Short Rate Expectations, Term Premiums, and Central Bank Use of Derivatives to Reduce Policy Uncertainty

SHORTRATEEXPECTATIONS,TERM PREMIUMS,ANDCENTRAL BANK USE OFDERIVATIVESTO REDUCE POLICYUNCERTAINTY P.A.Tinsley (cid:3) version: September1998 Abstract: The term structure of interest rates is the primary transmission channel of monetary policy. Under the expectations hypothesis, anticipated settings of the short-term interest rate controlled by the central bank are the main determinants of nominal bond rates. Historical experience suggests that bond rates may remain relatively high even if the short-term interest rate is reduced to zero, in part due to term premiumsreflectinguncertaintyaboutfuturepolicy. Termspreadsduetopolicyuncertaintymaybereduced by central bank trading desk options that provide insurance against future deviations from an announced interestratepolicy. Keywords: bondoptions, nominalratezerobound, termpremiums. JELclassification: E4,E5 (cid:3) Federal Reserve Board, Washington, D.C. 20551; after September 1998: Faculty of Economics and Politics, University of Cambridge, Cambridge CB3 9DD, United Kingdom, ptinsley@econ.cam.ac.uk. Views presented are thoseoftheauthoranddonotnecessarilyrepresentthoseoftheBoardofGovernorsorstaffoftheFederalReserve System. MythanksforthecommentsofG.Duffee,D.Kohn,andC.Thomas.

Theterm structureof interestrates is theprimarytransmissionchannel ofmonetarypolicy. Under the assumption that policy actions are summarized by movements in the one-period short rate, policy transmissions to the real economy are influenced more by market perceptions of monetary policythan by the current actions of the central bank trading desk in the spot market for the short rate. The main subject of this paper is the role of market perceptions of policy when expectations of long-term inflation are low and, consequently, the average level of nominal interest rates is significantlylowerthanthelevelofpostwarrates. Incriticalepisodes,suchasapersistentrecession where the short rate may be driven near zero, historical experience suggests long-term bond rates may remain well above zero. This paper indicates derivativesecurities issued by the trading desk maytightenconnectionsbetweenpolicyintentionsfortheshortrateandcurrentlong-terminterest rates. Discussionis directed principally at policyoptions to influence nominal yields on Treasury securities. Marketperceptionsembeddedinlong-termbondratesincludebothaveragesofexpectedfuture short-termratesanduncertaintyaboutfuturepolicyresponses,whichispricedinthetermpremium components of bond rates. Market short rate forecasts and variances are based on conjectured or average-history descriptions of future policy. In the case of the latter, estimated policy response functions help agents to quantify representative policy responses to current and future events. Data-based characterizations of policy range from reduced-form time series models of the short ratetostructuralfeedbackrules,suchasthecalibratedexampleofTaylor(1993),wherefutureshort rate responses are conditioned on forecast indicators of the economy such as inflation and output deviations from trend. Because policymakers are subject to the same difficulties in forecasting future states of the economy as are private sector agents, it is usually unrealistic for policymakers tomakeunconditionalforecasts oftheirfutureactions. However, in times of unusual economic stress, policymakers may wish to consider policy use of more explicit contingent contracts in order to transmit unambiguous signals to markets regardingthedirectionandvolatilityofthepolicy-controlledshortrateoveranear-termhorizonof some consequential length, say one or two years. Examples of such instances include episodes of unusualexcess ordeficient demandthat requireadeparture from average-historypolicies,suchas initiatedinOctober1979,andassurancetoprivatesectorsofasustainedcentralbankcommitment tosignificanttighteningorlooseningofborrowingrates. The fundamental assumption of this discussion is that complete information on future policy actions is not known, and agents are forced to make probabilistic judgments about future policy. This differs from the standard assumption in macroeconomics of model-consistent or “rational” expectations where private agents and policymakers share the same conditional information, 1

including valid structural descriptions of the economy and government policies.1 Even if a forward policy is announced by the central bank, it is assumed that such an announcement will bediscountedas“cheap talk”giventhepotentialforfuturepolicyreversals. There are two major differences in the present proposal from current operating procedures. First, instead of disclosing only the current policy setting of the short-term rate, the central bank also indicates one or more explicit upper or lower boundary points on the yield curve that will be enforcedbyfuturepolicy,presumablyattheshortendofthetermstructure. Second,thecredibility of the central bank policy is enforced by binding contractual arrangements with private sector agents,whowillbecompensatedforanyfuturedeviationsfromthepolicytermsdesignatedinthe contingentcontracts. The remaining discussion is organized as follows: Section I briefly discusses empirical evidence on the importance of long-maturity real rates, as opposed to the short-term real rate, as the primary transmission conduit of monetary policy to real economic activity. Discussion then documents that sizeable spreads can exist between the short rate and bond rates, even when the former is near zero. Without a model of agent expectations, it is impossible to determine the extent to which term spreads represent expectations of nonzero short-term rates in future periods or uncertainty about future short rates, as captured in the bond rate term premiums. Section II reviews bond rate forecasts from three alternative small models of monthly economic activity. Each model contains an average-history description of short rate policy. Historical bond rates are parsed into estimates of expected short rates and residual estimates of term premiums. One model is shown to imply theory-based term premium constructions that more closely match the corresponding residual estimates of bond rate term premiums. The parameters of this model can be used to formulate closed forms of simple option contracts. Section III briefly discusses issues in representing market expectations and term premiums that are omitted in certainty-equivalent simulationsofmacroeconomicmodelsandareconsequentialwhennominalshortrates arenearor atzero. SectionIVdiscussespolicyuseofcontingentsecuritiesdirectedatloweringbondrates,in partbyreducingmarketuncertaintyregardingtheshortrateintentionsofthecentralbank. Section Vconcludes. I.The MacroeconomicTransmissionEffects ofLong-Term Interest Rates It isoftenconvenienttoselect asingleinterestrateorinterestratespreadtosummarizetheeffects of monetary policy on aggregate demand. However, there is surprisingly little empirical work to 1The paper also departs from the standard use in macroeconomic model simulations of certainty-equivalent forecastswhichignoreinteractionsofconditionalmeansandvariances. 2

suggest which rate maturity,or maturities, best capture the aggregate demand effects of monetary policy. Although the short-term interest rate may be useful as a simple summary of monetary policy responses, as suggested by Bernanke and Blinder (1992), the initial portion of this section indicatesthattheshortrateisnotareliableindicatorofthetransmissioneffectsofmonetarypolicy onaggregatedemand. The remainder of the section notes that, historically, intermediate- and long-term bond rates have remained substantially above zero when the short nominal rate is near or at zero.2 Thus, it appears that moving the short rate to the zero boundary is not sufficient to lower bond rates near zero, even in the case of a severe depression in economic activity. Because bond rates dominate the short rate as more reliable indicators of the transmissioneffects of monetary policy, theconsequenceofelevatedlongrates is tomufflethecountercyclicalefforts ofthecentral bank. Selectingabenchmarkindicatorof theaggregatedemandeffectsof monetarypolicy. Given frictions, such as transactions costs and contractual agreements, the effective planning horizons for economic decisions to consume or invest extend over many years. In principle, the entire term structure of real interest rates is relevant in assessing the present value of utility or profits associatedwithplansforaggregatereal expenditures. However,it is often convenient toselect a singlematurityrate as a benchmarkindicatorof the transmission effects of monetary policy on aggregate demand. Table 1 presents results of using three alternative ex ante real rates as the benchmark real rate in an aggregate demand schedule. In the monthly regressions reported in Table 1, the dependent variable is output deviations from trend, represented by the capacity utilizationrate of manufacturing. The candidate regressors are: (cid:26) 1 ;t , the 1-month rate; (cid:26) 6 0 ;t , the 5-year bond rate; and (cid:26) 1 2 0 ;t , the 10-year bond rate. Each real rate requires an estimate of expected inflation over the relevant maturity horizon. For example, the 5-yearexanterealrate, (cid:26) 6 0 ;t (cid:17) r 6 0 ;t (cid:0) E t (cid:25) 6 0 ;t ,requiresanestimateoftheexpected5-yearinflation rate, E t (cid:25) 6 0 ;t . Expected inflation rates are constructed by a VAR model approximation of agent expectations,discussed in the next section. The sample span is 1967m1-1997m9,and sixlags are usedforeachregressor. Table 1 suggests that either the 10-year or the 5-year real bond rate is a better benchmark indicator of the effect of monetary policy on aggregate demand than the 1-month real rate. The first equation shown in Table 1 indicates the level of the 1-month real rate (cid:26) 1 , is not significant. 2Discussion in this paper is confined to yields on Treasury securities. The intent is to focus on implications of unpredictablemonetarypolicyforinterestratetermpremiums,abstractingfromadditionalcountercyclicalmovements incorporatebondtermpremiumsthatreflecttheriskofbankruptcy. 3

The second and fourth equations in Table 1 show that both the 5-year rate, (cid:26) 6 0 ;t , and the 10-year rate, (cid:26) 1 2 0 ;t , are significant determinants of cyclical deviations in output.3 The third and fifth equationsinTable1indicatethattermspreadsfrom the1-monthreal ratearenotsignificantwhen addedtoequationsalreadycontainingthelevelofeitherthe5-year or10-year real bondrates.4 Historicalbehaviorofbondrateswhen theshortrateisnear zero Of course, observation of the current short rate is not sufficient to determine current long-term bond rates. For zero-coupon bonds, an expectation-based interpretation of long-term bond rates contains two ingredients,an average of expected short rates over the maturityof the bond and the term premium. Thus,thelogyieldtomaturityonan n -periodbond, r n ;t ,mayberepresentedas r n ;t = 1 n (cid:0) n X =i 1 0 E t r 1 + ;t i + (cid:18) n ;t ; (1) where E t denotesexpectationsat thebeginningofperiod t ; r 1 + ;t i is theshortrate inthe i thmonth of the forecast horizon; and (cid:18) n ;t is the term premium required by agents to compensate for the covariationofbondrates withfuturestatesoftheeconomy. Eveninhistoricalepisodeswheretheshortratemightbeexpectedtoremainnearagivenlevel for some extended horizon, spreads between short- and long-term rates appear to be nontrivial. Oneexampleiswhenshort-termratesapproachedzerointhemid-1930s. WhereasU.S.short-term interestratesremainedbelow1%fromMarch1933throughJanuary1937,reachingalowof0.25% in mid-1935, yields on U.S. railroad bonds ranged from about 5.5% in March 1933 to 3.8% in January 1937, and were 4.1% in mid-1935.5 Discussion in Macaulay (1938, p.78) indicates that theminimummaturityoftherailroadbondsusedwas fourteenyears. Selecting a single year for snapshots of annual yields by asset maturity, the 3-monthTreasury rate was .14 % in 1935, whereas yields to maturity for that same year were 2.79 % for an 3The capacity utilization rate of manufacturing is stationary, suggesting that the mean utilization rate is the equilibrium or preferred rate of utilization. The associated equilibrium real rates for the 5-year and 10-year rates arelisted in thelast columnof Table 1. Thesearetherealratelevelsassociated with trendoutputgrowthwhen the utilization rate is at its preferredlevel. No adjustmenthas been made for the effects of tax deductibility of interest costssotheseequilibriumratesarehigherthanthoseassociatedwithequilibriumafter-taxrealrates. 4Thelasttwo equationsin Table1suggestthatthe5-yearrealbondratedominatesthe10-yearrealbondrateas a benchmarkindicator of monetary policy, at least when the manufacturingutilization rate is used as the proxy for aggregateoutputdeviationsfromtrend. Thespreadbetween the5-yearand 10-yearrates is significantwhen added to the equation containing the level of the 10-yearrate, but is statistically insignificantwhen added to the equation containingthe5-yearbondrate. 5Theshort-termrateisthecallmoneyrate(column1)andtherailroadbondyieldsareanindexofyieldstomaturity (column4)inTable10ofMacaulay(1938). 4

unweighted average of outstanding U.S. government bonds due or callable after 15 years, and 1.05%,2.37%,and3.00%forcorporatebondsof1-,5-,and10-year maturities. 6 Five years later, corporate spreads over the bill rate were reduced by about another 100 basis points in 1940, when the average 3-month Treasury rate was driven to 1 basis point, Bureau of Census (1975). Indeed, as noted by Friedman and Schwartz (1963), the Treasury bill rate was drivenbelowzero in1940.7 Unfortunately, it is impossible to know whether the non-negligible term spreads that remain whentheshortrateisnear,at,orevenbelowzeroareduetoexpectationsofareboundintheshort rate or term premiums reflecting, in part, uncertainty about the short rate intentions of monetary policy. Thenextsectionconsidersthreealternativemodelstorepresent agentexpectations. II.AlternativeModels ofExpected Short Ratesand Term Premiums8 ThreesmallVARmodelsofmonthlyeconomicactivityareconsideredasproxiesfortheshortrate expectations of agents. Each VAR contains three variables: a one-month nominal interest rate, r 1 , inflation, (cid:25) 1 , and the rate of capacity utilization, y . All interest rates referenced in this paper are end-of-month zero-coupon Treasury bond yields.9 Monthly inflation is measured by the BEA chain-weighted deflator for personal consumption expenditures, and monthly capacity utilization bytheFRB indexformanufacturing. Sixlagsare usedforeach regressor. ThealternativeVARmodelsdifferinassumptionsaboutthelong-runinterestrateandinflation rateexpectationsofagents. Thefixedendpointsmodelassumesthatbothinterestratesandinflation rates are stationaryand will return, in long-horizonsimulations,totheir respectivesample means. The moving-average endpoint model, by contrast, assumes that the first-differences of interest rates and inflation rates are stationary. As shown in Kozicki and Tinsley (1997), this assumption 6SeriesX451,X474,X487-9inU.S.BureauoftheCensus(1975).TheCPIinflationratefor1935wasabout1.4%. 7In principle, positive costs of storing accumulated cash are consistent with negative nominal interest rates on secure assets. “(B)anks presumably sought again to acquire bills. But the supply outside the Federal Reserve was sosmallthattheirattemptsservedonlytoreduceyieldsonbillstoalevelclosetozero....Indeed,yieldsonTreasury billswereoccasionallynegativein1940,whentheirpricewasbidupbypurchasersseekingtoconvertcashintoother assets for short periodsto reducetax liability under personalpropertytax laws.” Friedman and Schwartz (1963, p. 539).SeealsoCecchetti(1988)foradiscussionofconstraintsonTreasurycouponbondsthatinducednegativeyields asthebondsapproachedmaturity. 8MaterialinthissectionisdrawnfromKozickiandTinsley(1998). 9Interest rates for 1960m1-1991m1are from McCulloch and Kwon (1993). For comparability, these data were extendedto 1997m9by applyingthecubicsplineestimatordescribedin McCulloch (1975)to end-of-monthyields. MythankstoMarkFisherforhisgenerousassistance. 5

implies that long-horizon forecasts of interest rates and inflation rates will converge to weighted movingaveragesofrecentexperience. Bothofthesemodelsarestandardalternativespecifications in time series modeling. Generally, aggregate postwar data are not able to make a strong case against either the assumption that the short-term interest rate and inflation rate are stationary or the assumption that they are difference-stationary. However, as shown shortly, the difference in dynamic specifications is very consequential for long-horizon simulations, such as required for expectationsoflong-maturitybondrates. The shifting endpoints model is nonstandard. This model was developed to represent nonstationaryagent expectations associated withepisodic shifts in monetary policy.10 A common element of the long-run forecasts of the interest and inflation rates is shifting private sector perceptionsoftheinflationtargetofthecentralbank. Learningbyagentsisnonlinearbutgenerally slow, with average lags of 4-5 years in discerning shifts in the inflation target of monetary policy. Details on the specifications of the shifting endpoint movements of the interest rate and inflation rateare discussedinKozickiandTinsley(1998). Monthly predictions of 10-year bond rates by the three VAR models are shown in Figure 1. The influential roles of the alternative endpoint specifications are apparent. In the top panel, the effect of the fixed endpoints for variables is that the forecast movements of the 10-year bond rate are excessively damped relative to historical movements. Conversely, in the middle panel, the problem is reversed where the predicted 10-year bond rate is more volatile than the historical series,becausetheendpointoftheshort-termnominalrateisaweightedmovingaverageofrecent movements in the short rate. By contrast, the bottom panel of Figure 1 indicates that movements ofthe10-yearbondratepredictedbytheshiftingendpointsmodelmostcloselytrackthecontours ofthehistoricalseries. Residual estimates of the term premium for 10-year bond rates are provided by the difference between the historical bond rates and the VAR model predictions, which are simple averages of the predicted short rates over a 120-month forecast horizon. The difference between the true, unobserved term premium, (cid:18) n ;t , and the residual estimates of the term premium generated by a givenVARmodel, (cid:18) R n ;t ,is theerror inspecifyingthetruemodelofmarket expectations. ^(cid:18) R n ;t = = r [ n ;t ( 1 = (cid:0) n ) ( 1 (cid:0) n X =i = n 1 0 E ) (cid:0) 1 n X = 0 i 1 r t ;t ^r + 1 i + ;t (cid:0) ; i ( 1 = n ) (cid:0) n X =i 1 0 ^r 1 + ;t i ] + (cid:18) n ;t ; (2) 10A quarterly version of this model is incorporated in the staff model, FRB/US, to represent VAR expectations, Kozicki,Reifschneider,andTinsley(1996). 6

where ^r 1 + ;t i denotes a VAR model forecast at the beginning of period t of the short rate in the i th period of the forecast horizon. The second line in equation (2) is obtained by replacing the observed n-period bond rate, r n ;t , by the true (unobserved) model of the bond rate from equation (1). IfempiricalshortrateforecastsofaVARmodeldonotreplicatemarketexpectations,equation (2) indicates that the residual estimates of the term premium generated by that model will contain apotentiallylarge componentofspecificationerror,inadditiontotheactual termpremium, (cid:18) n ;t . It is apparent from the three panels in Figure 1 that time variation in residual estimates of term premiums depends criticallyon theparticular specificationof the VARused toproxymarket expectations. Thetoppanel,generated bythefixedendpointsVAR,suggeststhat almostall ofthe variation in bond yields comes from temporal variation in the term premiums. By contrast, in the middle panel of Figure 1, the long-average predictions by the moving average endpoints model closely track the recent history of the short rate. Because actual bond rates are smoother than the short rate, this model suggests term premiums vary considerably in a countercyclical fashion. In particular, the residual term premiums estimated by the moving average endpoints model vary with the long-short yield spread. Finally, the predictions in the third panel of Figure 1, generated by the shifting endpoints model, suggest that the lion's share of historical movements in yields reflects changing market expectations for future short rates. Thus, the shifting endpoints VAR model appears tosuggest a more reliable connectionbetween movementsinthe policy-controlled shortrateandmovementsinarepresentativebondratethandotheothertwoVARmodels. In principle, any of the three VAR models may be validrepresentations of agent expectations. Kozicki and Tinsley (1998) present evidence that the expectations of long-horizon inflation generated by the shifting endpoints model are closest to available survey estimates of expected inflation. Anadditionalselectionstandardistodetermineiftheresidualestimatesoftheterm premiums generatedbyeachVARmodelareconsistentwithacoherenttheoryofthemarketpricingofinterest raterisk. AsdevelopedinKozickiandTinsley(1998),atheory-basedestimateofthetermpremium willtaketheform ^(cid:18) T n ;t = (cid:0) n X =i 1 0 ^w +t i ^(cid:27) 2 r + ;t i ; (3) where ^(cid:27) 2 r + ;t i denotes the estimated conditional variance of the short-term interest rate in period t + i , and theweights, ^w +t i , are functionsof theestimatedparameters oftherelevant VARand an unobservedparameter, (cid:12) ,associatedwiththeriskaversionofagents. Note that even if one of the three VAR models provides a close approximation of market expectations, equation (3) will be unable to replicate the time-varying residual estimates of the 7

termpremiumsshowninFigure1iftheshortratevarianceisfixed. Thus,anexplicitmodelofthe time-varyingvolatilityoftheshort rate, (cid:27) 2 r + ;t i ,isrequired. There are two basic approaches to modeling the time-varying variances of variables. One is a shock-based (GARCH) model where conditional volatility is a function of recent shocks to the variable. The other is a level-based model where conditional volatility is a function of the level of the variable. In addition to historical studies associating the level and volatility of inflation,literaturelinkinginterestratestandarddeviationstothesquarerootofinterestratelevels iswell-known,includingMerton(1973)andCox,Ingersoll,andRoss (1985). Table 2 presents several screening regressions where the residual estimates of the term premiums, as shown in Figure 1, are regressed on 1-month predictions of the short rate, r 1 ;t . The predicted short rate appears to be a promising explanatory variable, explainingfrom 25% to 50% of monthly variations in the residual term premiums for the shifting and fixed endpoints models, respectively. Predictedshortratesdonotappeartobeagoodexplanatoryregressorforthemoving endpointsmodelfortheobviousreason thatrecent levelsofshort rates are alreadycaptured inthe bondratepredictionsofthismodelbythemovingaverageendpoint,as discussedearlier. Responses oftheVARpredictionerrorvariances oftheshortrate tomovementsinthelevelof the short rate are estimated in Table 3. These regression slopes will be used in the construction of theory-based estimates of the term premium, (cid:18) T t . Two parameters remain tobe determined; the impliedvariance ofthe short rate forzero levels ofthe shortrate, 0 , and themarket price of risk, (cid:12) . Thesetwoparametersareselectedbymatchingthemeanandvarianceoftheestimatedresidual term premiums. The residual-based estimates of the 10-year bond rate term premiums, (cid:18) R t , generated by each of the three VARs and the matching theory-based constructions of the term premiums, (cid:18) T t , are compared in Figure 2. If a particular VAR provides a coherent description of agent expectations, the theoretical term premium construction, based on parameters of the VAR, should be roughly alignedwiththeresidualterm premiumestimatedbythatVARmodel.11 As shown in the first panel of Figure 2, the theory-based term premium associated with the fixed endpoints model is quite responsive to movements in the short rate but the timing of these movements do not closely match the residual-based estimate of the term premium generated by thismodel. Although timing is improved for the theory-based construction of the term premium of the 11DuetothesmalldimensionsoftheVARmodels,onemayexpectthetheoreticaltermpremiumconstructionsto be smoother than the true term premiums. That is, it seems plausible that market agents use a much larger set of macroeconomicindicatorstorepresentthestochasticdiscountvaluationoffuturereturnsonfinancialassets. 8

moving average model, shown in the second panel of Figure 2, the signs of movements in the theoretical term premium are the reverse of those in the residual term premium. This mirror “reflection” problem is also summarized by the negativesign of the screening regression in Table 2 for the residual term premium of the movingaverage endpoints model. Essentially,the residual estimate of the term premium in this model must move against swings in the short rate in order to compensate for the exaggerated changes in the nominal rate endpoint imposed by the moving averagespecification. Thetheory-basedestimateof thetermpremiumof theshiftingendpointsmodel,inthebottom panelofFigure2,appearstomoresuccessfullymatchthecontoursofmovementsintheassociated residual-basedestimateofthetermpremium.12 Summarizing results in this section, the shifting endpoints model appears to provide a usable approximation of agent expectations of future short rates, along with an implied theory-based representation of bond rate term premiums. The ability to generate reasonable estimates of both first and second moments of interest rate expectations permits closed-form approximations of other financial market instruments, such as options contracts. As discussed later, derivative securities may be useful not only as market indicators of policy uncertainty but also as potential instruments of stabilization policy. Closed-form representations also enable modelers toexploretheeconomicdeterminantsandsignalingeffectsoftheseadditionalmarketinstruments. III.Issues inDescribingthe BehaviorofInterest Rates atZero A small, but growing, literature has emerged that uses empirical macroeconomic models to illustrate possible limitations on monetary stabilization policies when nominal interest rates are nearoratazerofloor. RecentcontributionsincludeLaxtonandPrasad(1997),FuhrerandMadigan (1997), Orphanides and Wieland (1998), and Tetlow and Williams (1998). Partly because each study used one or more different methods to impose the zero floor on nominal interest rates, conclusions are mixed. The two most recent studies suggest the zero lower bound for nominal rates is not a serious constraint for operating policy as long as policy is sufficiently aggressive in offsettingunfavorableshocksorthelong-runinflationtarget remainsmodestlypositive. Discussion in this section is less optimistic regarding the influence of the zero interest rate boundary on monetary stabilization policies when the average level of interest rates is low, due in part to a more conservative description of the formation of agent expectations. Several issues 12Thetheory-basedestimate ofthetermpremiumisconsiderablysmootherthantheresidualestimate, suggesting futureworkmightconsideracombinationof level-basedandshock-basedestimatesofthe shortratevariance,asin Brener,Harjes,andKroner(1996). 9

in representing the effects of the zero bound on nominal rates are discussed that have not been addressedinpreviousstudies. Abriefpreviewoftheseissuesincludes: (cid:15) Under normal distribution descriptions of interest rates, the zero bound induces a “positive” bias in expected nominal rates as the mode of the interest rate distribution approaches zero. Giventhehistoricalvolatilityoftheshortrate,thiseffectcanbesizeableandisnotcapturedby deterministicor“certainty-equivalent”simulations. (cid:15) Level-baseddescriptionsofvolatilityimplythatthispositivebiaswillalsopreventthevolatility of the short rate from falling to the level that would be associated with a zero mean expected rate. The effect of a binding lower bound on the nominal short rate necessarily increases the volatilityofreal activityandinflation. (cid:15) Largerperceptionsofshortratevolatility,duetotheindirecteffectofapositivebiasinexpected rates or to increased uncertainty regarding monetary policy during episodes that differ from “average-history” responses, are consistent with the existence of significant term premiums of bondrates evenwhenthecurrent shortrateis nearzero.13 (cid:15) Methodsusedbypreviousstudiestoimposethezerolowerboundoncounterfactualsimulations oftheshortraterequiredeparturesfromaverage-historydescriptionsofpolicyresponses,where the departures are specific to the size of shocks to macroeconomic indicators. Consequently, model-consistentor“rationalexpectations”simulationsarelikelytooverstatetheeffectiveness of rapidly adapting policies, whose changing response characteristics must be learned by market agents.14 In a situation where the current short rate has been driven near zero, whether or not long rates also move appreciably towards zero depends on two perceptions of market agents. The first is whether or not policy will continue to hold the short rate at zero for some extended horizon. The second is the degree of uncertainty agents have regarding policy forecasts. Both perceptions will generally require departures from standard certainty-equivalent and rational expectations descriptionsofagent expectations. Thepositivezerofloorbiasinexpectations Conventional representations of short rate expectations are not relevant if the zero boundary is 13One experiment briefly discussed in Fuhrer and Madigan (1997) illustrates a relative deterioration in policy effectivenessifthebondratecontainsatermpremium. 14Arelatedissueisthatmanypoliciessuggestedaseffectiveinneutralizingthelowerboundconstraintimplyprompt andsizeablepolicyresponsestounfavorableshocksandtoforward-lookingexpectationsoffuturebindingsatthezero boundary. Typically,thesesimulatedpoliciesaremoreaggressivethanhistoricalpolicyresponsesandareperceived withcertaintybybothpolicymakersandagents. 10

binding. In the case of standard linear feedback rules used to represent short rate responses, severalmethodshavebeenemployedtoimposeazerotruncationbarrieronthecertaintyequivalent forecast of the short rate and on the subsequent transmission of expected short rates to expected bondrates.15 However, in a reasonably wide neighborhood of zero, truncation of certainty-equivalent forecasts does not provide an accurate representation of the expected short rate. The effect of the zero lower boundaryon the mean expectationof the short rate is depicted in Figure 3. For the monthlysample,1982m10-1997m9,themeanand standarddeviationof the1-monthzero coupon rate are 6.06 and 1.80, respectively. The distributionof the short rate that is associated with these momentsis depictedbythenormaldistributioninFigure3centeredat 6.0.16 As the center of the normal distribution moves leftward towards zero, a larger portion of the distribution will extend over negative short rate outcomes. Because the support of the normal distributionruns from (cid:0) 1 to + 1 , a portion of any normal distributionwill always lap over into theregionofnegativeinterest rates. However,themagnitudeof theoverlappingdensityis modest whenthecenter ofthedistributionismorethantwostandarddeviationsabovezero. For comparison, the left-hand distribution in Figure 3 is centered at zero. In this instance, only the right half of the density is relevant in weighting the probabilities of nonnegative outcomes. Here, and in the remainder of the paper, the “+” superscript is used to indicate selection of only positive outcomes. The mean expectation of nonnegative short rates is now E t f r + g = ( 2 (cid:25) ) 1 2 (cid:27) r 1 (cid:25) : 8 (cid:27) r 1 . where (cid:27) r 1 denotes the standard deviation of the short rate. Thus, in the case of (cid:27) r 1 = 1 : 8 0 , the “bias” of the mean expectation of the short rate for the left-hand distributionis E t f r + g = 1 : 4 4 .17 Alternativecombinationsof shortrateexpectationsandtermpremiums The specification of level-based volatility, discussed earlier, implies that the positive bias at zero 15ThreealternativemethodsofimposingthezerolowerboundaryontheshortnominalrateareillustratedinFuhrer andMadigan(1997). 16The common assumption that bond prices are log normally distributed, implies that yield to maturity rates are normallydistributed. 17Forcaseswherethemodeofthenormaldistributionisnonzero, (cid:22)r =6 0 ,theexpectationsubjecttothezeroflooris: E f r + 1 g = (cid:22)r + ( (cid:30) ( (cid:8) (cid:0) ( z ) z ) ) (cid:27) r ,where z (cid:17) (cid:22)r = (cid:27) r and (cid:30) ( : ) and (cid:8) ( : ) denotethenormalprobabilitydensityandcumulative distribution functions, respectively, vid. Johnson and Kotz (1970). In the examples of this section, the standard deviationusedtocomputethezerofloorbiasinexpectationdoesnotvarywiththeleveloftherate,incontrasttothe levels-basedmodelofvolatilityusedforthetermpremium. Tocapturethelowervolatilityoftheshortrateinrecent years,thestandarddeviationusedtocalculatethezerofloorbiasinexpectationisbasedonthesampleafter1982m9. 11

has two potential channels of influence on bond rates, one preventing the weighted average of expected short rates from falling to zero and the other inhibiting reductions in the short rate conditional variance that is priced in the term premium. Effects of nontrivial expectational biases inducedbythezero lowerboundonthesetwocomponentsofbondrates are illustratedinTable4. The four columns of Table 4 illustrate the consequences for 5-year bond rates, r 6 0 ;t , of alternative market perceptions regarding the behavior of the short rate controlled by the central bank, r 1 ;t . The first row indicates alternative assumptions about the current short rate. Initially, we assume these expectations are assumed to persist indefinitely. Using terminology introduced earlier, the values in the first line are also the levels assumed for the long-run endpoint of the nominal 1-month rate. Thus, if the short rate is currently 6, the nominal rate endpoint is also 6. Moving to the fourth row of Table 4, the flat expectations of the short rate also imply that the 60-monthaverage ofexpectedshortrates is6. Continuingwiththeexampleinthefirstcolumn,thestandarddeviationoftheshortrateis1.80, asshowninthesecondline. UsingtheparametersoftheshiftingendpointsVARapproximationof agentexpectations,theestimatedtermpremiumforthe5-yearbondrateis2.14,showninthefifth lineofTable4. Summingthetwocomponents,the60-monthaverageoffirstmomentexpectations andtheterm premiumpricingofthesecondmomentexpectationssuggeststhe5-year rateis 8.04, showninthesixthrowofTable4.18 To illustratethe expectations effect of the zero bound, nowconsider the correspondingentries in the last column of Table 4. Here, both the certainty-equivalent (zero residuals) forecast of the current 1-month rate and the long-run endpoint of the 1-month rate are zero. However, as indicatedinthefourthrow,the60-monthaverageofexpected1-monthratesis1.44,whichreflects the asymmetric (half-normal) distribution of possible 1-month rates over the 60-month forecast horizon.19 Looking at the remaining entries in the fourth row of Table 1, it is apparent that the positive bias introduced by the zero bound is negligible for a short rate distribution centered at 6.0 (the first column), 6 basis points for a distribution centered at 4.0 (the second column), and 45 basis points for a short rate distribution centered at 2.0 (the third column). Returning to the fourthcolumnandagainusingtheparametersfromtheshiftingendpointsVAR,thetermpremium associated with expected 1-month rates at 1.44 is estimated to be 1.19, giving a total of 2.63 for 18It is encouragingto note that this estimate is close to the 1982m10-1997m9sample mean of about 7.9 for the zero-coupon5-yearrate. Thissuggeststhetermpremiummodelisatleastlocallyscalableforneighboringmaturities, eventhoughthemodelwascalibratedtofitonlythefirsttwomomentsofthe10-yearbondtermpremiums. 19Undertheconventionthatexpectationsarecalculatedatthebeginningofthecurrentperiod,thepositive“bias”is alsoaddedtothecurrentperiodforecastint. 12

the5-yearrate. Bondrateimplicationsofatemporarychangeintheshortrate Now consider a 1-month departure from standard expectations. This is depicted in the last block of rows in the top half of Table 4 (rows 7-9). At the beginning of the period, the central bank drivestheshortratetozero. Becausemarkettradersperceivethisasaone-timeevent,theexpected first and second moments of short rates in subsequent months revert to “average-history” values, such as the 6.0 and 1.80 assumptions used in the first column. Even in the first month, trader expectations for the remainder of the month are nondeterministic so the variance of the short rate is set equal to the average-history expectation. Consequently, in all four columns, the mean expectation of the 1-month rate in the current month is 1.44. In the example of the first column, the 60-month average is 5 6 9 0 6 : 0 + 1 6 0 1 : 4 4 = 5 : 9 2 . As might be expected, changes in the short rate thatareviewedastransientdeparturesfromaverage-historyexpectationshavelittleeffectsonlong bond rates. In contrast to results in preceding columns, results in the fourth column are unaltered becausetheaverage-historyassumptionregardingtheleveloftheshortrate isalreadyat zero. Effectsof apersistentchangeintheshortrate Toexploretheeffects ofmorepersistent expectationsregardingchanges intheshortrate, suppose the economy has entered a period of depressed demand and the central bank wishes to convey to markets a revised policy where the short rate will be held at zero for an extended period, such as 12 or 24 months. As indicated at the beginning of this section, the success of the new zero-rate policydepends ontwoperceptions of the market: (1) Howpersistentwill be the current short rate settingatzero? (2)Whatistheprobabilitythatthecentralbankmaydepartfromthispolicy? Three possiblescenarios foragent expectationsare showninthebottomhalfofTable4. The first block of rows inthe bottom half of Table4 (rows 10-12)addresses the issueof agent expectations regarding the persistence of the current policy action. An additional conditioning assumption is the average-history level to which the short rate is assumed to return after the new policy expires. In the first column, the short-rate is expected to revert to the average-history level of 6.0 after the initial 12 months, and the 5-year rate falls from 8.14 to 6.97. Note that the reductionof the 60-monthaverage of the 1-monthrate from 6.0 to5.09is smallerthanone would expect for a 20% drop in the level of the 1-month rate over the 5-year horizon ( : 8 (cid:2) 6 : 0 = 4 : 8 0 ). This is because agents remain uncertain as to possible stochastic departures from the announced target. Althoughthe zero boundary bias is inconsequential at the average-history assumptionof a short rate level at 6.0, it kicks in with an expected value of 1.44 at the alternative assumption of 13

a short rate level at zero. In other words, agents do not have certainty-equivalent forecasts when thedistributionofrateexpectationsmovesclosertothezeroboundary. Similarreasoningexplains the reductions shown in the rest of this block. Note that nothing is accomplished by the zero-rate policyannouncementinthelast columnbecauseagentexpectationsremainnondeterministic. Consequences ofreducedpolicyuncertainty Nowsupposepolicy,bysomemeansasyetunexplained,isabletoconvincemarketagentsthatthe newpolicyofzeroshortratesfor12monthsisacertainevent. Thenextblockofrows(rows13-15) indicates that reducing policyuncertainty to zero translates to an additional reduction of about 55 basis points in each of the four columns. The reason for the similarity in reductions is that, in each average-history column, the positive bias at zero in the first twelve months is eliminated (a reductionof29basispoints)alongwitheliminationofthetermpremiuminthefirst12months(an additionalreductionof26basispoints). The last block of rows in Table 4 (rows 16-18) illustrates an additional dimensionof potential policycommunications—thehorizonofapersistentalterationinpolicy. Inthisexample,thecertain reduction to zero of the 1-month rate is extended to a 24-month horizon. In the first column, the 5-year bondrate is reduced by341 (8.14 - 4.73) basis points. Evenin the case of the last column, wherethelong-runendpointofnominalratesiszero,the5-yearbondrateisreducedby109(2.63 -1.54)basispoints,anontrivialreductionofnearly60%. As noted earlier, historical bond rates have remained well above zero even when the short rate is near or at zero. This section has indicated that the higher levels of bond rates are due to a combination of expectations of an eventual resumption of the average-history level for the short rate, of which an important determinant is the perceived long-run policy target for inflation, and uncertain expectations regarding the level, volatility, and horizon of the new operating policy for the 1-month rate. Further reductions in bond rates are unlikely to be achieved by conventional operating policies that indicate only the current level of the short rate. Policies aimed at credible enforcement ofupperorlowerboundariesonsegmentsoftheterm structure,suchas illustratedin Table4,are discussedinthenextsection. IV.PolicyPuts Althoughmuchattentionispaidinthefinancialpresstothecurrentpolicytargetfortheshortrate, the results of section I indicatedthat bond rates accomplish most of the heavy liftingof monetary policy. Under expectation-based theories of the term structure, bond rates consist of averages of expected short rates and term premiums. In contrast to corporate bond rates where significant 14

portions of term premiums are compensation for risks of bankruptcy, term premiums of Treasury securitiesare principallymarket valuationsoftheuncertaintyofmonetarypolicy. In relatively quiescent periods, monetary policy appears to function well by communicating primarily the current setting of the short rate—a single point on the term structure. Market agents also interpret various public communications by the central bank, such as speeches or the historical record of instructions to the trading desk, to infer the most likely paths for future short rates,distributionsofpossibleoutcomesaboutthesetrajectories,andthepolicytargetforlong-run inflationthat,ultimately,is the singlecomponentcontrolledby policythat determines the average levelofnominalinterestrates. When the level of the short-term interest rate is driven near zero, the standard policy communicationchannel losesitsinformationcontentbecausethereis littleroom left forsignaling with the spot short rate. In these instances, it is desirable for the central bank to credibly communicate the outlook for short rates, including the level, volatility, and horizon of the new short rate policy. These characteristics of near-term policy essentially determine the distribution of yields associated with the initial portion of the term structure. In many instances, the central bankaimstooffsetalargeshockoraccumulatedshockstodemandorsupply. Thus,thenear-term policy objective is often asymmetric and directed at enforcing an upper or lower boundary on an initialsegmentoftheterm structure. Inanepisodeofdepresseddemandwiththeshortratealreadynearoratzero,itisunlikelythat long rates with sizeable term premiums will be substantially reduced unless policy is directed at enforcing a floor on the prices of shorter-maturity bonds. In the example of the previous section, policy wishes to persuade traders that prices of bonds with maturities of two years or less will not fall significantly below one. The remainder of this section discusses examples of derivative securities that can be written by the trading desk to communicate a specific policy of asymmetric intentions for future short rates. For simplicityof exposition,the examples will be European puts onTreasurybonds. Bondputsandinterestratecalls To enforce a floor on near-term bond maturities, derivative contracts that provide explicit policy signals over the next two years include the writing of short-horizon bond puts. For example, supposetheexpirationdateoftheoptionsisthreemonths,andputcontractsarewrittenonforward 9-monthand 21-monthbonds witha unit strike price of$1. Inthree months,thetrading deskwill be obligated to buy 9-monthand 21-month Treasury bonds from option owners at the strike price of $1 or, more likely, will settle in cash the difference between $1 and the existent bond prices, 15

0 < P 9 ;t + 2 ; P 2 1 + ;t 2 < 1 . The options will be exercised by market traders if the interest rates on thesebonds, r 9 + ;t 2 and r 2 1 + ;t 2 ,aregreater thanzero. For the purpose of public communications regarding policyoptions, a simple way to interpret the writing of bond puts is that the trading desk is issuing insurance policies against the risk of interestratesrisingabovezerooverthenexttwoyears. Intheexamplegiven,3-monthcalloptions are written,ineffect, on the9-monthand 21-monthinterest rates that will exist inthree months.20 Before turning to a discussion of broader issues raised by the policy use of derivative securities, it may be useful to review some simple mechanics of the market valuation of European calls on interestrates (or,equivalently,Europeanputsonbonds).21 Pricingtheinterest ratecall (bondput) As shown at the top of Table 5, and reproduced here, the value of a 3-month call option on the forward9-monthinterestrateis call t ( r 9 + ;t 2 ; r x ) = = E e (cid:0) t f r M 3 ;t = +t 4 0 2 0 ( ( r E 9 + ;t f r t 2 9 (cid:0) + ;t r 2 g ) x N + ( g d ; 1 ) (cid:0) r x N ( d 2 ) ) ; (4) where, as before, the “+” superscript selects only positive values between the forward 9-month rate, r 9 ;t + 2 andthe“strike”rate, r x . M +t 2 denotesthestochasticvaluationfactorwhichdetermines themarketvaluationofincomereceivedin t + 2 ;itsmainfunctionistodeterminethecompensation required by asset holders for covariation of asset income with expected variation in marginal utility such as due to changes in wealth. Turning to the second line in equation (4), the valuation of the call option is seen to be the product of the 3-month discount factor, e (cid:0) r 3 ;t = 4 0 0 , times a weighted spread between the forward 9-month rate at the date of option expiration and the strike rate. Because N ( : ) denotes the cumulative distribution of the normal distribution, the weights rangeinvaluefrom0to1. Theargumentsofthecumulativedistributionsare: 20Theexamplesareoptionsonforwardinterestratesatasinglepointintime. Asnotedlater,thetradingdeskcan writemorecomplexoptionsthatinsureagainstaveragedeviationsoftheshortratefromaprescribedtrajectoryover thenext12or24months. 21Indiscussingthefunctionaluseofbondoptions,itisimmaterialwhetherthebondoptionisformulatedinterms ofastrikeprice,suchasthebondput,orintermsofastrikerate,suchastheinterestratecall. Differencesmayexist in practicedueto expectationalassumptionsorinstitutionalconventions. Forexample,thevaluationofanoptionis notinvarianttoalternativeassumptionsthatthebondpriceorthebondinterestrateislognormallydistributed. Bond optionstradedinover-the-countermarketsareoftenformulatedininterestrateterms,whereasbondoptionstradedin exchangesareoftenexpressedinbondpriceterms. 16

d d 1 2 = = l d o 1 g (cid:0) ( E (cid:27) f r t 9 + ;t 9 + ;t 2 ; 2 (cid:27) g 9 = r + ;t x 2 ) + 0 : 5 (cid:27) 2 9 + ;t 2 ; where (cid:27) 9 + ;t 2 denotes thestandarddeviationofthelogoftheforward9-monthrate.22 Some illustrations of alternative market valuations of the 3-month call option on the forward 9-monthareshowninTable5. ThetophalfofTable5indicatesvariousconditioningassumptions. As in Table 4, four alternative assumptions are made for current agent expectations regarding the 9-monthratethatwilloccurinthreemonths, t + 2 ,rangingfrom 6.0toanear-zero valueof0.1. The initial column in the bottom half of Table 5 lists representative strike rates that could be selected by the writer of the call option. The neighboring 4 (cid:2) 4 array indicates market valuations associated with particular conditioning assumptions about agent expectations (in the top half of Table5) andthe strikerate selectedfor theoption. Thesmall numbersalongthemaindiagonal of this array are “at the money”valuations. That is, even when the forward 9-monthrate is expected to exactly match the strike rate, the value of the option is not zero due to perceived short rate uncertaintyrepresentedbythestandarddeviationofthelog9-monthrate, (cid:27) 9 + ;t 2 .23 Moving down each column from the main diagonals, the option is “in the money” when the trader expectation regarding the forward rate is above the relevant strike rate in the option. The option valuation is largest when the option is “deep in the money,” such as shown at the bottom of the first column where the mean expected forward rate is 6.0. In this example, the strike rate of 0.1 is sufficiently below the market forward rate expectation that traders are nearly certain that the option will be exercised. Thus, the value of the near-certain option is close to the product of the 3-month discount factor and the spread between the expected forward rate and the strike rate, : 9 8 5 ( 6 : 0 (cid:0) 0 : 1 ) = 5 : 8 1 . Alteringfundamentals In principle, the existence of well-developed derivatives markets increases the liquidity of asset marketsand,thus,tendstoreducebid-askspreadsonunderlyingassets,FedeniaandGrammatikos (1992). The valuations of derivativesare derived from contingent claims on underlying assets. In 22ThisoptionvaluationsolutionwasdevelopedbyBlack(1976).Morerefinedclosed-formsolutionsforinterestrate optionsareavailable,suchasMiltersen,Sandman,andSondermann(1997),butequation(4)appearstobecommonly referencedforinterestratederivatives,Hull(1997),andavoidsimputingdensitiesfornegativeinterestrates. 23Anapproximationofthesevaluationsis 4: (cid:27) 9 ;t+ 2 r x . Afirst-orderexpansionof N ( z ) = 0 5: + ( 2 (cid:25) ) (cid:0) 1 2 z ,andthe relevantdiscountfactorisgivenbythefourthlineofTable5. 17

turn, the values of these assets, such as equities and bonds, are due to the economic fundamentals that determine the present value of expected earnings generated by the relevant corporations or governments that issued the underlying stocks or bonds. Thus, apart from temporary increases in volatility in episodes of large price movements such as October 19, 1987, there is little empirical evidence to support a reliable causal connection between movements in the valuations of options andofunderlyingassets intheabsenceofeffects ofderivativesoneconomicfundamentals.24 By contrast,policyuseofderivativescan altertheperceptions ofeconomicfundamentalsheld by market agents. Given that the central bank has a monopoly on the supply of the domestic currency, it has the capacity to directly purchase or write options against any proportion of the outstanding Treasury debt. Private sector agents who exercise put options written by the trading deskarepaidinthedomesticcurrency.25 Of course, market expectations can be notoriously difficult to influence. In the case of a put option floor on Treasury bonds, a number of factors may influence the ability of the trading desk tomovethebondprice towardsthe selectedstrikeprice, especiallyifthe initialbondpriceis well below the strike price (the bond rate is well above the strike rate). Several issues regarding the operationalimplementationofpolicyputoptionsare brieflynoted: Itwillbedifficultforthetradingdesktomovethebondpriceup(thebondinterestratedown)if theprivatesectorbelievesthepolicycouldbereversedbeforetheexpirationdateoftheputoption. Ofcourse, themoreoptionsthat are writtenbythetradingdesk,themoremoneywouldbeleft on thetableifthecentralbankweretorenegeonitsannouncedpolicyofmaintainingalowshort-term rateforaspecifiedhorizon. Ironically,thepotentialofpolicyreversalstoundercuttheexpectations effects of policy options is more of a problem for a strong central bank than a weak central bank, if the latter is not able to absorb the large option losses that could accompany a significant policy reversal. On average, net receipts from market valuations of policy options will be positive if the options provide insurance value for investors against uncertain policy. As indicated in Table 5, lesspredictablepolicyis morecostlyfortheprivatesectortoinsureagainst. To the extent that relative supplies of assets are also a determinant of asset prices, the cash payoutofpolicyoptionsthatareunsuccessfulinreducingbondratesmayhaveapositiveinfluence 24DifferentviewsontheOctober1987consequencesofportfolioinsurancebasedondynamictradingofstocksare advancedinHull(1997)andMullins(1997). Thelatteralsoprovidesageneralreviewofempiricalliteratureonthe effectsofderivativesonassetmarkets. 25Note that this differs importantly from interventions in international currency markets, where the ability of a centralbanktosupportthepriceofthedomesticcurrencydependsonitsinitialsupplyofforeigncurrenciesrelative tothemonetarybaseorthecooperationofothercentralbanks. 18

on bond prices.26 In other words, if traders expectations are not consistent with the policy outlook specified in the policy options, bond prices may remain below the option strike prices. The consequence of the cash settlements when the options are exercised is roughly equivalent to a delayed open market operation of government security purchases, also aimed at lowering interest rates. Of course, the extent of interest rate pressure would depend on which assets were subsequentlypurchasedbyowners oftheexercisedoptions. Many economists, including Keynes (1930), have suggested that the central bank can operate directly on long bond rates by directly purchasing or selling bonds.27 As noted earlier, the effectivenessofsuchpolicyactionsrestsontheimperfectsubstitutabilityofportfolioassets. Ifthe privatesectorisnotindifferenttotherelativesuppliesofdifferentbondmaturities,thetradingdesk could duplicate the bond price effects of any particular put option contract by buying a sufficient quantity of outstanding bonds of the appropriate maturities. However, this may require extensive interventionsinTreasurymarkets. Mostimportantly,likeadedicatedbehaviorist,themarketwould have to examine trading desk operations closely to infer the ultimate target objectives of policy. Because traders would not be compensated for subsequent departures from the revised objectives ofthetradingdesk,itseemslikelytheywouldbecautiousinrevisingfutureshortrateexpectations. Bycomparison,theobjectivesofpolicyregardingfutureshortratesareimmediatelytransparentto traders from theexplicittermsoftheput options,as illustratedabove. Further, as noted by Merton (1995), an additional possible advantage of policy options is that they may continue to automatically work against unexpected shocks, without waiting for subsequent interventions by the trading desk. In the case of policy bond puts, option purchasers may conduct “delta” hedging to maintain a given level of risk exposure for their portfolios. If an unexpected shock serves to drive down the price of the bond of the maturity specified in the option, the value of the option rises (that is, the bond interest rate is higher). Since this increases the likelihood of exercising the option, the owners of the option will increase their holding of bonds to offset the larger delta (in absolute value)of the put option. Similarly,optionowners will decreasetheamountofbondsintheirportfoliosifashockincreasestherelevantbondprices. These actions have the same stabilizingeffects on the relevant bond prices as if the central bank trading desk were to buy or sell bonds of the appropriate maturities. In effect, owners of the options who 26Theprimarypolicytransmissionchannelemphasizedinthispaperisthatcurrentcentralbankactionsaffecttrader expectationsaboutfutureshortrates. Asecondchannelsometimesadvancedtoexplaindirecteffectsoftradingdesk purchasesisthatalternativefinancialassets, includingcurrency,areimperfectportfoliosubstitutessothepriceofan assetisexpectedtomoveinverselytoitsaggregatesupply. 27“ItshouldnotbebeyondthepowerofaCentralBank(internationalcomplicationsaside)tobringthelong-term market-rateofinteresttoanyfigureatwhichitisitselfpreparedtobuylong-termsecurities.”Keynes(1930,p.371) 19

conductdeltahedgesare delegatedagentsofthetradingdesk. Althoughtherecanbesomeshockvalueindestroyingtherelevanceofprevailingexpectations by sudden introductions of new policy instruments and objectives during crises, as occurred in October 1979, the probability of successful executions of policy options would most likely be higher if all participants, both the trading desk and asset markets, had prior experiences with policy use of options to supplement conventional policy trading. As an example, the trading desk couldwrite derivatives,suchas average rate collars, that wouldinsureoptionowners against short rate deviations from an announced policy range in intervals between FOMC meetings. If a mix of conventional trading and policy options were routinely familiar, it seems likely that whentheeconomyisconfrontedbyepisodeswhereconventionalpolicyoperationsarelikelytobe ineffectual,ashiftintheoperationalpolicymixtowardsmorerelianceonoptionswouldbeviewed as asensible,ifnotanticipated,alterationofpolicyoperations. Finally,ataminimum,explicitpolicyoptionsprovideatransparent,market-basedindexofthe credibilityoftheparticularpolicyscenarioembeddedintheterms oftheoptions. V.Concluding Remarks This paper indicates that bond rates are the primary transmission channel for monetary policy, and that bondrates contain bothaverages of trader expectations of future short rate actions by the central bankandterm premiumsreflectingtrader uncertaintyregardingfuturepolicy. In periods of routine fluctuations in economic activity and inflation, monetary policy appears to function well by setting the current level of the short rate. Given that future shocks to macroeconomic indicators are unknown, bond traders will discount policy announcements that thecentral bankis pre-committedtoa fixedpathofreal ornominalinterestrates. However, in situations where policy responses are apt to be less predictable, the central bank may wish to tighten connections between bond rates and the short rate. The example discussed in this paper is when the average level of interest rates is low, due to the perception that long-run inflation is near zero, and there is little room for extensive signaling by the central bank with the shortrate. Thecredibilityofsignalingregardingnear-termmovementsintheshortrateisapttobe improvedifthecentral banktradingdeskwrites putoptionsonTreasurybonds. Illustrations used in the previous section were simple European options on forward interest ratesatasinglepointintime. Thetradingdeskcanwritemorecomplexoptionsthatinsureagainst deviations of the short rate from a prescribed trajectory over a given horizon. For example, firms and households are likely to make decisions on the basis of average interest rates over a planning 20

period,notinterestratesatparticularpointsintimesuchasspecifiedintheEuropeanputexamples. Onealternativeisforthetradingdesktowriteaverage-valueoptionswheretheowneroftheoption iscompensatedforaveragedeviationsfromstrikepricesorratesoverthelengthoftheoptionrather thanforthedifferentialthatexistsonlyattheoptionexpirationdate. Becausethevolatilityoftime averages is smaller than the volatility at a single date, the cost of private sector insurance against unexpectedpolicyisalsosmaller.28 Discussioninthispaperhasbeenconfinedtopolicyuseofderivativesongovernmentsecurities. Subject to constraintsonadmissiblecentral banktrading,it maybe useful toexplorethepotential forpolicyderivativestoreduceothersourcesofuncertaintythatmayconfrontfirmsandhouseholds during periods of severe market stress, such as an unusual disruption of credit markets. The potentialflexibilityofoptionscontracts suggeststhat policyderivativescan beaversatileaddition tothetoolkitofcentral banks. 28Forvaluationsofaverage-valueoptions,seeTurnbullandWakemann(1991)anddiscussionofso-called“Asian” optionsinHull(1997). 21

References Bernanke, B., and A. Blinder, 1992, “The Federal Funds Rate and the Channels of Monetary Transmission,”AmericanEconomicReview,82(4),September,901-21. Black, F. 1976. “The Pricing of Commodity Contracts.” Journal of Financial Economics, 3, January/March,167-79. Brenner, R., R. Harjes, and K. Kroner, 1996, ”Another Look at Models of the Short-Term Interest Rate,” Journalof FinancialandQuantitativeAnalysis,31,85-107. Cecchetti, S., 1988. “The Case of the Negative Nominal Interest Rates: New Estimates of the Term Structure of Interest Rates during the Great Depression.” Journal of Political Economy,96,December, 1111-41. Cox, J, J. Ingersoll, and S. Ross, 1985. “A Theory of the Term Structure of Interest Rates.” Econometrica,53,March,385-407. Fedenia, M. and T. Grammatikos. 1992. “Options Trading and the Bid-Ask Spread of the UnderlyingStocks.” Journalof Business,65,July,335-51. Friedman, M. and A. Schwartz. 1963, A Monetary History of the United States, 1867-1960.” PrincetonUniversityPress: Princeton. Fuhrer,J.andB.Madigan. 1997. “MonetaryPolicywhenInterestRates areBoundedatZero.” Reviewof EconomicsandStatistics,79,November,573-85. Hull, J. 1997. Options, Futures, and Other Derivatives, 3rd edition, Upper Saddle River,NJ: Prentice-Hall. Miltersen, K., K. Sandmann, and D. Sondermann. 1997. “Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates.” Journal of Finance, 52, March, 409-30. Johnson,N.andS.Kotz. 1970. ContinuousUnivariateDistributions,NewYork: JohnWiley. Keynes,J.1930. ATreatiseonMoney,vol. II,London: Macmillan. Kozicki,S.,D.Reifschneider,andP.Tinsley. 1996. “TheBehaviorofLong-TermInterestRates in the FRB/US Model,” in The Determination of Long-Term Interest Rates and Exchange Rates andtheRoleof Expectations,Basle: BankforInternationalSettlements,215-50. Kozicki, S. and P. Tinsley. 1996. “Moving Endpoints and the Internal Consistency of Agents; ExAnteForecasts.” ComputationalEconomics,11,April,21-40. Kozicki,S.andP.Tinsley. 1998. “Term StructureViewsofMonetaryPolicy.” Federal Reserve Board/Federal Reserve BankofKansasCitystaffworkingpaper. Laxton, D. and E. Prasad. 1997. “Possible Effects of European Monetary Union on Switzerland: A Case Study of Policy Dilemmas caused by Low Inflation and the Nominal Interest Rate Floor,”IMFWorkingPaper,WP/97/23. Macaulay, F. 1938. The Movements of Interest Rates, Bond Yields and Stock Prices in the 22

UnitedStatessince1856,NationalBureau ofEconomicResearch: NewYork. McCulloch, H., 1975, “The Tax-Adjusted Yield Curve,” The Journal of Finance, 30, June, 811-30. McCulloch, H. and H. Kwon, 1993, “U.S. Term Structure Data, 1947-1991,” Ohio State UniversityWorkingPaper93-6,March. Merton, R. 1973. “An Intertemporal Capital Asset Pricing Model,” Econometrica, 41, September,867-87. Merton, R. 1995. “Financial Innovation and the Management and Regulation of Financial Institutions,”Journalof BankingandFinance,19,June,461-81. Mullins, D. 1997. “Challenges for Monetary Policy in the Evolving Financial Environment,” in I. Kuroda (ed.) Towards More Effective Monetary Policy, New York: St. Martin's Press, 99-129. Orphanides, O. and V. Wieland. 1998. “Price Stability and Monetary Policy Effectiveness whenNominalInterestRates areBoundedat Zero.” FRB staffworkingpaper. Taylor,J.1993. “DiscretionversusPolicyRulesinPractice,”inA.MeltzerandC.Plosser(eds.) Carnegie-Rochester Conference Series on Public Policy, 39, Amsterdam: North-Holland, 195-214. Tetlow,R.andJ.Williams. 1998. “ImplementingPriceStability: Bands,Bounds,andInflation Targeting.” FRB staffworkingpaper. Turnbull, S. and L. Wakeman. 1991. “A Quick Algorithm for Pricing European Average Options.” Journalof FinancialandQuantitativeAnalysis,26,September,377-89. U.S. Bureau of the Census. 1975. Historical Statistics of the United States, Washington D.C.: U.S. GovernmentPrintingOffice. 23

Table1: OutputDeviationsfromTrenda y t = c 0 + c 1 y t(cid:0) 1 + c 2 ( L ) (cid:1) y t(cid:0) 1 + c 3 (cid:26) x ;t(cid:0) 1 + c 4 ( L ) (cid:1) (cid:26) x ;t(cid:0) 1 + c 5 (cid:26) z ;t(cid:0) 1 + c 6 ( L ) (cid:1) (cid:26) z :t (cid:0) 1 + a t : realrate c 1 c 2 c 3 c 4 c 4 c 4 S E E (cid:22)(cid:26) x b (cid:26) x = (cid:26) 1 .966 .577 -.027 .107 .651 1.37 (109.6) (7.9) (-1.6) (1.4) (1.1) (cid:26) x = (cid:26) 6 0 .956 .505 -.102 .235 .625 3.23 (100.9) (6.9) (-3.7)*** (1.2) (10.0) (cid:26) x = (cid:26) 6 0 .956 .483 -.105 .226 -.028 .080 .626 (cid:26) z = (cid:26) 1 (cid:0) (cid:26) 6 0 (99.9) (6.4) (-3.8)*** (1.1) (-1.4) (0.9) (cid:26) x = (cid:26) 1 2 0 .962 .511 -.116 .170 .627 3.50 (107.3) (6.9) (-3.3)*** (0.7) (12.8) (cid:26) x = (cid:26) 1 2 0 .963 .510 -.117 .183 .006 -.016 .631 (cid:26) z = (cid:26) 1 (cid:0) (cid:26) 1 2 0 (105.9) (6.7) (-3.3)*** (0.8) (0.3) (-0.2) (cid:26) x = (cid:26) 6 0 .955 .504 -.086 .167 .049 -.173 .629 (cid:26) z = (cid:26) 1 2 0 (cid:0) (cid:26) 6 0 (95.7) (6.7) (-2.1)** (0.7) (0.5) (-0.3) (cid:26) x = (cid:26) 1 2 0 .955 .504 -.086 .167 -.134 .340 .629 (cid:26) z = (cid:26) 6 0 (cid:0) (cid:26) 1 2 0 (95.7) (6.7) (-2.1)** (0.7) (-1.7)* (0.7) aIn each regression, the dependent variable is monthly capacity utilization of manufacturing, y . Competing regressors are: 1-month real rate, (cid:26) 1 = r 1 (cid:0) E t (cid:25) 1 ; 5-year real rate, (cid:26) 6 0 = r 6 0 (cid:0) E t (cid:25) 6 0 ; and 10-yearrealrate, (cid:26) 1 2 0 = r 1 2 0 (cid:0) E t (cid:25) 1 2 0 . Polynomialsinthelagoperator, L ,arefifth-order.Thesamplespan is1967m1-1997m9. Confidencelevelsof90%(*),95%(**),and99%(***)areindicatedfortheboldface t -ratios. bEntriesinthelastcolumn, (cid:22)(cid:26) x ,denote“equilibrium”realrates,orlevelsoftherealinterestrateconsistent withtrendoutputgrowthatthemeanrateofcapacityutilization.

figure 1: VAR predictions of 10-year bond rates fixed endpoints VAR 16 14 12 10 8 6 4 1970 1975 1980 1985 1990 1995 moving average endpoints VAR 20 15 10 5 0 1970 1975 1980 1985 1990 1995 shifting endpoints VAR 20 15 10 5 0 1970 1975 1980 1985 1990 1995 historical predicted

Table2: RegressionAnalysisofResidual TermPremiumsa (cid:18) R t = c 0 + c 1 ^r 1 :t + a t : term premium source c 0 c 1 R 2 S E E fixed 1.74 1.95 endpointVAR (17.3) -1.98 .575 .51 1.36 (-10.0) (20.0) movingaverage 1.68 1.54 endpointVAR (20.8) 2.87 -.190 .09 1.47 (13.5) (-6.2) shifting 1.79 .486 endpointVAR (72.5) 1.12 .103 .26 .414 (18.4) (11.6) a (cid:18) R denotesresidualestimates ofthe termpremiumsfor10-yearbondsgeneratedbythefixed, movingaverage, andshiftingendpointVARs. TheregressorsarepredictionsbytherelevantVARsofthe1-monthnominalinterestrate, ^r 1 . Thesamplespanis1966m1-1997m9.

Table3: Level-Based Estimates ofConditional Variances ofOne-Month Bond Rate.a u 2 r ;t = (cid:30) 0 + (cid:30) 1 ^r t + a t : residual source (cid:30) 1 S E E fixedendpoints .237 1.21 VAR (9.3) movingaverage .240 1.19 endpointsVAR (9.6) shifting .241 1.18 endpointsVAR (9.6) a u r ;t denotestheone-periodforecasterrorof r t and ^r t theone-periodforecastoftheone-monthrate. Forecastsare generatedby 6-order, monthly VARs in r , (cid:25) , and y with fixed, moving-average,or shifting endpoints. The sample spanis1966m1-1997m9.

figure 2: Estimates of 10-year bond rate term premiums fixed endpoints VAR 8 6 4 2 0 -2 -4 1970 1975 1980 1985 1990 1995 moving average endpoints VAR 8 6 4 2 0 -2 -4 1970 1975 1980 1985 1990 1995 shifting endpoints VAR 4 2 0 1970 1975 1980 1985 1990 1995 residual-based estimate theory-based construction

Figure 3: AlternativeShort RateExpectations subject tothe Zero Boundary f(r) E[r+] E[r+] ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • •••••••••••••••• • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• •• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• r 0 6

Table4: EffectsofShortRateUncertaintyandtheZeroBoundonExpectedInterestRates alternative shortrateassumptionsa certainty equivalent expectation, E f r 1 g 6.0 4.0 2.0 0.0 standard deviation, (cid:27) 1 1.80 1.80 1.80 1.80 expectation underzerobound, E f r + 1 g 6.00 4.06 2.45 1.44 components of5-yearbondrate: expected shortrateaverage, 1 6 0 5 P =i 9 0 E f r + 1 + ;t i g 6.00 4.06 2.45 1.44 termpremium, (cid:18) 6 0 ;t 2.14 1.73 1.40 1.19 total, r + 6 0 ;t 8.14 5.80 3.85 2.63 additional assumption: initialmonthatzero(nondeterministic), r t = 0 : expected shortrateaverage, 1 6 0 5 P =i 9 0 E f r + 1 + ;t i g 5.92 4.02 2.43 1.44 termpremium, (cid:18) 6 0 ;t 2.11 1.72 1.39 1.19 total, r + 6 0 ;t 8.04 5.74 3.83 2.63 additional assumptions: initial12monthsatzero(nondeterministic), r 1 + ;t i = 0 ( i = 0 ; : : : ; 1 1 ) : expected shortrateaverage, 1 6 0 5 P =i 9 0 E f r + 1 + ;t i g 5.09 3.53 2.25 1.44 termpremium, (cid:18) 6 0 ;t 1.88 1.59 1.34 1.19 total, r + 6 0 ;t 6.97 5.13 3.59 2.63 additional assumptions: initial12monthsatzero(deterministic), r 1 ;t + i = (cid:27) 1 + ;t i = 0 ( i = 0 ; : : : ; 1 1 ) : expected shortrateaverage, 1 6 0 5 P =i 9 0 E f r + 1 + ;t i g 4.80 3.25 1.96 1.15 termpremium, (cid:18) 6 0 ;t 1.63 1.33 1.08 0.93 total, r + 6 0 ;t 6.43 4.58 3.04 2.08 additional assumptions: initial24monthsatzero(deterministic), r 1 ;t + i = (cid:27) 1 + ;t i = 0 ( i = 0 ; : : : ; 2 3 ) : expected shortrateaverage, 1 6 0 5 P =i 9 0 E f r + 1 + ;t i g 3.60 2.44 1.47 0.86 termpremium, (cid:18) 6 0 ;t 1.13 0.94 0.78 0.68 total, r + 6 0 ;t 4.73 3.37 2.25 1.54 aForthesample1982m10-1997m9,themeanandstandarddeviationofthe1-monthzero-couponratewere6.06and1.80, respectively.

Table 5: Hypothetical3-Month CallOptions onthe Forward9-Month Ratea call t ( r 9 + ;t 2 ; r x ) = = E e (cid:0) t f r M 3 ;t = +t 4 0 2 0 ( ( r E 9 + ;t f r t 2 9 (cid:0) + ;t r 2 g ) x N + ( g d : 1 ) (cid:0) r x N ( d 2 ) ) : alternativeforwardrateassumptionsb forward9-monthrate, E t f r 9 ;t + 2 g 6.0 4.0 2.0 0.1 standarddeviationoflogforwardrate, (cid:27) 9 + ;t 2 .312 .312 .312 .312 current 3-monthrate, r 3 ;t 6.00 6.00 6.00 6.00 discountfactor, e (cid:0) r 3 ;t = 4 0 0 .985 .985 .985 .985 alternativestrikerates: calloptionvalues r x = 6 : 0 .732 .068 0.0 0.0 r x = 4 : 0 2.038 .488 .004 0.0 r x = 2 : 0 3.940 1.974 .244 0.0 r x = 0 : 1 5.812 3.842 1.872 .012 a M t+ 2 denotesthestochasticvaluationfactorforyieldspayableinthreemonths;the“+”superscriptselectsonly positivevalues;andN( . ) indicatesthecumulativedistributionfunctionforthenormaldistribution. The d 1 and d 2 argumentsaredefinedinthetext. bFor the sample 1982m10-1997m9,means of the 3-month zero coupon rate and the forward 9-month rate were about6.3and7.0. Thestandarddeviationoftheforward9-monthlogratewasapproximately.312.