Probability Weighting of Rare Events and Currency Returns

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1 Probability Weighting of Rare Events and Currency Returns Fousseni Chabi-Yo and Zhaogang Song March 17, 2013

2 Currency Returns: Carry Trade Strategy Currency carry trade strategy: borrow in countries with low interest rates and invest in the currencies of countries that offer high interest rates.

3 Currency Returns: Carry Trade Strategy Currency carry trade strategy: borrow in countries with low interest rates and invest in the currencies of countries that offer high interest rates. The economic malaise of the E.U. and Japan has central bankers and bondholders worried. But it has currency speculators and arbitragers licking their lips at the prospect of riding the euro and yen on their downward slide, while at the same time making fat profits on high-yield investments and rising currencies in emerging markets...this sophisticated bit of financial wizardry is known as a carry trade, and its a popular way to offset moves in equity and bond markets while also producing a handsome return. (Time magazine, 1/26/11)

4 Carry Trade Returns 1 Understanding the role of peso effects in explaining carry trade payoffs (e.g., Jurek (2009), Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011)).

5 Carry Trade Returns 1 Understanding the role of peso effects in explaining carry trade payoffs (e.g., Jurek (2009), Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011)). 2 Identifying risk factors that determine average currency returns (e.g., Lustig, Roussanov, and Verdelhan (2011), Ang and Chen (2010), and Menkhoff, Sarno, Schmeling, and Schrimpf (2011), Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2009)).

6 Carry Trade Returns 1 Understanding the role of peso effects in explaining carry trade payoffs (e.g., Jurek (2009), Burnside, Eichenbaum, Kleshchelski, and Rebelo (2011)). 2 Identifying risk factors that determine average currency returns (e.g., Lustig, Roussanov, and Verdelhan (2011), Ang and Chen (2010), and Menkhoff, Sarno, Schmeling, and Schrimpf (2011), Farhi, Fraiberger, Gabaix, Ranciere, and Verdelhan (2009)). 3 Understanding whether carry trade returns exhibit time-varying predictability (Bakshi and Panayotov (2012)).

7 Currency Momentum Currency momentum: long in currencies with high past excess return ( winners ) and short in currencies with low past excess returns ( losers ). We find a significant cross-sectional spread in excess returns of up to 10% p.a. between past winner and loser currencies. This spread in excess returns is not explained by traditional risk factors...(menkhoff, Sarno, Schmeling, and Schrimpf (2012)).

8 Questions that still warrant reconciliation are: 1 Do carry trade returns exhibit time-series predictability? 2 What type of asset pricing model explains carry trade and currency momentum returns?

9 Probability Weighting of Rare Events in Currency Markets In currency markets, investors use non-expected utility models to maximize their utility Few examples of non-expected utility models: Rank-Dependent Expected Utility model in Quiggin (1993) The cumulative prospect theory: Tversky and Kahneman (1992), Barberis and Huang (2008). Main idea in currency markets: Investors overweight (underweight) the likelihood of extreme events in the tails of the distribution of exchange rates.

10 Finding We use options written on exchange rates to recover the amount by which investors, in currency markets, overweight or underweight the tails of the distribution of exchange rates. We construct probability weighting measures of rare events, in currency markets, that capture investors attitude toward extreme downside losses versus upside gains.

11 Finding 1 Probability weighting of rare events: 1 predict individual carry trade returns in both emerging and developed economies, carry trade portfolio returns (DOL and HML) 2 explain the cross-section and time variation in both carry trades and currency momentum returns.

12 Asimplemodel In currency markets, the representative investor maximizes the non-expected utility (RDEU): with max U [ω] ω W t+1 = ωr t+1 +(1 ω) Rt f,fcu. (1) U [ω] =E (u [W t+1 ] Z [P [. F t ]] F t ), (2) where R t+1 =(1+rt f,us ) S t+1, (3) S t S t denotes the spot exchange rate expressed as FCUs per US dollar. G [P] = Z [P [. F t ]]dp is the probability weighting measure. Expected Utility: Z [P [. F t ]=1

13 Asimplemodel Use the FOC to show u [W t+1 ]Z [P [. F t ]] = 1 1+r f,fcu t q [ St+1 S t p [ St+1 S t ] σt FX σ FX t ], (4) where q [ St+1 S t ] σt FX u [W t+1 ]= W 1 γ t+1 1 γ σ FX t [ neutral measure. p St+1 S t under the physical measure. with γ =0, 1, 2. =distribution of the exchange rate under the risk ] =distribution of the exchange rate

14 Can Probability Weighting of Rare Event Explain Currency Returns? Define the excess currency return as r e t+1 = R t+1 R f,fcu t, E [ r e t+1 ] 1 P0 = R f,us t δ L + Rt f,us (G [P] P) dp Rt f,us δ R, P 0 where the rare event measures δ L and δ R are defined as δ L = Implications P0 0 (G [P] P) dp and δ R = 1 1 P 0 (P G [P]) dp. δ R = E [ r e t+1] and δ L = E [ r e t+1]

15 Estimated Probability Weighting Function 1 Probability Weighting Function: Illustration 0.9 CORRECT PROBABILITY LINE δ L OVERWEIGHTING Probability Weighting Function δ R UNDERWEIGHTING Probability

16 Data Data of daily spot and forward rates. We start from daily spot and 1-month forward exchange rates versus the US dollars (USD), obtained from Barclays and Reuters (via Datastream). We follow Lustig et al. (2011) and select 32 different currencies.

17 Data Data of daily spot and forward rates. We start from daily spot and 1-month forward exchange rates versus the US dollars (USD), obtained from Barclays and Reuters (via Datastream). We follow Lustig et al. (2011) and select 32 different currencies. High-frequency spot exchange rates. To compute the exchange rate volatility σt FX at the daily frequency, we obtain data of high-frequency spot exchange rates from the Thomson Reuters Tick History database.

18 Data Data of daily spot and forward rates. We start from daily spot and 1-month forward exchange rates versus the US dollars (USD), obtained from Barclays and Reuters (via Datastream). We follow Lustig et al. (2011) and select 32 different currencies. High-frequency spot exchange rates. To compute the exchange rate volatility σt FX at the daily frequency, we obtain data of high-frequency spot exchange rates from the Thomson Reuters Tick History database. Data of currency options. We obtain daily over-the-counter (OTC) currency options data from JP Morgan for all the 32 currencies covered in our sample of January 1996 June 2012.

19 Estimated Probability Weighting Function 1 Estimates of G for AUD/USD over Probability Weighting Function γ=0 γ=1 95% Upper 95% Lower γ=2 45 o Line Cumulative Physical Distribution

20 Rare Event Measures δ L = P 0 0 (G [P] P) dp and δ R = 1 1 P 0 (P G [P]) dp Monthly δ L for AUD/USD over Monthly δ R for AUD/USD over δ L δ R

21 Currency Portfolios Sorted on Individual δ L and δ R δ: All Economies δ L Portfolio Avg. H/L Mean (%) (0.45) (0.85) (1.80) (1.41) (3.30) (1.66) (2.46) Sharp Ratio δ R δ Mean (%) (2.76) (1.49) (0.46) (1.47) (1.32) (1.66) (-2.13) Sharp Ratio

22 Cross Sectional Average of Individual Rare Event Measures δ L,G = 1 K K δk L and δ R,G δ G = 1 K k=1 K ( ) δ R k δ k k= δ L,G δ R,G -δ G

23 Return Predictability for Individual Currencies Return Predictability for Individual Currencies Predictability [ with x t = f t s t,δ L,G t rt+1 e = α + β x t + ε t+1, ] and xt = [ f t s t,δt R,G δt G The global left tail weighting measure δ L,G is a good predictor of individual carry trade returns in both developed and emerging economies after controlling for the forward discount. ]. The global right tail weighting measure δ R,G δ G performs well in predicting currency returns in emerging economies after controlling for the forward discount.

24 Return Predictability for Currency Portfolios Return Predictability for Currency Portfolios: All Economies 1 Predicting DOL returns r (DOL) t+1 = α (2.45) δl,g t (2.95) AFDt + ε t+1 ( δ R,G t δt G r (DOL) t+1 = α 2.50 ( 2.05) 2 Predicting HML returns ) (4.51) AFDt + ε t+1 r (HML) t+1 = α + 10 (4.07) δl,g t 0.13 ΔCRBt ( 1.31) (1.10) ΔσFX t (1.05) ΔLIQt + ε t+1 ( ) r (HML) t+1 = α 3.15 δ R,G t δt G +0.05ΔCRBt ( 2.41) (0.60) (1.02) ΔσFX t (0.37) ΔLIQt + ε t+1

25 Cross Section of Currency Returns Asset Pricing Results of Carry Portfolios: δ L,G versus HML, Volatility, and Skewness Risks Panel A: Δσ FX t DOL δ L,G Δσ FX t R 2 (%) χ 2 SH λ (SH) (1.66) (2.83) (-6.11) (0.35) Panel B: skew t DOL δ L,G skew t R 2 (%) χ 2 SH λ (SH) (1.66) (2.36) (-0.42) (0.64) Panel C: HML DOL δ L,G HML R 2 (%) χ 2 SH λ (SH) (1.67) (2.25) (6.97) (0.32)

26 Cross Section of Currency Returns Asset Pricing Results of Carry Portfolios: δ R,G δ G versus HML, Volatility, and Skewness Risks Panel A: Δσ FX t DOL δ R,G δ G Δσ FX t R 2 (%) χ 2 SH λ (SH) (1.66) (-2.75) (-6.16) (0.06) Panel B: skew t DOL δ R,G δ G skew t R 2 (%) χ 2 SH λ (SH) (1.64) (-1.79) (-0.60) (0.42) Panel C: HML DOL δ R,G δ G HML R 2 (%) χ 2 SH λ (SH) (1.66) (-1.73) (7.07) (0.05)

27 Cross Section of Currency Returns Fama-MacBeth Asset Pricing Results of Momentum Portfolios: δ L,G and δ R,G δ G versus HML, Volatility, and Skewness Risks Holding Period h 1 3 (f ) Δσ FX t skew t HML Δσ FX t skew t HML 1 δ L,G (SH) (3.24) (2.38) (2.78) (4.03) (3.92) (4.05) 3 δ L,G (SH) (5.18) (4.87) (5.10) (5.16) (5.16) (5.11) 1 δ R,G δ G (SH) (-2.23) (-1.62) (-1.64) (-4.41) (-4.55) (-4.44) 3 δ R,G δ G (SH) (-5.35) (-5.38) (-5.20) (-4.91) (-4.92) (-4.89)

28 Cross Section of Currency Returns Robustness Our cross-sectional results are robust to adjustment of bid-ask spread in computing excess returns. The global probability weighting measures δ L,G and δ R,G δ G explain US equity momentum portfolios and international sovereign bond portfolio returns.

29 Cross Section of Currency Returns Conclusion We use option on exchange rates, and exchange rates to recover probability weighting measures that capture investors attitudes toward extreme downside losses versus upside gains. We show that the probability weighting of rare events predict and explain currency returns. Results are robust to transaction costs. THANKS!

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