anomalies" is not a purely nominal one: "true" risk factors are not washed away by discovering them, as they remain the market compensation for receiving large payoffs in good states of consumption, and *vice versa*. Behaviourally driven irrationalities, instead, may be corrected by sufficiently well capitalised arbitrageurs (such as hedge funds); and as for institutional frictions, these can disappear at the stroke of a regulatory pen.
^{10} If one looks at a risky debt from an option-theoretical perspective (a put on the value of the assets), one would expect volatility, the interest rate level and the degree of in-the-moneyness to affect the value of the default option. These were the "fundamental" quantities.