where
g(s, u) =iu
ϕ(−i(Sr(s, T) +σI(s)))−ϕ(−iSn(s, T))−
ϕ(−i(Sr(s, Tk) +σI(s))) +ϕ(−iSn(s, Tk))
ξ(s, u) =iu
Sr(s, Tk)−Sn(s, Tk)−Sr(s, T) +Sn(s, T)
Changing to the nominal martingale measureQgives us ψIt(u, T, Tk, Tl) =
pr(t, Tk) pr(t, T)
pn(t, T) pn(t, Tk)
iu
× EtTl
exp
T
t h(s, u)vs−ds+ T
t k(s, u)dYs
× exp
A(T, Tk) +B(T, Tk)vT−
where h(s, u) =iu
ϕ(−i(Sr(s, T) +σI(s)))−ϕ(−iSn(s, T))−
ϕ(−i(Sr(s, Tk) +σI(s))) +ϕ(−iSn(s, Tk))
−ϕ(−iSn(s, Tl)) k(s, u) =iu
Sr(s, Tk)−Sn(s, Tk)−Sr(s, T) +Sn(s, T)
+Sn(s, Tl) Applying proposition 3 gives us the result.
2.11 Appendix: The Variance Gamma
Next, the Variance Gamma process arises from subordinating this process with a Gamma Process,G(t)∼Γ(a, b)
X(t) =Z(G(t)) =βG(t) +γW(G(t))
With respect to the Gamma process, Madan, Carr, and Chang (1998) show that it is sufficient to consider a gamma process wherea=b= 1/ν. In this case the Gamma process reflects an unbiased clock (i.e.E[G(t)] =t).
Using this specification, we say the processXisV G(β, γ, ν).
The VG process is a L´evy process (with infinite activity and finite variation) and has characteristic exponent given by
ϕ(u) =−1 νlog
1−iuβν+u2β2 2ν
Finally the VG process has mean, variance, skewness and kurtosis given as E[X(t)] =βt
Var (X(t)) = γ2+β2ν
t Skew (X(t)) =(3γ2+ 2β2ν)βν
(γ2+β2ν)3/2√ t
Kurt (X(t)) =(3γ4+ 12γ2β2ν+ 6β4ν2)ν (γ2+β2ν)2t
Essay 3
Inflation risk premia in the term structure of interest rates: Evidence from Euro area inflation swaps
Abstract
We estimate inflation risk premia in the Euro area using inflation swaps.
By proposing a no-arbitrage model for econometric analysis, and estimat-ing it usestimat-ing Markov Chain Monte Carlo, we find estimates of inflation risk premia, that on average show an upward sloping term structure, with 1 year risk premia of 18 bps and 10 year risk premia of 43 bps, however with fluctuation in risk premia over time. Our estimates suggest that surveys are important in identifying inflation expectations and thus inflation risk premia. We relate estimates of inflation risk premia to agents beliefs, and find that skews in short term inflation perceptions drive short term inflation risk premia, where beliefs on GDP growth drive longer term risk premia.
Keywords: Inflation risk premia, Inflation expectations, Inflation swaps, Surveys, Affine Term Structure Models, Markov Chain Monte Carlo JEL Classification:C11, C58, E31, E43, G12
1I would like to thank Thomas Werner, Jacob Ejsing, Kasper Lorenzen, Mads Stenbo Nielsen, Peter Feldh¨utter, Jesper Lund, Anne-Sofie Reng Rasmussen and participants at presentations at the Capital Markets/Financial Structure Division (ECB), Danmarks Nationalbank, Copenhagen Business School and the 2010 WHU Campus for Finance Conference for useful suggestions.
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3.1 Introduction
The ability to correctly estimate inflation risks are vital to investors, as well as central banks. One such measure is the Break Even Inflation Rate (BEIR), which is the difference in yield between a nominal and real bond.
Another measure is provided by inflation swaps. More precisely zero-coupon inflation indexed swaps, are swap agreements who at maturity pay the change in the reference index (the Consumer Price Index) as the floating leg and a pre-specified fixed payment as the fixed leg. The fixed leg is set, so that the contract has a value of zero at initiation. Hence the quotes of inflation swaps gives an additional measure of the BEIR. Typically inflation swaps require less capital to hold, than inflation linked bonds, making these contracts less prone to market distortions. In fact around the collapse of Lehman Brothers (end-2008), the spread between inflation swap rates and BEIRs from inflation indexed bonds, widened due to the financial crises and liquidity effects (see for instance Campbell, Schiller, and Viceira (2009) for an elaboration on this issue).
Recently a number of papers have tried to estimate inflation risk premia using various methodologies. On US-data the analysis have mainly been focused on using CPI data, surveys and/or US treasury inflation protected securities (TIPS) to estimate the inflation risk premia (see Ang, Bekaert, and Wei (2008), D’Amico, Kim, and Wei (2008), Chernov and Mueller (2008) and Christensen, Lopez, and Rudebusch (2008)). The only paper to use inflation swap data is Haubrich, Pennacchi, and Ritchken (2008), who use US inflation swap data.
With regard to Euro Area data, we are aware of three papers, namely Tristani and H¨ordahl (2007), Garcia and Werner (2010) and Tristani and H¨ordahl (2010). All papers extract real yields from inflation indexed bonds, and then estimate inflation expectations and inflation risk premia.
Overall only a few of these studies agree on the size of the inflation risk premia. Some papers have inflation risk premia of up to 300 basis points (Chernov and Mueller (2008)), where others show more moderate fluctu-ations (-50 to 50 basis points, see for instance Christensen, Lopez, and Rudebusch (2008)). These differences seem to arise from small differences in data periods and the data included, e.g. for instance the inclusion of sur-veys or not. Finally, only Tristani and H¨ordahl (2007) present confidence bands on their of estimates inflation risk premia. They find that their es-timate of inflation risk premia is statistically insignificant for most of the considered maturities.
In this paper we focus on Euro area inflation risk premia. However, instead
of using inflation indexed bonds to identify real yields, we use inflation swaps. We choose to use inflation swaps, since inflation swaps linked to the Euro area HICP have developed into a fairly liquid market.2 As mentioned above, swaps require less capital to hold, and swap rates are less likely to be distorted by market related issues compared to cash products. Furthermore inflation swap rates have the advantage, that they can be included directly into an estimation, making use of the data less prone to errors from inter-polation. Finally, to our knowledge, we are the first to conduct an analysis on inflation risk premia using inflation swaps.
Rather than trying to relate inflation risk premia to a large framework in-cluding agents, GDP, etc., we use a reduced form approach. The choice of a reduced form model is motivated by the large degree of disagreement on the inflation risk premia. We rely on the existing literature on continuous time term structure models (see Duffie and Kan (1996) and Dai and Single-ton (2002)), extended with an inflation process similar to D’Amico, Kim, and Wei (2008), although with slight differences. Thereby real zero-coupon bonds (and inflation swaps) can be priced through no-arbitrage methods.
To more easily identify inflation risk premia we follow Garcia and Werner (2010) and include the ECB survey of professional forecasters. Since we in this paper use a fairly short time series (data from 1999), we are likely to face a small-sample bias. The use of surveys may help to reduce such a bias and help identify the model. Furthermore, to derive the inflation risk premia, we want to model the inflation expectations of market participants.
With all likelihood one can construct a model, which fits realised inflation better than surveys, however such a model may not be representative of the actual inflation expectation, thus leading to wrong estimates of inflation risk premia.
We estimate our model using a Bayesian approach, namely Markov Chain Monte Carlo. This allows us to draw precise inference on derived variables such as inflation expectations and risk premia. By using draws from the Markov Chain Monte Carlo estimation, we examine the effect of includ-ing surveys, and find that surveys improve the identification of inflation expectations and thus inflation risk premia.
In terms of our estimate of inflation risk premia, we obtain estimates of average inflation risk premia that are increasing in time to maturity, with 1 year risk premia of 18 basis points and 10 year risk premia of 43 basis points.
These show significant fluctuations with 1 year inflation risk premia being
2In terms of US inflation linked markets, TIPS are still by far the most actively traded product, thus having a significant negative effect on the US inflation swap markets.
between -146 and 68 basis points, with the lowest value being in the time after the collapse of Lehman Brothers. Longer term inflation risk premia (10 year) show less variation, with inflation risk premia between -30 and 81 basis points.
Finally we relate the estimated risk premia to agents beliefs on the outcome of the economy. We find that short term inflation risk premia are mainly driven by the skewness of the distribution of inflation as measured by the ECB survey of professional forecasters, where longer term risk premia are driven by GDP expectations.
The paper is structured as follows: Section 3.2 describes the data and pro-vides an ad-hoc measure of inflation risk premia. Section 3.3 and 3.4 intro-duce the no-arbitrage model which we use to estimate inflation risk premia, and section 3.5 describes our estimation methodology. Section 3.6 describes the empirical results and finally section 3.7 concludes the paper.