Hedging Currency Risk: a Regret-Theoretic Approach

Transcription

1 Hedging Currency Rik: a Regret-Theoretic Approach Sébatien Michenaud and Bruno Solnik HEC Pari* Thi draft : october 4, 005 Abtract Contrary to the prediction of exiting normative currency-hedging model, a wide diverity of hedging policie i oberved among intitutional invetor. We propoe an alternative model of optimal currency-hedging choice baed on regret theory, a normative and axiomatic behavioral theory. With hindight, invetor may experience regret of not having taken the ex pot optimal hedging deciion; i.e. full hedging if the foreign currency depreciated, or no hedging if the foreign currency appreciated. Hence, invetor include expected future regret in their objective function. A a reult, our model feature two component of rik: traditional rik and regret. We derive cloed-form optimal hedging rule uing the Arrow-Pratt approach and highlight the difference with the traditional expected-utility reult. We find that difference in the level of regret averion among invetor may explain why we oberve uch a wide diverity of currency hedging policie among intitutional invetor. * HEC Pari, rue Libération, 7835 Jouy en Joa, Cedex, France and GREGHEC. Correponding author: Bruno Solnik, olnik@hec.fr, tel: We thank Fondation HEC and Inquire Europe for their financial upport. Part of thi reearch wa conducted while Bruno Solnik wa viiting the School of Banking and Finance of the Univerity of New South Wale. We benefited from comment by Bernard Duma, Ine Chaïeb, Chritian Gollier and eminar participant at the Univerity of New South Wale, HEC Pari, EFA meeting in Mocow, and GFC meeting in Dublin.

2 . Introduction I hould have computed the hitorical covariance of the aet clae and drawn an efficient frontier. Intead I viualized my grief if the tock market went way up and I wan t in it--or if it went way down and I wa completely in it. My intention wa to minimize my future regret, o I plit my [penion cheme] contribution 50/50 between bond and equitie. Harry Markowitz. Portfolio invetor have progreively accepted the argument that international diverification provide rik/return benefit. However, the currency dimenion ha remained an emotional iue and currency hedging i a enitive deciion. Attractive local-currency return on foreign aet market can be wamped by a depreciation of the foreign currency. Converely, the return on foreign currency can provide a major portion of the total return of international invetment when the dometic currency i weak. The currency hedging deciion i a imple one: what currency hedge ratio (proportion of foreign aet value hedged againt currency rik) hould be adopted. In other word, hould international aet be fully hedged againt currency rik, not hedged, or partially hedged? The currency hedging policie adopted by invetor eem quite divere. Figure give the ditribution of currency-hedging benchmark for intitutional invetor delegating the currency hedging deciion to overlay manager. 3 Data come from a urvey (Harri-004) conducted in 004 by Mellon/Ruell and cover 563 account of intitutional invetor worldwide. Looking at the worldwide average, 39% of invetor adopt a no-hedging policy, 34% of invetor adopt a 50% hedging policy, 4% of invetor adopt a 00% hedging policy and 3% of invetor adopt ome other hedge ratio. Becaue thee number reflect long-term policy benchmark, they cannot be explained by hort-term expectation on currency movement, but primarily by rik conideration or ome behavioral attitude. The wide diverity in hedge ratio i puzzling. One objective of thi article i to provide an explanation for the oberved diverity in hedging policie wherea exiting normative model upport uniform policie acro invetor. A quoted in Jaon Zweig, "How the Big Brain Invet at TIAA-CREF", Money, 7(), p4, January 998. In thi paper we will ue US invetor a dometic invetor. 3 The benchmark currency hedging policy drive the hedge ratio adopted by the currency overlay manager who can deviate baed on tactical currency expectation and rik aement. The fact that the data reported in Figure repreent benchmark aigned to currency overlay manager, not the actual hedge ratio, can introduce ome biae.

3 Figure : Ditribution of hedge ratio for invetor from everal bae currencie The figure give the ditribution of currency-hedging benchmark for intitutional invetor delegating the currency hedging deciion to overlay manager. Data come from a urvey (Harri-004) conducted in 004 by Mellon/Ruell and cover 563 account of intitutional invetor worldwide. Each column give the ditribution of the hedge ratio for invetor from a given bae currency, a well a the number of account in that bae currency. The lat column give the ditribution of all account. 00% 90% 80% 70% 60% 50% 40% Other 00% 50% 0% 30% 0% 0% 0% USA (304) Autralia (84) Japan (43) Euro (5) U.K. (7) Other (53) Total (563)

4 Traditional finance relie on expected utility maximization and ue the expected return/rik paradigm to earch for an anwer. Some reearcher have developed global market equilibrium model to derive optimal currency hedging rule. Solnik (974) and Adler and Duma (983) derive an international aet pricing model where invetor from different countrie ue their own currency a numeraire. Therefore invetor from different countrie view aet expected return and rik differently becaue of foreign exchange uncertainty. Becaue currency movement reflect much more than adjutment to inflation differential (purchaing power parity), currency rik i a real rik. Global equilibrium model conclude that all invetor hould hold a combination of their own rik-free aet (rik-free in home currency) and the world market portfolio partly hedged againt currency rik. Hence the riky portfolio i identical for all invetor and made of the equity market-capitalizationweighted portfolio optimally hedged againt currency rik. A major reult i that all invetor hould identically hedge their international invetment, whatever their nationality or their level of rik averion 4. The optimal hedge ratio i therefore "univeral". Unfortunately, the equilibrium hedge ratio depend on unobervable variable uch a relative rik averion and net foreign poition. Black (990) implifie the equilibrium model and come up with a univeral hedging rule. He etimate that international invetment hould be currency-hedged with a ratio of approximately 70%. Beide the difficulty of aeing the "exact" optimal hedge ratio, it appear that the normative implication of the global equilibrium model are not verified in the real world. Rather than holding the world market portfolio, invetor exhibit a trong home bia in their equity portfolio. In practice, many aet manager imply conduct an aet allocation mean-variance optimization baed on expected return and rik. Typically a two-tep approach i implemented. The aet allocation to international aet market i determined in a firt tep, and the amount of currency hedging i then decided for thi pecific aet allocation. So currency hedging i optimized auming that the global aet allocation i fixed. If currency rik premia are nil 5 and if aet return are uncorrelated with currency movement; then the optimal hedge ratio that minimize rik i 00% becaue currency rik i pure noie (ee Pérold and Schulman, 988). So the currency-rik-minimizing trategy i a hedge ratio of 00%. 6 Again, thi hedge ratio obtain for any (poitive) level of rik averion and hould therefore be "univeral". According to portfolio theory, deviation from thi fully-hegded policy can only be explained by belief in the correlation between aet return and currency movement and/or currency rik premia. Behavioral finance claim that invetor do not follow the traditional mean-variance paradigm and thi i clearly applicable to the currency hedging deciion (ee Statman, 005). We will rely on regret theory to propoe an alternative theory of the currency hedging deciion made by portfolio invetor. 4 The rik level i adjuted by the combination of the riky portfolio and the national rik-free aet. While the optimal hedge ratio for a pecific aet i identical for all invetor, different aet could have different optimal hedge ratio. 5 There i extenive theoretical and empirical dicuion about the exitence and magnitude of currency rik premia. Thi i beyond the cope of thi paper. Given the difficulty in etimating a forward-looking long-term currency rik premium, mot aet manager et a default value of zero. 6 Of coure, the optimal hedge ratio will differ from 00% if there i correlation between aet return and currency movement, and if the currency rik premium differ from zero. 3

5 Regret i a cognitively-mediated emotion of pain and anger when, with hindight, agent oberve that they took a bad deciion in the pat and could have taken one with better outcome. In financial market, agent will experience regret when their invetment yield, expot, a lower performance than an obviou alternative deciion that they could have choen. Contrary to mere diappointment, which i experienced when a negative outcome happen relative to prior expectation, regret i an emotion trongly aociated with a feeling of reponibility for the choice that ha been made, baed on a comparion of the actual outcome with the bet outcome that could have been achieved. 7 Regret i uch a powerful negative emotion that the propect of it future experience may lead individual to make eemingly ub-optimal, non-rational deciion relative to the expected utility paradigm. A the opening quote ugget, the anticipation of future regret wa trong enough to turn Harry Markowitz away from hi very own portfolio allocation theory when faced with a financial deciion on hi penion plan. We believe regret may alo apply to ophiticated fund manager when currency rik i concerned. Currency hedging i a dimenion where regret clearly applie. 8 For example, an American invetor who decided not to hedge currency rik would have incurred a currency lo of ome 40% on it eurozone aet from late 998 to late 000, with a vat regret of not having fully hedged. Converely a fully-hedged invetor would have mied the 50% appreciation of the euro from late 00 to late 004. Again, a vat regret of not having taken the "right" hedging deciion. 9 There i an extenive literature in experimental pychology and, to a leer extent, neurobiology that upport the aumption that regret influence deciion-making under uncertainty 0 beyond diappointment and traditional uncertainty meaure. Baed on thi concept of regret, Loome and Sugden (98) and Bell (98) derived independently an economic theory of regret. They propoed a theory of choice under uncertainty that explain many oberved violation of the axiom ued to build the traditional expected utility approach. Regret theory (RT) aume that agent are rational but bae their deciion not only on expected payoff but alo on expected regret. It predict Allai' paradox ("common conequence effect") and many other axiom violation reported in experiment by Kahneman and Tverky (979) and other. RT bear ome imilaritie with propect theory (Kahneman and Tverky, 979; Tverky and Kahneman, 99) a many reult of RT are conitent with the empirical obervation of human behavior that contitute the building block of propect theory. But propect theory i primarily decriptive while RT i a normative theory of rational choice under uncertainty (ection further dicue thi iue). RT incorporate regret into the utility function in addition to the traditional value function of total wealth. Invetor reach their invetment deciion by maximizing the expected value of thi modified utility. So invetor try to anticipate regret and take it into account in their invetment deciion in a conitent manner. RT i clearly relevant to invetor who compare the performance of their portfolio to foregone alternative that they could have choen, or to peer and benchmark portfolio whoe performance could have been achieved. 7 See Meller, Schwartz and Ritov (999). 8 See alo Gardner and Wuilloud (995), and Statman (005). 9 Furthermore, elling hort an appreciating foreign currency lead to cah loe on the forward poition that have to be covered by the ale of aet. A forced deciion that i painful. 0 For experimental pychology review ee Gilovich and Medvec (995), Zeelenberg et al., (000). For neurobiological experiment ee Damaio (994), Bechara, Damaio and Damaio (000), and Camille et al., (004). 4

6 In thi article, we ue RT to account for the oberved currency rik hedging behavior of fund manager with aet inveted in foreign market. While foreign currency hedging i an important deciion in it own right, it alo i imple enough to allow the modeling of regret in the utility function. We aume that the aet allocation ha been choen and focu on the currency hedging policy, a traditionally aumed in the hedging literature. Although, to our knowledge, thi i the firt attempt to apply RT to currency hedging deciion, the experience of regret in currency hedging i not new for the invetment world. Several practitioner have jutified a 50% naïve hedge ratio on uch intuitive ground. For example: "A partial hedging policy uch a 50/50 or 70/30 mean the invetor won t ever experience the major high of an unhedged portfolio, but won t be ubject to the lowet return either." "To Hedge or not to hedge", Simon Segal, SuperReview.com.au, march 003 "The 50% hedge benchmark i gaining in popularity around the world a it offer pecific benefit. It avoid the potential for large underperformance that i aociated with "polar" benchmark, i.e. being fully unhedged when the Canadian dollar i trong or being fully hedged when it i weak. Thi minimize the "regret" that come with holding the wrong benchmark in the wrong condition." " Managing Currency Rik: A Canadian Perpective", Gregory Chripin, State Street Global Advior, Eay and Preentation, March 3, 004. The 50% hedge ratio i the implet currency hedging policy that attempt to deal with regret. We propoe a theory that encompae all the cae of Figure for different level of rik and regret averion. We will now propoe a formal analyi of the optimal currency hedging deciion that incorporate regret in an expected utility optimization, and therefore deal imultaneouly with traditional rik (volatility of final wealth) and regret rik. The paper i tructured a follow. In Section, we introduce RT and apply it to the modeling of currency hedging in Section 3. Section 4 derive cloed-form hedging rule for currency rik minimization, while Section 5 derive cloed-form hedging rule in the general cae with expectation on currency movement and correlation between aet return and currency movement. Section 6 conclude thi paper.. Regret Theory Regret theory (RT) developed by Loome and Sugden (98) and Bell (98) i a theory of rational choice under uncertainty that i parimoniou yet can explain many of the oberved axiom violation of traditional expected utility theory. Thee author derive a modified utility function of final wealth x reulting from a given invetment choice, knowing that a different invetment choice would have led to a final wealth y: U( x, y) = v( x) + f( v( x) v( y)) () The other application of RT to financial deciion, that we are aware of, are Braun and Muermann (004) and Dodonova and Khorohilov (005). The former apply regret theory to demand for inurance, the latter to aet pricing. A mentioned in the introduction, RT predict Allai' paradox ("common conequence effect"), the "common ratio effect", the "iolation effect", the "preference reveral effect", the "reflection effect", and "imultaneou gambling and inurance". 5

7 where U(x,y) i the modified utility of achieving x, knowing that y could have been achieved. v(x) i the traditional utility function, alo called value function or choicele utility. It i the "value" or utility that an invetor would derive from outcome x if he experienced it without having to chooe. Thi value function i aumed to be monotonically increaing and concave (rik averion) a in traditional finance. The difference v(x) - v(y) i the value lo/gain of having choen x rather than a foregone choice y. It indicate the regret of having choen x, when y could have been choen. The regret function f(.) i monotonically increaing and decreaingly concave 3, with f(0) = 0. Thi modified utility U(.) i defined over the ex-pot (final) outcome of invetment choice; and rational invetor would make choice ex-ante by maximizing the expected value of thi modified utility. Loome and Sugden (98, 983) and Bell (98, 983) conclude that thi i a well-behaved parimoniou functional form that allow to take regret into account and i conitent with empirically-oberved deviation from traditional expected utility theory. Thi functional form ha been initially derived for pair-wie choice, but it can be extended (ee Quiggin, 994) to general choice et. Conider that an invetor can elect among variou invetment i (e.g, ome portfolio i), with outcome x i. The modified utility of chooing invetment i i given by: i i i [ i] where max[ i ] that the regret term vx v [ x] U( x ) = v( x ) + f( v( x ) v(max x )) () x i the bet ex-pot outcome that can be obtained among all invetment. Note ( i) (max i ) i alway non-poitive. Concavity of the regret function, f"<0, implie regret averion. Rational invetor chooe the optimal invetment portfolio by maximizing their expected modified utility of all poible invetment choice. So invetor try to anticipate regret and take it into account in their invetment deciion in a conitent manner. It can be ueful to highlight intuitively the difference with traditional expected utility. A oppoed to traditional von Neumann- Morgentern utility, which i only defined over the actual portfolio owned by the agent, the modified utility alo include a comparion with other portfolio that could have been choen but are not currently owned by the agent. Regretavere invetor do take into account traditional value but alo deviation from the (ex-pot optimal) benchmark. Note that the benchmark i not a predetermined paive benchmark, but rather a benchmark that will be determined ex-pot a the bet-performing invetment. So regret rik i different from tracking-error rik relative to a predetermined benchmark. In the mean-variance paradigm, invetor care about the expected return and volatility of their portfolio. A regret-avere invetor alo care about the expected return and volatility of their portfolio, but in addition the invetor alo care about the rik of deviation from the benchmark (regret rik). So there are two rik attribute in the utility function: volatility and regret rik. With the oberved evidence in favor of the influence of regret on deciion-making under uncertainty 4 a well a the axiomatic and normative appeal of RT for invetment choice, it i 3 Bell (98, 983) and Loome and Sugden (98,983) how that everal behavioral pattern which contradict traditional expected utility theory are predicted by regret theory with a function f(.) that i concave for negative value of the argument and with f"'>0, o that f(.) i decreaingly concave. 4 Connolly and Zeelenberg (00), page, tate that "the emotion that ha received the mot reearch attention from deciion theorit i regret". 6

8 urpriing that RT ha caught o little attention in the field of finance. Indeed, it i propect theory that ha been extenively ued in behavioral finance. The ucce of propect theory hold a much to it decriptive power a to the ability to handpick only ome feature that enable to explain elected puzzle in the field. Numerou author have ued propect theory in normative model of invetment choice, i.e. maximizing ome expected utility (e.g. Benartzi and Thaler, 995; Shefrin and Statman, 000; Barberi, Huang and Santo, 00, Berkelaar, Kouwenberg and Pot, 004, Gome, 005). But only a couple of feature of propect theory can be retained in uch model. Utility model inpired by propect theory typically include a diappointment term with a kink at the current invetment value (the "reference point") where the lope of utility i higher for loe than for gain ("lo averion"). Diappointment i eaier to model 5 a the benchmark expectation for a given invetment i uually et a a fixed number (poibly the current ituation or the current poition plu a rik-free rate)), while in RT the "bet" invetment trategy in the invetment deciion univere can only be determined ex-pot. In pite of the increaed complexity, RT i clearly relevant to invetor who compare the performance of their portfolio to foregone alternative that they could have choen, or to peer and benchmark portfolio whoe performance could have been achieved. It mut be treed that RT, although intuitively appealing, i difficult to apply becaue of the technical difficultie aociated with the optimization of an expected utility function with two attribute: value and regret. Indeed, applying RT to a general portfolio problem involving numerou aet eem a daunting technical tak. Thi i becaue, regret tem from a comparion of the actual return outcome of each portfolio with the actual return outcome of all other feaible portfolio. 6 In contrat, including diappointment in a utility function i much le intricate, a diappointment reult from a comparion, for each aet independently, of the actual return outcome to a preet expectation return (e.g. zero, or the rik-free rate, or ome other exogenou number). Thi technical difficulty probably explain the lack of application of RT to the field of finance. However, currency hedging deciion are imple enough to model in the framework of RT a the ex-pot optimal currency hedging choice i only one of two deciion: no hedging (if the foreign currency appreciated) or full hedging (if the foreign currency depreciated). To ummarize, a regret-theoretic approach preent an alternative way of introducing emotion in invetment choice. While it uffer from the criticim that it only decribe one apect of human behavior, it doe o in an elegant axiomatic way. Furthermore, it goe beyond modeling diappointment, a uually done in the literature, but deal with regret which eem an important pychological trait in portfolio choice, where invetor care about the outcome of their choice relative to other trategie they could have followed, paive benchmark and peer. 3. Currency Hedging: A Regret-Theoretic Framework We conider that the currency hedging deciion i a reidual one, once the global aet allocation ha been choen. Thi i the approach traditionally taken in the hedging literature and, accordingly, we do not claim to olve imultaneouly the general problem of individual 5 For example, Barberi and Huang (004) focu on narrow framing and lo averion by adding a piece-wie linear diappointment term for elected (narrow-framed) aet in a traditional utility function. There i a imple additive diappointment term for each of thee aet. Lo averion appear becaue the linear lope i higher for negative return on thee aet than for poitive return. The reference return i the rik-free rate. 6 In mean-variance analyi, we only compare the expected return and volatility of portfolio. 7

9 ecurity election, aet allocation and currency hedging. It i alo conitent with the behavioral approach of mental accounting, the human tendency to treat each type of invetment deciion in a eparate mental compartment (alo called "narrow framing") 7. A decribed in Kahneman and Lovallo (993), deciion maker are exceively prone to treat problem a unique; their evaluation of ingle riky propect neglect the poibility of pooling rik. Rather than looking at the whole portfolio a precribed by traditional expected utility theory, invetor tend to reach the bet deciion in each mental compartment. Thi feature i widely oberved a far a the currency expoure deciion i concerned. In global aet allocation, invetor clearly eparate the aet allocation deciion and the currency rik hedging deciion. Such a behavior i confirmed in a urvey 8 of Canadian penion plan. The vat majority of thee plan (94%) believe that the bet way to handle currency expoure i to decide firt on global aet allocation and then handle the currency expoure. Thi confirm that the currency hedging deciion i indeed taken a a reidual/eparate deciion from the invetment deciion that create the currency expoure. d f Of their initial wealth, W 0, invetor have allocated W 0 to dometic aet and W 0 to foreign aet: W = W + W d f All valuation are conducted in dometic currency (e.g. the dollar for American invetor). A in all currency-hedging reearch, we do not focu on the interaction between dometic aet and foreign currency and will make the implifying aumption that the final value of dometic aet, W d, i non-tochatic. Introducing tochatic dometic return in our model, would imply add to the complexity of the notation 9, but the analyi would not be affected ince the aet allocation deciion i already fixed. The dollar value of foreign aet i equal to the product of the foreign-currency value of the foreign aet time the exchange rate (dollar value of foreign currency). Uing log of price change a return, the final (dollar) value of the foreign poition, W f, i: f f W = W ( ) 0 + R + Where R i the return of the foreign aet in it local currency and i the percentage currency movement (e.g. change in the dollar value of the foreign currency). Invetor decide to hedge a proportion h of the foreign aet againt currency rik, by elling the foreign currency forward 0. Foreign aet are treated a an homogeneou aet cla with a ingle currency. Thi i equivalent to aying that American invetor care about an appreciation of the dollar againt all currencie (a drop in the weighted average dollar value of foreign currencie, where the weight are thoe of the elected foreign aet allocation). A hedge ratio of zero implie no currency hedge and a hedge ratio of one implie full currency 7 For a dicuion of narrow framing, or mental accounting, ee Tverky and Kahneman (98), Shiller (999), Thaler (999), Shefrin and Statman (000), Kahneman (003), Barberi and Huang (004). 8 William M. Mercer Invetment Conulting' Survey of Penion Plan On Currency Iue, September 000 (conducted in 000 with repone from more than 00 large fund). 9 R in equation (4) would imply be replaced by the weighted-average return on the global allocation. 0 We aume that interet rate are equal worldwide, o that the forward exchange rate i equal to the pot exchange rate. 8

10 hedging. A forward contract have a zero initial value, the initial wealth i unchanged by the hedging deciion. Given a hedge ratio h, the final wealth value i given by: W = W + W ( + R + ) hw d f f 0 0 [ ] W = W + W (+ R + h ) = W + W ( h) (3) d f H f 0 0 where W H refer to final wealth with full hedging. For our purpoe, the global aet allocation i fixed. Hence, the value (traditional utility) of final wealth, V(W), can be written a a function of h, the ole deciion variable, and of the two tochatic variable R and : [ ] VW ( ) = vr ( + h) f f f Note that derivative atify the condition v' = W0 V ' and v'' = W0 W0 V ''. The modified utility can be written a: UW ( ) = uhr (,, ) = vr ( + [ h ] ) + f( vr ( + [ h ] ) vr ( + max [ h ] )) (4) Where v(.) and f(.) are monotonically increaing and concave; f(.) i decreaingly concave (f"<0, f"'>0) and f(0)=0. We aume that invetor only exhibit regret on the currency dimenion, not on the aet allocation. Thi i a natural aumption of our model, a invetor only optimize the currency hedging dimenion. In addition experiment ( Harle, 99; Camille et al., 004) how that regret i only experienced if the outcome of unchoen option are "viible" (or "acceible"), and currency return are highly viible and emotional. Indeed, media talk daily about the fate of the dollar. In addition, all performance report eparate the currency gain/loe on the portfolio from other ource of return. Performance relative to peer or other imple hedging trategie are important. Furthermore, everyone, even outide the phere of finance, eem to have an opinion on the value of the dollar, epecially ex-pot. What might have been a reaonable hedging deciion ex-ante, can be eaily criticized ex-pot by a board of trutee. Currency hedging i eay to analyze within a regret-theoretic approach becaue, with hindight, the optimal hedging deciion can only be one of two poible choice. If the foreign currency appreciated, and whatever the poitive value of, the bet hedging policy would have been to tay unhedged (h=0). So for any poitive : [ ] max h = If the foreign currency depreciate by any amount, the bet hedging policy would have been to be fully hedged (h=). So for any negative : [ ] max h = 0 Equation (4) can be written a: uhr (,, ) = vr ( + h ) + f ( vr ( + h ) vr ( + )) + f ( vr ( + h ) vr ( )) (5) [ ] [ ] [ ] + Let u focu on the impact of a currency movement. The utility u(.) i continuou and twice differentiable except in =0. At =0, the left-hand derivative with repect to i equal to: 9

11 u = ( hv ) '( R ) + ( h) f'(0) v'( R ) The right-hand derivative i equal to: u = ( hv ) '( R ) hf'(0) v'( R ) At =0, the lope on the negative ide i greater than on the poitive ide, a the difference f '(0) v'( R ) i alway poitive. A a reult, the utility function u(.) preent a kink at = 0. Furthermore, the function u(.) i concave with repect to (ee Appendix A). The current exchange rate i a reference point and invetor are more enitive to reduction in financial wealth than to increae in financial wealth. Thee are common feature in propect theory. Here we have "currency lo averion", to coin a term frequently ued in behavioral finance. Thi reult from the regret averion on currency loe. In Figure, we illutrate the modified utility with regret averion by auming two imple functional form for v(.) and f(.). Invetor have a logarithmic value function and f(x) i a negative exponential in the form e X. For thi illutrative purpoe focuing on currency rik, we aume that the return on foreign aet R i non-tochatic. Without lo of generality, we take R=0. Wealth i aumed contant except for the impact of a currency movement from the current exchange rate. So the utility depicted in Figure i a function of the exchange rate movement but alo of the hedging deciion that i taken. To contrat thi modified utility with the traditional value function, we alo plot the utility function without regret in dahed line. Let u firt conider Figure a which aume that no currency hedging take place (h=0). For poitive value of, utility increae with becaue of the increae in the value function and the abence of regret (no hedging i ex-pot optimal when the foreign currency appreciate). For negative value of, utility decreae with - becaue of the decreae in the value function but alo becaue invetor experience regret of not having hedged. Let u now conider the cae, illutrated in Figure b, where the invetor undertake ome currency hedging (h=0.5). For poitive value of, on the one hand, the value function increae with becaue of the wealth increae caued by 50% of the currency appreciation (half of it i hedged), but there i regret to have only half of the poition hedged while the currency appreciate. A a reult, the lope become maller than for the utility function without the regret term. For negative value of, the invetor uffer a reduction in wealth (maller than when h=0), but alo a regret to have only hedged 50% of the poition when it would have been optimal to have hedged fully. Figure c illutrate the cae of full hedging. For poitive value of, the invetor' wealth remain unchanged (full hedge), but he experience regret to have mied the currency appreciation. For negative value, the value function remain unchanged (a the currency depreciation i fully hedged) and there i no regret a the optimal hedging deciion wa taken. Note that in all three cae there i currency lo averion, with a kink at =0 : the lope of the utility function i larger for negative value of than for poitive. The combination of the two function yield a very imple form for the modified utility. It i equal to a contant plu: log(+ ( h)) /(+ ( h)) for negative value of and log(+ ( h)) /(+ ( h)) /(+ ( h)) for poitive value of. 0

12 Figure : Utility function with regret averion for variou hedging deciion Utility i given a a function of. Dahed line repreent value function alone while full line repreent utility function with regret averion Figure a: h=0 (No hedging) Figure b: h=0.5 (Partial hedging) Figure c: h= (Full hedging)

13 4. Derivation and Reult: Currency rik minimization The optimal hedge ratio i obtained by maximizing the expected modified utility with repect to h. It can be noted that u(h,r,) in (5) i concave with repect to h (ee Appendix B). To derive optimal hedging rule, we need to make pecific aumption on the function v(.) and f(.) to be ued a well a on the ditribution of. If f(.) i linear (no regret averion), then the problem reduce to traditional expected utility maximization, a the maximization with repect to h of the expected utility given in (5) reduce to the maximization of Ev( R + [ h] ). With a linear regret function, RT alway reduce to traditional expected utility theory. In general f(.) i aumed concave (regret averion). Except for very particular and implitic function v(.) and f(.), we cannot derive explicit hedging rule and would have to reort to numerical olution with little generality. The problem already arie in the cae of maximizing expected traditional utility in portfolio theory, but there exit ome intereting cae where explicit rule can be worked out. 3 In our model, the problem i compounded by the preence of a piece-wie regret function defined over a value function. An ad-hoc aumption, that would make the model a bit more tractable, could be to model the regret term a piece-wie linear and defined over payoff, not valuation of payoff. 4 But thi implification would not be conitent with RT and we would loe the theoretical and empirical appeal of thi approach. An intereting alternative i to ue the two-moment approximation propoed by Pratt (964) to conduct hi analyi of rik averion for mall rik. We ue a Taylor expanion of (5) and take it expected value, ignoring moment higher than two. We then maximize with repect to h and are able to derive explicit hedging rule with intereting economic interpretation. Thi two-moment Arrow-Pratt approximation i very imilar in pirit and reult to the multivariate normality aumption for return ditribution that wa introduced in the finance literature 5. In both cae, we end up with model relying olely on the firt two moment of return ditribution. In traditional finance model, the normality aumption implie that for wellbehaved utility function, expected utility Eu(.) can be expreed a a function of the mean and covariance. So parameter of the utility function only affect invetment choice to the extent that they affect the rik averion parameter in the expected utility function. In our model, the modified utility function i complex with two attribute, rik and regret. To allow for economic interpretation, we wih to explicitly retain the parameter of the modified utility in the optimal hedging rule derived under maximization of the expected modified utility. Thi cannot be done by imply auming normality of return. However, we can do it in the An unattractive alternative i to aume that v i linear (no rik averion) and f quadratic. 3 When the utility function belong to the HARA cla and aet return are multivariate normally ditributed, there i a linear relation between optimal portfolio weight and the wealth level. f(( v R h ) v( R max h ) would be replaced by h for poitive value of and ( h ) for negative value of.thi would be imilar in pirit to the piecewie linear implification in the diappointment approach of Barberi and Huang (00, 004). It would imply that the regret term i only concave at the kink =0.; invetor would not exhibit regret averion elewhere. For example a lo of x i jut twice a unpleaant a a lo x. One can check that the rik-minimizing hedge ratio would not be affected by regret and would be imilar to what obtain under traditional expected utility optimization (a hedge ratio of one). 5 Or lognormality in the cae of continuou-time model,. 4 So + [ ] + [ ]

14 cae of the Arrow-Pratt approach. 6 Mot of the hedging literature ha been uing the twomoment aumption of multivariate normal ditribution for R and, where the firt two moment of the ditribution are ufficient to characterize the whole ditribution. A we will compare our reult to thi traditional mean-variance optimization, we are quite atified with making an equivalent two-moment aumption. A mentioned before, our primary focu i on the currency-rik-minimizing behavior of invetor, where invetor earch for the optimal hedge ratio in the abence of expectation on currency movement and of pecific aumption about the correlation between foreign aet return and currency movement. Hence, we will now detail the derivation under the implifying aumption that the return on foreign aet i non-tochatic that the expected currency return i zero, and that the ditribution i ymmetric. To implify notation, and without lo of generality, we et R=0. We will provide a dicuion relaxing thoe aumption in the next ection. The expectation of (5) under thoe aumption can be written a: [ ] [ ] [ ] Eu = Ev( h ) + E f ( v( h ) v( )) + E f ( v( h ) v(0)) (6) + A mentioned above, our problem i well-behaved a firt derivative of u(.) are well-defined and continuou, except in =0, and u(.) i concave in h and. Eu(.) i concave in h a hown in Appendix B. The optimal hedge ratio atifie the firt order condition: Eu = 0 h Becaue Eu(.) i concave in h, thi firt-order condition i neceary and ufficient for optimality. In the abence of regret averion (f(.) i linear) the optimization problem reduce to the traditional expected utility optimization MaxhEv( [ h] ). Here i a pure rik (no expected return) and -h i non negative, therefore any rik avere invetor will attempt to eliminate that rik by etting h equal to. Thi i the typical full-hedging rik-minimization reult. We now derive the Arrow-Pratt approximation in the preence of regret averion (f"<0). For a given hedge ratio h, the Taylor expanion 7 of v [ ] v( [ h] ) = v(0) + ( h) v '(0) + ( h) v"(0) + o( ) Hence, the expected value function: ( h ) around =0 i: 6 Strictly peaking, the Arrow-Pratt approximation i valid for mall rik. The quality of the two-moment approximation depend on the actual return ditribution and the hape of the utility function. Thi ha been extenively dicued in the literature, ee Samuelon (970), Loitl (976), Levy and Markowitz (979), and Kroll, Levy and Markowitz (984). We thank Chritian Gollier for hi upport in getting a clearer view of thi approach. 7 Our derivation could be made a bit more formal by taking =ξ' and letting ξ become very mall, where ' i a normal ditribution. Thi i a direct application of the "compact" derivation of the approximation by Samuelon (970). 3

15 Ev( [ h ] ) v(0) + ( hev ) ( ) '(0) + ( h) E ( ) v"(0) (7) The expected regret function, over +: E+ f(( v [ h] ) v()) f(0) + E+ v( [ h] ) v() f '(0) + E+ v( [ h] ) v() f ''(0) over -: E f( v( [ h] ) v(0)) f(0) + E v( [ h] ) v(0) f '(0) + E v( [ h] ) v(0) f ''(0) With: Ev ( ([ h ] ) v ( )) hev ( ) '(0) + ( h) E ( ) v"(0) Ev ( ([ h ] ) v(0)) ( hev ) ( ) '(0) + ( h) E ( ) v"(0) Let' drop the argument 0 in the derivative. Let' denote = E( ) = 0, + = E ( + ), = E ( ), Σ = E( ), Σ + = E+ ( ), Σ = E ( ). Note that f(0)=0, = + + =0 and that Σ = σ =Σ + +Σ. With ymmetric (e.g. normal) ditribution we have Σ + =Σ = Σ. The expected value function (7) become: Ev( [ h] ) v(0) + ( h) Σv" (8) Dicarding moment higher than two, we get for the expected regret function over +: + E+ f(( v [ h] ) v()) h+ v' + ( h) Σ v" f ' + h Σ+ v' f '' (9) Similarly for -: E f( v( [ h] ) v(0)) ( h) v' + ( h) Σ v" f ' + ( h) Σ v' f '' (0) The expected utility i the um of three term: Eu = (8) + (9) + (0) () The expected utility () can be rewritten by grouping the term in variou power of h a: Eu v(0) + v" Σ + v ' f ' + f ' v" f ' Σ + v ' f '' Σ hv" Σ hv" f '( Σ + +Σ ) hv ' f '' Σ () + hv" Σ + hv" f'( Σ + +Σ ) + hv' f''( Σ + +Σ ) 4

16 Where Σ + +Σ =Σ. We now compute the firt order condition for optimal hedging by etting the derivative of Eu(.) with repect to h equal to zero. Hence: h 0 v" v" f ' v' f '' hv" hv" f ' hv' f '' * = Σ Σ Σ + Σ + Σ + Σ (3) v f v f v f v f = = v f v f v f v f "( + ') Σ + ' '' Σ "( + ') + ' ''( Σ / Σ) "( + ') Σ + ' '' Σ "( + ') + ' '' We can rearrange h *, noting that h * Σ = Σ : v' f '' = = θ v"( + f ') + v' f " The optimal hedge ratio i equal to one, a would obtain in rik minimization without regret, minu a term linked to regret averion. Note that v and f are poitive (invetor prefer more wealth and le regret), v i negative (rik averion) and f i alo negative (regret averion), o θ i generally poitive and leer than one. Let' now introduce traditional rik averion λ = v" v' and, following Bell (983), define regret averion ρ a: v' f " ρ = (5) + f ' We can now rewrite θ a : ρ θ = ρ + λ. A mentioned previouly, when the regret function f(.) i linear (f"=0), we are back to traditional utility maximization and an optimal hedge ratio of. Since rik averion and regret averion are both poitive, the optimal hedge ratio i alway between 50% and 00%. Ceteri paribu, the lower the regret averion ρ, the higher the optimal hedge ratio. When regret averion ρ i very mall relative to rik averion λ (value dominate regret and θ i cloe to zero), h * goe to a ρ goe to 0, a can be expected from traditional expected utility theory. Converely, when regret averion i large relative to rik averion (θ cloe to one), the optimal hedge ratio get cloe to 50%. We call infinite regret averion the cae where regret averion i very large relative to rik averion. 8 With thi aumption of infinite regret averion, we believe that our regret-theoretic model yield imilar reult to the minimax regret deciion rule of Savage (954). In thi early model, Savage aumed that agent conider, for all poible deciion, the maximum regret that they may carry ex-pot. Agent then elect the deciion that carrie the maller uch maximum regret. Note that thi deciion i taken irrepective of the likelihood that uch a (4) 8 Note that thi doe not necearily imply that ρ i infinite. It could alo be that regret averion dominate traditional rik averion and that ρ i finite but λ i equal to zero. What really matter i the ratio ρ / λ. 5

17 regret may actually occur (provided that the probability i trictly poitive). Intuitively, in our model with infinite regret averion, invetor care excluively about the higher level of regret attained for any hedging deciion, whether in the region of gain or in the region of loe. When they hedge fully, invetor anticipate that the maximum regret aociated with a trong appreciation of the foreign currency, though unlikely, i o high that they reject uch a hedging deciion. Converely, if they do not hedge at all, the regret aociated with a trong depreciation in the foreign currency i again perceived a extremely high, even though it may be very unlikely, and i again rejected. A we aumed that the ditribution of the foreign currency value i ymmetric, the naïve 50% hedging policy will alway be wrong and exhibit regret ex-pot. However, the maximum amount of regret will be cut in half whether it i attained in the region of gain or in the region of loe. To ummarize the cae of pure currency-rik minimization, a regret avere invetor will alway hedge le than 00%, the optimum hedging of a traditional expected-utility maximizer. However an optimal hedge ratio of 50% will only obtain for infinite regret averion. In general the optimal hedge ratio will be between 00% and 50%, depending on regret averion. 5. Derivation and Reult: General cae We will now conider the general cae where the return on foreign aet i tochatic and where the expected currency return can be non-zero. The derivation follow the previou methodology and are given in the Appendix C. The expected value to be maximized with repect to h i: [ ] [ ] [ ] EuhR (,, ) = EvR ( + h ) + E fvr ( ( + + h ) vr ( + )) + E f(( v R + h ) v( R )) where R and are tochatic with mean R and, o that R = R+ r, where r i a random variable with zero mean, Σ = E ( ) and Σ = Er ( ). r Traditional utility with no regret To compare with the exiting literature on hedging, let' firt conider the pecial cae where there i no regret. Then for mall rik 9, we get by developing around R : Eu( h, R, ) E( r + [ h] ) v'( R) + E( r + [ h] ) v"( R) Eu( h, R, ) [ h] v '( R) + Σ r + ( h)cov( r, ) + ( h) Σ v"( R) (7) The optimal hedge ratio i obtained by etting to zero the derivative of (7) with repect to h. We obtain: [ ] 0 = v '( R) + cov( r, ) Σ + hσ v"( R) (6) 9 The derivation can be made more formal by etting =ξ', R=ξR' and letting ξ become very mall, where ' and R'' are multivariate normal ditribution. Thi i the pirit of the approach of Samuelon (970). 6

18 * v'( R) cov( r, ) cov( r, ) h = + + = + Σ v"( R) Σ Σ λ Σ Where λ i the Arrow-Pratt meaure of local rik averion, -v"/v'. Thi reult i the traditional one in the hedging literature. We will refer to it a the mean-variance cae. It would be exact if the value function wa quadratic or the ditribution multivariate normal. Ceteri paribu, a poitive expectation on the foreign currency movement reduce the optimal hedge ratio (peculative term). The leer the rik averion, the lower the hedge ratio (invetor peculate more). Similarly, a negative covariance 30 between foreign aet return and currency movement (the local price of the foreign aet tend to go up when the foreign currency depreciate) reduce the optimal hedge ratio (covariance term). Modified utility with regret The derivation for the general cae in preence of regret i given in Appendix C. The optimal hedge ratio in the general cae i: * Σ+ ρ cov( r, ) λ h = + (9) Σ ρ + λ Σ ρ+ λ Σ λ+ ρ The hedge ratio i equal to 00% (the rik-minimizing term) minu three term: a) regret term: h Σ ρ Σ ρ + λ + regret = Thi i imilar to the expreion that Σ+ (8) (0) θ of (4) in the currency rik-minimizing cae, except will generally differ from Σ, if the expectation of differ from zero3. Hence, the previou dicuion applie with one caveat. If > 0, Σ+ will be greater than Σ (for a ymmetric ditribution) and invetor will hedge le becaue they anticipate to experience le regret if they decide not to hedge than in the rik-minimizing cae. Thi i becaue they ue the current exchange rate a reference point, while they anticipate the future exchange rate to appreciate. Converely they would hedge more if they anticipate the foreign currency to depreciate. b) peculative term: v'( + f ') hpecul = = () Σ v"( + f ') + v' f '' Σ ρ + λ A in the traditional mean-variance cae (8), a poitive expectation on the foreign currency movement reduce the optimal hedge ratio. The leer the rik averion λ, the lower the hedge 30 The term cov( r, ) can be thought a the elaticity, or beta, of aet return to currency movement. Σ 3 Thi would alo be the cae if the ditribution of i not ymmetric. 7

19 ratio (invetor peculate more). But thi i a modified rik averion that take regret into account. Regret averion will, overall, add to traditional rik averion: Without concavity in the regret function f, the rik averion would be imilar to the traditional one. In general, regret add to rik averion becaue f" i negative. Ceteri paribu, regretavere invetor will tend to "peculate" le on their anticipation of currency movement. However, a argued above, thi effect will be mitigated by the regret term. h c) covariance term cov( r, ) v"( + f') cov( r, ) λ = = Σ v"( + f ') + v' f '' Σ λ + ρ cov Where γ = λ/( λ+ ρ) i poitive. In the trivial cae of a linear regret function, γ = λ/( λ+ ρ) = and the covariance hedging term i identical to that in the traditional meanvariance cae a hown in (8). In general, invetor will take into account the covariance between aet return and currency movement, or more preciely the elaticity of aet return to currency movement, in their hedging deciion. However, in the preence of regret, γ i le than one and invetor tend to deviate le from their rik-minimizing hedging policy. The intuitive explanation i traightforward. A negative correlation between foreign aet return and currency movement implie that aet return tend to often the impact of currency rik at the portfolio level; but regret i only meaured on the currency movement itelf, not on aet return. The value function in the modified utility take into account total portfolio rik and ugget a lower hedge ratio becaue of the negative correlation, but thi i partly dampened by regret averion on currency loe. Note that with infinite regret averion, the peculative and covariance term are equal to zero, and the optimal hedging policy i a 50% hedge ratio. In the limit, with infinite regret averion, the covariance term will no longer influence the hedging deciion. Thi ugget that our reult from previou ection on infinite regret avere invetor i robut to aumption on the covariance between foreign aet and currency movement and on the expected movement of the foreign currency. Explaining variou hedging policie Let u now return to the quetion aked in the introduction. How can our model explain the variou hedging policie oberved in Figure : A proportion of invetor adopt a full hedging policy. Thee are traditional invetor experiencing no regret. Currency expectation do not affect their benchmark policy and they conider that the covariance between aet return and currency movement i null. A ignificant proportion of invetor adopt a 50% hedging policy. They could be regarded a invetor with infinite regret averion. A far a currencie are concerned, they wih to minimize regret. A dicued above, thee invetor try to minimize the ize of ex-pot regret. The no-hedging policy can have everal explanation. Our model ugget two poible explanation that can alo be advanced for traditional invetor. 3 Firt invetor could () 3 Becaue currency return are a ignificant component of the total return on foreign aet, it i not urpriing to remain unhedged on foreign aet that have been elected to enhance return. But it i clear that explanation for a no-hedging hedge ratio can alo be found outide the cope of our model. Some invetor are not allowed to 8

20 hold expectation that their own currency i overvalued and that foreign currencie will appreciate in the long run. If the expectation i large enough, thi will ugget keeping a full currency expoure on foreign aet becaue of the peculative term () and the regret term (0). Another traditional explanation i that invetor conider equitie a "real" aet. Conider for example a high-inflation country where all price, including nominal equity price, go up at the rate of inflation. Then, the local currency alo depreciate at that inflation rate. Thi would ugget a trong negative correlation between r and and the term cov( r, ) in the covariance term () would Σ be equal to minu one. Invetor with no regret (ρ = 0) would choe a zero hedge ratio in the abence of currency expectation a can be een in equation (8). Indeed, Froot (993) ugget that inflation get built up in exchange rate and equity price over the long run, implying a trong negative covariance term. Invetor who adopt another hedging policy (3% of all invetor) could do o becaue they have ome level of regret averion between zero and infinity. The interaction between rik and regret averion We now dicu how the relative level of regret and rik averion affect hedging policie. We provide in Figure 3 the optimal hedge ratio and it three component (regret term, peculative term and covariance term) for different level of regret averion. We arbitrarily et rik averion at one, and let regret averion vary from zero to 5. A an illutration we ue different aumption on market expectation. In all cae, the tandard deviation of percentage currency movement i et at 0% per year (o Σ = % ). Figure 3a, preent the cae where invetor hold no expectation about currency movement o that = 0 and cov( r, ) = 0. A dicued previouly, we find that the rik-minimizing hedge ratio i 00% in the abence of regret averion. The optimal hedge ratio reache 75% when regret averion equal rik averion and drop to 50% when regret averion dominate rik averion. Figure 3b introduce a poitive expectation on the foreign currency with = % per year. In the abence of regret, the optimal hedge ratio 33 i 0%. A regret averion increae, the optimal hedge ratio increae. Thi i caued by a trong increae in the peculative term in () a ρ increae: remaining unhedged create the potential for a trong regret that i not offet by the utility of the higher expected return. The optimal hedge ratio i 5% when regret averion equal rik averion, and reache 50% when regret averion dominate rik averion. Figure 3c introduce a poitive expectation on the foreign currency a well a a negative covariance between aet return and currency. In other word, the price of the foreign aet, meaured in foreign currency, tend to rie when the foreign currency depreciate. Thi i an attractive feature from a rik viewpoint a it reduce the impact of a currency lo; it lead to a ue derivative or find the adminitrative cot and rik of hedging not worth it. Other wih to get a currency expoure becaue currencie bring an element of diverification to their dometic portfolio. Thi i typically the cae if foreign aet repreent a mall portion of the overall portfolio, ee Jorion (999). Finally, the data in Figure are benchmark aigned to currency overlay manager not the actual "final" hedge ratio. When the benchmark i no hedging, a currency overlay manager might engage in currency-hedging to attain both a lower rik level and a higher return than the benchmark. 33 The ratio of expected currency movement to variance i equal to one when = Σ = %. For a rik averion of one, thi tranlate into a peculative term of 00% for the hedge ratio. 9