The Quanto Theory of Exchange Rates Lukas Kremens Ian Martin April, 2018 Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 1 / 36
It is notoriously hard to forecast exchange rates Much of the literature is organized around the uncovered interest parity (UIP) benchmark, which predicts that exchange rate movements should offset interest rate differentials on average, and thereby equalize expected returns across currencies Hansen–Hodrick (1980), Fama (1984), and others: UIP fails badly Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 2 / 36
Three appealing properties of UIP Based on asset prices alone: observable in real time; no reliance on 1 infrequently updated, imperfectly measured macro statistics No free parameters: nothing to estimate, so no in-sample / 2 out-of-sample issues Straightforward interpretation: represents the expected currency 3 appreciation perceived by a risk-neutral investor #1–#3 explain why UIP is such an important benchmark #3 also explains why it should never have been expected to work empirically: risk neutral expectation E ∗ t � = true expectation E t Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 3 / 36
This paper We propose an alternative benchmark, the quanto theory , that has the three appealing properties, but also allows for risk aversion . . . and performs well empirically Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 4 / 36
Theory (1) Start from a fundamental equation of asset pricing, � � M t + 1 � E t R t + 1 = 1 ◮ E t : expectation conditional on time- t information ◮ M t + 1 : SDF that prices dollar payoffs ◮ � R t + 1 : any gross dollar return Since E t M t + 1 = 1 / R $ f , t , we can write this as � � E t � R t + 1 − R $ f , t = − R $ M t + 1 , � f , t cov t R t + 1 Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 5 / 36
Theory (2) Currency trade: take a dollar, convert to euros, invest at the (gross) euro riskless rate, R e f , t , and then convert back to dollars e t : price of a euro in dollars, so e 1 = $ e t and $ 1 = e 1 / e t Return on currency trade is R e f , t e t + 1 / e t Setting � R t + 1 = R e f , t e t + 1 / e t and rearranging, � � R $ e t + 1 M t + 1 , e t + 1 f , t − R $ = f , t cov t (1) E t R e e t e t f , t � �� � ���� risk adjustment UIP forecast Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 6 / 36
Theory (3) Sometimes convenient to use risk-neutral notation, 1 time t price of a claim to $ X t + 1 = E ∗ t X t + 1 = E t ( M t + 1 X t + 1 ) R $ f , t The identity (1) can be rewritten R $ e t + 1 f , t E ∗ = t e t R e f , t Reduces to UIP in a risk-neutral world in which E ∗ t = E t Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 7 / 36
Theory (4) The UIP forecast is the expected appreciation perceived by a risk-neutral investor—but this is a very unrealistic perspective What about an investor with log utility? Answer: depends on the investor’s financial wealth, background risk, human capital, etc. . . But if the investor is unconstrained, with wealth fully invested in the market, � e i , t + 1 � R $ e i , t + 1 + 1 f , t = cov ∗ , R t + 1 E t t R i R $ e i , t e i , t f , t f , t where R t + 1 is the return on the market Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 8 / 36
Theory (5) Result (An identity) More generally, � e i , t + 1 � � � R $ e i , t + 1 + 1 M t + 1 R t + 1 , e i , t + 1 f , t = cov ∗ , R t + 1 − cov t E t t R i R $ e i , t e i , t e i , t f , t f , t � �� � ���� � �� � residual UIP forecast quanto-implied risk premium (2) where R t + 1 is an arbitrary dollar return, and the first covariance term is computed using the risk-neutral probability distribution Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 9 / 36
Theory (6) Relies only on absence of arbitrage: in particular, must hold in any equilibrium model We do not assume complete markets We do not assume existence of a representative agent We do not assume everyone is rational We do not assume everyone is unconstrained We do not assume lognormality Must hold even for pegged or tightly managed exchange rates Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 10 / 36
Theory (7) Tension between two goals: want to choose R t + 1 (i) to make the second term measurable; and (ii) to make the third term small (ideally, negligible) We will set R t + 1 equal to the return on the S&P 500 index Then the second term is measurable given quanto forward prices on S&P 500 index The third term is zero from the log investor’s point of view because M t + 1 = 1 / R t + 1 Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 11 / 36
Measuring risk-neutral covariance Conventional forward Quanto forward A commitment to pay $ F t in A commitment to pay e Q t in exchange for value of S&P 500 exchange for value of S&P 500 index in dollars, $ P t + 1 . Payoff index in euros, e P t + 1 . Payoff is is $( P t + 1 − F t ) at time t + 1 e ( P t + 1 − Q t ) , or equivalently $ e t + 1 ( P t + 1 − Q t ) , at time t + 1 To make value equal to zero at To make value equal to zero at initiation, Q t = E ∗ t e t + 1 P t + 1 initiation, F t = E ∗ t P t + 1 E ∗ t e t + 1 It follows that � e t + 1 � Q t − F t 1 = cov ∗ , R t + 1 t R e R $ e t f , t P t f , t Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 12 / 36
The log investor Result The exchange-rate appreciation anticipated by a log investor who holds the S&P 500 index can be computed from asset prices via the equation R $ e i , t + 1 + Q i , t − F t f , t − 1 = − 1 E t R i R i e i , t f , t P t f , t � �� � � �� � IRD i , t QRP i , t � �� � ECA i , t Equivalently, the currency risk premium anticipated by such an investor is revealed by QRP: R $ e i , t + 1 f , t − = QRP i , t E t e i , t R i f , t Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 13 / 36
Beyond the log investor We view the log investor as a benchmark Well suited for out-of-sample forecasting: no free parameters But also allow for nonzero second covariance term in various ways ◮ Intercept (captures potential dollar effect) ◮ Fixed effects (captures currency-specific but time-invariant effects) ◮ Other proxies (both currency-specific and time-varying) ⋆ IRD i , t ⋆ QRP i , t ⋆ Average forward discount, IRD t (Lustig, Roussanov and Verdelhan, 2014) ⋆ Log real exchange rate, RER i , t (Dahlquist and Penasse, 2017) Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 14 / 36
Theory: summary Intuition: currencies that perform poorly when marginal value of a dollar is high (‘bad times’) are risky and must earn a risk premium Thinking from the perspective of the log investor, the notion of ‘bad times’ is revealed by the return on the market Currencies with positive (risk-neutral) covariance with the market are risky Quantos reveal this risk-neutral covariance Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 15 / 36
Data Monthly data on quanto forwards ( Q i , t ) and conventional forwards ( F t ) on the S&P 500, obtained from Markit ◮ Australian dollar (AUD) ◮ Canadian dollar (CAD) ◮ Swiss franc (CHF) ◮ Danish krone (DKK) ◮ Euro (EUR) ◮ British pound (GBP) ◮ Japanese yen (JPY) ◮ Korean won (KRW) ◮ Norwegian krone (NOK) ◮ Polish zloty (PLN) ◮ Swedish krona (SEK) Maturities of 6, 12, and 24 months, Dec 2009 to Oct 2015 Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 16 / 36
Currency forecasts, 2yr horizon Expected currency appreciation (ECA) Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 17 / 36
Currency forecasts, 2yr horizon Expected excess returns (QRP) Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 17 / 36
IRD 1 CHF CHF JPY JPY 3 QRP - 1 EUR EUR 1 2 GBP GBP CAD CAD DKK DKK SEK SEK - 1 KRW KRW NOK NOK - 2 - 3 AUD AUD PLN PLN - 4 IRD and QRP negatively correlated in time series and cross section High interest rates ← → high risk premia: carry trade is profitable Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 18 / 36
Testing the model R $ − 1 = Q i , t − F t e i , t + 1 f , t Log investor: E t − 1 + e i , t R i f , t P t R i f , t � �� � � �� � QRP i , t IRD i , t We test the model by forecasting R $ ◮ currency excess return: e i , t + 1 f , t e i , t − R i f , t ◮ currency appreciation: e i , t + 1 e i , t − 1 Stylized facts from the literature ◮ High-interest-rate currencies have high excess returns (eg, Hansen–Hodrick, 1980; Fama, 1984) ◮ Hard to forecast currency appreciation (eg, Meese–Rogoff, 1983) Bootstrapped covariance matrices Kremens & Martin (LSE) The Quanto Theory of Exchange Rates April, 2018 19 / 36
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