The Real Exchange Rae, Real Ineres Raes, and he Risk Premium Charles Engel Universiy of Wisconsin
How does exchange rae respond o ineres rae changes? In sandard open economy New Keynesian model, increase in ineres rae leads o real appreciaion Under uncovered ineres pariy (UIP), currency hen depreciaes Bu famous UIP puzzle says when ineres rae goes up, currency expeced o appreciae (bu says nohing abou he level) Wha is acual behavior of level and change? Can models explain i? Imporan for moneary policy: o Firs, jus o undersand impac of policy o Second, o undersand economic forces a work 2
Definiion of Terms (Log of) nominal exchange rae: s (Price of Foreign currency increase means Home depreciaion) (Log) of Home and Foreign prices: p, p Home and Foreign nominal ineres raes: i, i Home and Foreign inflaion: π + = p + p, π + = p + p (Log of) real exchange rae: q= s+ p p (Increase means Home real depreciaion) Home and Foreign real ineres raes: r = i Eπ +, r = i Eπ + Uncovered Ineres Pariy: i + Es+ s = i Equivalenly (adding Eπ Eπ o each side): + + r + E q q = r + Risk premium : λ i + Es+ s i = r + Eq+ q r 3
Sandard models assume UIP: r + Eq q = r Rearrange and subrac off uncondiional means: q = ( r r r) + Eq + Ierae forward: where q = ( r r r) E ( r r r) + Eq + + + 2 = = R + limeq + j j ( + i + i ) i= 0 R E r r r + Now assume also long run purchasing power pariy: limeq + j j = q Then q q = R 4
Ineres rae differenials are posiively serially correlaed (AR for example). When r, we find R Under UIP, q q = R, so when r, we find q The UIP model is consisen wih one well known fac: When a counry s real ineres rae moves up, is currency ends o srenghen in real erms. 5
Uncovered Ineres Pariy 0 Monhs -2-4 Exchange Rae under UIP -6-8 -0-2 -4 6
A second well known facs in inernaional finance: The forward premium puzzle: Counry wih high ineres rae has high excess reurn. Regression: s s = β + β ( i i ) + u. + 0 + UIP says β 0 = 0 and β =. Typical finding: β < (ofen β < 0), β 0 = 0. This means λ i + Es+ s i = r + Eq+ q r negaively correlaed wih i i. Paper shows ha λ is negaively correlaed wih r r also. (!!!!) Risk premium models have: λ =, γ > γ ( r r ) 7
To resae hings: Eq + q = r r + λ. How do you explain Eq + q when r r? In he models in he lieraure, λ =, γ >. γ ( r r ) Bu wha does his imply abou how q changes when r r? I implies q q = ( γ ) R so q when r r 8
Models of Risk Premium 5 0 Exchange Rae in Risk Premium Models 5 0 Monhs -5 Exchange Rae under UIP -0-5 9
Here is he problem: Mos macro models cenral banks use assume UIP: When r r q (True) When r r Eq + q (False) Models ha are used o explain UIP puzzle: When r r q (False) When r r Eq + q (True) Wha do he daa say? When r r q When r r Eq + q 0
Real Exchange Rae in Daa 20 0 Exchange Rae in Risk Premium Models 0-0 Monhs -20 Exchange Rae under UIP -30 Exchange Rae in Daa -40-50 -60
Real ineres raes and real exchange raes. Rewrie: q E q r r r λ λ + = ( ) ( ) Ierae forward o ge: where q lim( E q + ) = R Λ j j ( + j + j ) Λ E λ+ j j= 0 j= 0 R E r r r ( λ ) R prospecive real reurn differenial Λ level risk premium We find when r r ha Eq+ q (implies λ ) bu q and Λ 2
Wha model accouns for his behavior? I don know When r r ha Eq+ q (implies λ ) bu q and Λ Risk Premium? Overall Home shor erm deposis expeced o be less risky ( Λ ) Bu in he shor run hey are more risky ( λ ) None of our models can accoun for his. Liquidiy Premium? Same problem shor erm and long erm premiums move in opposie direcions. Marke Dynamics Overreacion plus herding. Undersanding his is a challenge bu i maers! 3
Figure plos slope coefficiens from he following regressions (Daa are monhly, ineres raes are monh, end of monh. For his slide, U.S. relaive o weighed average of res of G7): q r r k αqk βqk( ) = + + R + k = αrk + βrk( r r ) (If UIP held, we d have q = k R.) +, k (Real ineres raes hemselves are esimaes) Difference beween Λ = R q. + k + k + k R + kand q + kis Λ + k: 4
q = k α + qk βqk( r + r ), R + k = αrk + βrk( r r ), and Model q lime q + k + j j G6 Levels 20 0 0 Monhs -0-20 -30-40 -50-60 5
Daa U.S., Canada, France, Germany, Ialy, Japan, U.K., and G6 Figure plos slope coefficiens from he following regressions (Daa are monhly, ineres raes are monh, end of monh. For his slide, U.S. relaive o weighed average of res of G7): q r r + k = αqk + βqk( ) R + k = αrk + βrk( r r ) (If UIP held, we d have q = k R.) +, k (Real ineres raes hemselves are esimaes) Difference beween Λ = R q. + k + k + k R + kand q + kis Λ + k: 6
q = k α + qk βqk( r + r ), R + k = αrk + βrk( r r ), and Model q lime q + k + j j G6 Levels 20 0 0 Monhs -0-20 -30-40 -50-60 7
Exchange raes las day of monh (noon buy raes, NY) Prices consumer price indexes Ineres raes 30 day Eurodeposi raes (las day of monh) Monhly, June 979 Ocober 2009 (Uni roo es for real exchange raes, uses daa back o June, 973) 8
Preliminary uni roo ess Counry ADF DF GLS Canada.77.077 France 2.033 2.036 Germany 2.038 2.049 Ialy.888.94 Japan 2.07 0.70 Unied Kingdom 2.765 2.076 G6 2.052.846 Model No Covariaes Wih Covariaes Panel Uni Roo Tes, 973:3 2009:0 Esimaed % 5% 0% Coefficien 0.0705 0.0299 0.0697 0.0485 0.0688 0.0226 0.0638 0.04 9
Fama Regressions: s s = β + β ( i i ) + u 979:6 2009:0 + 0 + Counry ˆβ 90% c.i.( ˆβ ) Canada.27 ( 2.564,0.220) France 0.26 (.298,0.866) Germany.09 ( 2.335,0.53) Ialy 0.66 ( 0.206,.528) Japan 2.73 ( 4.4,.285) U.K. 2.98 ( 3.646, 0.750) G6.467 ( 2.783, 0.5) 20
VAR mehodology Two differen VAR models: Model : Model 2: q, i i, i π ( i π ) q, i i, π π Esimae VAR wih 3 lags. Use sandard projecion measures o esimae r = i Eπ +, r = i Eπ +, and ( + j + j ) j= 0 R E r r r Then λ is consruced as λ r + E q + q r Λ esimae is consruced from Λ = R q 2
Fama Regression in Real Terms: q q = β + β ( rˆ rˆ ) + u + 0 + Counry ˆβ 90% c.i.( ˆβ ) Canada 0.38 (.222,.498) (.908,.632) (.800,.676) France 0.576 ( 2.269,.7) ( 2.240,0.79) ( 2.602,.25) Germany 0.837 ( 2.689,.05) ( 3.458,0.33) ( 3.49,0.4) Ialy 0.640 (.056,2.336) (.36,2.087) (.328,2.358) Japan.34 ( 2.860,0.232) ( 3.300,0.254) ( 3.44,0.379) Unied Kingdom.448 ( 3.042,0.46) ( 3.64, 0.27) ( 3.846, 0.039) G6 0.933 ( 2.548,0.682) ( 2.932,0.409) ( 3.005,0.527) 22
Regression of q on rˆ ˆ r : q = β + β ( rˆ rˆ ) + u + 0 Counry ˆβ 90% c.i.( ˆβ ) Canada 48.57 ( 62.5, 34.88) ( 94.06, 3.4) ( 40.54, 27.34) France 20.632 ( 32.65, 8.62) ( 44.34,.27) ( 54.26,.75) Germany 52.600 ( 67.02. 38.8) ( 85.97, 25.35) ( 05.29, 9.38) Ialy 39.0 ( 5.92, 26.28) ( 67.63, 6.36) ( 90.0, 3.70) Japan 9.708 ( 29.69, 9.72) ( 42.0,.05) ( 46.53, 4.33) Unied Kingdom 8.955 ( 3.93, 5.98) ( 40.9, 3.08) ( 55.94,4.08) G6 44.204 ( 55.60, 32.80) ( 73.7, 23.62) ( 82.87, 2.74) 23
Regression of Λ ˆ on rˆ ˆ r : Λ ˆ = β + β ( rˆ rˆ ) + u + Counry ˆβ 90% c.i.( ˆβ ) Canada 23.60 (5.2,32.0) (2.62,5.96) (.96,63.7) France 3.387 (.06,25.72) ( 2.56,36.25) ( 6.98,42.40) Germany 34.722 (9.66,49.78) (9.34,57.59) (3.68,69.36) Ialy 27.528 (7.58,37.48) (4.98,48.32) (2.5,58.54) Japan 5.20 (4.76,25.66) ( 0.45,37.08) 0 (0.9,38.87) Unied Kingdom 4.093 (0.33,27.86) (0.39,34.46) ( 8.70,46.45) G6 3.876 (20.62,43.3) (6.89,54.62) (6.78,60.89) 24
Figure plos slope coefficiens from he following regressions (Daa are monhly, ineres raes are monh, end of monh. For his slide, U.S. relaive o weighed average of res of G7): q r r k αqk βqk( ) = + + R + k = αrk + βrk( r r ) (If UIP held, we d have q = k R.) +, k (Real ineres raes hemselves are esimaes) Difference beween Λ = R q. + k + k + k R + kand q + kis Λ + k: 25
q = k α + qk βqk( r + r ), R + k = αrk + βrk( r r ), and Model q lime q + k + j j G6 Levels 20 0 0 Monhs -0-20 -30-40 -50-60 26
Implicaions: cov(, r r ) 0 λ < (Fama regression in real erms) cov( Λ, r r ) = cov( Eλ+ j, r r ) > 0 j= 0 (from VAR esimaes) cov( E j, r r ) 0 λ + > for some j (as in previous figure) Explaining cov(, r r ) 0 λ < and cov( E j, r r ) 0 r λ + > is a challenge for risk premium models when r is high, he home currency is boh riskier han average and expeced o be less risky han average. 27
Edˆ ( ˆ ˆ + k = βdk r r ),.5.0 0.5 E rˆ rˆ = rˆ rˆ, and ( + k + k) βrk( ) G6 E ˆ ( ˆ ˆ λ+ k = β λ k r r ) r(+j) d(+j) l(+j) 0.0 0.5 4 7 0 3 6 9 22 25 28 3 34 Monhs.0.5 2.0 2.5 28
In fac, models are buil o accoun for β < 0 in Fama regression and have Eqˆ ˆ ˆ ˆ + q = β( r r ) imply q lim ˆ ˆ Eq+ j = β E( r+ j r+ j), j j= 0 So ha a higher home real ineres rae depreciaes he currency. Tha goes agains evidence, agains he way we hink abou exchange raes. Bu ha is wha gives level effec in he model line in Figure he wrong sign. 29
Models assume a single facor for risk premium and real ineres differenial: Eqˆ qˆ + = aφ λ = g φ ˆ We need var( λ ) > var( Eq + q) (Fama, 984) If we normalize a > 0, we need g > a > 0. 30
q = k α + qk βqk( r + r ), R + k = αrk + βrk( r r ), and Model q lime q + k + j j G6 Levels 20 0 0 Monhs -0-20 -30-40 -50-60 3
We need a leas wo facors: Eqˆ qˆ + = aφ + aφ λ = gφ + gφ 2 2 ˆ 2 2 We need g a 0 > > 0 < g2 < a2 (and also need he second facor o be more persisen in order o mainain var( λ ) > var( Eq + q).) No a big deal if hese were all jus arbirary. The challenge is when we relae hem o preferences and consumpion. 32
Review of foreign exchange risk premium m+, m+ are logs of home, foreign sochasic discoun facors λ = 2 m+ m+ (var var ) Ed = E( m m ) + + + So we have o have Ed + = E( m+ m+ ) = aφ + a2φ 2 λ = (var m var m ) = gφ + gφ 2 + + 2 2 The challenge is ha we have carefully designed models of preferences and he consumpion process o give us g > a > 0. This gives us a volaile risk premium, which we need for los of asse pricing puzzles. Now we also need 0 < g2 < a2. Goes opposie direcion 33
Campbell Cochrane preferences (Verdelhan, 200): Eq + q = γ ( φσ ) σ is deviaion of consumpion from exernal habi level γ > 0 is risk aversion, 0< φ < is persisence of σ Bs, where B is a parameer assumed o be < 0. λ =[ γ ( φ) ] Hence, λ responds more han Eq + q o changes in s. 34
Long run Risks (Bansal Yaron (2004), Bansal Shaliasovich (200)) m = ( η /2) w + φ w ε (+ oher suff) + + w is he variance of persisen componen of consumpion growh. The model assumes Epsein Zin preferences. η and φ are deermined by he parameers of risk aversion and ineremporal elasiciy of subsiuion. Under Epsein Zin preferences, a preference for early resoluion of risk 2 ends up giving us η < φ Eq q = E( m m ) = ( η /2)( w w ) λ (var var ) ( /2)( ) + + + 2 = 2 m+ m+ = φ w w 35
Conclusion on risk premium models: They will no explain he evidence of his paper. They have been reverse engineered o deliver response of var m + o sae variables is greaer han response of Em +. This feaure is needed no only for UIP puzzle bu for oher asse pricing puzzles in equiy and bond markes. These puzzles require large and variable risk premiums. 36
Delayed overshooing Models wih a leas wo ypes of agens (BvW, AAK) One ype is slow o reac o shocks The oher ype moves quickly bu is risk averse When r r is high, q falls as in UIP. I keeps falling in period + because he slow reacing agens move. This leads o iniial underreacion of q o increase in r r. Some weaking may work bu markes are no efficien here. 37
q = k α + qk βqk( r + r ), R + k = αrk + βrk( r r ), and Model q lime q G6 Levels + k + j j 20 0 0 Monhs -0-20 -30-40 -50-60 38
Conclusions: A new puzzle. Our models for he UIP puzzle don seem adequae. This maers. Policy consideraions:. We need o know how q reacs o changes in moneary policy. r r o assess 2. If indeed here is overreacion, momenum rading, or delayed overshooing in foreign exchange markes, his inefficiency ranslaes ino resource misallocaion hrough is effecs on relaive goods prices. 39