The Forward Premium Puzzle in a Two-Country World Ian Martin ian.martin@stanford.edu Stanford GSB and NBER October 20, 2010 Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 1 / 45
The forward premium puzzle Uncovered interest parity relates next year’s spot exchange rate, e t +1 , to today’s spot exchange rate, e t , and 1-yr interest rates in each country, i 1 , t and i 2 , t : ? E log e t +1 = log e t + i 1 , t − i 2 , t “If country 2 has a lower interest rate than country 1 then (surely??) this should be compensated by the expected appreciation of its currency” Using covered interest parity, this is equivalent to ? E log e t +1 = log f t Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 2 / 45
The forward premium puzzle UIP fails badly in the data Regress exchange rate changes on interest rate differentials ∆ log e t +1 = a 0 + a 1 ( i 1 , t − i 2 , t ) + ε t +1 UIP is said to hold if a 0 = 0 and a 1 = 1 Empirically, a 1 is around zero or even negative Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 3 / 45
The forward premium puzzle UIP fails badly in the data Regress exchange rate changes on interest rate differentials ∆ log e t +1 = a 0 + a 1 ( i 1 , t − i 2 , t ) + ε t +1 UIP is said to hold if a 0 = 0 and a 1 = 1 Empirically, a 1 is around zero or even negative Carry trade generates positive excess returns: borrow in low interest-rate currencies, invest in high interest-rate currencies. Currency movements do not undo the interest differential, and may even help the trade Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 3 / 45
Related papers Vast empirical literature: Frankel (1980), Hansen and Hodrick (1980), Fama (1984), Brunnermeier, Nagel and Pedersen (2008), Jurek (2009), Jorda and Taylor (2009) Responses ◮ Segmented markets: Alvarez, Atkeson and Kehoe (2009) ◮ Disasters: Farhi-Fraiberger-Gabaix-Ranciere-Verdelhan (2009) ◮ Country size, nontradables: Hassan (2009) ◮ Long-run risk: Colacito and Croce (2010), Bansal-Shaliastovich (2010) ◮ Habits: Verdelhan (2010) Cole and Obstfeld (1991), Zapatero (1995), Pavlova and Rigobon (2007), Stathopoulos (2009) ◮ All rely on log utility and unit elasticity of substitution between goods Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 4 / 45
Setup The structure of this paper is comparatively simple Infinite horizon, continuous time Two countries, producing outputs D 1 t and D 2 t at time t Assets are priced by a representative agent Two important caveats: I have nothing to say about nominal issues—model is fully real 1 spatial issues—countries are distinguished only in that the rep agent 2 views their outputs as imperfect substitutes Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 5 / 45
Setup Preferences Assets priced by a representative agent with power utility � ∞ e − ρ t C 1 − γ t 1 − γ dt E 0 Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 6 / 45
Setup Preferences Assets priced by a representative agent with power utility � ∞ e − ρ t C 1 − γ t 1 − γ dt E 0 Consumption is a CES aggregator of goods produced by two countries � � η η − 1 η − 1 η − 1 w 1 /η D + (1 − w ) 1 /η D η η C t ≡ 1 t 2 t Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 6 / 45
Setup Preferences Assets priced by a representative agent with power utility � ∞ e − ρ t C 1 − γ t 1 − γ dt E 0 Consumption is a CES aggregator of goods produced by two countries � � η η − 1 η − 1 η − 1 w 1 /η D + (1 − w ) 1 /η D η η C t ≡ 1 t 2 t So units matter: there is an intratemporal relative price (exchange rate between the two goods) and each good has its own interest rate Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 6 / 45
Setup Preferences � � η η − 1 η − 1 η − 1 w 1 /η D + (1 − w ) 1 /η D η η C t ≡ 1 t 2 t 1 t D 1 − w η = 1: Cobb-Douglas case, C t ∝ D w 2 t η = ∞ : Perfect substitutes case, C t = D 1 t + D 2 t Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 7 / 45
Setup Technology Output growth is i.i.d. over time, but may be correlated across assets; formally, (log D 1 , t , log D 2 , t ) is a L´ evy process Allows for dividends to follow geometric Brownian motions, or compound Poisson processes, or a combination of both, or many other possibilities (Will mean that the log exchange rate follows a random walk) Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 8 / 45
Setup Technology Output growth is summarized by the cumulant-generating function, c ( θ 1 , θ 2 ), defined by �� D 1 , t +1 � θ 2 � � θ 1 � D 2 , t +1 c ( θ 1 , θ 2 ) = log E D 1 , t D 2 , t Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 9 / 45
Setup Technology Output growth is summarized by the cumulant-generating function, c ( θ 1 , θ 2 ), defined by �� D 1 , t +1 � θ 2 � � θ 1 � D 2 , t +1 c ( θ 1 , θ 2 ) = log E D 1 , t D 2 , t Example: if the two output streams are independent GBMs, c ( θ 1 , θ 2 ) = µ 1 θ 1 + µ 2 θ 2 + 1 1 + 1 2 σ 2 1 θ 2 2 σ 2 2 θ 2 2 Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 9 / 45
Setup Technology Output growth is summarized by the cumulant-generating function, c ( θ 1 , θ 2 ), defined by �� D 1 , t +1 � θ 2 � � θ 1 � D 2 , t +1 c ( θ 1 , θ 2 ) = log E D 1 , t D 2 , t Example: if the two output streams are correlated GBMs, c ( θ 1 , θ 2 ) = µ 1 θ 1 + µ 2 θ 2 + 1 1 + κσ 1 σ 2 θ 1 θ 2 + 1 2 σ 2 1 θ 2 2 σ 2 2 θ 2 2 ∂ 2 c Notice: positively correlated fundamentals iff ∂θ 1 ∂θ 2 > 0 Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 9 / 45
Setup Technology Output growth is summarized by the cumulant-generating function, c ( θ 1 , θ 2 ), defined by �� D 1 , t +1 � θ 2 � � θ 1 � D 2 , t +1 c ( θ 1 , θ 2 ) = log E D 1 , t D 2 , t Example: if the two output streams are also subject to jumps, µ 1 θ 1 + µ 2 θ 2 + 1 1 + κσ 1 σ 2 θ 1 θ 2 + 1 2 σ 2 1 θ 2 2 σ 2 2 θ 2 c ( θ 1 , θ 2 ) = 2 + + ω 1 ( e µ J , 1 θ 1 + 1 1 − 1) + ω 2 ( e µ J , 2 θ 2 + 1 2 σ 2 J , 1 θ 2 2 σ 2 J , 2 θ 2 2 − 1) Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 9 / 45
Setup Technology Output growth is summarized by the cumulant-generating function, c ( θ 1 , θ 2 ), defined by �� D 1 , t +1 � θ 2 � � θ 1 � D 2 , t +1 c ( θ 1 , θ 2 ) = log E D 1 , t D 2 , t Example: if the two output streams are also subject to jumps, 0 . 02 θ 1 + 0 . 02 θ 2 + 1 1 + 0 · 0 . 1 2 θ 1 θ 2 + 1 20 . 1 2 θ 2 20 . 1 2 θ 2 c ( θ 1 , θ 2 ) = 2 + + 0 . 02( e − 0 . 2 θ 1 + 1 2 0 . 1 2 θ 2 1 − 1) + 0 . 02( e − 0 . 2 θ 2 + 1 2 0 . 1 2 θ 2 2 − 1) Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 9 / 45
Some identities Intratemporal price of good 2 in units of good 1 e t ≡ u 2 ( D 1 t , D 2 t ) u 1 ( D 1 t , D 2 t ) Stochastic discount factor depends on units M 1 , t +1 ≡ e − ρ u 1 ( D 1 , t +1 , D 2 , t +1 ) M 2 , t +1 ≡ e − ρ u 2 ( D 1 , t +1 , D 2 , t +1 ) u 1 ( D 1 t , D 2 t ) u 2 ( D 1 t , D 2 t ) Backus, Foresi and Telmer (2001), Brandt, Cochrane and Santa-Clara (2006) start from e t +1 = M 2 , t +1 e t M 1 , t +1 I follow these papers and refer to e t as the exchange rate Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 10 / 45
Some identities Start from e t +1 = M 2 , t +1 e t M 1 , t +1 Take logs, then expectations: E t ∆ log e t +1 = E t log M 2 , t +1 − E t log M 1 , t +1 “Ignore risk”: E t ∆ log e t +1 “ ≈ ” log E t M 2 , t +1 − log E t M 1 , t +1 = i 1 , t − i 2 , t Allowing for risk, E t ∆ log e t +1 = i 1 , t − i 2 , t + log E t M 1 , t +1 − E t log M 1 , t +1 � �� � L t ( M 1 , t +1 ) − (log E t M 2 , t +1 − E t log M 2 , t +1 ) � �� � L t ( M 2 , t +1 ) Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 11 / 45
Two assets, Cobb-Douglas case, is easy 1 t D 1 − w u ( D 1 t , D 2 t ) ∝ ( D w ) 1 − γ / (1 − γ ) 2 t � � � ∞ e − ρ t u 1 ( D 1 t , D 2 t ) P 10 = u 1 ( D 10 , D 20 ) · D 1 t dt E t =0 � � � D 1 t � w (1 − γ ) − 1 � D 2 t � (1 − w )(1 − γ ) � ∞ e − ρ t = E · D 1 t dt D 10 D 20 t =0 D 10 = ρ − c ( w (1 − γ ) , (1 − w )(1 − γ )) Details aren’t important Cobb-Douglas is tractable because everything is multiplicative (consumption aggregator, log dividend growth i.i.d.) Price-dividend ratio is constant—intratemporal price adjusts Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 12 / 45
Two assets are hard if η > 1 Consider the log utility, perfect substitution case, C t = D 1 t + D 2 t : � C t � − γ � ∞ e − ρ t P 1 , 0 = · D 1 , t dt E C 0 0 � � � ∞ D 1 t e − ρ t E = ( D 10 + D 20 ) dt D 1 t + D 2 t 0 � ∞ � � 1 e − ρ t E = ( D 10 + D 20 ) dt 1 + D 2 t / D 1 t 0 Cochrane, Longstaff, Santa-Clara (2008), Martin (2009) Intratemporal price is constant—valuation ratios adjust Ian Martin (Stanford GSB) The Forward Premium Puzzle October 20, 2010 13 / 45
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