Optimizing Yield Curve Positioning for Multi-Asset Portfolios

QUANTITATIVE RESEARCH AND ANALYTICS • JULY 2019

Optimizing Yield Curve Positioning for Multi-Asset

AUTHORS Portfolios

Executive Summary

•• Many investors buy U.S. Treasuries to hedge equity risk. But do long Josh Davis Treasuries deliver the optimal risk/return trade-off? Managing Director Portfolio Manager Head of Client Analytics •• Our research investigates this question by analyzing the optimal allocation of duration along the yield curve.

•• We find that a dynamic swap overlay strategy – positioned along the yield curve to account for carry and the stage of the business cycle – has the potential to deliver sizable Sharpe ratio and drawdown improvements.

•• On average, the “belly” of the curve – around five years – maximizes the Cristian Fuenzalida diversification benefit relative to a benchmark portfolio. Quantitative Research Analyst

Investors often justify the allocations to long with a benchmark portfolio. This framework Treasuries in their portfolios by noting that may be particularly relevant for managers that these allocations have historically offered do not have an exact liability matching or active positive returns and diversification of equity leverage constraint. risk. We expand on this idea by investigating the particular point on the yield curve where an INTRODUCTION investor obtains the best combination of return If you are an investor with exposure to equities, and diversification. We set forth a dynamic in all probability your portfolio contains some swap overlay strategy to position along the form of diversifier of equity risk. A widely curve, depending on the carry and roll-down, accepted way to offset equity risk is to add and the position of the economy across the bonds because of their propensity for negative business cycle. The dynamic swap positioning correlation and positive yield. Within the fixed delivered sizable Sharpe ratio and drawdown income asset class, Treasuries and interest rate improvements. On average, the intermediate swaps can offer hedging power without level of duration – around five years – exposing the portfolio to industry credit risk or maximized the diversification benefit compared significant counterparty risk. If you are 2 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS interested in maximizing the bang-for-your-buck of equity Another important decision a public pension plan makes is how hedging benefits along the yield curve, how should you evaluate to allocate the remaining 30% of risk to nonequity assets. This the risk/return trade-offs of using duration to hedge against bucket typically consists of some combination of fixed income equity risk? and alternatives, such as hedge funds, that is constructed to seek to deliver positive returns yet exhibits low correlation to In recent decades, government bonds and swaps have exhibited equities. In times of severe stress in equity markets, this under- a negative correlation with equities that has served to strengthen allocation could offset some losses and provide a hedge to the case for allocating risk to these financial instruments. equity risk; in normal times, it could serve as a tailwind to returns Furthermore, regardless of the mathematical sign of the to help achieve the return target. In a separate paper, we discuss correlation (positive or negative), Treasury bonds’ excess returns how to optimize a basket of these strategies (Baz, Davis and have been positive in nine of the past 10 recessions. This feature Rennison 2017). Several strategies are generally thought to of bonds – being a positive return diversifier of equity risk – is a provide a good basis for this allocation: long Treasuries within mainstay of strategies like risk parity, which, under certain fixed income, and trend-following and global macro strategies correlation and expected return conditions, can be better than a within the alternatives space. traditional market-cap-weighted portfolio when leverage constraints are relaxed (Asness et al. 2012). In this paper, we turn our attention to the fixed income allocation of a representative large pension fund. In particular, While the role of government bonds and their associated interest we focus on the Treasury exposure, which can be approximated rate swaps as a hedge to equity risk is well accepted in modern by a long Treasury index. Treasuries are among the most liquid portfolios, one question that remains open is the particular point instruments; carry minimal credit risk; have been, on average, on the yield curve where an investor obtains the best negatively correlated with equities for the past 20 years; and combination of return and diversification benefit. This paper have historically offered positive yields – all key factors attempts to provide a simple framework for answering this supporting their inclusion in equity-heavy portfolios. question in the context of both a standard 60/40 portfolio of Interestingly, public pension plans have chosen to target their stocks and bonds and a risk-balanced portfolio consisting of Treasury allocations to longer maturities, perhaps because equal risk allocations to bonds and equities. We found that the many face leverage constraints. Although this may be the optimal point on the curve changes based on several factors, optimal point on the curve at certain times, we will show that including whether the investor is using Treasuries or swaps, the historically this has not been the optimal point for maximizing steepness of the yield curve and the dynamic nature of the returns per unit of duration. On the contrary, the optimal point correlation between stocks and bonds. has been moving across time, depending on the economy’s position along the business cycle. A simple framework RELEVANCE FOR PUBLIC PENSIONS incorporating aspects like carry and relative value captures the Based on a panel of publicly available data (Willis Towers Watson attractiveness of different points on the yield curve and can be 2018), the median public pension plan in the seven most used to more optimally obtain duration exposure. important pension markets (Australia, Canada, Japan, Netherlands, Switzerland, the U.K. and the U.S.) in 2017 had FIRST GLANCE: EXPLORATORY ANALYSIS OF roughly 46% in equities, with the balance of 54% allocated to EQUITY BETAS lower-volatility and/or diversifying assets such as fixed income, One simple way to analyze hedging effectiveness along the curve alternatives and cash. In a risk context, most of the variation of is to take every point of the curve scaled by duration and returns was driven by the allocation to public or private equity calculate its hedging beta with respect to the equity market. The and alternatives, while only a small fraction was driven by the lower the beta, the better hedging properties it will provide. remaining portfolio allocations. Based on this risk assessment, the fraction allocated to equities is one of the most important The appendix shows the results of a simple regression analysis decisions a public pension plan makes, as this drives a large on the beta to the equity market of different points along the fraction of the overall risk. Ultimately, the large allocation to curve. First, we focus on zero-coupon Treasury returns. equity is driven by the need to deliver upon expected returns over Maturities of between four and seven years provide the best the long run (typically 7.5% for a public pension fund). hedge to equities. The “belly” of the curve offers the best beta per unit of duration for the period January 1960 to December 2018 in . (1)




. .


(1) (1)


. (1) . (1)










































We use this decomposition to start our analysis by discussing the carry plus roll-down component of the expected return of these instruments. Within the expected return component WeWe use use this this decomposition decomposition to to start Westart useour our thisanalysis analysis decomposition by by discussing discussing to start the the carry ourcarry analysis plus plus roll-down roll-down by discussing the carry plus roll-down We use this decomposition to start our analysis by discussing the carry plus roll-down , the carry plus roll-down excess return can be thought of as the excess return of an asset, componentcomponent of of the the expected expected return returncomponent of of these these ofinstruments. instruments. the expected Within Within return the the of expected theseexpected instruments. return return component component Within the expected return component component of the expected return of these instruments. Within the expected return component
, ,the the carry carry plus
assumingplus
roll-down roll-down that marketexcess excess, the return pricesreturn carry can canstay plus be be constant, roll-down thought thought of orof excess as as the the returnexcess excess. You can return couldreturn be thoughtofthink of an an asset, ofasset, of carry as the plus excess roll-down return asof an asset, , the carry plus roll-down excess return can be thought of as the excess return of an asset,
the part of the expected
return that comes simply from
the passage of time. Hence, by definition, assumingassuming

that that market market prices prices stay stay
assuming constant,
constant, that or or market .prices .You You could staycould
constant, think think
of of carry orcarry plus plus. roll-down Youroll-down could as asthink of carry plus roll-down as





assuming
that market prices stay constant, or . You could think of carry plus roll-down as
carry is a value that can be observed

in advance. The second part
of the expected return comes thethe part part of of the the expected expected
return return thethat that part comes comes of thesimply simply expected from from

returnthe the passage passage that comes of of time. time. simply Hence,
Hence,
from by by thedefinition, definition, passage of time. Hence, by definition,




FirstFirst glance: glance: Exploratory Exploratory analysis analysisFirst of ofequity glance: equity betas Exploratorybetas analysisthe of part equity of the betas expected return that comes simply

from the passage of time. Hence, by definition, carrycarry is is a a value value that fromthat can can expected be be observed observedcarry price isin inappreciation a advance. advance.value that The The can second second be observed: part part of of in the the advance. expected expected The return return second comes comes part of the expected return comes carry is a value that can be observed in advance. The second part of the expected

return comes from expected price appreciation
: OneOne simple simple way way to analyzeto analyze hedging hedgingOne effectiveness effectiveness simple way along to along analyze the the curve curvehedging is tois taketoeffectiveness take every every point pointalong of ofthe the the fromcurvefrom expected expectedis to take price price every appreciation appreciation point of the : :

from expected price appreciation : (2)






curve scaled by duration and calculate its hedging beta with respect to

the equity market.
The




curvecurve scaled scaled by byduration duration and and calculate calculate its itshedging hedging beta beta with with respect respect to theto the equity equity market. market. The The




(2)













(2) (2)


lower the beta, the better hedging properties it will provide. (2)


lowerlower the the beta, beta, the the better better hedging hedging properties properties it will it will provide. provide.

































































First glance: Exploratory analysis of equity betas The above decomposition has hidden simplifications that deserve to be made explicit. An TheThe above above decomposition decompositionimportant has onehas hidden The hiddenis that above simplificationsthesimplifications decompositionmodels used that that to hasdeserve capturedeserve hidden to theto simplificationsbe be expected made made explicit. explicit. return that An Anunder deserve constant to be made prices explicit. and the An The above decomposition has hidden simplifications that deserve to be made explicit. An TheThe appendixOne appendix simple shows wayshows theto theanalyze results results ofhedging ofaThe simple a simpleappendix effectiveness regression regression shows analysisalong the analysis results the on curve onthe of the abeta is simple betato to take theto regression theevery equity equity point market analysis market of the on the beta to the equity market importantimportant one one is is that that the the models modelsimportant used used to to capture onecapture is that the the theexpected expected models return return used under tounder capture constant constant the expected prices prices and and return the the under constant prices and the important one is that the models used to captureexpected the returnexpected under return price under changes constant must prices be correctly and the identified. If so, the decomposition is an of ofdifferentcurve different scaled points points by along durationalong the the curve. and curve. calculate ofFirst, differentFirst, we itswe focus hedging pointsfocus on onalongzero-coupon beta zero-coupon thewith curve. respect Treasury Treasury First, to the wereturns. equityreturns. focus Maturities onmarket. Maturities zero-coupon The Treasury returns. Maturities JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS expectedexpected return return under identity.under price price Omissions changes changesexpected must tomust thereturn be be price correctly correctly under appreciation price identified. identified. changes model If 3If mustso, so, will the the be bedecomposition decompositioncorrectly irrelevant identified. in is isour an an framework If so, the decomposition if they is an of between four and seven years provideexpected the return best hedge under toprice equities. changes The must “belly” be correctly of the curve identified. If so, the decomposition is an of ofbetweenlower between the four fourbeta, and and theseven sevenbetter years yearshedging provide provide properties the the best best ithedge will hedge provide.to equities.to equities. The The “belly” “belly” of ofthe the curve curve identity.identity. Omissions Omissions to to the the price priceidentity. appreciation appreciation Omissions model model to will will the be pricebe irrelevant irrelevant appreciation in in our our modelframework framework will be if if they irrelevantthey in our framework if they identity. Omissions to the price appreciationare model evenly will applied be irrelevant to all swap in our tenors. framework Indeed, if giventhey that we are focusing on the comparison of offersa monthly the frequency.best beta per If we unit extend of duration. the analysis If we to theextend swap the curve analysis they to theare evenlyswap curve applied across to all swap tenors. Indeed, given that we offersoffers the the best best beta beta per per unit unit of ofduration. duration. If weIf we extend extend the the analysis analysis to theto the swap swap curve curve across across areare evenly evenly applied applied to to all all swap swap tenors. aretenors. evenly Indeed, Indeed, applied given given to that allthat swap we we are aretenors. focusing focusing Indeed, on on the giventhe comparison comparison that we are of of focusing on the comparison of are evenly applied to all swap tenors. Indeed,expected given thatSharpe we areratio focusing signals onusing the acomparison particular model,of if the same term is omitted in two given severalacross severaldeveloped developed market market countries, countries, we obtain we obtain results results in the in sameare spirit, focusing with on some the comparison differences. of expected Sharpe ratio signals severalseveral developed developed market market countries, countries, we we obtain obtain results results in thein the same same spirit, spirit, with with some some differences. differences.expected expected Sharpe Sharpe ratio ratio signals signals using usingexpected a a particular particular Sharpe ratiomodel, model, signals if if the the using same same aterm termparticular is is omitted omitted model, in in two iftwo the given given same term is omitted in two given the same spirit, with some differences.expected The absolute Sharpe value ratio of signals the using a particulartenors model,model, the difference if if the the same same will term term remain is is omitted omitted constant, in intwo two leavinggiven given the relative ranking untouched. The absolute value of the beta is maximized at an intermediate maturity for the U.S., Europe and TheThe absoluteThe absolute appendix value value showsof ofthe the thebeta beta results is maximizedis maximized of a simple at anat regression anintermediate intermediate analysis maturity maturity on the for betafor the the U.S.,to theU.S., Europeequity Europe marketandtenors tenorsand the the difference difference will will remain remaintenors constant, constant, the difference leaving leaving the thewill relative relative remain ranking rankingconstant, untouched. untouched. leaving the relative ranking untouched. beta is maximized at an intermediate maturitytenors the for difference the U.S., will remaintenors constant, the difference leaving willthe remainrelative constant, ranking untouched.leaving the relative thethe U.K.,of U.K., different although although points it shifts it alongshifts to theto the curve.10-yeartheEurope 10-year U.K., First, contract. and contract.although we the focusU.K., The The italthough onJapanese shifts Japanesezero-coupon to it theshiftsswap swap 10-year tomarket, Treasury themarket, 10-yearcontract. in returns.contrast,in contract.contrast, The Maturities Japanese ranking swap market, untouched. in contrast, exhibitsof between greater four diversification and seven years benefitexhibitsThe provide Japanese per greater unit the swapbestof diversification duration hedgemarket, toat in equities. thecontrast, benefit 30-year The exhibits per contract, “belly” unit greater of ofdurationreflecting the curve at the the 30-year contract, reflecting the exhibits greater diversification benefit per unit of duration at the 30-year contract, reflecting the When appliedWhen to bond applied returns, to webond define returns, carry we as definethe return carry as the return assuming that the entire term diversification benefit per unit of duration at the 30-year contract, When applied to bond returns, we define carry as the return assuming that the entire term uniqueuniqueoffers dynamics dynamics the best of beta ofthe the Japaneseper Japanese unit of bond uniqueduration. bond market. market. dynamics If we extend of the the Japanese analysis bond to the market. swap curve across WhenWhenassuming applied applied thatto to bond bond the entirereturns, returns, term we we structure define define carry carryof rates as as thestaysthe return return constant. assuming assuming that that the the entire entire term term reflecting the unique dynamics of theWhen Japanese applied bond to bondmarket. returns, we define carrystructure as the of return rates assumingstays constant. that the Under entire this term definition, carry becomes the sum of the excess yield several developed market countries, we obtain results in the same spirit, with some differences.structurestructureUnder of of this rates rates definition, stays stays constant. constant. carry structurebecomes Under Under this ofthisthe rates definition, sumdefinition, staysof the constant. carry excesscarry becomes becomes yield Under the thisthe sum sumdefinition, of of the the excess carryexcess becomes yield yield the sum of the excess yield structure of rates stays constant. Under thisover definition, the risk-free carry becomesrate (slope) the plussum theof the roll-down excess yield derived from stepping from one maturity on the The analysis centered on beta has obvious limitations, as it does over the risk-free rate (slope) plus the roll-down derived from The absolute value of the beta is maximized at an intermediate maturity for the U.S., Europeoverover and the the risk-free risk-free rate rate (slope) (slope) plus plusover the thethe roll-down roll-downrisk-free ratederived derived (slope) from from plus stepping stepping the roll-down from from one one derived maturity maturity from on on steppingthe the from one maturity on the over the risk-free rate (slope) plus the roll-downcurve toderived a shorter from maturity. stepping Iffrom one maturity is the price on theof a bond at time t with maturity T when the not account correctly for the volatility of each contract. What stepping from one maturity curveon the to curve a shorter to a shorter maturity. maturity. If If is the price of a bond at time t with maturity T when the TheThe analysisthe analysis U.K., centered although centered on it on betashifts beta has to has obvioustheThe obvious 10-year analysis limitations, limitations, contract. centered as The asiton does itJapanesebeta does not has not account obviousswap account market, correctly limitations, correctly in. contrast,for for asthe (1) itthe does curve curve not to to aaccount a shorter shorter correctlymaturity. maturity. forIf If the is is the the price price
of of a
a bond bond at at time time t twith with maturity maturity T T when when the the good is a contract with a negative betacurve if it is to also a shorter very volatile? maturity. If isis thethe priceprice of of a abond bond at at time time t with t with maturity maturity

T when T when the
the

yield for maturity

T

at time t is
given


by
and
is the risk-free rate, then we calculate carry volatility of each contract. What good is a contract with a negative
beta
if it is also very volatile?yield



for maturity T at time t is given by and is the risk-free rate, then we calculate carry volatilityvolatilityexhibits of ofeach greater each contract. contract.diversification What What good Wegoodbenefit iswould ais contract pera becontract unitbuying withof withaduration longa negative a hedgenegative at the positionbeta
30-yearbeta if it ifat is
itthe contract, alsois expensealso
very very reflecting volatile? of volatile? yieldyield theyield forfor maturityfor maturity maturity T T at Tat timeat time time t tis ist isgiven given given by byby and and isis is thethe the risk-free risk-free risk-free



rate,
rate,
then then we we calculate calculate carry carry

yield

for
maturity

T at time t is
given
by and is the

risk-free





rate, then we calculate carry




plus roll-down as




Weincreased would volatility. be buying Because a long of hedge this, in position the next atsection the expense we of increasedrate, then volatility. we calculate Because carry ofplus this, roll-down

as



WeWe wouldunique would be dynamics bebuying buying a of long a the long hedgeJapanese hedge position position bond atmarket. theat the expense expense of ofincreased increased volatility. volatility. Because Because of ofthis, this,

plus roll-down

as



plusplus roll-down roll-down
as as



present a framework that takes into accountplus roll-down the global as



in the next section we present a framework that takes into account the global improvement
in the
in thein the next next section section we we present present a framework a framework that that takes takes into into account account theWe the global use global this improvement decompositionimprovement in thein to the start our analysis

by discussing the carry plus
roll-down (3)


(3) improvement in the Sharpe ratio derived from shifting
the









(3) Sharpe ratio derived from shifting the position along the curve.








(3)
(3)

















SharpeSharpe ratio ratio derived derived from from shifting shifting the the position position along along the the curve. curve. component of the expected return of these instruments.











Within








the expected


return




component










position along the curve.











(3)














When

estimating




expected










returns,





we model

as the sum
of two


































When




estimating

expected


returns,


we
model


as the sum of two pa
rts: the anticipated return for The analysis centered on beta has obvious limitations, as it does not
,account the
carry correctly
plus roll-down for the





excessparts:



the
return
anticipated



can be returnthought for ofconstant as

the marketexcess pricesreturn (carryof an asset,








EXPECTED RETURNS: CARRY AND ROLL-DOWN IN constant market prices (carry plus roll-down 

) and the expected return
coming from volatility of each contract. What good is a contract with a negative
assuming

beta if that it is market also very prices volatile? stayplus constant, roll-down or

.) andYou the could expected think returnof carry coming plus fromroll-down as SWAP CONTRACTS

When estimating expected
returns, we model
as the sum of two parts: the anticipated return for Expected returns: Carry and roll-down
in swap contractsreturns returns converging converging
to to their their mean mean values values .. ExhibitThis
assumes 1 shows that all the predicable component ExpectedExpectedWe would returns: returns: be buying Carry Carry a and long and roll-down hedge roll-down position in inswap swapat the contracts contractsexpense of increasedthe part ofvolatility. the expected Because return of this,that comes simply

from
the passage of time.

Hence,
by definition, Beyond the discussion of the beta exposure to equities, we a graphicalconstant decomposition market prices between (carry plus these roll-down
two
parts. This ) and the expected return coming from in the next section we present a framework that takes into accountcarry the is globala value improvement that can be observed inof the comes in advance. from meanThe second reversion. part Toof the
model
expected

this mean
return reversion comes of bond returns, we consider Beyond the discussion of the beta exposure to equities, we should analyze the best dynamic
BeyondBeyond the the discussion discussion of ofthe the beta beta exposureshould exposure analyze to equities,to equities, the best we dynamicwe should should allocationanalyze analyze the of the swapbest best dynamic contracts, dynamic assumes

returns that converging all the predicable to their component mean values of comes. This from assumes that all the predicable component the dynamic convergence of the yield curve to its expected


value as given by an autoregressive Sharpe ratio derived from shiftingtaking the into position account along not only the thesecurve. diversificationfrom expected benefitsprice appreciation but
mean reversion.: To model this mean reversion of
bond
returns, allocationallocation of ofswap swap contracts, contracts, taking takingallocation into into account account of swap not not only contracts, only these these diversification taking diversification into account benefits benefits not but only but also these also diversificationof comes benefits from meanbut also reversion. To model
this mean reversion of bond returns, we consider model.




also the estimated expected return and volatility of the strategy. A we consider


the dynamic convergence of the yield curve to its the estimated expected return and volatility of the strategy. A useful
the starting dynamic
point convergence is the self- of the yield curve to its expected value as given by an autoregressive thethe estimated estimated expected expected return return and and volatilityuseful volatility starting of ofthe pointthe strategy. strategy. is the Aself-evident usefulA useful starting statementstarting point point that is the isthe the self-return self- expected
(2) value as given by an autoregressive model. evident statement that the return of any asset
can be
decomposed
Exhibit into 1: Return its expected decomposition and between carry plus roll-down and repricing evidentevident statement statement that that the the return return of ofany anyany asset asset asset can can can be be bedecomposed decomposed decomposed into into into
its its its expectedexpected expected
andand
and model. Expected returns: Carry and roll-down in swap contracts











unexpected components:
Exhibit 1: Return decomposition
between carry plus unexpectedunexpected components: components:
unexpected
components:






. (1)


. (1) . (1)






roll-downExhibit 1:and Return repricing decomposition between carry plus roll-down and repricing



Beyond the

discussion
of the beta exposure
to equities,
we should analyze the best dynamic (1)






The
.above (1) decomposition has hidden simplifications that deserve to be made explicit. An



























Yield allocation of swap contracts, taking into account not
only these
important diversification
one is benefits that the butmodels also used to capture the expected returnEC (runder) constant prices and the
t

We use this decomposition



to start

our
analysis by discussing t the estimated expected return and volatility of the strategy. A useful starting point is the self- the carry plus roll-downWe component use this decompositionexpected of the expected return to underreturn start priceofour analysis changes by must discussing be correctly the carry identified. plus roll-downP If so, the decomposition is an We use this decomposition to start our Weanalysis use this by decompositiondiscussing the carryto start plus our roll-down analysis by discussing the carry plus roll-down E (rt ) Δ t + 1 evident statement that the returnthese of instruments. any asset can WithinWe be use decomposedthe thisexpected decompositionidentity. returninto Omissions its component expected to start to theour priceanalysis, and appreciation by discussing model the will carry be plusirrelevant roll-down in our framework if they component of the expected return of thesecomponent instruments. of thecomponent Within expected the of return expected the expected of these return returninstruments. component of these Within instruments. the expected Within return the expected component return component the carry plus roll-down excess return can be thought of as the unexpected components:

, the carry plus roll-down excess
return can be thoughtcomponent, theof as carry of arethe the evenlyexcessplus expected roll-down appliedreturn return of toexcess an allof
asset,
theseswap
return
instruments.tenors. can be Indeed, thought Within given of theas that the expected we excess are focusing returnreturn componentof on an the asset, comparison of
excess return, the of carryan asset, plus assuming roll-down that excess market return prices can stay be thought of as the excess return of an asset, t – 1
expected Sharpe ratio signals using a particular model, if the same term is omitted in two given assuming
that market
prices stay constant,constant, or . YouassumingYou
could could, the think thatcarry think marketof plus carryof carry roll-down pricesplus plus roll-down stay roll-down excess constant, as return as or can be .thought You could of as think the excess of carry return plus ofroll-down an asset, as


assuming

that market

prices stay constant, or . You could think of carry plus roll-down as


the part of the expectedtheassuming return
part of that that thetenors comes marketexpected the simply prices difference return from stay that the
constant,will comes remain simply or constant, from
. You theleaving couldpassage the think relativeof time. of carry rankingHence, plus by untouched.roll-down definition, as the
part of the expected return that comesthe simplypart
of the
from
expected
the
passage return of thattime. comes Hence, simply by definition, from
the passage

of time. Hence, by definition, passage of time. Hence, by definition, carry is a value that

can be
τ – 1 τ Maturity carry is a value that can be observed in advance. The secondcarrythe part is part aof value the of expectedthe that expected can returnbe observed return that comescomes in advance. simply Thefrom
second the passage part of of the time. expected Hence, return by definition, comes observedcarry in is advance. a value Thethat second can be partobserved of the inexpected advance. return The secondFor
part illustrative
of the purposes expected only. return Source: comes PIMCO carry is a value that can be observed in advance. The second part of the expected return comes from expected price appreciation comesfrom from: expected expected fromprice price expectedappreciation appreciation price appreciation: : When expected returns are modeled using the decomposition



from expectedWhen price applied
appreciation
to bond returns,
: we define carry as the return assuming that the entire term





between carry plus roll-down and curve mean reversion, the (2)



(2) structure of (2) rates stays constant.
Under this definition, carry becomes the sum of the excess yield







curve steepness becomes the focus of analysis. The slope of the





(2)














The above
decomposition


has

hidden

over
simplifications
the


risk-free

ratethat (slope) plusyield the curve, roll-down which givesderived the from implied stepping bond risk from premium, one maturity is related on the














deserve to be made explicit.
An important
one is that
the models For illustrative purposes only. Source: PIMCO




curve

to


a shorter

maturity. If to the term is the premium price of by a itsbond own at definition. time t with Fama maturity and Bliss T when (1987) the used to capture the expectedThe above return decomposition under constant has prices hidden and simplifications
document
that the predictabilitydeserve to be of made bond explicit.returns by An the spread The above decomposition has hidden simplificationsThe above decomposition that deserve has to behidden made simplifications explicit. An that deserve

to be made explicit. An the expected return under price changesyield formust maturity be correctly T at time t Whenis
given

For expected
by illustrative and returns purposes isare the modeled risk-free only. using rate,Source: the then decompositionPIMCO we calculate carrybetwe en carry plus roll-down and The above decomposition has hidden simplificationsbetween forward that deserve rates andto be one-year made explicit. rates; Cochrane An and important one is that the models used toimportant capture theone expected isimportant that the return modelsone isunder that used constantthe to models capture prices used the andexpected to capturethe return the expected under
constant return
underprices constantand the prices and the identified. If so, the decompositionplus is an roll-down identity. Omissions as to curvePiazzesi mean (2005) reversion,
refine
the this curve to model steepness bond returnsbecomes with the a focusmeasure of analysis. The slope of the yield important one is that the models used to capture the expected

return under constant prices and the expected return under price changesthe must priceexpected beappreciation correctly returnexpected modelunderidentified. willprice return be If changes irrelevant so, under the decompositionmust pricein our be changesframework correctly mustis if anidentified. be correctlyWhen If so, expected identified. the decomposition returns If so, are the modeled decompositionis an using the is andecomposition between carry plus roll-down and
curve,related which to the gives concavity the implied of the yieldbond curve. risk premium, is related to the term premium by its own expected return under price
changes must be correctly identified. If so, the decomposition is an identity. Omissions to the price appreciationidentity. model Omissions willidentity. be to irrelevant the Omissions price in appreciation our to frameworkthe price model appreciation if theywill be irrelevantmodelcurve will mean in be our(3) reversion,irrelevant framework inthe ourif curve they framework steepness if becomesthey the focus of analysis. The slope of the yield





definition.



Fama and Bliss (1987) document the predictability of bond returns by the spread identity. Omissions
to the
price


appreciation










model




will be irrelevant in our framework if they are evenly applied to all swap tenors. Indeed,are evenly given applied thatare we evenlyto areall swapfocusing applied
tenors. to
on all theIndeed, swap comparison tenors.given that Indeed, of
we are givencurve, focusing that which we on are givesthe focusing comparison the implied on the of bond comparison risk premium, of is related to the term premium by its own









between
forward rates and one-year rates; Cochrane and Piazzesi (2005) refine this to model are evenly applied to all swap tenors. Indeed, given that we are focusing on the comparison of expected Sharpe ratio signals using a particularexpected model,Sharpeexpected ifratio the signalssame Sharpe term using ratio is aomitted signalsparticular in using two model, agiven particular if the samedefinition. model, term if is theFama omitted same and term inBliss two is (1987) omittedgiven document in two given the predictability of bond returns by the spread bond returns with a 
measure related to the concavity of the yield curve. tenors the difference will remain constant,tenors leaving the difference thetenorsexpected relative willthe rankingSharpe differenceremain
ratiountouched. constant, will signals remain leaving using constant, thea particular relative leavingbetween rankingmodel, the relative ifforwarduntouched. the same ranking rates term and untouched. is one-yearomitted in rates; two Cochranegiven and Piazzesi (2005) refine this to model tenors the difference will remain constant, leavingbond the returns relative with ranking a measure untouched. related to the concavity of the yield curve.

When applied to bond returns, we define carry as the return assuming that the entire term 
When applied to bond returns, we defineWhen carry applied as the toreturn bond assuming returns, we that define the entire carry term as the return
assuming that the entire term structureWhen applied of rates to bondstays returns,constant. we Under define this carry definition, as the return carry assumingbecomes thethat sum the entireof the termexcess yield structure of rates stays constant. Understructure this definition, of rates carry stays becomes constant. the Under sum thisof the definition, excess yield carry becomes the sum of the excess yield 
structure of rates stays constant. Under this definition, carry becomes the sum of the excess yield over the risk-free rate (slope) plus the roll-downover the risk-free derivedover ratefrom the (slope) risk-freestepping plus ratefrom the (slope) roll-downone maturity plus derived the on roll-down the from
stepping derived fromfrom onestepping maturity from on one the maturity on the over the risk-free rate (slope) plus the roll-down derived from stepping from one maturity on the curve to a shorter maturity. If iscurve the priceto a shorter of acurve bond maturity. toat atime shorter If t with maturity. maturity is the If price T when of thea is bond the price at time of ta withbond maturity at time tT with when maturity the T when the



curve to a shorter
maturity.
If

is the price of a bond at time t with maturity T when the yield for maturity T at time t is

given



byyield andfor maturity isyield the T risk-free atfor time maturity t rate,is
given Tthen at
bytime we calculatet isand
given


carry isby the risk-free and israte, the then risk-free we calculate rate, then carry we calculate carry











yield for maturity T at time
t is given by
and
is the risk-free rate, then we calculate carry plus roll-down as plus
roll-down
plus as roll-down as















plus roll-down
as






(3)










(3)



















(3)


















































































(3)









































































4 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS

The yield curve tends to flatten late in the business cycle. This Exhibit 2: Average carry plus roll-down across vanilla movement occurs in tandem with the central bank raising rates interest rate swap contracts beyond expectations in order to curb inflation. The yield curve will Average carry + roll-down steepen when a recession hits, as the short end will reflect an When estimating expected returns, we model as the sum of2-year two parts: the5-year anticipated return10-year for 30-year easing monetary policy. This dynamic can serve as a basis for 34 39 27 11 constant market prices (carry plus roll-down ) and the expected return coming from implementing a basic mean-reversion model to capture the For illustrative purposes only. Source: PIMCO and Bloomberg as of 31
December 2018. Average carry plus roll-down (in basis points) for different portion of expected returns driven by a change in prices, .
returns converging to their mean values

.
Thistenors assumes of U.S. interest that all rate the swap predicable contract returns component adjusted per year of duration.

of comes from mean reversion. To
model


this mean reversion of bond returns, we consider MAXIMIZING
CARRY
AND ROLL-DOWN IN THE TIME SERIESthe
dynamic convergence of the yield curve to its expectedThe picture value painted as given by the by carry an autoregressive plus roll-down time series is model. more nuanced. Carry has fluctuated along different maturities Based only on carry and roll-down, the maximum average return depending on the shape of the swap curve. In 86% of the sample is located at the five-year portion of the curve. In line with the Exhibit 1: Return decomposition between carry plus periodroll-down 1995–2018, and repricing either the two-year or the five-year point current results, carry plus roll-down seems to pinpoint the maximized carry plus roll-down. The most prolonged period of duration range with the best performance. Exhibit 2 shows the higher carry on the long end of the curve was between March average return collected from carry plus roll-down per unit of 2006 and May 2017, corresponding to a late-cycle period with an duration for each point in the curve analyzed here. Exhibit 3 inverted yield curve. illustrates the time series of carry and roll-down per year of duration for the time period 1995-2018.

Exhibit 3: Carry plus roll-down per year of duration 2Y 5Y 10Y 30Y 140 120 100 80 60 40 20 Basis points 0

-20 -40 For illustrative purposes only. Source: PIMCO -60 When expected returns are modeled using the decomposition between carry plus roll-down and 2011-08 2011-02 2017-08 2017-02 1997-08 1997-08 1997-02 2012-08 2012-08 2012-02 2012-02 2015-08 2015-02 2013-08 2013-02 2018-08 2018-02 2016-08 2016-02 2001-08 2001-02 2001-02 2014-08 2014-02 2010-08 2010-02 1995-08 1995-08 1995-02 1998-08 1998-08 2007-08 1998-02 2007-02 2007-02 1996-08 1996-08 1996-02 1999-08 1999-08 1999-02 1999-02 2002-08 2002-02 2002-02 2005-08 2005-02 2005-02 2003-08 2003-08 2003-02 2003-02 2008-08 2008-08 2008-02 2008-02 2006-08 2006-02 2006-02 2009-08 2009-02 2009-02 2000-08 2000-08 2000-02 2000-02 2004-08 2004-02 2004-02 curve mean reversion, the curve steepness becomes the focus of analysis. The slope of the yield Source: PIMCO and Bloomberg as of 31 December 2018. Carry plus roll-down time series (in basis points) for different tenors of U.S. interest rate swap contract returns adjustedcurve, per year which of duration. gives the implied bond risk premium, is related to the term premium by its own definition. Fama and Bliss (1987) document the predictability of bond returns by the spread between forward rates and one-year rates; Cochrane and Piazzesi (2005) refine this to model MEAN REVERSION OF THE YIELD CURVE AND THE ECONOMIC CYCLE bond returns with a measure related to the concavity of the yield curve. Our mean-reversion model consists of an AR(1) equation Treasuries versus swaps. The swap curve typically is shaped projected on a horizon of 12 months in an expanding window. similar to the Treasury yield curve, and, as Exhibit 4 indicates, We use a minimum window of 10 years of data to initialize the the slopes of both curves are highly correlated. Consequently, expanding window. Although modeling the swap curve directly 
we choose to estimate the mean-reversion model on the would most
accurately characterize the mean reversion at play Treasury yield curve. in this trade, there is a large difference of data availability for JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS 5

Exhibit 4: The relationship between Treasury and swap curves

3.5

3

2.5

2

1.5 y = 0.89x + 0.1396 1 R2 = 0.93931

Swap 2/10 slope 0.5

0

-0.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 Treasury 2/10 slope

Hypothetical example for illustrative purposes only. Source: PIMCO and the Federal Reserve Board as of 31 December 2018. The exhibit shows the close relationship between the slope of the two- over the 10-year Treasury versus the slope along the swap curve for the same tenors. The swap 2/10 slope has been calculated by subtracting the plain-vanilla interest rate swap rates at the 10 and two tenor, while the Treasury 2/10 slope subtracts the Treasury yields computed by Gürkaynak et al. (2006) for zero-coupon bonds of those respective maturities. The yield basis is annual.

Our AR(1) model is taken on the slope of two-year, five-year positively sloped curve implies an expected upward turn for the and 10-year over the 30-year point on the yield curve and economy, as it indicates the bottom of the contraction of the takes the form cycle and the beginning of the recovery.

(4) Exhibit 5 shows the time series of the slope of the 30-year over 




the two-year Treasury bond versus the U.S. unemployment rate.























The slope of the yield curve historically has been a relatively This plot underlines two observations. First, there is a clearly reliable leading indicator of recessions and the economy’s identifiable cycle in the slope of the curve that suggests a mean- position in the business cycle. The long end of the curve reflects reversion pattern in this market. Second, the larger oscillations in long-term expectations about the monetary policy rate. Given the slope cycle are tied to the economic cycle, with rises in that nominal rates comprise expectations of both real rates and unemployment associated with steepening of the curve, and vice inflation, the long end will be informative about the beliefs about versa. The sample correlation of slope and unemployment over long-term inflation and the economy’s growth rate. A sharply the period 1985–2017 amounts to 71%.

Exhibit 5: Yield curve slope versus unemployment rate Slope Unemployment 5 12

4 10

3 8

2 6

1 4

0 2 Percentage points (slope) (slope) points Percentage -1 0 Percentage points (unemployment) (unemployment) points Percentage 2011-12 1997-12 2017-10 2014-11 2007-11 1993-11 1986-11 2010-10 1989-10 1996-10 1990-12 2011-05 1991-07 2000-11 2000-11 2003-10 2003-10 2004-12 1997-05 1987-06 2012-07 2012-07 2017-03 2016-01 2018-05 2015-06 1995-01 2013-02 2013-09 1998-07 2016-08 2001-06 1988-01 1992-02 1992-09 2010-03 1995-08 1995-08 2014-04 1990-05 1985-02 1985-02 1989-03 1985-09 1985-09 1988-08 2007-04 1993-04 1994-06 1996-03 1999-02 1999-02 1986-04 1999-09 2002-01 2005-07 2002-08 2009-01 2008-06 2008-06 2004-05 2003-03 2003-03 2006-02 2006-02 2006-09 2009-08 2000-04 Source: PIMCO and Haver Analytics as of 31 December 2018. The green line shows the headline unemployment rate, which is measured on the right vertical axis (in percentage points). The blue line shows the slope of the yield curve between the two-year and 30-year points, measured on the left vertical axis. 6 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS

The forecast derived from this model is implemented by where the expected return derived from mean reversion is the rebalancing to one year of duration in the swap at the end of the highest. We compare these returns by calculating the returns month. The return time series coming from mean reversion per obtained by betting on the expected change in the slope of the unit of duration is plotted in Exhibit 6. We add this mean two-year, five-year and 10-year points of the yield curve versus reversion to our strategy by moving into the portion of the curve the 30-year point.

Exhibit 6: Expected return from mean reversion

2Y 5Y 10Y 50 40 30 20 10 0 -10 -20 -30 -40 -50 2011-10 2013-11 2018-11 2015-12 1998-11 2016-10 2001-10 2010-12 1995-12 1996-10 2011-05 2003-11 2003-11 2008-11 2008-11 2005-12 2006-10 2000-12 2000-12 2017-08 2017-03 1997-08 1997-08 1997-03 2015-07 2013-01 2018-01 2012-08 2012-08 2012-03 2012-03 2016-05 2010-07 2013-06 2018-06 2001-05 1995-07 2015-02 1998-01 2010-02 2014-09 1995-02 1996-05 1998-06 2007-08 2014-04 2007-03 2007-03 1999-09 1999-09 1999-04 1999-04 2005-07 2003-01 2003-01 2002-08 2008-01 2008-01 2002-03 2002-03 2006-05 2000-07 2000-07 2003-06 2003-06 2008-06 2008-06 2005-02 2005-02 2009-09 2009-09 2000-02 2000-02 2004-09 2009-04 2004-04 2004-04

Hypothetical example for illustrative purposes only. Source: PIMCO and the Federal Reserve Board as of 31 December 2018. The expected return from the mean- reversion signal is calculated by projecting the fitted AR(1) model described by Equation 4 into a horizon of 12 months. We assume that the 30-year point remains unchanged and infer the return on each swap contract based on the forecasted slope. The expected return is reported in basis points on an annual basis.

The mean-reversion model presented in this paper captures only Exhibit 7: Average allocations of pension plans according the business-cycle factor, in a naive manner. Sophisticated to size models for bond risk premia in general and for the swap curve in Pension plan size Asset allocation ≤ 5M$ ≤ 50M$ > 50M$ particular must explicitly tackle the more important challenges of Equity 49% 50% 48% modeling the swap spread itself and the speed of mean reversion Fixed income 24% 22% 22% across different business cycles. Real estate 7% 6% 9% Private equity 5% 8% 10% OPTIMAL DURATION POSITIONING IN THE Hedge fund 7% 6% 8% CONTEXT OF A REPRESENTATIVE PUBLIC Commodity 4% 3% 2% Alternatives 2% 2% 1% PENSION FUND Cash 1% 2% 1% In the context of a large pension fund, we are interested in Other 1% 0% 0% Total 100% 100% 100% both the expected return and the volatility of a representative portfolio that includes a swap overlay to gain exposure to Source: PIMCO and the Center for Retirement Research at Boston College as of 31 December 2018 duration, as well as the particular hedging properties of the curve location with respect to this benchmark portfolio. Exhibit 8 shows a decomposition of an average pension fund Exhibit 7 shows the average asset distribution of U.S. portfolio, and a detail on the fixed income allocation. Fixed pension plans according to their size. Pension fund income is allocated approximately 60% to U.S. aggregate, 10% to allocations do not vary greatly across fund size; the most global investment grade, 10% to high yield, 10% to U.S. long notable divergence is a greater private equity exposure for Treasuries, 5% to emerging market debt and 5% to securitized the largest funds. The breakdown for the main asset classes mortgages. These allocations are distributed into the total 21.7% is almost identical. In large pension funds, with assets fixed income in the total portfolio, resulting in the percentages greater than $50 million, slightly under half of the allocations shown in the exploded pie chart. are directed toward the equity market and 22% are allocated to fixed income instruments. JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS 7 level of duration achieved by the benchmark Treasury portfolio so we can assess the level of duration achieved by the benchmark Treasury portfolio so we can assess the Exhibit 8: Average pension plan allocation to asset classes achieved by the benchmark Treasuryeffectiveness portfolio of so each we cancontract at reaching the same target. A separate and very interesting effectiveness of each contract at reaching the same target. A separate and very interesting 1% assess the effectiveness of eachquestion contract is theat reaching amount the of durationsame to take to offset equity risk – a question we do not answer in 10%question is the amount of durationtarget. A to separate take to andoffset very equity interesting risk – question a question is thewe amountdo not answer in 9.5% 12% this exercise. 7.6% of duration to take to offset equity risk – a question we do not 9.2% this exercise. answer in this exercise. 13.0%

AN OPTIMAL SWAP TENORAn FRAMEWORK:optimal swap tenor framework: Expected Sharpe ratio maximization 21.7% An optimal swap tenor framework:EXPECTED Expected SHARPE Sharpe RATIO ratio MAXIMIZATION maximization 2.2% First, we estimate expected returns.First, We we characterize estimate expected expected returns. We characterize expected returns by adding the carry plus First, we estimate expected returns. We characterize expected returns by adding the carry plus 48.1% 2.2% returns by adding the carry plusroll-down roll-down to thethe expected return coming from price changes , which in our roll-down to the expectedexpected return return coming coming from from price price changeschanges
, , which inin our

2.2% framework is uniquely
informed by the mean-reversion model. When we add
both effects, we
1.1% our framework is uniquely informed by the

mean-reversion




framework

is uniquely
1.1%
informed by the mean-reversion obtainmodel. the When resulting
we
add
time both series effects, of expected we returns for each maturity, shown in Error! model. When we add both effects, we obtain the resulting time Equity Commodity U.S. aggregateobtain theU.S. resultinglong Treasury time series of expected returns for eachReference maturity, source shown not in found.Error!. Total expected returns exhibit some cyclicality, with Real estate Alternatives Global IG EM debt series of expected returns for each maturity, shown in Exhibit 11. Private equity Cash High yieldReferenceSecuritized source mortgages not found.Total. Total expected expected returns returns exhibit exhibit somesynchronized cyclicality, some cyclicality, lows with synchronizedin the with periods immediately before the dot-com and financial crisis recessions. Hedge fund lows in the periods immediately before the dot-com and financial synchronized lows in the periods immediately before theThe dot-com volatility and offinancial the front crisis end recessions.of the curve is, in general, larger than that of the long end. On Source: PIMCO, Bloomberg and the Center for Retirement Research at Boston crisis recessions. The volatility of the front end of the curve is, in College as of 31 December 2018. The proxy for equitiesThe volatility is the SPX TR of Index. the front end of the curve is, in general, larger than that of the long end. On general, larger than that of the longaverage, end. Onthe average, front and the intermediate front sections of the curve provide the highest expected return per We use the Bloomberg Barclays US Aggregate Bondaverage, Index, J.P. the Morgan front Global and intermediate sections of the curve provide the highest expected return per Aggregate Bond Index, Bloomberg Barclays US Corporate High Yield Total and intermediate sections of theunit curve of durationprovide the among highest the considered duration levels, although there is considerable variability Return Index Value Unhedged USD, Bloomberg Barclays US Treasury Index, Bloomberg Barclays Emerging Markets Hard Currencyunit Aggregateof duration Index, among SXI the expectedconsidered return duration per unit levels, of duration althoughacross among time. there the If is weconsidered considerable decompose variability our sample among the precrisis period 1995–2007, the financial Real Estate Funds Broad Total Return, S&P Listed Private Equity Index, HFRI duration levels, although there is considerable variability across Fund Weighted Composite Index, Bloomberg Commodityacross Indextime. and If Morningstar we decompose our sample among the precrisiscrisis periodperiod 2008–20111995–2007, and the thefinancial post-financial-crisis period 2012–2018, we see that the time. If we decompose our sample among the precrisis period Diversified Alternatives Index to approximate thecrisis returns period of the U.S. 2008–2011 aggregate, and the post-financial-crisis period 2012–2018, we see that the global investment grade, high yield, long Treasury, emerging markets debt, real 1995–2007, the financial crisis differenceperiod 2008–2011 in favor and of theshort-tenor-swap post- average returns widens almost fourfold during the estate, private equity, hedge fund, commodity anddifference alternatives, inrespectively. favor of The short-tenor-swap average returns widens almost fourfold during the proxy for cash is one-month LIBOR, and the proxy for securitized mortgages is financial-crisis period 2012–2018,financial we see crisis that theand difference reverses inin the period after the crisis, when, on average, the highest return is the Bloomberg Barclays US MBS Index Total Return Value Unhedged. financial crisis and reversesfavor in the of periodshort-tenor-swap after the crisis, averagelocated when, returns inon thewidensaverage, 10-year almost the contract highest (seereturn Exhibit is 9). located in the 10-year contractfourfold (see during Exhibit the 9). financial crisis and reverses in the period after To evaluate the optimal contract used to obtain the target the crisis, when, on average, theExhibit highest 9: return Average is located returns in the and volatility of different swap tenors duration exposure, we consider the following exercise: We Exhibit 9: Average returns 10-yearand volatility contract of (see different Exhibit swap 9). tenors replace the 10% U.S. long Treasury within the fixed income allocation with a Libor exposure and add a swap overlay Exhibit 9: Average returns and volatility of different swap tenors leveraged to obtain the same duration as a representative U.S.





long Treasury index. Given that we aim to determine
which


2-year 5-year
10-year 
30-year
 
 
 


position in the swap curve provides the maximum benefit to the 
 
Mean  
0.18% 0.17%  
0.13% 
 
0.10%  
 
 
 

1995–2007 portfolio, we will calculate the expected Sharpe ratio to serve as a  
 
Volatility  
0.57% 0.78%  
0.62%  

0.43%  
 
 
 
Mean 0.49% 0.36%
0.24% 0.19% ranking criterion for different contracts across time. Each month 
2008–2011 
 
 
 
 
 
 
 

Volatility 0.44% 0.46% 0.71% 0.90% (see appendix for quarterly rebalancing), we select
the swap  
 
Mean  
0.03% 0.11%  
0.13% 

0.10% 
 
 
 
2012–2018 
tenor with the highest expected Sharpe ratio to optimally position 

Volatility  
0.29% 0.24%  
0.30%  
 
0.43% 
 

 

along the curve.  
Total sample
mean  
0.58% 0.64% 
 
 
 0.54%  
0.43%
 
 
 
 
Total sample volatility 1.02% 0.86% 0.75% 0.75%  
 

 
 
 
 
The analysis that follows relies on leveraging swap contracts at Hypothetical example for illustrative purposes only. Source: PIMCO and 
a particular point of the yield curve. For that reason, this Bloomberg as of 31 December 2018. The
mean return corresponds to the annualized average monthly
returns of the respective swap contract for the time framework tends to be most relevant for
strategies that are not period specified in the first column. Volatility corresponds to the annualized heavily constrained on leverage positions. Likewise, we choose sample standard deviation of monthly returns. to compare the strategy with the same level of duration  
 
 
 
 
 
 

Hypothetical example for illustrative purposes only. Source: PIMCO and Bloomberg as of 31 December 2018 The mean return corresponds to the annualized average monthly returns of the respective swap contract for the time period specified in the first column. Volatility corresponds to annualized the sample standard deviation of monthly returns.

Exhibit 10: Total expected return from carry plus roll down and mean reversion
8 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS

Exhibit 10: Total expected return from carry plus roll-down and mean reversion  
2Y 5Y 10Y 30Y  
120 
100 
80 

60


40 
20  
 
0

 
-20

-40  
 
 








 
 
 
 
 
 
 








 
 
 
 
 
 
 








2011-10 2018-11 2013-11 2015-12 1998-11 2016-10 2001-10 2010-12 1995-12 1996-10 2011-05 2003-11 2003-11 2008-11 2008-11 2005-12 2006-10 2000-12 2000-12 2017-08 1997-08 1997-08 2017-03 1997-03 2015-07 2013-01 2018-01 2012-08 2012-08 2012-03 2012-03 2016-05 2001-05 2010-07 2013-06 2018-06 1995-07 2015-02 1998-01 2010-02 2014-09 1995-02 1996-05 1998-06 2007-08 2014-04 2007-03 2007-03 1999-09 1999-09 1999-04 1999-04 2005-07 2003-01 2003-01 2002-08 2002-03 2002-03 2008-01 2008-01 2006-05 2000-07 2000-07 2003-06 2003-06 2008-06 2008-06 2005-02 2005-02 2009-09 2009-09 2000-02 2000-02 2004-09 2009-04 2004-04 2004-04
Hypothetical example for illustrative purposes only. Source: PIMCO and the Federal Reserve Board as of 31 December 2018 Hypothetical example for illustrative purposes only. Source: PIMCO and the Federal Reserve Board as of 31 December 2018 To compute the Sharpe ratio corresponding to each point of the We now turn to the denominator of our projected Sharpe ratio. curve, we need to include the return of the benchmark portfolio in Given that we are implementing a swap overlay on the To computethe numerator. the Sharpe Although ratio corresponding a predictive model to each based point on of a thefactor curve, webenchmark need to include portfolio, the the total volatility will be determined by that specification could be used to render a better forecast of the of the particular swap tenor used to achieve the target volatility, return of the benchmark portfolio in the numerator. Although a predictive model based on a expected return of the benchmark, we fix its expectation at a scaled by the multiple needed to achieve the target duration of factor specification could be used to render a better forecast of the expected return of the long-term average calculated using the full sample from 1995 the benchmark. Differences in volatility among combinations of benchmark,through we 2018.fix its In expectation this time frame, at a long-termthe average average return ofcalculated the usingthe the benchmark full sample with from different points on the curve will arise from 1995 throughbenchmark 2018. is In 8.1%, this which time frame,corresponds the average to the long-termreturn of theaverage benchmark 1) differences is 8.1%, which in the volatility of different swap contracts and 2) correspondsof the to portfolio the long-term that results average from of replacing the portfolio the U.S. that long results Treasury from replacingdifferent the correlations U.S. long of those contracts with the benchmark. component with Libor. The total expected return of the portfolio From an econometric point of view, we model portfolio volatility Treasury component with Libor. The total expected return of the portfolio with swap overlay with swap overlay corresponds to this long-term average plus the by computing the historical standard deviation of returns of the correspondsexpected to this return long-term model ofaverage each swap plus pointthe expected in the curve, return model of eachportfolio swap with point the in swap the overlay, using an expanding window. curve, . Exhibit 10 plots the time series of the total


expected
return for
each swap tenor.

Exhibit 11: Rolling (36-month) volatility 
swap returns per unit of duration
2Y 5Y 10Y 30Y 0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0 2011-01 2011-01 2011-07 2011-07 2017-01 2017-01 2017-07 2017-07 2012-01 2012-01 2012-07 2012-07 2015-01 2015-07 2013-01 2018-01 2013-07 2018-07 2016-01 2016-07 2001-01 2001-07 2014-01 2014-07 2010-01 2010-07 1998-01 1998-07 2007-01 2007-07 1999-01 1999-01 1999-07 1999-07 2002-01 2002-07 2005-01 2005-07 2003-01 2003-01 2003-07 2003-07 2008-01 2008-01 2008-07 2008-07 2006-01 2006-07 2009-01 2009-07 2000-01 2000-01 2000-07 2000-07 2004-01 2004-07 Source: PIMCO and Bloomberg as of 31 December 2018. Calculations based on the standard deviation of swap returns scaled per unit of duration on a rolling window of 36 months. JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS 9

(Using a rolling window between 12 and 48 months does not As Exhibit 12 illustrates, volatility profiles across tenors have generate significant changes in total performance.) Alternative gone through regime changes over time. From 2008 onward, the specifications using more sophisticated volatility modeling, such 30-year contract has become the most volatile per unit of as a GARCH (1,1) process, yield results with no material duration, although the difference with the two-year contract has improvement in this case. As Exhibit 11 makes apparent, the diminished since the peak in the middle of the financial crisis. ranking of volatilities among different swap tenors has changed Exhibit 13 shows the optimal swap contract according to the across time. Although the 30-year contract displays the lowest maximum expected Sharpe ratio criterion. In line with the volatility before 2007, during the post-quantitative-easing (QE) regression evidence presented previously, this criterion indicates era, after 2009, the two-year contract is the least volatile by a that it is optimal to position the portfolio in the five-year tenor wide margin. 54% of the time, followed by the two-year 28.1% of the time and Exhibit 12: Average volatility of duration-adjusted returns the 10-year 12.4%. It is remarkable that the front end of the curve across regimes for different swap tenors was an optimal positioning only during the pre-crisis period and

19952007 2002011 2012201 from 2016 onward. In the decade 2006–2015, the two-year 001 contract was chosen only 2.5% of the time. During the period of 0009 financial turmoil and subsequent quantitative easing in the 000 decade following 2007, the two-year contract was not optimal 0007 according to the yield curve mean-reversion model, which 000 predicts a normalization of the slope to historical levels. One 0005 challenge to be tackled by a more sophisticated mean-reversion 0004 0003 model is to capture the extraordinarily long mean reversion of the 0002 yield curve experienced in the QE era; it greatly exceeds that of 0001 previous business cycles. While we can qualitatively describe the 0 exceptional regime of the past decade as related to the 2 5 10 30 quantitative easing policies implemented by the Federal Reserve Source: PIMCO as of 31 December 2018. Calculations of the annualized standard deviation of swap returns are adjusted per unit of duration for each and central banks around the world, a systematic way to capture specified time interval. this phenomenon would render an obvious advantage to the forecasting power of the expected Sharpe ratio used to switch among contracts.

Exhibit 13: Optimal contract according to dynamic swap strategy

35

30

25

20

15 Contract tenor

10

5

0 2011-12 2017-10 1997-10 2012-10 2012-10 2014-11 2016-12 2001-12 1999-11 1999-11 2007-10 1996-12 2011-07 2011-07 2002-10 2002-10 2009-11 2011-02 2004-11 2006-12 2017-05 1997-05 2012-05 2012-05 2016-07 2001-07 2014-01 2013-08 2018-08 2013-03 2015-09 2018-03 2016-02 2015-04 2007-05 2014-06 2001-02 2001-02 1996-07 1999-01 1999-01 2010-09 1998-08 1998-08 2010-04 1998-03 1996-02 1999-06 2002-05 2006-07 2009-01 2004-01 2003-08 2003-08 2005-09 2003-03 2003-03 2008-08 2008-08 2008-03 2008-03 2009-06 2006-02 2006-02 2005-04 2004-06 2000-09 2000-09 2000-04

Hypothetical example for illustrative purposes only. Source: PIMCO as of 31 December 2018. This exhibit shows the optimal contract as suggested by the maximum expected Sharpe ratio criterion proposed in this paper.The vertical axis indicates the tenor of the swap contract to leverage to obtain the target duration. 10 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS

Under what conditions does the optimality of the medium- The dynamic swap switching strategy compares favorably with tenor part of the curve break down? If we experience a the benchmark, displaying a total Sharpe ratio of 0.915 versus structural change in the slope of the curve, either because of a 0.621 in the benchmark pension fund portfolio. Moreover, the repricing of inflation risk or because of an anticipated new 36-month rolling marginal Sharpe ratio over the benchmark, inflation regime, the model will mean revert to the incorrect illustrated in Exhibit 14, was positive for 99% of the sample target. To derive the best benefit from this approach, the before 2012 and dropped to 63% from 2012 onward. A notable manager will have to keep her attention on the correct model benefit of this strategy is shown in Exhibit 15, where we observe of yield curve mean reversion. Different views of the target the improved protective effects with respect to those offered by values of the yield curve will inform different optimal the Treasury bond exposure in the benchmark. As illustrated in positioning strategies. the exhibit, the maximum drawdown experienced in both the dot-com crash and the financial crisis is substantially lower in the portfolio with swap overlay.

Exhibit 14: Rolling 36-month Sharpe ratio for portfolio with dynamic swap overlay versus benchmark

Dynamic Benchmark 3

2.5

2

1.5

1

0.5

0

-0.5

-1 2017-12 2012-12 2015-11 2010-11 2013-10 2018-10 2007-12 2002-12 2002-12 2005-11 2011-09 2000-11 2000-11 2011-04 2003-10 2003-10 2008-10 2008-10 2017-07 2017-07 2017-02 2012-07 2012-07 2015-01 2012-02 2012-02 2018-05 2013-05 2015-06 2010-01 2007-07 2016-09 2010-06 2001-09 2014-08 2016-04 2014-03 2001-04 2007-02 2007-02 1999-08 1999-08 1999-03 1999-03 2002-07 2005-01 2002-02 2002-02 2003-05 2003-05 2008-05 2008-05 2000-01 2000-01 2005-06 2006-09 2000-06 2000-06 2009-08 2009-03 2009-03 2006-04 2004-08 2004-03 2004-03

Hypothetical example for illustrative purposes only. Source: PIMCO as of 31 December 2018. We calculate the Sharpe ratio by averaging the annualized excess returns of each portfolio on a rolling window of 36 months. Excess returns are calculated with respect to one-month Libor. The denominator is calculated using the sample standard deviation of the annualized portfolio returns.

Exhibit 15: Drawdown for portfolio with dynamic swap overlay versus benchmark

Dynamic Benchmark 0 -0.02 -0.04 -0.06 -0.08 -0.1 -0.12 -0.14 -0.16 -0.18 -0.2 2011-08 2011-02 2017-08 2017-02 1997-08 1997-08 1997-02 2012-08 2012-08 2012-02 2012-02 2015-08 2013-08 2015-02 2013-02 2018-02 2018-08 2016-08 2016-02 2001-08 2001-02 2001-02 2014-08 2014-02 2010-08 2010-02 1998-08 1998-08 2007-08 2007-08 1998-02 2007-02 2007-02 1996-08 1996-08 1996-02 1999-08 1999-08 1999-02 1999-02 2002-08 2002-02 2002-02 2005-08 2003-08 2003-08 2005-02 2003-02 2003-02 2008-08 2008-08 2008-02 2008-02 2006-08 2006-02 2006-02 2009-08 2009-02 2009-02 2000-08 2000-08 2000-02 2000-02 2004-08 2004-02 2004-02 Hypothetical example for illustrative purposes only. Source: PIMCO and Bloomberg as of 31 December 2018. This exhibit shows the drawdown patterns for the benchmark strategy versus the optimal swap switching (“dynamic”) strategy suggested in this paper for the time period 1996–2018. The drawdown is the peak-to- trough decline in the portfolio value, quoted as a fraction, between the peak and subsequent trough. JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS 11

Exhibit 16: Value of $100 invested in March 1996 (dynamic swap strategy versus benchmark)

Dynamic Benchmark $1,200

$1,000

$800

$600

$400

$200

$0 2011-12 2017-10 1997-10 2012-10 2012-10 2014-11 2016-12 2001-12 1999-11 1999-11 2007-10 1996-12 2011-07 2011-07 2002-10 2002-10 2009-11 2011-02 2004-11 2006-12 2017-05 1997-05 2012-05 2012-05 2016-07 2001-07 2014-01 2013-08 2015-09 2018-03 2018-08 2013-03 2016-02 2015-04 2007-05 2014-06 2001-02 2001-02 1996-07 1999-01 1999-01 2010-09 1998-08 1998-08 2010-04 1998-03 1999-06 1999-06 1996-02 2002-05 2006-07 2009-01 2004-01 2003-08 2003-08 2005-09 2003-03 2003-03 2008-08 2008-08 2008-03 2008-03 2009-06 2006-02 2006-02 2005-04 2004-06 2000-09 2000-09 2000-04

Hypothetical example for illustrative purposes only. Source: PIMCO and Bloomberg as of 31 December 2018. This exhibit shows the drawdown patterns for the benchmark strategy versus the optimal swap switching (“dynamic”) strategy suggested in this paper for the time period 1996–2018. The drawdown is the peak- to-trough decline in the portfolio value, quoted as a fraction, between the peak and subsequent trough. Model performance figures do not reflect the deduction of investment advisory fees. Performance would be lower if fees were applied.

CONCLUSION REFERENCES

We have presented a framework for considering the optimal Asness, Clifford S., Andrea Frazzini and Lasse H. Pedersen. positioning along the yield curve to hedge a generic portfolio “Leverage Aversion and Risk Parity.” Financial Analysts Journal, against equity drawdowns. By replacing the typical long January/February 2012 Treasury exposure in a representative pension fund portfolio Baz, Jamil, Josh Davis and Graham A. Rennison. “Hedging for with a leveraged swap overlay, we are able to offer an improved Profit: A Novel Approach to Diversification.” PIMCO Quantitative Sharpe ratio and drawdown profile. The optimal positioning Research, October 2017 depends on the carry and roll-down of each contract along the curve, plus a suitable mean-reversion model for different Cochrane, John H. and Monika Piazzesi. “Bond Risk Premia.” segments of the yield curve. American Economic Review, March 2005

In this paper, we have presented a simplified mean-reversion Fama, Eugene F. and Robert R. Bliss. “The Information in Long- model of the yield curve that relies only on autoregressive Maturity Forward Rates.” American Economic Review, September properties to find the speed of mean reversion. In practice and 1987 in our historical sample of interest, we observe substantial variation in the mean reversion of the swap curve, as well as Gürkaynak, Refet S., Brian Sack and Jonathan H. Wright. “The irregularity in the business-cycle duration, the bond market U.S. Treasury Yield Curve: 1961 to Present.” Journal of Monetary response to it and the dynamics of the swap spread. The astute Economics, November 2007 manager will find it useful to insert more sophisticated models Moore, James and Vijendra Nambiar. “The Journey to LDI and for the mean reversion in the swap curve into the general Beyond.” Journal of Investing, Summer 2012 framework presented here in order to improve the performance and hedging effectiveness of his fixed income exposures. Sargent, Thomas J. The Conquest of American Inflation. Princeton, NJ: Princeton University Press, 1999

Thinking Ahead Institute. Global Pension Assets Study 2018. Willis Towers Watson, 2018

We would like to thank Mike Cudzil and Sudi Mariappa for the invaluable discussions that led to the development of the ideas in this paper. APPENDIX APPENDIX 12 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS APPENDIXAPPENDIXAPPENDIX A simplified analysis using GSW zero-coupon data A simplified analysis usingAPPENDIX GSW zero-coupon data For concreteness, we translate the previous results into a AA Asimplified simplifiedsimplified analysis analysisanalysis using usingusing GSW GSWGSW zero-coupon zero-couponzero-coupon data datadata 60/40 portfolio conditional on an equity market drop of 25%. WeA simplified first carry outanalysis a simplified using GSW analysis zero-coupon using data data from Gürkaynak et al., henceforth referred to We first carry out a simplified analysis using data from Gürkaynak et al., henceforth referredWe to plot in the horizontal axis the theoretical duration of a WeWeWe first firstfirst carry carrycarry out outout a asimplifieda as simplifiedsimplified GSW, (2006)analysis analysisanalysis to using calculate usingusing data datadata yieldsfrom fromfrom Gürkaynak for GürkaynakGürkaynak zero-coupon et et etal., al.,al., Treasuryhenceforth henceforthhenceforth bonds. referred referredreferred We tointroduce toto this analysis as GSW, (2006) to calculateWe yields first for carry zero-coupon out a simplified Treasury analysis bonds. using We introducedata from this analysiszero-coupon Treasury and in the vertical axis the portfolio asasas GSW, GSW,GSW, (2006) (2006)(2006) to toto calculate calculateascalculateGürkaynak a first yields approximationyieldsyields et al.,for forfor henceforthzero-coupon zero-couponzero-coupon to the referred problemTreasury TreasuryTreasury to asbecause bonds. GSWbonds.bonds. (2006),WeGSW WeWe introduce introduce introduceyields to are this thisthis imperfectlyhedging analysis analysisanalysis benefit linked conditional to tradable on a 25% drop in equity prices. as a first approximation to the problem because GSW yields are imperfectly linked to tradable asasas a afirsta firstfirst approximation approximationapproximationassets.calculate to toto the thethe problem yields problemproblem for because zero-coupon becausebecause GSW GSWGSW Treasuryyields yieldsyields are arearebonds. imperfectly imperfectlyimperfectly We linked linkedlinked to toto Thetradable tradabletradable benefit is defined as excess return with respect to the assets. assets.assets.assets. introduce this analysis as a first approximation to the problem 10-year duration, which is used as the benchmark and because GSW yields are imperfectly linked to tradable assets. therefore corresponds to a benefit of zero. Exhibit 18 shows

that being in the duration point between four and seven years To determine which point of the yield curve provides the best To determine which point of the yield curve provides the best hedgewill increaseto equities, our wehypothetical normalize portfolio performance by 50 To determine which point ofhedge the yield to equities, curve provides we normalize the best the hedgeexcess to zero-coupon equities, we bond normalize ToToTo determine determinedetermine which whichwhich pointthe pointpoint excess of ofof the thethe zero-coupon yield yieldyield curve curvecurve providesbond providesprovides returns the thethe best by bestbest their hedge hedgehedge analytical to toto equities, equities,equities, duration we wewe normalize normalizebasisnormalizeand estimate points compared the beta regression with the benchmark. the excess zero-coupon bondreturns returns by by their their analytical analytical duration duration and and estimate estimate the the beta beta regression thethethe excess excessexcess zero-coupon zero-couponzero-couponasregression bond bondbond returns returnsreturns as by byby their theirtheir analytical analyticalanalytical duration durationduration and andand estimate estimateestimate the thethe beta betabeta Exhibitregression regressionregression 18: Incremental hedging benefit with respect to a as asasas benchmark of a position in a 10-year zero-coupon bond





















002




































































001






InIn this this equation,equation,
the

beta term measures thethe sensitivitysensitivity toto equity returns0 of a normalized 10- In this equation, the beta term measures the sensitivity to equity returns of a normalized 10- InInIn this thisthis equation, equation,equation, the thethe beta equitybetabeta term termterm returns measures measuresmeasures of a normalized the thethe sensitivity sensitivitysensitivity 10-year-duration to toto equity equityequity returns portfolio returnsreturns of of of thata anormalizeda normalized normalized -00110- 10-10- year-duration portfolio that uses

bonds of maturity and cash. For example, a portfolio of year-duration portfolio that uses
bondsbonds ofof maturity and cash. For example,example, a a portfolio portfolio of -002 year-durationyear-durationyear-duration portfolio portfolioportfolio100% that
thatthat allocateduses usesuses
bonds

bondsbonds to ofthe ofof maturity 10-yearmaturitymaturity point and andand in cash. cash.thecash. zero-yieldFor ForFor example, example,example,
curve a aportfolioa will portfolioportfolio have of aofof theoretical duration of 10 of 100% allocated


to the 10-year point in the zero-yield
curve -003 100% allocated to the 10-year point in the zero-yield

curve will have a theoretical duration of 10 100%100%100% allocated allocatedallocated to toto the thetheand will10-year 10-year10-year exposurehave pointa pointpointtheoretical to in the inin the the themarket zero-yield duration zero-yieldzero-yield of
of

curve
10
.curvecurve and willis will willexposurethe have intercepthavehave a toatheoreticala theoretical theoreticalthe of the regression duration durationduration -004of, andofof 10 1010 is the regression and exposure to the market of . is the intercept of the regression, and is the regression -005 andandand exposure exposureexposure to toto the thethe marketerror. marketmarket Weof ofof est imate. .. is isthis isisthe the thethe regressionintercept intercept interceptintercept
of of byofof thethe
thefeasiblethe regressionregression, regressionregression generalized, and,, andand isleast isisthe thethe squares regression regressionregressionon 25 rop year (FG LS)
to correct for




-00
error. We estimate this regressionthe 
regression by
feasible error. generalized We estimate least this squares regression (FGLS) by
feasible to correct for














error.error.error. We WeWe est estestimateimateimate this thisthisheteroscedasticity regression regressionregression






by
byby

feasible

feasiblefeasible




and
autocorrelation generalized generalizedgeneralized least leastleastin thesquares squaressquares error (FGseries. (FG(FGLS)
LS)LS)

to toto correct correctcorrect onitional eneit ein ortolio for -007forfor heteroscedasticity and autocorrelationgeneralized in least the errorsquares series. (FGLS) to correct for 0 5 10 15 20 25 30 heteroscedasticityheteroscedasticityheteroscedasticity and andand autocorrelation autocorrelationautocorrelation in inin the thethe error errorerror series. series.series. Term years heteroscedasticity and autocorrelation in the error series. Hypothetical example for illustrative purposes only. Source: PIMCO and the Federal Reserve Board as of 31 December 2018. The exhibit shows The results show that maturities ranging from four to seven the incremental hedging benefit in terms of additional returns obtained by The results show that maturities ranging from four to seven years provided the best hedge to years provided the best hedge to equities, while the short and changing the zero-coupon bond used to hedge against equity risk. We perform The results show that maturities ranging from four to seven years provided the best hedge toa simulation of a 25% decline in the stock price index and compare this with TheTheThe results resultsresults show showshow that thatthatequities, maturitieslong maturitiesmaturities ends while wereranging rangingranging the less short from effective. fromfrom and four fourfour long toAs toto seven seen endssevenseven inyearswere yearsExhibityears lessprovided providedprovided 17, effective. hedging the thethe best As is bestbest seenhedge hedgehedgethe inbaseline to Error! toto scenario Reference of taking asource position in a 10-year zero-coupon bond. equities, while the short and long ends were less effective. As seen in Error! Reference source equities,equities,equities, while whilewhile the thethe short shortshortnotmaximized and found. andand long longlong below, atends endstheends werelowest hedging werewere less lessbetaless iseffective. effective.effective.value,maximized which As AsAs seenat happens seenseen the in lowestinin Error! Error!Error! at the beta Reference ReferenceReference value, which source sourcesource happens at the “belly” not found. below, hedging is maximized at the lowest beta value, which happens at the “belly”The diversification benefit of Treasury bonds depends notnotnot found. found.found. below, below,below, hedging hedginghedging“belly” is of isismaximized the maximizedmaximized curve. at at atthe thethe lowest lowestlowest beta betabeta value, value,value, which whichwhich happens happenshappens at at atthe thethe “belly” “belly”“belly” of the curve. crucially on the correlation between stocks and bonds. This of the curve. ofofof the thethe curve. curve.curve. Exhibit 17: Beta of zero-coupon bonds as a function correlation has shifted significantly over the decades. Exhibit Exhibitof maturity 17: Beta of zero coupon bonds as a function of maturity Exhibit 17: Beta of zero coupon bonds as a function of maturity 19 shows the rolling three-year inflation versus the rolling Exhibit 17: Beta of zero coupon bonds as a function of maturity ExhibitExhibitExhibit 17: 17:17: Beta BetaBeta of ofof zero zerozero coupon01 couponcoupon bonds bondsbonds as asas a afunctiona functionfunction of ofof maturity maturitymaturity three-year stock-bond correlation; each decade has been 01 underlined with a different color. What we discover in this 014 exhibit is that the correlation has shifted significantly across 012 the decades, with the 1960s–1990s in the positive range and 01 00 the new century turning negative. Interestingly, this shift has followed a generalized decrease in inflation levels. Higher 00 

004 
average inflation levels were associated with a positive 002 



10-year onstant ration eta correlation (at decreasing rates); low inflation levels reverted



0 0 5 10 15 20 25 30 the sign. Term years Source: PIMCO and the Federal Reserve Board as of 31 December 2018. The exhibit depicts the term of a zero-coupon bond versus the beta obtained in a regression with the returns per unit of duration against the S&P 500 TR index for the period from January 1960 to December 2018, in monthly frequency. The beta has been scaled to match 10 years of duration instead of one year. JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS 13

Exhibit 19: Trailing 3-year inflation versus trailing 3-year stock-bond correlation

2010 2000 1990 190 1970 190 1950 0

0

04

02

0

-02

-04

-0 Trailin 3-year sto-onTrailin orrelation -0 0 003 00 009 012 Trailin 3-year inlation Source: PIMCO and Haver Analytics as of 31 December 2018. The exhibit plots the three-year trailing inflation built on the headline CPI index versus the trailing correlation between the S&P 500 TR index and the return of the 10-year zero-coupon bond, calculated using the methodology of Gürkaynak et al. (2006).

The changing correlation between stocks and equities In conclusion, the four- to 10-year range of the U.S. yield curve translates into a different trade-off across bond durations. has shown the greatest equity diversification benefit. However, Thus, while the intermediate maturity (“belly”) of the curve while equity beta has been minimized in this range, the sign minimizes the beta for the 1970s, 1980s and 2000s, this shifts based on different economic regimes. In particular, the relationship was broken in the past decade. Exhibit 20 plots the beta of Treasuries to the S&P 500 TR index shifted from positive beta to the equity market of Treasury bonds along the yield to negative in the late 1990s, which might reflect increased curve for each decade since 1970. Although each previous central bank credibility with respect to inflation.1 Since 2008, the decade favored either intermediate or short-duration bonds, the front end of the curve has been less of an equity diversifier due to analysis for the decade after the last financial crisis shows that post-financial-crisis monetary loosening, which has kept the the beta is most negative at the longest maturities. short rate near the zero lower bound.

Exhibit 20: Beta of different maturities of Treasury bonds, Although using zero-coupon bond analysis provides a shortcut per decade to proxy duration effects, using these model-generated yields is 2010s 2000s 1990s 1980s 1970s not exempt from problems. First, the GSW methodology is 05 difficult to replicate. It is based on filtering out bonds with callable 04 03 features (Treasury-issued callable bond series before 1985), 02 bonds shorter than three months in duration (they behave 01 “oddly”), Treasury bills (out of concern about segmented 0 markets), 20-year bonds after 1996 (liquidity issues or high -01 Treasry on eta -02 coupons decrease demand for tax-related reasons in the U.S.), -03 “on the run” and “first off the run” bonds (liquidity and repo -04 market special status) and ad hoc modifications on a case-by- 0 5 10 15 20 25 30 Term years case basis (example: a May 2013 10-year note was priced at a sustained premium near repo and therefore deleted from the Source: PIMCO and the Federal Reserve Board as of 31 December 2018. We compute the returns of zero-coupon bonds on a daily basis to generate the beta sample). All these ad hoc, case-by-case data selections might profile across different maturities. Betas are calculated with a linear regression containing observations for each respective decade. The data series for the make an extension of the GSW methodology to different data 2010s ends in September 2018. frames challenging.

1 The Federal Reserve formulates policy weighting inflation and output gap. As a result of the successful “conquest of American inflation” (Sargent 1999) and the adoption of inflation targeting, the emphasis on inflation might have decreased. In a low inflation environment, the Fed has shifted its focus to economic activity, providing the market with an equity market put. During the same period, we observed that the correlation between equities and bonds has shifted from positive to negative. Post-2008, the expansive monetary policy implemented with QE and the existence of a zero lower bound on nominal rates have anchored the front end of the curve. In the past decade, the low volatility of short rates decreased their hedging power against equities, as the correlation of any variable with a constant is zero. At the same time, in the past 10 years the low term premium has made the return series of bonds with maturities over 15 years similar to one another. 14 JULY 2019 • QUANTITATIVE RESEARCH AND ANALYTICS

ANALYSIS WITH SWAP CONTRACTS Exhibit 21 shows the results of time-series regressions of duration-adjusted swap rates for a universe of four developed The analysis in the previous section serves as only a first market economies (Europe, the U.S., Japan and the U.K.) along approximation to our problem. If we are to use fixed income the two-, five-, 10- and 30-year contracts on their respective local instruments to hedge against equities, we find swap contracts equity market return. All the estimated market betas are offer greater market depth for practical implementation: In April significant at 0.1%. The table also shows the estimated beta 2016, the average turnover associated with interest rate swaps coefficient and, below, its associated estimated standard and options was $2 trillion, versus a quarter of that magnitude deviation. As Exhibit 22 shows, the absolute magnitude of the associated with government bonds and options. Internationally, hedging beta is maximized at an intermediate maturity for more than 27 currencies offer access to active over-the-counter Europe, the U.S. and the U.K. Interestingly, the Japanese swap markets for interest rate swaps, whereas only eight have active market exhibits greater diversification benefit per unit of duration government bond futures.2 Since the 1990s, the evidence has for the longest contract, 30 years. pointed toward a generalized shift of the market benchmark from government bonds to interest rate swaps, possibly motivated by Exhibit 22: Beta per unit of duration in swap contracts, basis risk (Kreicher et al. 2017). major economies

rope .. apan .. Historically, the swap curve shape has been shaped much like 2 3 4 5 7 9 10 11 12 13 14 15 1 17 1 19 20 21 22 23 24 25 2 27 2 29 30 the Treasury yield curve, and the distance between the two (the 000 -002 swap spread) has been positive. Swap spreads represent the -004 markup for the credit spread between interbank lending and the -00 U.S. Treasury. In this section, we move away from nontradable -00 -010 zero-coupon rates to actual quoted on-the-run swap contracts. -012 -014 This analysis has the advantage of analyzing a more realistic -01 context for a trading strategy, at the cost of shrinking our sample -01 to 1995–2018. -020 Source: PIMCO and Bloomberg as of 31 December 2018. The exhibit presents Exhibit 21: Betas of 10-year duration swap returns to the the beta per unit of duration of swap contracts at the two-, five-, 10- and 30-year local equity market, 1995-2018 tenors for four advanced economies.

Pension plan size If we decompose our sample into two subsamples, the first Europe U.S. Japan U.K. being 1995–2009 and the second 2010–2018, a more nuanced -0.075 -0.111 -0.019 -0.071 2-year (0.005) (0.012) (0.003) (0.009) picture emerges. The highest absolute beta per unit of duration -0.101 -0.161 -0.042 -0.111 in the first part of the sample is always of a shorter maturity than 5-year (0.005) (0.013) (0.003) (0.009) in the second part. In the case of the U.S. and the U.K., the -0.118 -0.174 -0.059 -0.121 10-year highest-magnitude beta is achieved at the five-year contract, (0.006) (0.012) (0.004) (0.009) -0.112 -0.170 -0.076 -0.102 versus the long 30-year contract for the post-financial-crisis era. 30-year (0.008) (0.013) (0.005) (0.008) We can see these results in Exhibit 23, where we show the profile of hedging betas for the four economies in both estimations. As Source: PIMCO and Bloomberg as of 31 December 2018. The table shows regression betas to swap returns per unit of duration, scaled to a 10-year a result of the new low rate environment of the second decade of duration. For each swap tenor, we report the coefficient value and its standard the 21st century, we observe that the hedging profiles have deviation. Values marked in bold indicate the minimum beta in a particular country across different tenors. shifted toward longer durations.

2 We define as “active” a market with an average turnover of more than $1 billion per day. Exhibit 23: Betas of 10-year duration swap returns to the local equity market

1995–2009 2010–2018 Europe U.S. Japan U.K. Europe U.S. Japan U.K. -0.100 -0.144 -0.034 -0.082 -0.042 -0.042 -0.005 -0.055 2-year (0.007) (0.017) (0.004) (0.014) (0.006) (0.008) (0.002) (0.008) -0.114 -0.166 -0.061 -0.098 -0.084 -0.149 -0.022 -0.128 5-year (0.007) (0.018) (0.006) (0.012) (0.007) (0.013) (0.003) (0.012) -0.123 -0.156 -0.079 -0.092 -0.112 -0.212 -0.040 -0.159 10-year (0.008) (0.016) (0.007) (0.011) (0.008) (0.015) (0.003) (0.012) -0.076 -0.136 -0.096 -0.075 -0.143 -0.242 -0.056 -0.139 30-year (0.014) (0.017) (0.009) (0.010) (0.009) (0.016) (0.005) (0.011)

Source: PIMCO as of 31 December 2018. The table shows the betas and associated standard deviations of the regression of duration-adjusted daily returns across different advanced economies for two subsamples: 1995–2009 (left) and 2010–2018 (right).

Exhibit 24: Beta of different maturities of Treasury bonds, per decade

uroe 2010201 S 2010201 aan 2010201 2010201 uroe 19952009 S 19952009 aan 19952009 19952009 2 3 4 5 7 9 10 11 12 13 14 15 1 17 1 19 20 21 22 23 24 25 2 27 2 29 30 0

-005

-01

-015

-02

-025

-03

Source: PIMCO and Bloomberg as of 31 December 2018. The exhibit presents the beta per unit of duration of swap contracts at the two-, five-, 10- and 30-year tenors for four advanced economies for two subsamples: 1995–2009 and 2010–2018. pimco.com blog.pimco.com

This paper contains hypothetical analysis. Results shown may not be attained and should not be construed as the only possibilities that exist. The analysis reflected in this information is based upon data at time of analysis. Forecasts, estimates, and certain information contained herein are based upon proprietary research and should not be considered as investment advice or a recommendation of any particular security, strategy or investment product. Hypothetical and simulated examples have many inherent limitations and are generally prepared with the benefit of hindsight. There are frequently sharp differences between simulated results and the actual results. There are numerous factors related to the markets in general or the implementation of any specific investment strategy, which cannot be fully accounted for in the preparation of simulated results and all of which can adversely affect actual results. No guarantee is being made that the stated results will be achieved. Target and expected returns are for illustrative purposes only and are not a prediction or a projection of return. Return targets and expectations are an estimate of what investments may earn on average over the long term. Actual returns may be higher or lower than those shown and may vary substantially over shorter time periods. Figures are provided for illustrative purposes and are not indicative of the past or future performance of any PIMCO product. It is not possible to invest directly in an unmanaged index. Past performance is not a guarantee or a reliable indicator of future results. All investments contain risk and may lose value. Investing in the bond market is subject to risks, including market, interest rate, issuer, credit, inflation risk, and liquidity risk. The value of most bonds and bond strategies are impacted by changes in interest rates. Bonds and bond strategies with longer durations tend to be more sensitive and volatile than those with shorter durations; bond prices generally fall as interest rates rise, and low interest rate environments increase this risk. Reductions in bond counterparty capacity may contribute to decreased market liquidity and increased price volatility. Bond investments may be worth more or less than the original cost when redeemed. Certain U.S. government securities are backed by the full faith of the government. Obligations of U.S. government agencies and authorities are supported by varying degrees but are generally not backed by the full faith of the U.S. government. Portfolios that invest in such securities are not guaranteed and will fluctuate in value. Swaps are a type of derivative; swaps are increasingly subject to central clearing and exchange-trading. Swaps that are not centrally cleared and exchange-traded may be less liquid than exchange-traded instruments. Derivatives may involve certain costs and risks, such as liquidity, interest rate, market, credit, management and the risk that a position could not be closed when most advantageous. Investing in derivatives could lose more than the amount invested. There is no guarantee that these investment strategies will work under all market conditions or are suitable for all investors and each investor should evaluate their ability to invest long-term, especially during periods of downturn in the market. Investors should consult their investment professional prior to making an investment decision. The Sharpe Ratio measures the risk-adjusted performance. The risk-free rate is subtracted from the rate of return for a portfolio and the result is divided by the standard deviation of the portfolio returns. The correlation of various indexes or securities against one another or against inflation is based upon data over a certain time period. These correlations may vary substantially in the future or over different time periods that can result in greater volatility. This material contains the current opinions of the manager and such opinions are subject to change without notice. This material is distributed for informational purposes only and should not be considered as investment advice or a recommendation of any particular security, strategy or investment product. It is not possible to invest directly in an unmanaged index. Information contained herein has been obtained from sources believed to be reliable, but not guaranteed. PIMCO provides services only to qualified institutions and investors. This is not an offer to any person in any jurisdiction where unlawful or unauthorized. | Pacific Investment Management Company LLC, 650 Newport Center Drive, Newport Beach, CA 92660 is regulated by the United States Securities and Exchange Commission. | PIMCO Europe Ltd (Company No. 2604517) and PIMCO Europe Ltd - Italy (Company No. 07533910969) are authorised and regulated by the Financial Conduct Authority (12 Endeavour Square, London E20 1JN) in the UK. The Italy branch is additionally regulated by the Commissione Nazionale per le Società e la Borsa (CONSOB) in accordance with Article 27 of the Italian Consolidated Financial Act. PIMCO Europe Ltd services are available only to professional clients as defined in the Financial Conduct Authority’s Handbook and are not available to individual investors, who should not rely on this communication. | PIMCO Deutschland GmbH (Company No. 192083, Seidlstr. 24-24a, 80335 Munich, Germany), PIMCO Deutschland GmbH Italian Branch (Company No. 10005170963), PIMCO Deutschland GmbH Spanish Branch (N.I.F. W2765338E) and PIMCO Deutschland GmbH Swedish Branch (SCRO Reg. No. 516410-9190) are authorised and regulated by the German Federal Financial Supervisory Authority (BaFin) (Marie- Curie-Str. 24-28, 60439 Frankfurt am Main) in Germany in accordance with Section 32 of the German Banking Act (KWG). The Italian Branch, Spanish Branch and Swedish Branch are additionally supervised by the Commissione Nazionale per le Società e la Borsa (CONSOB) in accordance with Article 27 of the Italian Consolidated Financial Act, the Comisión Nacional del Mercado de Valores (CNMV) in accordance with obligations stipulated in articles 168 and 203 to 224, as well as obligations contained in Tile V, Section I of the Law on the Securities Market (LSM) and in articles 111, 114 and 117 of Royal Decree 217/2008 and the Swedish Financial Supervisory Authority (Finansinspektionen) in accordance with Chapter 25 Sections 12-14 of the Swedish Securities Markets Act, respectively. The services provided by PIMCO Deutschland GmbH are available only to professional clients as defined in Section 67 para. 2 German Securities Trading Act (WpHG). They are not available to individual investors, who should not rely on this communication. | PIMCO (Schweiz) GmbH (registered in Switzerland, Company No. CH-020.4.038.582- 2), Brandschenkestrasse 41, 8002 Zurich, Switzerland, Tel: + 41 44 512 49 10. The services provided by PIMCO (Schweiz) GmbH are not available to individual investors, who should not rely on this communication but contact their financial adviser. | PIMCO Asia Pte Ltd (8 Marina View, #30-01, Asia Square Tower 1, Singapore 018960, Registration No. 199804652K) is regulated by the Monetary Authority of Singapore as a holder of a capital markets services licence and an exempt financial adviser. The asset management services and investment products are not available to persons where provision of such services and products is unauthorised. | PIMCO Asia Limited (Suite 2201, 22nd Floor, Two International Finance Centre, No. 8 Finance Street, Central, Hong Kong) is licensed by the Securities and Futures Commission for Types 1, 4 and 9 regulated activities under the Securities and Futures Ordinance. The asset management services and investment products are not available to persons where provision of such services and products is unauthorised. | PIMCO Australia Pty Ltd ABN 54 084 280 508, AFSL 246862 (PIMCO Australia). This publication has been prepared without taking into account the objectives, financial situation or needs of investors. Before making an investment decision, investors should obtain professional advice and consider whether the information contained herein is appropriate having regard to their objectives, financial situation and needs. | PIMCO Japan Ltd (Toranomon Towers Office 18F, 4-1-28, Toranomon, Minato-ku, Tokyo, Japan 105-0001) Financial Instruments Business Registration Number is Director of Kanto Local Finance Bureau (Financial Instruments Firm) No. 382. PIMCO Japan Ltd is a member of Japan Investment Advisers Association and The Investment Trusts Association, Japan. Investment management products and services offered by PIMCO Japan Ltd are offered only to persons within its respective jurisdiction, and are not available to persons where provision of such products or services is unauthorized. Valuations of assets will fluctuate based upon prices of securities and values of derivative transactions in the portfolio, market conditions, interest rates and credit risk, among others. Investments in foreign currency denominated assets will be affected by foreign exchange rates. There is no guarantee that the principal amount of the investment will be preserved, or that a certain return will be realized; the investment could suffer a loss. All profits and losses incur to the investor. The amounts, maximum amounts and calculation methodologies of each type of fee and expense and their total amounts will vary depending on the investment strategy, the status of investment performance, period of management and outstanding balance of assets and thus such fees and expenses cannot be set forth herein. | PIMCO Taiwan Limited is managed and operated independently. The reference number of business license of the company approved by the competent authority is (107) FSC SICE Reg. No.001. 40F., No.68, Sec. 5, Zhongxiao E. Rd., Xinyi Dist., Taipei City 110, Taiwan (R.O.C.), Tel: +886 (02) 8729-5500. | PIMCO Canada Corp. (199 Bay Street, Suite 2050, Commerce Court Station, P.O. Box 363, Toronto, ON, M5L 1G2) services and products may only be available in certain provinces or territories of Canada and only through dealers authorized for that purpose. | PIMCO Latin America Av. Brigadeiro Faria Lima 3477, Torre A, 5° andar São Paulo, Brazil 04538-133. | No part of this publication may be reproduced in any form, or referred to in any other publication, without express written permission. PIMCO is a trademark of Allianz Asset Management of America L.P. in the United States and throughout the world. ©2019, PIMCO.

63863 CMR2019-0405-389677