RDP 2011-01: Estimating Inflation Expectations with a Limited Number of Inflation-indexed Bonds Appendix B: The Mathematics of Our Model
March 2011
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We first give some general results regarding affine term structure models, then relate these results to our specific model and its interpretation.
B.1 Some Results Regarding Affine Term Structure Models
Start with the latent factor process
Given x_{t} we have, for s > t (see, for example, p 342 of Duffie (2001))
where denotes equality in distribution and with
Further, if we define
then since is normally distributed,
with
where we have used a stochastic version of Fubini's theorem to change the order of integration (see, for example, p 109 of Da Prato and Zabczyk (1992)). Evaluating the inner integral of line (B2), using Itô's Isometry (see, for example, p 82 of Steele (2001)) and making the change of variable s = t + τ − u we have
where for x a vector we define x^{2} = x′^{2} as the vector dot-product x′x. Hence
Now for we have,
while
and
where from Kim and Orphanides (2005) for example,
Putting this together we have
with
Equivalent formula are available in Kim and Orphanides (2005).
B.2 Bond Price Formula
If we model the SDF according to
then the price of a zero-coupon bond at t paying one dollar at t + τ is given by (see, for example, Cochrane (2005))
where is like π_{s} in Equation (B6) above but with
where and (Here is the ‘risk neutral’ version of π_{s}.) Hence we can price bonds via Equation (B3) using K^{*} and μ^{*} in place of K and μ in Equations (B4) and (B5). We can write Equation (B7) as
In terms of the inflation yield from Equation (A5) this can be written as
B.3 Inflation Forecast Formula
Inflation expectations are reported in terms of percentage growth in the consumer price index, not average inflation (the two differ by a Jensen's inequality term). As such, expectations at time t of how the CPI will grow between time s > t and time s + τ in the future correspond to a term of the form
where the last line follows since Here and are equivalent to α_{τ} and β_{τ} from Equations (B4) and (B5) respectively but with the market price or risk λ_{τ} set to zero and using −ρ_{0} and −ρ in place of ρ_{0} and ρ. So if the CPI is expected to grow by 3 per cent between s and s + τ for example, we would have