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DEFLATION RISKMatthias FleckensteinFrancis A. LongstaffHanno LustigWorking Paper 19238http://www.nber.org/papers/w19238NATIONAL BUREAU OF ECONOMIC RESEARCH1050 Massachusetts AvenueCambridge, MA 02138July 2013
We are grateful for the comments of Chris Downing, Xavier Gabaix, Huston McCulloch, James Mauro,Mike Rierson, Bradley Yim, and seminar participants at New York University and the Wharton Schoolat the University of Pennsylvania. All errors are our responsibility. The views expressed herein arethose of the authors and do not necessarily reflect the views of the National Bureau of Economic Research.NBER working papers are circulated for discussion and comment purposes. They have not been peerreviewed or been subject to the review by the NBER Board of Directors that accompanies officialNBER publications. 2013 by Matthias Fleckenstein, Francis A. Longstaff, and Hanno Lustig. All rights reserved. Shortsections of text, not to exceed two paragraphs, may be quoted without explicit permission providedthat full credit, including notice, is given to the source.
Deflation RiskMatthias Fleckenstein, Francis A. Longstaff, and Hanno LustigNBER Working Paper No. 19238July 2013JEL No. E31,G13ABSTRACTWe study the nature of deflation risk by extracting the objective distribution of inflation from the marketprices of inflation swaps and options. We find that the market expects inflation to average about 2.5percent over the next 30 years. Despite this, the market places substantial probability weight on deflationscenarios in which prices decline by more than 10 to 20 percent over extended horizons. We find thatthe market prices the economic tail risk of de- flation very similarly to other types of tail risks suchas catastrophic insurance losses. In contrast, inflation tail risk has only a relatively small premium.De- flation risk is also significantly linked to measures of financial tail risk such as swap spreads, corporatecredit spreads, and the pricing of super senior tranches. These results indicate that systemic financialrisk and deflation risk are closely related.Matthias FleckensteinHanno LustigUCLA Anderson School of ManagementUCLA Anderson School of Management110 Westwood Plaza110 Westwood Plaza, Suite C413Los AngelesLos Angeles, CA email@example.com email@example.com A. LongstaffUCLAAnderson Graduate School of Management110 Westwood Plaza, Box 951481Los Angeles, CA 90095-1481and firstname.lastname@example.com
Deation has played a central role in the worst economic meltdowns experiencedin U.S. history. Key examples include the deations associated with the Panicof 1837, the Long Depression of 18731896, and the Great Depression of the1930s. In light of this, it is not surprising that deation is now one of themost-feared risks facing participants in the nancial markets. In recent years,the nancial press has increasingly raised concerns about a global deationaryspiral and has used terms such as nightmare scenario or looming disasterto describe the growing threat.1 Furthermore, addressing the risk of deation isone of the primary motivations behind a number of actions taken by the FederalReserve in the past several years such as the quantitative easing programs.2Despite the severe potential eects of deation, relatively little is knownabout how large the risk of deation actually is, or about the economic andnancial factors that contribute to deation risk. The primary reason for thismay simply be that deation risk has traditionally been very dicult to measure.For example, as shown by Ang, Bekaert, and Wei (2007) and others, econometricmodels based on the time series of historical ination perform poorly even inestimating the rst moment of ination. In addition, while surveys of inationtend to do better, these surveys are limited to forecasts of expected ination overshorter horizons and provide little or no information about the tail probabilityof deation.This paper presents a simple market-based approach for measuring deationrisk. This approach allows us to solve directly for the markets assessment of theprobability of deation for horizons of up to 30 years using the prices of inationswaps and options. In doing this, we rst use standard techniques to infer therisk-neutral density of ination that underlies the prices of ination calls andputs. We then use maximum likelihood to estimate the ination risk premiumembedded in the term structure of ination swap rates using methods familiarfrom the ane term structure literature. Finally, we solve for the actual or1
For example, see Coy (2012), Five Charts that Show that Deation is a Growing Threat, www.businessweek.com/articles/1012-06-05/ve-charts-that-showdeation-is-a-growing-threat, Lange (2011), Nightmare Scenario: U.S. Deation Risks Rising, www.reuters.com/article/2011/10/16/us-economy-deationidUSTRE79P7FV201110126, and Carney (2010), Deation: Looming Disasterfor Banks, www.cnbc.com/id/39648704/Deation Looming Disaster for Banks.2
For example, see Bernanke (2012), Monetary Policy since the Onset of theCrisis, www.federalreserve.gov/newsevents/speech/bernanke20120831a.htm.1
objective distribution of ination by inverting the risk-premium-adjusted characteristic function of the risk-neutral density. A key advantage of this approachis that we recover the entire distribution of ination rather than just the rstmoment or expected ination. This is important since this allows us to measurethe probability of tail events such as deation.The shape of the distribution of ination varies signicantly for shorter horizons, but is much more stable for longer horizons. Ination risk premia areslightly negative for horizons of one to ve years, but increase to about 30 basispoints for a 30-year horizon.We nd that the market expects ination of close to 2.5 percent for horizonsfrom 10 to 30 years. The volatility of ination is roughly two percent for shorterhorizons, but is about one percent or less for horizons of ten years or more.Thus, the market views ination as having a strongly mean reverting nature.The distribution of ination is skewed towards negative values and has longertails than a normal distribution.We solve for the probability of deation over horizons ranging up to 30 yearsdirectly from the distribution of ination. The empirical results are very striking.We nd that the market places a signicant amount of weight on the probabilitythat deation occurs over extended horizons. Furthermore, the market-impliedprobability of deation can be substantially higher than that estimated by policymakers. For example, in a speech on August 27, 2010, Federal Reserve ChairmanBen S. Bernanke stated that Falling into deation is not a signicant risk for theUnited States at this time.3 On the same date, the market-implied probabilityof deation was 15.11 percent for a two-year horizon, 5.36 percent for a veyear horizon, and 2.84 percent for a ten-year horizon. These probabilities areclearly not negligible. On average, the market-implied probability of deationduring the sample period was 11.44 percent for a two-year horizon, 5.34 percentfor a ve-year horizon, 3.29 percent for a ten-year horizon, and 2.33 percentfor a 30-year horizon. The risk of deation, however, varies signicantly andthese probabilities have at times been substantially larger than the averages. Inparticular, the probability of deation exhibits jumps which tend to coincide withmajor events in the nancial markets such as the ratings downgrades of Spain in2010 or the downgrade of U.S. Treasury debt by Standard and Poors in August2011.Deation is clearly an economic tail risk and changes in deation risk mayreect the markets fears of a meltdown scenario.4 Thus, a natural next step is3
See Bernanke (2010), The Economic Outlook and Monetary Policy, www.federalreserve.gov/newsevents/speech/2010speech.htm.4
Note that we are interpreting tail risk as including more than just event risk or2
to examine whether deation risk is related to other serious types of tail risk inthe nancial markets or in the macroeconomy in general. Focusing rst on thepricing of deation risk, we nd that the ratio of the risk-neutral probability ofdeation to the objective probability of deation is on the order of three to one.This ratio is very similar to that of other types of tail risk. For example, Froot(2001) nds that the ratio of the price of catastrophic reinsurance to expectedlosses ranges from two to seven. Driessen (2005), Berndt, Due, Douglas, Ferguson, and Schranz (2005), Giesecke, Longsta, Schaefer, and Strebulaev (2011)estimate that the ratio of the price of expected losses on corporate bonds toactual expected losses is on the order of two to three. These ndings are alsoconsistent with models with rare consumption disasters, such as that pioneeredby Rietz (1989) and further developed by Longsta and Piazzesi (2004), Barro(2006), and Gourio (2008), which were explicitly engineered to produce highrisk-neutral probabilities for rare consumption disasters such as the Great Depression. Recently, Barro has argued that this class of models can account forthe equity premium when calibrated to the 20th Century experience of developedeconomies. Gabaix (2012) and Wachter (2013) have extended these models toincorporate a time-varying intensity of consumption disasters. This extensiondelivers bond and stock market return predictability similar to what is observedin the data.We next consider the relation between deation risk and specic types ofnancial and macroeconomic tail risks that have been described in the literature. In particular, we consider a number of measures of systemic nancial risk,collateral revaluation risk, sovereign default risk, and business cycle risk and investigate whether these are linked to deation risk. We nd that a number ofsystemic risk variables are signicantly related to the probability of deation.For example, the risk of deation increases as the price of protection on supersenior tranches increases. This is intuitive since the types of economic meltdownscenarios that would result in losses on super senior tranches would likely beassociated with sharp declines in the level of prices. Similarly, we nd that deation risk increases as the credit and liquidity risks faced by the nancial sectorincrease. Thus, economic tail risk increases as the nancial sector becomes morestressed. We also nd that the risk of deation increases as the unemploymentrate increases. This is consistent with a number of classical macroeconomic theories about the relation between prices and employment. Overall, these resultsprovide support for the view that the risk of severe macroeconomic shocks inwhich deation occurs is closely related to tail risks in nancial markets. Thus,jump risk. Event or jump risks are adverse economic events that occur relativelysuddenly. In contrast, tail risk can also include extreme scenarios with severeeconomic consequences which may unfold over extended periods. The modelingframework used in this paper is consistent with both types of risks.3
option prices are highly informative about the probability the market imputesto these rare disaster states, arguably more informative than quantity data (see,for example, recent work by Backus, Chernov, and Martin (2011) using equityoptions). Our ination option ndings imply that market participants mostlyexpect deation in the U.S. in these disaster states. This is consistent with U.S.historical experience in which depressions/deationary spirals have been associated with major collapses in the nancial system.Finally, we also compute the probabilities of ination exceeding variousthresholds. The results indicate that while the probability of ination in thenear term is relatively modest, the long-term probabilities of ination are muchhigher. Interestingly, we nd that the ratio of the probability of ination exceeding ve percent under the risk-neutral measure is only about 1.4 times thatunder the actual measure. Thus, ination tail risk is priced much more modestlythan is deation tail risk.Our results also have important implications for Treasury debt management. In particular, whenever the Treasury issues Treasury Ination ProtectedSecurities (TIPS), the Treasury essentially writes an at-the-money deation putand packages it together with a standard ination-linked bond. The returns onwriting these deation puts are potentially large because of the substantial riskpremium associated with deation tail risk. If the Treasury is better suited tobear deation tail risk than the marginal investor in the market for inationprotection, then providing a deation put provides an extra source of revenue forthe Treasury that is non-distortionary. There are good reasons to think that theTreasury is better equipped to bear deation risk, not in the least because theTreasury and the Federal Reserve jointly control the price level.5This paper contributes to the extensive literature on estimating the properties of ination. Important papers on estimating ination risk premia andexpected ination include Hamilton (1985), Barr and Campbell (1997), Evans(1998, 2003), Campbell and Viceira (2001), Bardong and Lehnert (2004), Buraschi and Jiltsov (2005), Ang, Bekaert, and Wei (2007, 2008), Adrian and Wu(2007), Bekaert and Wang (2010), Chen, Liu, and Cheng (2010), Christensen,Lopez, and Rudebusch (2010, 2011), Gurkaynak and Wright (2010), Gurkaynak, Sack, and Wright (2010), Pueger and Viceira (2011a, 2011b), Chernovand Mueller (2012), Haubrich, Pennachi, and Ritchken (2012), Faust and Wright(2012), Grishchenko and Huang (2012), and many others. Key papers on deation include Hamilton (1992), Fisher (1933), Cecchetti (1992), Atkeson and5
Since the ratio of risk-neutral to actual probabilities is much larger for deation than for high-ination scenarios, this same logic is not as applicable to thestandard ination protection built into TIPS.4
Kehoe (2004), Kilian and Manganelli (2007), and Campbell, Sunderam, and Viceira (2013).Two important recent papers have parallels to our work. Christensen, Lopez,and Rudebusch (2011) t an ane term structure model to the Treasury realand nominal term structures and estimate the value of the implicit deationoption embedded in TIPS prices. Our research signicantly extends their resultsby estimating deation probabilities for horizons out to 30 years directly usingmarket ination option prices. Kitsul and Wright (2012) also use ination optionsto infer the risk-neutral density for ination, but do not formally solve for theobjective density of ination. Our results corroborate and extend their innovativework.The remainder of this paper is organized as follows. Section 2 briey discusses the history of deation in the United States. Section 3 provides an introduction to the ination swap and options markets. Section 4 presents the inationmodel used to value ination derivatives. Section 5 discusses the maximum likelihood estimation of the ination model. Section 6 describes the distribution ofination. Section 7 considers the implications of the results for deation probabilities and the pricing of deation risk. Section 8 examines the relation betweendeation risk and other types of nancial and macroeconomic tail risks. Section9 presents results for the probabilities of several ination scenarios. Section 10summarizes the results and makes concluding remarks.2. DEFLATION IN U.S. HISTORYThe literature on deation in the U.S. is far too extensive for us to be able toreview in this paper. Key references on the history of deation in the U.S. includeNorth (1961), Friedman and Schwartz (1963), and Atack and Passell (1994). Wewill simply observe that deation was a relatively frequent event during the19th Century, but has diminished in frequency since then. Bordo and Filardo(2005) report that the frequency of an annual deation rate was 42.4 percentfrom 18011879, 23.5 percent from 18801913, 30.6 percent from 19141949, 5.0percent from 19501969, and zero percent from 19702002. The nancial crisisof 20082009 was accompanied by the rst deationary episode in the U.S. since1955.Economic historians have identied a number of major deationary episodes.Key examples include the crisis of 18151821 in which agricultural prices fell bynearly 50 percent. The banking-related Panic of 1837 was followed by six years ofdeation in which prices fell by nearly 30 percent. The post-Civil-War greenbackperiod experienced a number of severe deations and the 18731896 period has5
been called the Long Depression. This period experienced massive amounts ofcorporate bond defaults and Friedman and Schwartz (1963) estimate that theprice level declined by 1.7 percent per year from 1875 to 1896. The U.S. suereda severe deationary spiral during the early stages of the Great Depression in19291933 as prices rapidly fell by more than 40 percent.Although Atkeson and Kehoe (2004), Bordo and Filardo (2005), and othersshow that not all deations have been associated with severe declines in economicoutput, a common thread throughout U.S. history has been that deationaryepisodes are typically associated with turbulence or crisis in the nancial system.3. THE INFLATION SWAPS AND OPTIONS MARKETSIn this section, we begin by reviewing the ination swaps market. We thenprovide a brief introduction to the relatively new ination options market.3.1 Ination SwapsAs discussed by Fleckenstein, Longsta, and Lustig (2012), U.S. ination swapswere rst introduced in the U.S. when the Treasury began auctioning TIPSin 1997 and have become increasingly popular among institutional investmentmanagers. Pond and Mirani (2011) estimate the notional size of the inationswap market to be on the order of hundreds of billions.In this paper, we focus on the most widely-used type of ination swap whichis designated a zero-coupon swap. This swap is executed between two counterparties at time zero and has only one cash ow which occurs at the maturity dateof the swap. For example, imagine that at time zero, the ten-year zero-couponination swap rate is 300 basis points. As is standard with swaps, there are nocash ows at time zero when the swap is executed. At the maturity date of theswap in ten years, the counterparties to the ination swap exchange a cash owof (1 + .0300)10 IT , where IT is the relative change in the price level betweennow and the maturity date of the swap. The timing and index lag constructionof the ination index used in an ination swap are chosen to match precisely thedenitions applied to TIPS issues.The zero-coupon ination swap rate data used in this study are collectedfrom the Bloomberg system. Ination swap data for maturities ranging from oneto 30 years are available for the period from July 23, 2004 to October 5, 2012.Data for ination swaps with maturities of 40 and 50 years are available beginninglater in the sample. Recent research by Fleming and Sporn (2012) concludes thatthe ination swap market appears reasonably liquid and transparent despitethe markets over-the-counter nature and modest activity. They estimate that6
realized bid-ask spreads for customers in the ination swap market are on theorder of three basis points. Conversations with ination swap traders conrmthat these instruments are fairly liquid with typical bid-ask spreads consistentwith those reported by Fleming and Sporn. To guard against any possibility ofusing illiquid or stale prices in the sample, however, we only include an inationswap rate when that rate has changed from the previous day. Table 1 presentssummary statistics for the ination swap rates.As shown, average ination swap rates range from 1.758 percent for oneyear ination swaps, to a high of 2.903 percent for 30-year ination swaps. Thevolatility of ination swap rates is generally declining in the maturity of thecontracts. The dampened volatility of long-horizon ination swap rates suggeststhat the market may view ination as being strongly mean-reverting in nature.Table 1 also shows that there is evidence of deationary concerns during thesample period. For example, the one-year swap rate reached a minimum of4.545 percent during the height of the 2008 nancial crisis amid serious fearsabout the U.S. economy sliding into a full-edged depression/deation scenario.3.2 Ination OptionsThe ination options market had its inception in 2002 with the introduction ofcaps and oors on the realized ination rate. While trading in ination optionswas initially muted, the market gained considerable momentum as the nancialcrisis emerged and total interbank trading volume reached $100 billion.6 Whilethe ination options market is not yet as liquid as, say, the stock index optionsmarket, the market is suciently liquid that active quotations for ination capand oor prices for a wide range of strikes have been readily available in themarket since 2009.In Europe and the United Kingdom, insurance companies are among themost active participants in the ination derivatives market. In particular, muchof the demand in ten-year and 30-year zero percent oors is due to pension fundstrying to protect long ination swaps positions. In contrast, insurance companiesand nancial institutions that need to hedge ination risk are the most activeparticipants on the demand side in the U.S. ination options market.The most actively traded ination options are year-on-year and zero-couponination options. Year-on-year ination options are caps and oors that pay thedierence between a strike rate and annual ination on an annual basis. Zerocoupon options, in contrast, pay only one cash ow at the expiration date ofthe contract based on the cumulative ination from inception to the expiration6
For a discussion of the ination derivatives markets, see Jarrow and Yildirim(2003), Mercurio (2005), Kerko (2005), and Barclays Capital (2010).7
date. To illustrate, assume that the realized ination rate over the next ten yearswas two percent. A ten-year zero-coupon cap struck at one percent would paya cash ow of max(0, 1.020010 1.010010) at its expiration date. In this paper,we focus on zero-coupon ination options since their cash ows parallel those ofzero-coupon ination swaps.As with ination swaps, we collect ination cap and oor data from theBloomberg system. Data are available for the period from October 5, 2009 toOctober 5, 2012 for strikes ranging from negative two percent to six percent inincrements of 50 basis points. We check the quality of the data by insuring thatthe cap and oor prices included satisfy standard option pricing bounds such asthose described in Merton (1973) including put-call parity, monotonicity, intrinsicvalue lower bounds, strike price monotonicity, slope, and convexity relations. Toprovide some perspective on the data, Table 2 provides summary statistics forcall and put prices for selected strikes.As illustrated, ination cap and oor prices are quoted in basis points, orequivalently, as cents per $100 notional. Interestingly, ination option prices arenot always monotonically increasing in maturity. This may seem counterintuitivegiven standard option pricing theory, but is it important to recognize that theination rate is a macro variable rather than the price of a traded asset.7 Formost maturities, we have about 25 separate cap and oor prices with strikesvarying from negative two percent to six percent from which to estimate therisk-neutral density of ination.4. MODELING INFLATIONIn this section, we present the continuous time model used to describe the dynamics of ination under both the objective and risk-neutral measures. We alsodescribe the application of the model to the valuation of ination swaps andoptions.4.1 The Ination ModelWe begin with a few key items of notation. For notational simplicity, we willassume that all ination contracts are valued as of time zero and that the initialprice level at time zero is normalized to one.8 Furthermore, time zero values ofstate variables are unsubscripted. Let It denote the relative change in the price7
We observe that similar nonmonotonic behavior occurs with interest rate optionssuch as interest rate caps, oors, and swaptions; see Longsta, Santa-Clara, andSchwartz (2001).8
Since the initial price level equals one, we will further simplify notation by not8
dI = I X dt + I V dZI ,dX = (Y X) dt + dZX ,
dY = ( Y ) dt + dZY ,
dV = dt + s dZV .
dI = I X dt + I
V dZI ,
dX = (Y X) dt + dZX ,
dY = ( Y ) dt + dZY ,dV = dt + s dZV ,
where the parameters , , and that now appear in the system of equations allow for the possibility that the market incorporates time-varying ination-relatedrisk premia into asset prices. In particular, the model allows the risk-neutral distributions of X, Y , and I to dier from the corresponding distributions underthe objective measure. Thus, the model permits a fairly general structure forination risk premia. On the other hand, the model assumes that variation inthe state variable V is not priced in the market. This assumption appears to be amodest one and has the important advantage of making the analysis much moretractable. We acknowledge, however, that more general types of risk premiumspecications are possible.Finally, let rt denote the nominal instantaneous riskless interest rate. Wecan express this rate as rt = Rt +Xt where Rt is the real riskless interest rate andXt is expected ination. For tractability, we also assume that Rt is uncorrelatedwith the other state variables It , Xt , Yt , and Vt .4.2 Valuing Ination SwapsFrom the earlier discussion, an ination swap pays a single cash ow ofIT F at maturity date T , where F is the ination swap price set at initiationof the contract at time zero. Note that F = (1 + f)T where f is the inationswap rate. The Appendix shows that the ination swap price can be expressedin closed form asF (X, Y, T ) = exp (A(T ) B(T )X C(T )Y ) ,where21T2T(1 eT (1 e)+)
(T (1 eT ) +(1 e2T ))
( )
111(1 e(+)T )) (T (1 eT ) (1 eT ) +
+212(1 e2T )) 2 (T (1 eT ) ++ 222 ( )
111(1 eT ) (1 eT ) +(1 e(+)T ))
21(1 e2T ))+ 2 (T (1 eT ) +
(( )T (1 eT ) + (1 eT )),( )
(1 eT ),
(1 eT ) (1 eT ),( )
where D(T ) is the price of a riskless discount bond with maturity T , and theexpectation is taken with respect to the adjusted risk-neutral measure Q forination dened by the following dynamics,
dI = I X dt + I V dZI ,dX = ( (Y X) + 2 B(T t) + C(T t)) dt + dZX ,
The adjustment to the risk-neutral measure arises because the ination rateis correlated with the riskless interest rate and allows us to discount the optioncash ow outside of the expectation.10 This adjusted measure has been referred to10
See Jamshidian (1989) and Longsta (1990) for a discussion of this adjustmentto the risk-neutral measure.11
ln IT =
Xs ds
T1 TVds+Vs dZV .s02 0
ln IT = uT + wT ,ln IT = vT + wT ,
h(z) = ,2aq1/2 Kq (d a2 b2 ) ( d2 + (z c)2 )1/2q
best t to the set of cap and oor prices, while requiring that the model exactlymatch the corresponding ination swap rate.11 With this latter condition, thereare essentially four free parameters that can be optimized to t the cross-sectionof option prices. To value the options, we numerically integrate the product ofthe option payo and the density. The optimization algorithm solves for theparameter vector that minimizes the sum of squared pricing errors, where eachoption receives equal weight. We then repeat this process for each day in thesample period and for each horizon of option expirations, one, two, three, ve,seven, ten, 20, and 30 years.12 Although not shown, the algorithm is able to tthe ination cap and oor prices very accurately. In particular, the model pricesare typically within several percent of the corresponding market prices and wouldlikely be well within the actual bid-ask spreads for these options.5.2 Maximum Likelihood EstimationAs shown in Equation (9), the closed-form solution for ination swap pricesdepends only on the two state variables X and Y that drive expected ination.An important advantage of this feature is that it allows us to use standard aneterm structure modeling techniques to estimate X and Y and their parametersunder both the objective and risk-neutral measures. In doing this, we applythe maximum likelihood approach of Due and Singleton (1997) to the termstructure of ination swaps for maturities ranging from one to 30 years (but notfor the 40 and 50 year maturities).Specically, we assume that the two-year and 30-year ination swap ratesare measured without error. Thus, given a parameter vector , substitutingthese maturities into the log of the ination swap expression in Equation (9)results in a system of two linear equations
in the two state variables X and Y . This means that X and Y can be expressed as explicit linear functions of the two ination swap prices F (X, Y, 2)11
This latter condition implicitly requires that the moment generating functionfor the density be nite. This requirement places some mild restrictions on theparameters which are incorporated in the tting algorithm.12We solve for the density of each option expiration horizon separately since themodel allows for a general ination specication rather than a specic representation. Thus, we place no a priori restrictions on the term structure of risk-neutraldensities possible at a specic date.14
and F (X, Y, 30). Let J denote the Jacobian of the mapping from the two swaprates into X and Y .At time t, we can now solve for the ination swap rate implied by the modelfor any maturity from the values of Xt and Yt and the parameter vector .Let
t denote the vector of dierences between the market value and the modelvalue of the ination swaps for the other maturities implied by Xt , Yt , and theparameter vector . Under the assumption that
t is conditionally multivariatenormally distributed with mean vector zero and a diagonal covariance matrix with main diagonal values vj (where the subscripts denote the maturities ofthe corresponding ination swaps), the log of the joint likelihood function LLKtof the two-year and 30-year ination swap prices and
t+t conditional on theination swap term structure at time t is given by(Xt+t Xt )21= ln(2X Y 1 XY ) 22(1 2XY )X 2(Xt+ Xt )(Yt+t Yt )(Yt+t Yt )+, 2XYXYY2
2 2 ofwhere the conditional moments Xt , Yt , X , Y , and XY = XY / XYXt+t and Yt+t are given in the Appendix. The total log likelihood function isgiven by summing LLKt over all values of t.We maximize the log likelihood function over the 22-dimensional parametervector = {, , , , , , , , , v1 , v3 , v4 , v5 , v6 , v7 , v8 , v9 , v10 , v12 , v15 , v20 , v25 }using a standard quasi-Newton algorithm with a nite dierence gradient. As arobustness check that the algorithm achieves the global maximum, we repeat theestimation using a variety of dierent starting values for the parameter vector.Table 3 reports the maximum likelihood estimates of the parameters and theirasymptotic standard errors. The tting errors from the estimation are all relatively small with the typical standard deviation ranging from roughly six to toten basis points, depending on maturity.5.3 Solving for the Objective DistributionIn solving for the objective distribution of ination, we follow Heston (1993),Due, Pan, and Singleton (2000), and many others by inverting the characteristic function of the objective distribution. Let (x; ) denote the characteristicfunction for the density function h(x),(x; ) =
Recall from the earlier discussion that the log of the relative price level can beexpressed as uT + wT under the risk-neutral measure, and as vT + wT under theobjective measure, where wT is independent of uT and vT . Using the propertiesof characteristic functions, it is easily shown that
(vT + wT ; ) =
(uT + wT ; ) (vT ; ).(uT ; )
Thus, given the densities for uT and vT , once we can identify the characteristicfunction of the price uT +wT under the risk-neutral measure, we can immediatelysolve for the characteristic function of the log of the relative price level uT + wTunder the objective measure. Given this characteristic function (vT + wT ), wecan recover the cumulative density function (ln(IT )/T ) of the realized inationrate using the Gil-Pelaez inversion integral,11(z) = 2
Im[eiz (vT + wT ; )]d,
where Im[ ] represents the imaginary component of the complex-valued argument. Once the cumulative distribution function for the ination rate z =ln(IT )/T is determined, the cumulative distribution function for the relativeprice level IT is obtained by a simple change of variables.6. THE DISTRIBUTION OF INFLATIONAs a preliminary to the analysis of deation risk, it is useful to rst present theempirical results for ination risk premia, expected ination, ination volatility,and the higher moments of ination.6.1 Ination DensitiesTo provide some perspective on the nature of the ination density under theobjective measure, Figure 1 plots the time series of densities of ination forseveral horizons. As shown, there is considerable variation in the shape of theination distribution for the shorter horizons. In contrast, the distribution ofination for longer horizons is more stable over time.6.2 Ination Risk PremiaWe measure the ination risk premium by simply taking the dierence betweenthe tted ination swap and expected ination rates. This is the way in whichmany market participants dene ination risk premia. When the ination swap16
rate is higher than expected ination, the ination risk premium is positive,and vice versa. There is no compelling theoretical reason why the ination riskpremium could not be negative in sign. In this case, the risk premium might wellbe viewed as a deation risk premium.Table 4 presents summary statistics for the average ination risk premiafor horizons ranging from one year to 30 years. Figure 2 plots the time seriesof ination risk premia for a number of horizons. As shown, the average riskpremia are slightly negative for horizons out to ve years, but are positive forlonger horizons and reach a value of about 30 basis points at the 30-year horizon.The ination risk premia vary signicantly through time, although the volatilityof ination risk premia for longer horizons is slightly higher than for shorterhorizons.These ination risk premia estimates are broadly consistent with previousestimates obtained using alternative approaches by other researchers. For example, Haubrich, Pennachi, and Ritchken (2012) estimate the ten-year and 30-yearination risk premia to be 51 and 101 basis points, respectively. Buraschi andJiltsov (2005) and Campbell and Viceira (2001) estimate the ten-year inationrisk premium to be 70 and 110 basis points, respectively. Ang, Bekaert, and Wei(2008) estimate the ve-year ination risk premium to be 114 basis points. Inaddition, the fact that all of the estimated risk premia take negative values atsome point during the sample period is consistent with the ndings of Campbell,Shiller, and Viceira (2009), Bekaert and Wang (2010), Campbell, Sunderam, andViceira (2013), and others.Finding that ination risk premia change signs through time is an intriguingresult. Intuitively, one way to think about why the risk premium could changesigns is in terms of the link between ination and the macroeconomy. For example, when ination risk is perceived to be counter-cyclical, the market priceof ination risk should be positive. This is the regime in which aggregate supplyshocks (e.g. oil shocks) account for most of the variation in output: high inationcoincides with low output growth. Thus, investors pay an insurance premiumwhen buying ination protection in the market.But this line of reasoning suggests that the sign of the ination risk premiumcould be negative when ination risk was perceived to have become pro-cyclical.In that regime, ination innovations tend to be positive when the average investors marginal utility is high. Hence, a nominal bond provides insuranceagainst bad states of the world, whereas a real bond does not. In this case,investors receive an insurance premium when buying ination hedges. When aggregate demand shocks account for most of the variation in output growth, thenwe would expect to see pro-cyclical ination: high ination coincides with highoutput growth.17
The variation in the sign of the ination risk premium also raises a number of interesting questions about optimal monetary policy. Since the Treasuryseems constrained to issue mostly nominal bonds, it strictly prefers lower ination risk premia. Higher ination risk premia increase the costs of governmentdebt nancing, funded by distortionary taxes. In particular, when ination riskpremia are negative, issuing nominal bonds is very appealing. However, since thelevel of expected ination is related to the size of the ination risk premium, thegovernment may have the ability to inuence the risk premium to some degreeby how it targets ination.6.3 Expected InationTo solve for the expected ination rate for each horizon, we use the inationswap rates observed in the market and adjust them by the ination risk premiumimplied by the tted model. Table 5 presents summary statistics for the expectedination rates for the various horizons.The results indicate that the average term structure of ination expectationsis monotonically increasing during the 20042012 sample period. The averageone-year expected ination rate is 1.776 percent, while the average 30-year expected ination rate is 2.597 percent. The table also shows that there is timevariation in expected ination, although the variation is surprisingly small forlonger horizons. In particular, the standard deviation of expected ination rangesfrom 1.348 percent for the one-year horizon to less than 0.20 percent for horizonsof ten years or longer. To illustrate the time variation in expected ination moreclearly, Figure 3 plots the expected ination estimates for the ve-year, ten-year,and 30-year horizons.It is also interesting to contrast these market-implied forecasts of inationwith forecasts provided by major ination surveys. As discussed by Ang, Bekaert,and Wei (2007), these surveys of ination tend to be more accurate than thosebased on standard econometric models and are widely used by market practitioners. Furthermore, these ination surveys have also been incorporated intoa number of important academic studies of ination such as Fama and Gibbons (2004), Chernov and Mueller (2012) and Haubrich, Pennachi, and Ritchken(2012).We obtain ination expectations from four surveys: the University of Michigan Survey of Consumers, the Philadelphia Federal Reserve Bank Survey of Professional Forecasters (SPF), the Livingston Survey, and the survey of market participants conducted by Bloomberg. The sample period for the forecasts matchesthat for the ination swap data in the study. The Appendix provides the background information and details about how these surveys are conducted.
Table 6 reports the average values of the various surveys during the sampleperiod and the corresponding average values for the market-implied forecasts.These averages are computed using the month-end values for the months inwhich surveys are released. Thus, monthly averages are compared with monthlyaverages, quarterly averages with quarterly averages, etc. As shown, the averagemarket-implied forecasts of ination tend to be a little lower than the surveyaverages for shorter horizons. The market-implied forecasts, however, closelyparallel those from the surveys for longer horizons. While it would be interestingto compare the relative accuracy of the market-implied and survey forecasts, oursample is too short to do this rigorously.6.4 Ination Volatility and Higher MomentsTable 7 reports the average values of the estimated volatility, skewness, and excesskurtosis of the continuously-compounded ination rate for horizons ranging fromone year to 30 years. The average ination volatility estimates range from ahigh of about 2.258 percent at the two-year horizon to a low of about 0.693percent at the 30-year horizon. The dampened volatility at the longer horizonsis consistent with a scenario in which ination is anticipated to follow a meanreverting process.The distribution of ination is typically negatively skewed for all horizons.The negative skewness is particularly pronounced for horizons of less than tenyears, but is still evident in the distribution of ination over a 30-year horizon.The median excess kurtosis coecients are all positive (with the exception of the30-year horizon), indicating that the distribution of ination has heavier tailsthan a normal distribution.7. DEFLATION RISKWe turn now to the central issue of measuring the risk of deation implied bymarket prices and studying the properties of deation risk. First, we presentdescriptive statistics for the implied deation risk. We then examine how themarket prices the tail risk of deation and contrast the results with those foundin other markets.7.1 How Large is the Risk of Deation?Having solved for the characteristic function for the ination distribution, wecan apply standard inversion techniques to solve for the cumulative distributionfunction for ination. In turn, we can then directly compute the probabilitythat the average realized ination rate over a specic horizon is less than zero,which represents the risk of deation. Table 8 provides summary statistics for19
link between deation risk and major nancial risk more formally later in thepaper.Although not shown, we also calculate the partial moment in which we takethe expected value of ination conditional on the ination rate being less thanor equal to zero. This partial moment provides a measure of the severity ofa deation, conditional on deation occuring over some horizon. For example,nding that this partial moment was only slightly negative would argue that adeationary episode was likely to be less severe, while the opposite would be truefor a more negative value of this partial moment. The results indicate that theexpected severity of a deation is typically very substantial with these conditionalmoments increasing from about 1.60 percent for a one-year deation, to 1.85for a ve-year horizon, and then decreasing to 1.15 percent for a 20-year horizon.On average, the expected value of deation over all of the horizons is 1.56percent. Note that a deation of 1.56 percent per year would translate intoa decline in the price level of 7.6 percent over a ve-year period, 15.5 percentover a ten-year period, and 27.0 percent over a 20-year period. These wouldrepresent protracted deationary episodes comparable in severity to many ofthose experienced historically in the U.S.7.2 Pricing Deation Tail RiskAlthough we have solved for the ination risk premium embedded in inationswaps earlier in the paper, it is also interesting to examine how the market pricesthe risk that the tail event of a deation occurs. This analysis can provide insightinto how nancial market participants view the risk of events that may happeninfrequently, but which may have catastrophic implications.A number of these types of tail risks have been previously studied in theliterature. For example, researchers have investigated the pricing of catastrophicinsurance losses such as those caused by hurricanes or earthquakes. Froot (2001)nds that the ratio of insurance premia to expected losses in the market forcatastrophic reinsurance ranges from about two to seven during the 1989 to 1998period. Lane and Mahul (2008) estimate that the pricing of catastrophic risk ina sample of 250 catastrophe bonds is about 2.69 times the actual expected lossover the long term. Garmaise and Moskowitz (2009) and Ibragimov, Jaee, andWalden (2009) oer both empirical and theoretical evidence that the extremeleft tail catastrophic risk can be signicantly priced in the market.The default of a corporate bond is also an example of an event that is relatively rare for a specic rm, but which would result in an extremely negativeoutcome for bondholders of the defaulting rm. The pricing of default risk hasbeen considered in many recent papers. For example, Giesecke, Longsta, Schaefer, and Strebulaev (2011) study the pricing of corporate bond default risk and21
nd that the ratio of corporate credit spreads to their actuarial expected loss is2.04 over a 150-year period. Similarly, Driessen (2005) and Berndt, Due, Douglas, Ferguson, and Schranz (2005) estimate ratios using data for recent periodsthat range in value from about 1.8 to 2.8.Following along the lines of this literature, we solve for the ratio of the riskneutral probability of deation to the objective probability of deation. Thisratio provides a simple measure of how the market prices the tail risk of deationand has the advantage of being directly comparable to the ratios discussed above.Table 9 presents the means and medians for the ratios for the various horizons. As shown, the mean and median ratios range from between one and two toslightly higher than ve. The overall average of the ratios is 3.321 and the overallmedian of the ratios is about 3.166. These values are in the same ballpark asthose for the dierent types of tail risk discussed above. These ratios all indicatethat the market is deeply concerned about nancial and economic tail risks thatmay be dicult to diversify or may be strongly systematic in nature.8. WHAT DRIVES DEFLATION RISK?A key advantage of our approach is that by extracting the markets assessment ofthe objective probability of deation, we can then examine the relation betweenthese probabilities and other nancial and macroeconomic factors. In particular,we can study the relation between the tail risk of deation and other types oftail risk that may be present in the markets.In doing this, we will focus on four broad categories of tail risk that havebeen extensively discussed in the literature. Specically, we will consider the linksbetween deation risk and systemic nancial system risk, collateral revaluationrisk, sovereign default risk, and business cycle risk.The link between systemic risk in the nancial system and major economiccrisis is well established in many important papers including Bernanke (1983),Bernanke, Gertler, and Gilchrist (1996), and others. Systemic risk in the nancialsystem is widely viewed as having played a central role in the recent globalnancial crisis and represents a motivating force behind major regulatory reformssuch as the Dodd-Frank Act. We use a number of measures of systemic risk inthe analysis.First, we use a measure of the ight-to-liquidity risk in the market whichis computed as the spread between a one-year zero-coupon Refcorp bond and acorresponding maturity zero-coupon Treasury bond. This variable is introducedin Longsta (2004) as a measure of the premium that market participants place22
on Treasuries because of their role as the safest asset in the nancial marketsduring episodes when investors fear that massive losses will occur in less liquidmarkets. We obtain the data for the ight-to-liquidity spread from the Bloombergsystem.Second, we use data on the pricing of super senior tranches on a basket ofcorporate debt to measure the risk of a major systemic collapse in the creditmarkets. Specically, we collect data on the points-up-front pricing on the veyear 10-15 percent tranche on the CDX IG index. This index is computed as anaverage of the ve-year CDS spreads for 125 U.S. rms with investment graderatings. The 10-15 percent tranche would only experience losses if the totalcredit losses on the index exceeded 10 percent of the total notional of an equallyweighted basket of the underlying debt obligations of these rms. As shown byCoval, Jurek, and Staord (2009) and Giesecke, Longsta, Schaefer, and Strebulaev (2011), a major meltdown that would produce losses of this magnitude in aportfolio of investment grade corporate debt would be a very rare tail event. Thepoints-up-front price for this tranche represents the market price to compensateinvestors for taking the risk of this extreme scenario.14 We obtain data on thepricing of the super senior tranche from the Bloomberg system.Third, we use the spread between the one-year Libor rate and a one-yearTreasury bond as a proxy for the systemic credit and liquidity risk embedded inthe Libor rate. The data are from the Bloomberg system.Fourth, we use the ve-year swap spread as a measure of the systemic creditand liquidity stresses on the nancial system. As discussed by Due and Singleton (1997), Liu, Longsta, and Mandell (2006), and others, the swap spreadreects dierences in the relative liquidity and credit risk of the nancial sectorand the Treasury. We obtain ve-year swap spread data from the Bloombergsystem.We also considered a number of other measures of systemic risk such as theaverage CDS spread for both major U.S. and non-U.S. banks and nancial rms.These measures, however, were highly correlated with the other measures suchas swap spreads and provided little incremental information.Recent economic theory has emphasized the role that the value of collateralplays in propagating economic downturns. Key examples include Kiyotaki andMoore (1997) who show that declines in asset values can lead to contractionsin the amount of credit available in the market which, in turn, can lead tofurther rounds of declines in asset values. Bernanke and Gertler (1995) describe14
For a description of the CDX index tranche markets and the pricing of CDOtranches, see Longsta and Rajan (2008) and Longsta and Myers (2013).23
similar interactions between declines in the value of assets that serve as collateraland severe economic downturns. Collateral revaluation risk, or the risk of abroad decline in the market value of leveraged assets, played a major role inthe Great Depression as the sharp declines in the values of stock and corporatebonds triggered waves of defaults among both speculators and banks. A similarmechanism was present in the recent nancial crisis as sharp declines in real estatevalues led to massive defaults by underwater mortgagors. In the context ofthis study, we explore the relation between deation probabilities and valuationsin several major asset classes that may represent important sources of collateralin the credit markets: stocks and bonds. In addition, we also include measuresof the volatility of these asset classes since these measures provide informationabout the risk that large downward revaluations in these forms of collateral mayoccur.The rst of these proxies for collateral revaluation risk is the VIX index ofimplied volatility for options on the S&P 500. This well-known index is oftentermed the fear index in the nancial press since it reects the markets assessment of the risk of a large downward movement in the stock market. We collectVIX data from the Bloomberg system.The second measure is the Merrill Lynch MOVE index of implied volatilities for options on Treasury bonds. This index is essentially the xed incomecounterpart of the VIX index. This index also captures the markets views ofthe likelihood of a large change in the prices of Treasury bonds, which are widelyused as collateral for a broad variety of credit transactions. This data is alsoobtained from the Bloomberg system.The third measure is simply the time series of daily returns on the S&P 500index (price changes only). This return series reects changes in the value of oneof the largest potential sources of collateral in the macroeconomy. We computethese returns from S&P 500 index values reported in the Bloomberg system.The fourth measure is the spread between the yield for the Moodys Baarated index of corporate bonds and the yield on ve-year Treasury bonds. Variation in this credit spread over time reects changes in the markets assessmentof default risk in the economy as well as the pricing of credit risk. We collectdata on the Baa-Treasury spread from the Bloomberg system.Another major type of economic tail risk stems from the risk that a sovereigndefaults on its debt. As documented by Reinhart and Rogo (2009) and manyothers, sovereign defaults tend to be associated with severe economic crisis scenarios.As a measure of the tail risk of a sovereign default by the U.S., we include24
in the analysis the time series of sovereign CDS spreads on the U.S. Treasury.Ang and Longsta (2012) show that the U.S. CDS spread reects variation inthe valuation of major sources of tax revenue for the U.S. such as capital gainson stocks and bonds. This data is also obtained from the Bloomberg system.Finally, to capture the eect of traditional types of business cycle risk oreconomic downturn risk, we also include a number of key macroeconomic variables that can be measured at a monthly frequency. In particular, we include themonthly percentage change in industrial production as reported by the Bureauof Economic Analysis, the monthly change in the national unemployment rateas reported by the Bureau of Labor Statistics, and the change in the ConsumerCondence Index reported by the University of Michigan. The link between thebusiness cycle and its eects on output and employment are well establishedin the macroeconomic literature and forms the basis of many classical theoriesincluding the Phillips curve.Since these measures of systemic, collateral, and sovereign default risk areall available on a daily basis, we begin our analysis by regressing daily changes inthe deation probabilities on daily changes in these variables (the macroeconomicvariables, which are only observed monthly, will be included in later regressions).In doing this, it is important to note that while these variables were chosen as ameasure of a specic type of tail risk, most of these variables may actually reectmore than one type of tail risk. Thus, the eects of the variables in the regressionshould be interpreted carefully since the dierent types of tail risk need not bemutually exclusive.Table 10 presents summary statistics for the regression results. The resultsindicate that the variables proxying for systemic risk are often statistically signicant for a number of the horizons. In particular, the coecient for the supersenior tranche price is signicant for four of the horizons. The sign of this coecient is uniformly positive in sign for all but the longest horizons, which isclearly consistent with the intuition that an increase in the extreme type of tailrisk reected in the tranche price would be associated with an economic meltdown in which deation occured. The Libor spread is positive and signicant forthe two shortest horizons. The positive sign of this eect is also consistent withour intuition about the eects of a systemic nancial crisis on the macroeconomy.The ve-year swap spread has some of the strongest eects in the regression. Inparticular, it is signicant for ve of the ten horizons. Interestingly, the signsof the signicant coecients are all positive, with the exception of the shortest horizon. This is again consistent with an economic scenario in which stressin the nancial sector leads to an increase in the perceived risk of an adversemacroeconomic shock in which price levels decline.Surprisingly, Table 10 shows that neither of the two volatility variables has25
much explanatory power for changes in deation risk. In contrast, both S&P500 index returns and changes in the Baa credit spread are often signicant. Forexample, the stock market return is signicant for the one-year, two-year, and30-year horizons. In each of these three cases, the sign is negative, indicatingthat an increase in the stock market reduces fears about deation. This is againvery intuitive and completely consistent with a simple collateral revaluation interpretation. The Baa credit spread is also signicant for three of the horizons.The signs of the signicant coecients for the one-year and two-year horizonsare both positive, indicating that deation risk increases as credit fears in theeconomy increase. It is also interesting to note that the sign of the coecients forthis variable become negative for all longer horizons. Thus, the long-run eectsof increased credit risk on deation risk are somewhat counterintuitive.Finally, Table 10 shows that the eect of an increase in U.S. CDS spreadson deation risk is relatively limited. The coecient for changes in U.S. CDSspreads is only signicant for the one-year and 15-year horizons. In addition, thesigns of these two signicant coecients dier from each other.The overall R2 s from the regressions are fairly modest, ranging from justunder two percent to more than nine percent. Note, however, that these results are based on daily changes in these variables. Thus, given the challengesin measuring tail risks, the explanatory power of these regressions is far fromnegligible.Turning now to regressions in which we include the macroeconomic variables,we observe that since our sample period is relatively short, it is important to usea parsimonious specication. Accordingly, we regress monthly changes in thedeation probabilities (using the last deation probability for each month) on aselected set of variables. Specically, as proxies for systemic risk, we include onlymonthly changes in the super senior prices and the ve-year swap spread. As theproxy for collateral revaluation risk, we include only the change in the Baa creditspread. We then include the monthly percentage change in industrial production,the change in the unemployment rate, and the change in the Michigan ConsumerCondence Index. The regression results for horizons ranging from one to 20years (there are too few observations for the 30-year horizon) are summarized inTable 11.The results in Table 11 are consistent with those reported in Table 10. Inparticular, several of the proxies for systemic and collateral revaluation tail riskare again signicant. The coecient for the super senior tranche variable issignicant for three of the horizons, with those for the one-year and two-yearhorizons again having a positive sign. The coecient for the Baa credit spreadis signicant for four of the horizons. All four of these coecients are positive insign, indicating that increases in credit risk are associated with an increased risk26
of deation.The results for the macroeconomic variables are somewhat surprising. Farfrom being strongly related to deation risk, none of the coecients for changesin industrial production or the consumer condence index are signicant. Theonly macroeconomic variable that is signicantly related to changes in deationrisk is the change in the unemployment rate which is signicant and positive forthree of the horizons. The positive sign of these coecients is intuitive since itindicates that deation risk increases as the unemployment rate increases. Thisis also consistent with classical macroeconomic theory about the relation betweenprice levels and unemployment such as the Phillips curve.In summary, the empirical results indicate that there is a strong relationbetween tail risk in nancial markets and the risk of deation. In particular,a number of the proxies for systemic nancial risk and the value of potentiallycollateralizable nancial assets are signicantly linked to deation risk. Theseresults underscore the importance of understanding the role that the nancialsector plays in economic downturns such as the recent nancial crisis that beganin the subprime structured credit markets. In contrast, these results suggest thatmore traditional macroeconomic variables such as industrial production may playless of a role in the risk of economic tail events such as a deationary spiral.9. INFLATION RISKAlthough the focus of this paper is on deation risk, it is straightforward toextend the analysis to other aspects of the distribution of ination. As one lastillustration of this, we compute the probabilities that the ination rate exceedsvalues of four, ve, and six percent using the techniques described earlier. Table12 reports summary statistics for these probabilities.As shown, the market-implied probabilities of experiencing signicant ination are uncomfortably large. Specically, the average probability of inationexceeding four percent is less that 15 percent for the two shortest horizons, butincreases rapidly to nearly 30 percent for horizons ranging from ten to 30 years.Figure 5 plots the time series of probabilities that ination exceeds four percentfor several horizons. These plots also show that the probability of ination inthe long run appears substantially higher than in the short turn. This is exactlythe opposite from the risk of deation which tends to be higher in the short run.Table 12 also shows that the average probability of an ination rate in excessof ve percent is substantial. An average ination rate of ve percent or moreover a period of decades would rival any inationary scenario experienced bythe U.S. during the past 200 years. Finally, the results show that the market27
cial and macroeconomic tail risks. We study the relation between deation riskand a number of measures of systemic nancial risk, collateral revaluation risk,sovereign credit risk, and business cycle risk. We nd that there is a signicantrelation between deation risk and measures capturing stress in the nancial system and credit risk in the economy. These results support the view that the riskof economic shocks severe enough to result in deation is fundamentally relatedto the risk of major systemic shocks in the nancial markets.
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Friedman, Milton and Anna Schwartz, 1963, A Monetary History of the UnitedStates, 18671960. Princeton University Press, Princeton.Froot, Kenneth A., The Market for Catastrophe Risk: A Clinical Examination,Journal of Financial Economics 60, 529-571.Gabaix, Xavier, 2012, Variable Rare Disasters: An Exactly Solved Frameworkfor Ten Puzzles in Macro-Finance, The Quarterly Journal of Economics 127,645-700.Garmaise, Mark J., and Tobias Moskowitz, 2009, Catastrophic Risk and CreditMarkets, Journal of Finance 64, 657-707.Ghysels, Eric, and Fangfang Wang, 2011, Some Useful Densities for Risk Management and their Properties, Working paper, University of North Carolina atChapel Hill.Giesecke, Kay, Francis A. Longsta, Stephen Schaefer, and Ilya Strebulaev, 2011,Corporate Bond Default Risk: A 150-Year Perspective, Journal of FinancialEconomics 102, 233-250.Gourio, Francois, 2008, Disasters and Recoveries, American Economic Review98, 68-73.Grishchenko, Olesya V., and Jing-zhi Huang, 2012, Ination Risk Premium: Evidence from the TIPS Market, Finance and Economics Discussion Series, Divisionsof Research & Statistics and Monetary Aairs, Federal Reserve Board.Gurkaynak, Refet, Brian Sack, and Jonathan H. Wright, 2010, The TIPS YieldCurve and Ination Compensation, American Economic Journal: Macroeconomics 2, 70-92.Gurkaynak, Refet, and Jonathan H. Wright, 2010, Macroeconomics and the TermStructure, Journal of Economic Literature, forthcoming.Hamilton, James D., 1985, Uncovering Financial Market Expectations of Ination, Journal of Political Economy 93, 1224-1241.Hamilton, James D., 1992, Was the Deation During the Great Depression Anticipated? Evidence from the Commodity Futures Market, American EconomicReview 82, 157-178.Haubrich, Joseph, George Pennacchi, and Peter Ritchken, 2012, Ination Expectations, Real Rates, and Risk Premia: Evidence from Ination Swaps, Review ofFinancial Studies 25, 1588-1629.Heston, Steven L., 1993, A Closed-Form Solution for Options with Stochastic33
Longsta, Francis A., and Arvind Rajan, 2008, An Empirical Analysis of thePricing of Collateralized Debt Obligations, Journal of Finance 63, 509-563.Longsta, Francis A., Pedro Santa-Clara, and Eduardo S. Schwartz, 2001, TheRelative Valuation of Caps and Swaptions: Theory and Empirical Evidence, TheJournal of Finance 56, 2067Mercurio, Fabio, 2005, Pricing Ination-Indexed Derivatives, Quantitative Finance 5, 289-302.Merton, Robert C., 1973, Theory of Rational Option Pricing, Bell Journal ofEconomics and Management Science 4, no. 1., 141-183.North, Douglass C., 1961, The Economic Growth of the United States, 17901860, Prentice Hall.Pueger, Carolin E., and Luis M. Viceira, 2011a, Ination Indexed Bonds and theExpectations Hypothesis, Annual Review of Financial Economics, forthcoming.Pueger, Carolin E., and Luis M. Viceira, 2011b, An Empirical Decomposition ofRisk and Liquidity in Nominal and Ination-Indexed Government Bonds, Working paper, Harvard University.Rietz, Thomas A., 1988, The Equity Risk Premium: A Solution, Journal ofMonetary Economics 22, 117-131.Pond, Michael, and Chirag Mirani, 2011, Barclays Capital, Market StrategyAmericas, June 3, 12-14.Roll, Richard, 1996, U.S. Treasury Ination-Indexed Bonds: The Design of aNew Security, Journal of Fixed Income (December), 9-28.Roll, Richard, 2004, Empirical TIPS, Financial Analysts Journal 60 (Jan-Feb),31-53.Wachter, Jessica, 2013, Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?, Journal of Finance 68, 987-1035.Wright, Jonathan H., 2011, What Does Monetary Policy Do at the Zero LowerBound?, Working paper, Johns Hopkins University.
APPENDIXA.1. The Ination Swap Rate.From Equation (1), the relative price level index at time T can be expressed asIT = exp
TTXs ds exp 12 0 Vs ds + 0 Vs dZV (s) .
which implies
T TE Q exp 12 0 Vs ds + 0 Vs dZV (s)
,F (X, Y, T ) =TE Q exp 0 Xs ds=
.TQE exp 0 Xs ds
TLet H(X, Y, ) denote the value of the expectation E Q [exp( t Xs ds)], where = T t. Standard results imply that this expectation satises the partialdierential equation36
These three equations are readily solved by the use of an integrating factor anddirect integration. Substituting the solutions into the expression for H(X, Y, ),substituting H(X, Y, ) into Equation (A5), and then evaluating as of time zero( = T ) gives the expression for the ination swap price in Equation (9).A.2 Ination Option PricesLet C(X, Y, V, T ) denote the price at time zero of a European call option on theprice level at time T with strike K. The cash ow at the option expiration dateis max(0, IT (1 + K)T ). The present value of this cash ow can be expressedas
TE Q exp 0 rs ds max(0, IT (1 + K)T ) ,
(A12)subject to the terminal condition N(I, X, Y, V, 0) = max(0, IT (1 + K)T ). Weconjecture that the solution is of the form
(A14)From Equations (A7) through (A9) the term in brackets multiplying M is zero.Without the M term in the partial dierential equation, however, the solutionto the partial dierential equation can be expressed as
where the expectation is taken with respect to the density of IT implied by thedynamics,
V dZI ,2
dX = ( (Y X) + B( ) + C( )) dt + dZX ,dY = ( Y + 2 C( ) + B( )) dt + dZY ,
TD(T ) = E exp 0 rs ds ,
T= E Q exp 0 Rs ds exp(A(T ) + B(T )X + C(T )Y ).Q
Note that under this measure, the expected value of the price level equals theination swap price F . This follows since the cash ow from an ination swapat time T is IT F . Under the Q measure, however, the present value of this
cash ow is given by D(T )E Q [IT F ]. Since the initial value of the ination
Xt dt + wT ,
where wT represents the terms that involve Vt . Let uT denote the value of theintegral of Xt is the above expression under the objective measure P . Under P ,solving the stochastic dierential equation for Yt givesYt = Y et + (/)(1 et ) + et
dZY (s).
es Ys ds + et0
es dZX (s).
Substituting Equation (A24) into the above equation, interchanging the order ofintegration, and evaluating terms gives the following expression for Xt
(Y /)(et et )Xt = Xet + (/)(1 et ) +
tt+ /( ) 0 et es et es dZY (s) + et 0 es dZX (s). (A26)39
+(Y /)((1 eT )/ (1 eT )/)
T + /( ) 0 (1 e(T t))/ (1 e(T t) )/ dZY (t)T(A27)+ 0 (1 e(T t) )/ dZX (t).
and variance
( )2 2 2
( )
11222 2
(1 eT ) 2+( )2 2 ( )2 2(1 e2T )+2( )2 2212 222 211(1 eT ) 3 ++
( )2 2
( ) 2
2+(1 e2T ) 2+2( )2 232 ( )222 2(1 e(+)T ).+(A29)+( + )( )2( + )( )
X(1 e
)/ +
2 2 2
T (1 eT )/
2 eTeT 2 eTY + 2++
++ 2
( )( + )( + )2 ( )TT)/ (1 e)/(1 eeT 2 eTeT 2eT++
+ 2( )( + )( + )222( ) T(e 1)/ (1 eT )/eT 22 eT++2 2 ( + )( ) ( )( + ) T(A30)(e 1)/ (1 eT )/ .
112222 211+
( )2 2 2
12122 2
(1 eT ) 2+( )2 2 ( )2 2(1 e2T )+2( )2 212112 222 2
(1 eT ) 3 ++( )2 2
2+(1 e2T ) 2+2( )2 232 ( )222 2(1 e(+)T ).+(A31)+( + )( )2( + )( )
Yt = Yt et + (/)(1 et ),
2(1 e2t ),Y =(A34)212 2212t(+)t2t(1 e(1 e(1 eX =))+)( )2 2+2112(+)t2t))(1 e(1 e+ +22(A35)+(1 e2t ),2 2112t(+)t))XY =(1 e(1 e 2+
+(A36)(1 e(+)t).+A.5 The Ination SurveysThe data from the University of Michigan Survey of Consumers consist of one andve year ahead ination forecasts. The series is released at monthly frequencyand reports the median expected price change over the next twelve months andthe next ve years, respectively. A detailed description of how the survey isconducted is available at http:// www.sca.isr.umich.edu/documents.php?c=i. Incontrast to the Livingston survey and the Survey of Professional Forecasters,the participants in the University of Michigan Survey of Consumers are actualconsumers (households) and not professionals. The time between the conductof the survey and release is up to three weeks. The University of Michiganreports that a review of the estimates of ination expectations indicated thatfor comparisons over time, the median, rather than the mean, may be a morereliable measure of the central tendency of the response distribution due to thechanging inuence of extreme responses. Therefore, we use the median surveyforecasts throughout our analysis.The Philadelphia Federal Reserve Bank Survey of Professional Forecasters isconducted on a quarterly basis. The questionnaires are sent to the participantsat the end of January, at the end of April of the second quarter, at the endof July for the third quarter, and at the end of October for the fourth quarter. The survey results are published in the middle of February, May, August,and November, for the rst, second, third, and fourth quarter, respectively. Incontrast to the Livingston survey, participants in the SPF forecast changes in42
the quarterly average CPI-U levels. A detailed description of how the survey is conducted is available at http://www.philadelphiafed.org/research-anddata/real-time-center/survey-of-professionalforecasters/spf-documentation.pdf.The Livingston survey is conducted twice a year, in June and in December, usually in the middle of the month. Participants include economists from industry,government, and academia. The surveys taken in June consist of two annual average CPI forecasts: for the current year, and for the following year. The Decembersurveys include three annual average forecasts: for the current year, for the nextyear, and for the year after. The participants forecast non-seasonally-adjustedCPI level six and twelve months in the future. A detailed description of how theLivingston survey is conducted is available at: http://www.philadelphiafed.org/research-and-data/real-time-center/livingston-survey/livingston-documentation.pdf. For the Livingston surveys, there is a lag of up to four and three weeksbetween the time the survey and when the results are disseminated.Finally, Bloomberg provides one-year-ahead forecasts at the monthly frequencycompiled from more than 80 professionals. Participants include economists fromBank of America, BNP Paribas, JP Morgan, and many others. Detailed information on the composition of each forecast index can be found in the Bloombergsystem under US CPI Forecast Index.
Figure 1. Ination Densities. This gure plots the time series of inationdensities for horizons of one year (upper left), two years (upper right), ve years(lower left), and ten years (lower right).
Basis Points
Figure 2. Ination Risk Premia. This gure plots the time series of inationrisk premia for horizons of one year (upper left), ve years (upper right), ten years(lower left), and 30 years (lower right).
Figure 3. Expected Ination. This gure plots the time series of expectedination for horizons of one year (upper left), ve years (upper right), ten years(lower left), and 30 years (lower right).
Figure 4. Deation Probabilities. This gure plots the time series of deation probabilities for horizons of one year (upper left), two years (upper right),ve years (lower left), and ten years (lower right).
Figure 5. Ination Probabilities. This gure plots the time series of probabilities that ination is greater than or equal to four percent for horizons of oneyear (upper left), two years (upper right), ve years (lower left), and ten years(lower right).
Table 1Summary Statistics for Inflation Swap Rates. This table reports summary statistics for the inflation swap rates for the indicatedmaturities. Swap maturity is expressed in years. Inflation swap rates are expressed as percentages. The sample consists of daily observationsfor the period from July 23, 2004 to October 5, 2012.
Table 2Summary Statistics for Inflation Caps and Floors. This table reports the average values for inflation caps and floors for the indicatedmaturities and strikes. The average values are expressed in terms of basis points per $100 notional. Option Maturity is expressed in years.Ave. denotes the average number of caps and floors available each day from which the risk-neutral density of inflation is estimated. N denotesthe number of days for which the risk-neutral density of inflation is estimated. The sample consists of daily observations for the period fromOctober 5, 2009 to January 23, 2012.
Table 3Maximum Likelihood Estimation of the Inflation Swap Model. This table reports the maximum likelihood estimates of the parameters of the inflation swap model along with their asymptoticstandard errors. The model is estimated using daily inflation swap prices for the period from July23, 2004 to October 5, 2012.
Table 4Summary Statistics for Inflation Risk Premia. This table reports summary statistics for the estimated inflation risk premia for theindicated horizons. Horizon is expressed in years. The inflation risk premia are measured in basis points. The inflation risk premia areestimated using the period from July 23, 2004 to October 5, 2012.
Table 5Summary Statistics for Expected Inflation. This table reports summary statistics for the expected inflation rate for the indicated horizons.Horizon is expressed in years. Expected inflation rates are expressed as percentages. The sample consists of daily observations for the periodfrom July 23, 2004 to October 5, 2012.
Table 6Comparison of Survey Forecasts with Market-Implied Forecasts. This table reports theaverage values of the survey forecasts for the indicated forecast horizon along with the correspondingaverage of the market-implied expected inflation for the same horizon. The averages of the marketimplied expected inflation estimates are taken using month-end values for months in which surveysare released. Inflation forecasts are expressed as percentages. The sample period is July 2004 toSeptember 2012.
1 Year
2 Years
3 Years
5 Years
10 Years
Table 7Summary Statistics for the Volatility, Skewness, and Kurtosis of the Inflation Distribution. This table reports the average values of the standard deviation and skewness coecient, andthe median excess kurtosis coecient for the annualized ination rate for the indicated horizons.The standard deviation is expressed as a percentage. Horizon is expressed in years. The sampleconsists of daily observations for the period from October 5, 2009 to January 23, 2012.
Table 8Summary Statistics for Deflation Probabilities. This table reports summary statistics for the probability of the average inflation ratebeing below zero for the indicated horizons. Horizon is expressed in years. Probabilities are expressed as percentages. The sample consists ofdaily observations for the period from October 5, 2009 to January 23, 2012.
Table 9Summary Statistics for the Pricing of Deflation Tail Risk. This table reports the meansand medians of the ratio of the probability of deflation under the risk-neutral Q measure divided bythe probability of deflation under the actual P measure. Horizon is expressed in years. The sampleconsists of daily observations for the period from October 5, 2009 to January 23, 2012.
Table 10Results from the Regression of Daily Changes in Deflation Probabilities on Financial Tail Risk Variables. This table reportsthe results from the regression of daily changes in the deflation probabilities for the indicated horizon on the daily changes in the followingvariables: the flight-to-liquidity spread (the one-year Refcorp-Treasury yield spread), the super senior tranche price (the points-up-front pricefor the 10-15 percent CDX IG index tranche), the Libor spread (the one-year Libor-Treasury spread), the five-year swap spread, the VIX index,the Merrill Lynch MOVE index of implied Treasury volatility, the return on the S&P 500 index, the Baa spread over the five-year Treasuryrate, and the spread for a five-year CDS contract on the U.S. Treasury. Horizon is expressed in years. The t-statistics are based on theNewey-West estimator of the covariance matrix (five lags). The superscript denotes significance at the five-percent level; the superscript denotes significance at the ten-percent level. The sample consists of monthly observations for the period from October 2009 to January 2012.
Flight toLiquidity
SP 500Return
Table 11Results from the Regression of Monthly Changes in Deflation Probabilities on Financial and Macroeconomic Variables. Thistable reports the results from the regression of monthly changes in the deflation probabilities for the indicated horizons on the monthly changesin the following variables: the super senior tranche price (the points-up-front price for the 10-15 percent CDX IG index tranche), the five-yearswap spread, the Baa spread over the five-year Treasury rate, the percentage change in industrial production, the unemployment rate, and theMichigan consumer confidence index. Horizon is expressed in years. The t-statistics are based on the Newey-West estimator of the covariancematrix (two lags). The superscript denotes significance at the five-percent level; the superscript denotes significance at the ten-percentlevel. The sample consists of monthly observations for the period from October 2009 to January 2012.
Table 12Summary Statistics for the Probabilities of Inflationary Scenarios. This table reports summary statistics for the probability of theaverage inflation rate being above the indicated thresholds for the respective horizons. Horizon is expressed in years. Probabilities are expressedas percentages. The sample consists of daily observations for the period from October 5, 2009 to January 23, 2012.
Table 13Summary Statistics for the Pricing of Inflation Tail Risk. This table reports the means andmedians of the ratio of the probability of inflation exceeding five percent under the risk-neutral Qmeasure divided by the corresponding probability under the actual P measure. Horizon is expressedin years. The sample consists of daily observations for the period from October 5, 2009 to January23, 2012.


text:
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ShadowProtect Desktop
Download ShadowProtect Desktop 126.96.36.199 from our software library for free. Our built-in antivirus checked this download and rated it as virus free. The size of the latest downloadable installation package is 49.4 MB. The most popular versions of the program 4.1, 4.0 and 3.5.
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text:
Lot 16 Walker, FergusonMLS ® 2435303
- N/A Beds |
- N/A Baths |
- N/A sqft |
- Land Only |
- N/A built |
- N/A lot sqft
Vacant Land
Located in the heart of Ferguson, this lot is big enough for a small cabin or just camp in a tent or camper. There is year round back-country activities in this remote area. With close to 8 feet of snow in the winter, snow sports are at the top of everyone's to do list. Just step off the property, for summer quadding and hiking. Only a 10 minute drive to Trout Lake with it's great fishing & swimming. (id:26763)
British Columbia
Mountain view
Small single dwelling
Rural Setting, Quiet Area
Royal Lepage Selkirk Realty
The listing data is provided under copyright by the Canadian Real Estate Association. The listing data is deemed reliable but is not guaranteed accurate by the Canadian Real Estate Association nor Estateblock.com


text:
Dinges Mid-Century Modern Beige And Brown Fabric 5-Piece Dining Set Find
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I seem to see Shadow People a lot when I was a young mom in my twenties living by myself.
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- How can I make my husband throw away stuff / stop hoarding all kinds of things?
- How much do u think would cost to make a house.?
- how does Jysk normally measure things?
- What furniture will i need if i just move into an apartment?
- should i match my dishes to my kitchen or my dining room ?
- Long counter in the kitchen where people can eat, what is it called?
- Hi!! i have one large room which i have to make an all in one?
- how should i paint my home?
Where to Find Dinges Mid-Century Modern Beige And Brown Fabric 5-Piece Dining Set
this valuable specific offer is 20-70% off. We recommend Dinges Mid-Century Modern Beige And Brown Fabric 5-Piece Dining Set. quality pricing from the model is special . surefire customer satisfaction. Integrated! Price Dinges Mid-Century Modern Beige And Brown Fabric 5-Piece Dining Set starting at April 2019 regarding you to receive goods from a great many other stores, you need to to update on Sunday-December-8-2019
04 Comments
Georgianna Mischke2019-09-21
Easy to put together, I am 61 yrs. And I was done in 2 hrs. The only problem was when I ready to screw the final wood screws to add the seats, it was not marked which side needed to be up , and I had to redo three chair's . The seats sit nice and firm, that i really like., Pros: sturdy, great color easy to erect and seats are nice and firm Cons: the seat needs to be marked which side is top and or bottom so when you attend attach the cushion you own it correct. Dinges Mid-Century Modern Beige And Brown Fabric 5-Piece Dining Set
Maribel Menjivar2019-10-08
Love it! Just what I necessary for my small space. I couldn't put it together by myself so I asked my father and he put it together in less than an hour. A supplementary group of hands definitely had a need to put it together.
Laine Menjivar2019-11-20
Great value, beautiful workmanship, simply perfect for my room. Ordered as a combination storage piece, filler for a big room and use by the end of my sofa. Color is really as shown online - not distressed or "Shabby Elegance" but instead a rich deep black finish which is even and glossy. Glas shelves are thick and smooth on the edges and fit into backets that hold them securely. All of the hardware lines up and is a superb mixture of modern and classic. Arrived in two large flat boxes, with plenty of padding and protection for the painted surfaces. Arrived promptly with tracking available anytime I needed to check on it. Easy to put together with the included instructions - all the fasteners were bagged and marked and each piece was labeled. Instructions included pictures and words which were easy to understand. I would suggest this piece to anyone.
Samatha Damron2019-12-01
Think it's great! I ordered the table with the natural legs and received the walnut legs. Called Wayfair instantly and they took care of it for me. It was a little difficult to figure out how to put it together and you definitely need 2 visitors to do so. Other than that, it fits perfect in my kitchen nook.