be the log excess return. As above, the payoff to this strategy is
Xt+k = (1 - eρt+k )e(io→k +po -pk )
and the asset pricing equation reads
0 = Et [emt,t+k Xt+k ] (3)
Let
μρ = log Et[eρt+k ]
be the logarithm of the expected excess return. Let
r0→k = i0→k + p0 - pk
be the real short rate. The asset pricing equation (3) can be rewritten as
SRρ,k ≡ — = -corrt(mt,t+k,pt+k)σm - corrt(ro→k,ρt+k)σr0→k (4)
σρ
where corrt(∙, ∙) denotes conditional correlation, where σm, σρ and σr0→k are
the conditional standard deviations of mt,t+k, ρt+k and r0→k. This equation
defines the Sharpe ratio SRρ,k, expressed in terms of log returns. For equity
(and the reverse of the strategy described here, i.e. for going long on equity
and borrowing at the short rate) and an investment horizon of one year,
Sharpe ratios of 0.3 to 0.5 are common, as is well-known from the literature.
For a discussion in the context of DSGE models, see e.g. Uhlig (2004).
We thus evaluate the Sharpe ratio SRρ,k for the hedging strategy ρk from
the perspective of a Bayesian investor, who is able to “insure” against all
other current and future shocks, but remains uncertain about the precise
impact of monetary policy shocks on the forward discount premium due to
uncertainty regarding the reduced-form dynamics of the economy as well as
uncertainty regarding the precise nature of monetary policy shocks.
10
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