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What can bond investors borrow from equity quants?

Benson Durham*

SQA Meeting

December 16, 2015

* The views in this presentation do not reflect those of any other person at Brevan Howard US Investment Management LP. The information has been obtained or derived from sources believed by the presenter to be reliable. However, the presenter does not make any representation or warranty, express or implied, as to the information’s accuracy or completeness, nor does the presenter recommend that the information serve as the basis of any investment decision. This presentation does not constitute an offer or solicitation of an offer, or any advice or recommendation, to purchase any securities or other financial instruments, and may not be construed as such.

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Slide Two

Disclaimer: Remember, empirical replication is the finest form of flattery

Agenda: Expect (demand) even treatment from Bad Santa:

o Five ?s for betting against beta (BAB) o Six ?s (that’s one more) for momentum (MOM)

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Why should bond investors be able to borrow from equity quants? (Equity anomalies seem to get more attention, of the academic variety at least.)

“True” behavioral patterns should be ubiquitous

o Exploitable shortcomings of “the representative investor”

o Predictable interactions among heterogeneous investors

“Limits to arbitrage” must be pervasive (e.g., “leverage aversion theory”

applies to all assets, not just across them or within a few)

(Before tackling BAB and MOM, what are the bogeys for bonds?)

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Bogey #1: Term Premiums Just as “anomalous” equity returns don’t load on “the” equity premium…

“The” (?) term premium is “the spread between the model-implied yield

and the average expected short rate”

But, more precisely, it’s just expected excess return, related to the quantity and price of risks, as in (for a -year, zero-coupon U.S. Treasury security):

0

,, 1ˆ USTt s

s

USTtE rxy E r

“Term” connotes only the “level” of the term structure, but the bond

premium is also compensation for “slope,” “curvature,” and other factors

A problem: it’s a model-dependent estimate, with many alternatives to choose from (e.g., Kim-Wright, ACM, etc.)1

1 See “Another View on U.S. Treasury Term Premiums,” Journal of Fixed Income, 2015, pp. 5–21, and http://libertystreeteconomics.newyorkfed.org/2014/06/do-expected-in-addition-to-spot-us-treasury-term-premiums-matter.html.

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Bogey #2: The Principal Components of the Term Structure As with equities and the CAPM, at a(n) (unsatisfactory) minimum, do not load on PC1!

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Anomaly #1: BAB using government bonds (BABgov) 2

Risk parity (RP) based on a single but long sample (and Leverage Aversion Theory)3

2 See “Betting against Beta (and Gamma) using Government Bonds,” 2015; http://www.newyorkfed.org/research/staff_reports/sr708.pdf; and http://libertystreeteconomics.newyorkfed.org/2015/04/please-read-this-before-betting-against-government-bond-betas.html 3 Asness, Cliff, Andrea Frazzini, and Lasse Pedersen, 2012, “Leverage Aversion and Risk Parity,” Financial Analysts Journal, vol. 70, pg. 56.

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Visual inspection of USTs and BAB4 Lower SRs with “risk,” but leverage constraints “Everywhere?”

4 Ilmanen, Antti, 2012, Expected Returns on Major Asset Classes (http://www.cfapubs.org/doi/pdf/10.2470/rf.v2012.n1.1), CFA Institute, pg. 52.

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? #1: Wait, who’s the marginal investor in “low risk?” Is RP across asset classes really consistent with BAB within them?

RAB/RP says that leverage constraints flatten the security market line

Meet two investors with binding leverage limits, Pete and Mary:

1. Pete is a UST-only investor who buys the long end to goose returns

2. Mary with a balanced mandate buys equities, and sells USTs, to do so

So, BAB says Pete buys duration at the same time RP says Mary sells it!

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What’s the untold segmentation story about heterogeneous investors?

5 Asness et al. (2012, pg. 49) argue that under BAB “safer short-maturity USTs offer higher risk-adjusted returns than do riskier long-maturity ones” but “(w)ith respect to RP, bonds are the low-beta asset and stocks the high-beta asset, and the benefit that we documented is another empirical success of the theory.”

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BAB might not confirm RP out-of-sample…6 But it sure seems to work!

6 See Andrea Frazzini and Lasse Heje Pedersen, 2014, “Betting against Beta,” Journal of Financial Economics, vol. 111, pp. 1–25.

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What’s the BABgov trade?7 Go long (short) high (low) SRs, and lever up and down such that = 1 on both sides.

r ( ) is the return ( s) vector,8 fr is the risk-free rate, and returns follow

1

1

1

1 1ˆ ˆ1 1gov

T TL t H t

long sho

T TBAB f ft L t t

rt

H t tr w r r w rw w

r

Rank on s,9 itz rank , but “damp” weights to limit leverage

And, rank only on s

7 That is,

L Hw k z z where 1'2 1nk z z

, 1n denotes a vector of ones of length n, and x+(-) indicate the positive (negative) elements of the vector x. See F&P (2014), pg. 12. 8 The vector for BABgov corresponds to maturity categories along the term structure. 9 The s are estimated w.r.t. the “market” comprised of the asset class index, e.g., USTs in the case of BABgov.

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? #2: Is BABgov an anomaly, or just a (bull and bear) steepener? BABgov, (arbitrage-free) affine term structure models (ATSMs), and risks

F&P note that multi-factor ATSMs capture inverted SR schedules10

But, if BABgov is consistent with ATSMs, it’s just not anomalous!

Low (duration) bonds must be higher risk on slope, curvature, etc.

Short the front based on ATSM SRs. But, the hedges outlaw a free lunch.11

10 See also Duffee, Gregory, 2010, “Sharpe Ratios in Term Structure Models,” working paper (http://www.econ2.jhu.edu/people/Duffee/duffeeSharpe.pdf). 11 For example, the hedge ratios (i.e., instead of leveraging to 1 ) with two factors, i.e., from 0

L SP Px x

, follows:

1

1 1

1 2 11

2 2 21

1 2 21

S S L

s

s S S L

n

n n n

P P Px x xP P Px x x

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PC-based constrained strategies wipe out excess BABgov returns Try a long (short) position in low (high) beta, but hedge multiple PC exposures ex-ante

1960 1965 1971 1976 1982 1987 1993 1998 2004 2009 2015-6

-4

-2

0

2

4

6

8x 10

-4

Betting Against Beta: U.S. Treasuries: CRSP Data (Daily) (05/28/1964-12/31/2013)Level-Slope-Curve Beta Equality Constraint

BAB Sharpe Ratio (0.17704), Market Sharpe Ratio (0.47461)BAB Sharpe Ratio: (01/31/1992-12/31/2013) (0.08361)

Lag Length (Months) = (36) (Note: Beta Shrinkage follows Vasicek (1973))

BAB Ex. Ret. 60-month Mean Ave. Ex. Ret. Ave. Market Ex. Ret.

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? #3: Have BABgov returns waned over time? Yes, somewhat. 12 Why? LSAP (MEP) effects? Forward guidance that pins down the front end?

1960 1965 1971 1976 1982 1987 1993 1998 2004 2009 2015-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3x 10

-3

Betting Against Beta: U.S. Treasuries: CRSP Data (Daily) (05/28/1964-12/31/2013)BAB (F&P 2014)

BAB Sharpe Ratio (1.0089), Market Sharpe Ratio (0.47461)BAB Sharpe Ratio: (01/31/1992-12/31/2013) (0.73965)

Lag Length (Months) = (36) (Note: Beta Shrinkage follows Vasicek (1973))

BAB Ex. Ret. 60-month Mean Ave. Ex. Ret. Ave. Market Ex. Ret.

12 The results are robust to daily data, market cap weights, alternative look-backs, 8 maturity buckets (including 1 to 9 months, 9 months to 2 years, 2 to 3 years, 3 to 4 years, 4 to 5 years, 5 to 6.5 years, 6.5 years to 12 years, and 12+ years from CRSP), etc.

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? #4: Does it work anywhere else? (Cross-sectional evidence) SR schedules suggest that it might not…

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Sharpe Ratio Schedules: Selected Government Bond Market (Estimates based on Bloomberg data)Sample Averages (Jan. 1993--Nov. 2014)

Maturity (years)

Shar

pe R

atio

US Germany Japan UK Switzerland

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Indeed, no case produces results as promising as for USTs E.g., Germany

1993 1995 1998 2001 2004 2006 2009 2012 2015-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Germany (Monthy) (01/31/1995-11/27/2014)BAB (F&P 2014)

BAB Sharpe Ratio (0.19065), Market Sharpe Ratio (0.22879)Lag Length (Months) = (36) (Note: Beta Shrinkage follows Vasicek (1973))

BAB Ex. Ret. 60-month Mean Ave. Ex. Ret. Ave. Market Ex. Ret.

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Why are the results for non-USTs less compelling? BABgov may just not be robust

The data are limited to the mid-1990s, when arguably the effect waned

However, the results for USTs are fairly compelling nonetheless

Is near-term Fed policy uncertainty greater? Are USTs a “safe-haven?”

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? #5: Does BABgov compensate for skew risks? “Low” in “Low-risk” investing refers to the 2nd, not the 3rd moment

The 3-moment CAPM implies 0 and 0 from

3ˆ ˆe

i i i mE r sign M

where coskew is (the “scaled marginal contribution to portfolio skew”), which is analogous to , follows

2

3

covˆ i m m

i

m m

r r

E r

Early studies suggest that is a “source” for —e.g., so BABS&P might

reflect investors’ demand for “lottery-like” payoffs of high-beta stocks13

13 See Kraus and Litzenberger, 1976, “Skewness Preference and the Valuation of Risk Assets,” Journal of Finance, vol. 31, pp. 1085–1100.

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The H0 for Coskew (Harvey and Siddique, 2000, pg. 1271)14

14 See Campbell Harvey and Akhtar Siddique, 2000, “Conditional Skewness in Asset Pricing Tests,” Journal of Finance, vol. 55, no. 3, pp. 1263–1295.

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So, why might skew risks be priced in USTs?

(BABgov coskew risks may be distinct from, say, BABS&P)

The Zero (Nominal?) Lower Bound and “Pull to par” imply asymmetry

For USTs, just as (obviously) increases with maturity, so does

And, UST market returns have positive skew (unconditionally)

Plus, BABgov returns have comparatively low

So, are UST investors leverage averse, or just keen for coskew insurance?15

15 Fujiwara et al. (2013) report conditional negative skew using daily government bond excess returns, particularly however at shorter maturities. See Ippei Fujiwara, Lena Mareen Korber, and Daisuke Nagakura, 2013, “Asymmetry in Government Bond Returns,” Journal of Banking and Finance, vol. 37, pp. 3218–3226. Durham (2008) reports a positive relation between Eurodollar-option-implied skew and GATSM-based term premiums, by some small miracle. “Implied Interest Rate Skew, Term Premiums, and the ‘Conundrum,’” Journal of Fixed Income, Spring 2008, pp. 88–99.

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BABgov and coskew: alternative trading strategies “BAB, but hedge coskew risks ex-ante”

Linear-programming (long side) follows

*

*,

* 1, , ,

1

ˆmin

.t .

ˆ ˆ 1

L

T

L t tw

nT

L t t i t i ti

w

s

w n w

(parallel constraints for the short side)

Therefore, low- and high-beta portfolios have the same UST market skew

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based constraints damp returns notably, but SRs are not unfavorable16

1960 1965 1971 1976 1982 1987 1993 1998 2004 2009 2015-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-3

Betting Against Beta: U.S. Treasuries: CRSP Data (Daily) (05/28/1964-12/31/2013)Gamma Constrained Gov. Bonds (Linear)

BAB Sharpe Ratio (0.57945), Market Sharpe Ratio (0.47461)BAB Sharpe Ratio: (01/31/1992-12/31/2013) (0.41329)

Lag Length (Months) = (36) (Note: Beta Shrinkage follows Vasicek (1973))

BAB Ex. Ret. 60-month Mean Ave. Ex. Ret. Ave. Market Ex. Ret.

16 Excess return with linear gamma constraints with respect to the UST market is about 1.7 bps per day, compared to 10.6 bps for the unconstrained strategy. The SR is about 0.58, compared to 1.27 for the unconstrained portfolio.

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But, don’t necessarily “beta against gamma (BAG)” … 1. Coskew w.r.t what?(!) Results are ambiguous w.r.t. s of stocks and USTs + stocks

2. Tests are problematic for pricing (consistent with Taleb, 2004)17

00.5

11.5

22.5

3

00.5

11.5

22.5

2

2.2

2.4

2.6

2.8

3

x 10-4

Beta

Betting Against Beta: U.S. Treasuries: CRSP Data (Daily) (06/15/1961-12/31/2013)Lag Length (Months) = (13051)

Market Proxy: Gov. BondsNote: Beta Shrinkage follows Vasicek (1973)

Gamma

Exc

ess R

etur

ns

17 See Nassim Taleb, 2004, “Bleed or Blowup? Why Do We Prefer Asymmetric Payoffs?” Journal of Behavioral Finance, vol. 5, pp. 2–7.

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Anomaly #2: Momentum Some rudiments…

1. Excess returns persist…until they don’t, e.g.:

o U.S. stocks with the greatest returns about 2 to 12 months ago have

subsequent abnormal returns (Jegadeesh and Titman, 1993)18

o But, shares with the lowest returns between 12 and 60 months ago have positive excess returns (DeBondt and Thaler, 1985)19

2. Warning: An unlimited number of trading rules and finite data produce

“predictable” patterns with near certainty!

3. So, we must ask…

18 Narasimhan Jegadeesh and Sheridan Titman, 1993, “Returns to Buying Winners and Selling Losers: Implications for Stock Market Efficiency,” Journal of Finance, vol. 48, pp. 699–720. 19 See Werner DeBondt and Richard Thaler, 1985, “Does the Stock Market Overreact, Journal of Finance, vol. 40, pp. 793–805.

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Why might momentum (and reversal) strategies work? Besides spurious luck, two general strands to convey in 45 seconds:

1. The “representative investor” suffers from cognitive biases, e.g.:

o “Disposition effect:” investors maximize “realization utility” to generate

“positive investment episodes” (Frazzini, 2006, Frydman et al., 2012)20

2. Bounded, heterogeneous investors create inefficiencies, e.g.: o News-watchers “under-react” and momentum traders “extrapolate” to

generate persistence and reversal patterns (Hong and Stein, 1999)21

3. Beware retrofitted stories, but it works “Everywhere”

20 See Andrea Frazzini, 2006, “The Disposition Effect and Underreaction to News,” Journal of Finance, vol. 61, pp. 2017–46 and Frydman, Cary, Nicholas Barberis, Colin Camerer, Peter Bossaerts, and Antonio Rangel, 2013, “Testing Theories of Investor Behavior Using Neural Data,” mimeo. 21 See Harrison Hong and Jeremey Stein, 1999, “A Unified Theory of Underreaction, Momentum Trading, and Overreaction in Asset Markets, Journal of Finance vol. 54, pp. 2143–2184.

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U.S. Treasury term structure momentum22 This is neither time-series23 nor cross-sectional country bond market momentum24

Among 6 buckets (1-3, 3-5, 5-7, 7-10, 10-20, and 20+ years), form the

portfolio with the greatest return m to n months ago, subject to constraints Formally, I use linear programming as opposed to portfolio sorts:

1. maximize : ,

Toptt t m t n tw R , with 2 9 6 12m n

2. subject to Topt indext t tw D d , ,0 1opt

i tw , and 6

,1

1opti t

i

w

Rebalance each month

22 See http://libertystreeteconomics.newyorkfed.org/2014/05/can-investors-use-momentum-to-beat-the-us-treasury-market.html as well as “Can Long-only Investors use Momentum to Beat the U.S. Treasury Market,” 2015, Financial Analysts Journal, September/October, vol. 73, pp. 57–74. 23 See Tobias J. Moskowitz, Yao Hua Ooi, and Lasse H. Pedersen, 2011, “Time Series Momentum,” Journal of Financial Economics, vol. 104, pp. 228–50. 24 See Clifford S. Asness, Tobias J. Moskowitz, and Lasse Pedersen, 2013, “Value and Momentum Everywhere,” Journal of Finance, vol. 62, pp. 929–985.

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Excess returns (over the benchmark) are meaningful Without a behavioral story, results may reflect “true” risk, nonetheless…

An equally-weighted strategy generates excess returns of about 34 bps (per

year), with an IR of about 0.46 from March 1985 through December 2013 Returns and IRs appear safely statistically significant, largely insensitive to

20 alternative window look-back lengths, larger than cross-sectional country-bond momentum patterns, stationary (i.e., through the crisis)

Transaction costs are a minimal drag (e.g., bid-ask spreads of 6 to 10 bps

required to wipe out statistical significance), and it’s not “picking up nickels in front of steamrollers” (with skew > 0)

The loadings, if any, are “favorable” w.r.t. S&P 500 returns, credit spreads, volatility, and UST liquidity premiums, no seasonality, low R2 values, etc.

Alpha also remains after controlling for yield curve factors (term premiums, forward rates, principal components, etc.)

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Cumulative excess returns are “consistent.” (Relative-excess-return “hit rates” at the 3-year horizon are favorable.)

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Turnover is also modest

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Some bad news:

1. No relation to signal size (i.e., more momentum relative to the index does not correlate with greater realized excess returns in subsequent periods)

2. No lagged market-dependence, unlike for stocks (Cooper et al., 2004)25

3. Mixed breadth: Less compelling results in Japan and Canada, nothing in the UK? Is there a behavioral explanation that fits Treasuries?

4. What is the reference or purchase price w.r.t. any “disposition effect?”

5. Why does momentum work at such a low frequency?26

6. Why is there no (corresponding) contrarian effect?

25 See Michael J. Cooper, Roberto C. Gutierrez Jr., and Allaudeen Hameed, 2004, “Market States and Momentum,” Journal of Finance, vol. 59, pp. 1345–1365. But, there is some contemporaneous correlation with “up markets.” (100-bp return on the index corresponds to a 7-bp boost to momentum returns.) 26 See Pierluigi Balduzzi, Edwin J. Elton, and T. Clifton Green, 2001, “Economic News and Bond Prices: Evidence from the U.S. Treasury Market,” Journal of Financial and Quantitative Analysis, vol. 36, pp. 523–43.

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Closing comments

Bond investors should be able to borrow freely from equity quants

This “survey” covered BAB and MOM, but not “value” or “carry”

Some broader motivations for active investment management:

Are Market Forecasts Right?

Are Market Prices Fair? Yes, No Yes, Yes No, No No, Yes

The southeast quadrant is not the only source for