New Evidence on the Puzzles. Results from Agnostic Identification on Monetary Policy and Exchange Rates.

be the log excess return. As above, the payoff to this strategy is

X_t+_k = (1 - e^ρt+^k )e(ⁱo→k ^+po ^-pk )

and the asset pricing equation reads

0 = Et [e^mt,t^+k Xt+k ]                              (3)

Let

μρ = log Et[e^ρt⁺k ]

be the logarithm of the expected excess return. Let

r0→k = i_0→k + p₀ - p_k

be the real short rate. The asset pricing equation (3) can be rewritten as

SRρ,k ≡ — = -corrt(mt,t+k,pt+k)σ_m - corrt(ro→k,ρt+k)σr₀→_k     (4)

^σρ

where corr_t(∙, ∙) denotes conditional correlation, where σ_m, σρ and σ_r0→_k are
the conditional standard deviations of m_t,_t+_k, ρ_t+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.

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