The Swiss franc safety premium
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Abstract
This paper applies a recent method proposed by Maggiori (The U.S. Dollar Safety Premium, 2013) to estimate the Swiss franc safety premium. The results show that the three-step instrumental variable approach as used by Maggiori does not work for the Swiss franc exchange rates. The price of risk estimates take unrealistic, negative values. One possible explanation is that the approach as it is used by Maggiori suffers from a measurement error for the expected exchange rate which represents a potential source of imprecision. By using the prediction of an augmented Fama regression to measure the expected exchange rate change, this measurement error can be avoided and the safety premium estimates become more realistic and closer to those obtained with a maximum likelihood-estimated GARCH approach. Overall, however, the GARCH approach still seems to be preferable to the instrumental variable approach.
Keywords
Exchange rates Safe haven currency Swiss francJEL Classification
F31 G12 G15Introduction
The recent financial crisis and the subsequent European sovereign debt crisis provoked a large flight to quality among investors and caused strong upward pressure on the Swiss franc (CHF). It appreciated against the euro (EUR) by almost 40% within the relatively short time span of only 3 years. Major exchange rate interventions by the Swiss National Bank did not lead to the desired tension release, so that in September 2011, fearing an overvaluation of its currency, the Swiss central bank announced a lower bound of 1.20 on the EUR/CHF exchange rate.
A currency that has a general tendency to appreciate during episodes of intense crisis and offer hedging value against global risk is a currency that we would expect to earn a safety premium, defined as the compensation that investors require to short a safe currency and invest in a basket of foreign currencies. And as the willingness to short a safe currency decreases during risky episodes, we would expect this safety premium to be time-varying and to reach its highest values during periods of crises. One objective of this paper is to examine whether the Swiss franc earns a safety premium and to give an idea about its approximate size. Studying the potential safety premium of the Swiss franc might help to understand the dynamics of the Swiss franc exchange rate. Given Switzerland’s strong trade linkages with the rest of the world, variations in the Swiss franc exchange rate are not only an important factor in determining the profitability of Switzerland’s major export-oriented sector, but are also an important factor in determining domestic inflation. Hence, its dynamics will have major implications for monetary policy makers. Furthermore, the safety premium is a priced factor that can be reflected in many internationally traded assets.
While the focus of the recent empirical literature mainly lies on the analysis of unconditional safety premiums and ex-post currency excess returns, I make an attempt to calculate the time-varying Swiss franc safety premium, based on the conditional version of an International Capital Asset Pricing Model. An obvious way to estimate such a model would be to use a multivariate GARCH process, as has for example been done by De Santis and Gérard (1998). In a recent paper, however, Maggiori (2013) calculates the US dollar (USD) safety premium using another estimation methodology, based on the three-step instrumental variable approach developed by Duffee (2005). As compared to a GARCH approach, such a setup imposes less structure on the dynamics of the conditional covariance and has a higher flexibility. Maggiori (2013) presents promising results. In a first step, he calculates an estimate for the conditional covariance between a USD exchange rate index and the MSCI stock market return index. His results clearly show that this conditional covariance peaks in times of crisis, meaning that investors expect strong appreciations of the USD after negative stock market shocks. In a second step, Maggiori then estimates the risk price coefficient and finds a positive and significant value. By multiplying the conditional covariance by this risk price estimate, he gets an estimate for the USD safety premium, he finds the monthly USD safety premium to be around 10% in crisis times.
Applying Maggiori’s procedure to a trade-weighted Swiss franc exchange rate index and the EUR/CHF exchange rate, I expected to find similar patterns, but found results that are unsatisfactory. The conditional covariance estimates indeed confirm that investors expect the CHF to appreciate in times of crises, in other words, that they consider the CHF to be safe. At the same time, however, my results suggest that this safety is priced negatively, which is highly unrealistic. Given that investors are on average risk averse, theory and common sense tell us that the price of risk should be positive. Consequently, there seem to be some limitations in the methodology. So, a further objective of this paper is to provide a possible explanation and solution to these limitations. I argue that a potential problem lies in the construction of the dependent variable of the model: By definition, the safety premium is equal to the sum of the interest rate differential and the expected exchange rate change, which is unobservable. Maggiori suggests using the actual ex-post exchange rate change instead, which incorporates the prediction error made by investors. Given that forecasting exchange rates is difficult, this prediction error and hence the measurement error in the dependent variable are likely to be big. The solution to get around this problem is simple: The measurement error in the dependent variable can be avoided by choosing a different measure for the expected exchange rate change. I test two alternatives: The first one is to set the expected exchange rate change equal to zero, and the second one to set it equal to the prediction of an augmented Fama regression. For the Swiss franc, both options improve the results, but the second seems to dominate the first one. When using the prediction of the Fama regression, the risk price estimates become more realistic and closer to what I get when estimating the model, for comparison, with a multivariate GARCH specification, the Dynamic Conditional Correlation model by Engle (2002). Once a potential structural break in the relationship between Swiss franc exchange rate returns and equity returns in early 1999 is taken into account, these results reveal that the CHF safety premium is indeed time-varying, highest in times of crisis, and was equal to around 4.5% with peaks of up to 12.5% during the recent financial crisis, supporting the view of the CHF acting as a safe haven during periods of high risk.
Overall, my contribution shows that the three-step instrumental variable procedure proposed by Maggiori does not work for the Swiss franc and reveals a potential source of imprecision. I suggest a slight modification in the procedure that helps to improve the results for the Swiss franc. In my opinion, however, the instrumental variable approach still has some shortcomings compared to the maximum likelihood-estimated GARCH models: While maximum likelihood allows to estimate the model elegantly in one single step, the need for three separate steps to estimate it with instrumental variables is a source of impreciseness. Each individual step adds some uncertainty. Furthermore, an instrumental variable approach can only lead to convincing results when the available instruments are strong, which, at least in my sample, appears not to be the case.
The structure of the paper is as follows. The “Related literature” section gives an overview of the empirical literature on exchange rate returns and currency risk premiums. In the “Some descriptive evidence” section, I provide some descriptive evidence on the relationship between Swiss franc exchange rate changes and stock market returns. The “Safety premium model” section discusses the theoretical safety premium model. In the “Estimation strategy” section, I present the three-stage instrumental variable approach and in the “Data” section the data. The “Results” and “Time-varying price of risk” sections discuss the results and some extension. Finally, the “Comparison to a multivariate GARCH-in-mean specification” section compares the results to results obtained when using a GARCH specification and “Conclusions” section concludes and summarizes the main findings.
Related literature
After the famous results by Fama (1984), the literature on risk premiums in the foreign exchange market experienced a first boom. The goal of testing for the presence of a time-varying currency risk premium was to deliver an alternative explanation for the failure of uncovered interest parity, as opposed to the explanation of simple failure of market efficiency. Prominent contributions were made by Hansen and Hodrick (1983), Hodrick and Srivastava (1984) and Domowitz and Hakkio (1985), to name only a few. All the three of them used the two-country model developed by Lucas (1982) as a theoretical foundation. Comprehensive surveys of the early literature on currency risk premiums are provided by Hodrick (1987) and Engel (1996). Overall, this evidence that departures from uncovered interest parity might be driven by risk premiums was rather mixed and received with considerable scepticism. The empirical contribution by Lustig and Verdelhan (2007) helped to increase the popularity of currency risk premium models again. Using a version of the consumption-based capital asset pricing model, they argue that risk associated with aggregate consumption growth can account for the differences in expected returns across different currency portfolios that are formed based on the size of the interest differential towards the US dollar. Burnside (2011), however, points out some critical features in Lustig and Verdelhan’s methodology and argues that one cannot reject the null hypothesis that their model explains none of the cross-sectional variation of the expected returns. Further examples of asset pricing models of currency returns with systematic deviations from UIP are estimated by Burnside et al. (2011), Lustig and Verdelhan (2011) and Lustig et al. (2014). The work by Maggiori (2013) and this paper differ from this recent literature insofar as the focus of the latter is mainly on the cross section of currency returns and unconditional moments. Maggiori, on the other hand, suggests a procedure to study the time-series properties of currency returns and allows to estimate the conditional currency safety premium.
An alternative approach for studying the properties of currencies and exchange rates is provided by the so-called factor models, where the sensitivity of ex-post currency returns to a set of risk factors is analysed. Burnside et al. (2006) find no significant covariance with a wide array of risk factors when analysing the returns to carry trade. Work using factor models to study the safe haven properties of currencies and in particular the Swiss franc appears to be more successful. Ranaldo and Söderlind (2010) estimate linear and non-linear factor models to study high-frequency exchange rates. Using risk factors that measure the performance of stock and bond markets as well as proxies for market volatility and liquidity, they find that the Swiss franc clearly exhibits the typical pattern of a safe haven currency as it tends to appreciate when there is an increase in risk. In addition, they document that there is some non-linearity in this pattern insofar as the appreciation of the Swiss franc is more than proportional to increases in risk and particularly strong during crisis episodes. Another study in this field is Hoffmann and Suter (2010), who examine the role of global and country-specific risk factors for exchange rates of the Swiss franc and find that it acts as a safe haven against some currencies, but not all. Grisse and Nitschka (2015) analyse the relationship between Swiss franc exchange rate returns and risk factors by estimating augmented UIP regressions. They find that the CHF exhibits safe haven characteristics against most other currencies. Furthermore, they also find significant time variation in the relationship between Swiss franc returns and the risk factors, with this link becoming stronger in times of stress. Finally, there is also empirical work analysing ex-post currency returns that has its main focus on the fact that an exchange rate’s comovement with falling markets might differ from its comovement with rising markets. Hossfeld and Macdonald (2015) for example document major differences in correlations between currency returns and global stock market returns conditional on the level of financial stress. Based on these observations, they explicitly distinguish between low and high stress regimes by estimating a threshold model. Controlling for the impact of carry trade reversal, they provide further evidence that the Swiss franc qualifies as a safe haven currency. A possible explanation for the relevance of this differentiation between rising and falling markets is provided by the literature on investors’ loss aversion (see for example Kahneman and Tversky (1979) and Gul (1991)). Ang et al. (2006) find evidence for the existence of a significant risk premium for holding stocks with high sensitivities to downside market movements and Atanasov and Nitschka (2014) find that downside risk is also priced in bilateral exchange rates.
Some descriptive evidence
where e_{ t } corresponds to the (log) exchange rate defined as the price of the foreign currency in terms of home currency units and △e_{t+1} to the change in it. \(r^{\omega }_{t+1}\) is equal to the stock market return, and Open image in new window is a dummy equal to one when stock market returns are below the sample average and zero otherwise^{2}. The coefficient δ thus measures the extent of asymmetry in the exchange rate reaction to high versus low equity returns. The model is estimated by OLS, using Newey-West standard errors in order to account for serial correlation. A Wald test is then performed to test the hypothesis of β and δ being jointly equal to zero. While this hypothesis is not rejected for the USD index, it is rejected at the 5% level for the CHF exchange rates, confirming the guess that the CHF reacts asymmetrically to high versus low stock market returns. The corresponding fitted lines are plotted in the CHF scatterplots. In the USD scatterplot, due to the insignificance of the asymmetry terms, the predicted line based on the model without dummy is plotted. The sample average of stock market returns is indicated by the red vertical line^{3}.
A Chow (1960) test performed on Eq. (1) to detect a potential structural break at the introduction of the euro in January 1999 finds supportive evidence at the 5% level for the EUR/CHF exchange rate^{4}. Altogether, this break test and the results of the rolling estimation suggest that for further analyses incorporating the Swiss franc exchange rate, it might be necessary to split my sample into two subperiods^{5}.
Correlation table
\(Corr(r_{t+1}^{\omega }, \triangle e_{t+1})\) | \(Corr(r_{t+1}^{\omega }, \triangle e_{t+1})\) | \(Corr(r_{t+1}^{\omega }, \triangle e_{t+1})\) | |
---|---|---|---|
for \(r_{t+1}^{\omega } < \bar {r}\) | for \(r_{t+1}^{\omega } \geq \bar {r}\) | ||
USD Index | |||
all | 0.18*** | 0.16* | 0.23*** |
CHF Index | |||
all | 0.32*** | 0.23** | −0.02 |
T<1999 | 0.25** | 0.44*** | 0.03 |
T≥1999 | 0.37*** | 0.13 | 0.15 |
EUR/CHF | |||
all | 0.27*** | 0.24** | −0.09 |
T<1999 | 0.13 | 0.46*** | −0.03 |
T≥1999 | 0.38*** | 0.17 | 0.12 |
To summarize, the descriptive evidence provided in this section confirms that the CHF has clear safe haven tendencies vis-à-vis a trade-weighted basket of currencies and the euro in particular given the positive correlation of the exchange rate returns with the stock market returns. Hence, if investors care about risk, we would expect CHF exchange rates to, at least on average, incorporate a safety premium. However, there is also evidence that the relationship between stock market returns and exchange rate returns changed over time, namely, that it was subject to a structural break. This, in turn, suggests that it might be necessary to split my sample into two subperiods.
Safety premium model
M_{t+1} denotes the home stochastic discount factor (SDF), R_{t+1} any asset return in the home country and \(R_{t+1}^{*}\) a corresponding return in the foreign country (rest of the world). \(\mathcal {E}_{t}\) is the exchange rate which is defined as the price of the foreign currency in terms of home currency units. An increase in \(\mathcal {E}_{t}\) therefore corresponds to a depreciation of the home currency. According to Eq. (2), a home investor should expect a zero discounted excess return of investing abroad by shorting a home asset.
The lower case letters denote natural logarithms. The left-hand side of Eq. (3) is the expected excess log return of investing in the foreign risk-free asset by shorting the home risk-free asset plus Jensen’s inequality term. It defines the log currency safety premium csp_{ t } that holders of the home currency have to pay. A currency is judged to be safe if it appreciates in times of economic distress and this is exactly what the right-hand side of Eq. (3) tells us: Times of economic distress are characterized by high marginal utility growth and thus a high stochastic discount factor. csp_{ t } is positive if \({Cov}_{t} \left (m_{t+1}, \triangle e_{t+1} \right)\) is negative, thus if the home currency appreciates when the stochastic discount factor increases.
The size of the currency safety premium at a certain point in time t is determined by two components: The conditional covariance between the market return and the exchange rate change measures the time-varying quantity of risk an investor faces when investing into the foreign risk-free asset. The coefficient b_{ t } tells us how much investors care about this risk and can thus be interpreted as the price of risk. It corresponds to the size of the safety premium in a case where the conditional covariance is equal to one. Theoretically, this price of risk can vary through time, but Maggiori takes it to be constant (b_{ t }=b), an assumption which I adopt. It will later be relaxed (see “Time-varying price of risk” section).
Estimation strategy
Three-stage instrumental variable approach
To estimate a currency’s safety premium according to Eq. (4), Maggiori (2013) suggests a procedure that relies on the three-stage instrumental variable approach developed by Duffee (2005) and is closely related to the instrumental variable approach of Campbell (1987) and Harvey (1989). The advantage of Duffee’s three-stage methodology is that it imposes only little structure on the dynamics of the conditional covariance and is therefore very flexible. The first two stages aim at estimating the conditional covariance \({Cov}_{t} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\). The goal of the third stage is to estimate the price of risk, b_{ t }.
where the barred values denote sample averages.
\(\widetilde {Var_{t}} \left (\triangle e_{t+1} \right) \equiv \left (\hat {\epsilon }_{t+1}^{e}\right)^{2} \) is the ex-post estimate of the variance of the exchange rate. In order to measure the unobservable conditional expectation of the exchange rate change \(E_{t}\left [\triangle e_{t+1}\right ]\), Maggiori uses the actual ex-post exchange rate change △e_{t+1}. In the next section, I will discuss this approach in more detail and make two alternative suggestions for how to proxy the expected exchange rate change.
The first-stage regression is estimated by OLS and corrects for possible heteroscedasticity and serial correlation by using Newey-West standard errors with the maximum lag order set equal to T^{1/2}, i.e. the square root of the sample size (16 months for the full sample, 10 for the first and 12 for the second subsample). The zero-stage regression and second-stage regression are estimated jointly by GMM which allows the standard errors of the second-stage regression to incorporate not only the uncertainty deriving from the first-stage regression (as is common IV setups), but also the one from the zero-stage regression.^{8} Standard errors are based on the Newey-West estimate of the covariance matrix, with the maximum number of lags corresponding to the values of the first-stage regression.
Alternative measures of the expected exchange rate change
In the empirical literature on risk and safety premiums, but also in the uncovered interest parity literature, there is a dependent variable that incorporates the expected exchange rate change, a variable that unfortunately is not observable. A common practice is to assume that the rational expectations hypothesis holds, so that forecast errors are uncorrelated with any time-t information. This allows to use the actual (ex-post) exchange rate change △e_{t+1} in place of the unobservable expected exchange rate change \(E_{t} \left [ \triangle e_{t+1} \right ]\) to construct the dependent variable, with e_{t+1} being the future spot exchange rate. In cross-sectional studies where only the unconditional value of variables is of interest, the time-series average of the actual ex-post exchange rate return is likely to be a good approximation for its unconditional expected value. Namely, under the assumption that the exchange rate change is stationary and ergodic, a strong law of large numbers can be applied stating that the time-series average will almost surely converge to the unconditional expectation (see Karlin and Taylor (1975)): \(T^{-1} \sum \triangle e_{t+1} \rightarrow E \left [ \triangle e_{t+1}\right ]\). In this case, there is no measurement error in the dependent variable. The situation is different, however, in time-series studies: Each observation consists of a single point in time, and using the actual ex-post exchange rate in place of the expected exchange rate implies that the dependent variable contains a measurement error consisting of the spread between the actual and the expected exchange rate change at a single point in time, the exchange rate prediction error \(\eta _{t+1}^{e} = \triangle e_{t+1} - E_{t}\left [ \triangle e_{t+1}\right ]\). And this prediction error is likely to be big given that forecasting exchange rates is difficult. In the empirical literature, it has proven to be hard to find models that beat a simple random walk when considering short horizons. The prediction error and hence the measurement error that is incorporated in the dependent variable if the actual ex-post exchange rate is used to measure the expected exchange rate are therefore likely to be of similar magnitude as the actual exchange rate change itself.
What could be alternative and more appropriate measures for the expected exchange rate change in view of the actual exchange rate change generally being a bad proxy for the investors’ “real” exchange rate expectations and therefore potentially leading to imprecision? One option is to set the expected exchange rate change simply equal to zero (E_{ t }[△e_{t+1}]=0), which would imply that investors believe that the exchange rate is following a random walk. Another option is to use the prediction of an augmented Fama model as it is estimated in the zero-stage regressions (\(E_{t}\left [\triangle e_{t+1}\right ] = \widehat {E_{t}\left [\triangle e_{t+1}\right ]}\), see Eq. (7)). In the next sections, I am going to follow the estimation procedure suggested by Maggiori to estimate the conditional safety premium model, first using the actual exchange rate change as a proxy for the expected exchange rate change, and then using these two different measures.
Data
Maggiori analyses monthly returns with his sample covering the period from January 1970 to March 2010. My analysis covers the much shorter period of January 1990 to August 2011, stopping at the time the Swiss National Bank introduced the lower bound for the EUR/CHF exchange rate^{9}. This gives a total of 259 observations of which 107 are attributed to the first subsample covering the episodes before January 1999, while the remaining 152 observations are attributed to the second subsample. Maggiori builds exchange rate and interest rate differential indices for the US dollar using the MSCI World country weights incorporating 24 developed economies. These weights are constructed based on the market capitalization of the partner economies. This might be a reasonable weight when analysing the properties of the US dollar. When building these indices from the perspective of Switzerland, however, trade-flow shares might be an equally valid if not even better weight: Given Switzerland’s status as a heavily export-oriented economy, the policy-relevant exchange rate and thus the one of interest is rather the trade-weighted one. Moreover, the US dollar gets a weight of roughly 50% in the case of market capitalization-based weights and therefore definitely dominates the behaviour of such an index, which seems to be an undesirable property in the light of Switzerland’s strong linkages with the Euro area.
For this reason, I calculate trade-weighted indices for the Swiss franc. For the restricted period of 2006 to 2013, MSCI (Morgan Stanley Capital International) provided me with data on their MSCI World Index country weights comprising 24 developed economies which I all include in my index for the US dollar. By averaging over this sample, I create time-invariant weights^{10}. Trade data for the construction of trade weights for the Swiss franc is taken from UN Comtrade. I create time-invariant trade weights for Switzerland by averaging over the whole sample period, equally weighting imports and exports. Only the five biggest trading partners are included in the index: The Euro area, the USA, the UK, China including Hong Kong, and Japan. Taken together, they make up for almost 80% of Switzerland’s trade volume. Euro area data which obviously is only available since the introduction of the Euro 1999 is merged with trade-weighted indices of exchange rates and interest rates of a sample of selected future Euro area countries to complete the series. A list of all countries considered in the construction of the indices can be found in the Appendix.
All data collected are end of period values. Bilateral spot exchange rates for building the indices are taken from IMF’s International Financial Statistics (IFS). For the risk-free interest rates, I use 1-month interbank rates from Datastream. I received early data on the STIBOR (Stockholm Interbank Offered Rate) from the Swedish National Bank directly. The S&P 500 Total Return Index, available on Datastream, and the Swiss Performance Index (SPI) Total Return Index, obtained from SIX Swiss Exchange, are taken to measure the benchmark market return of a US and Swiss investor, respectively. I chose to work with local stock market indices instead of global stock market indices in order avoid direct exchange rate effects that by construction appear in large international indices. Under the, given the financial openness of both the US and Switzerland not entirely unrealistic, assumption of complete markets, investors should anyway be indifferent about where to invest. Still, results based on the MSCI World index will be provided as a robustness test. The correlations between the local stock market indices and the MSCI World index (converted into the corresponding currency) are relatively high: 0.97 for the US and 0.83 for Switzerland. This suggests that the choice of the representative investor’s benchmark portfolio should in the end not make a big difference. Finally, I use the MSCI Dividend Yield Index to proxy for the dividend-price ratio (dividend yield is synonym for dividend-price ratio) of both these stock market indices. I obtained the corresponding data from MSCI directly.
Descriptive statistics
USD Index | CHF Index | EUR/CHF | |
---|---|---|---|
Mean ExR (ann.) | 1.38% | −0.54% | −0.30% |
Mean up to 2007m7 | 0.93% | 1.10% | 1.34% |
StD ExR | 7.44% | 5.51% | 5.38% |
Mean △e_{t+1} | 1.16% | −2.23% | −2.10% |
Mean \(r^{*}_{f,t+1}-r_{f,t+1}\) | 0.22% | 1.69% | 1.80% |
Max (mon.) | 5.77% | 4.57% | 4.76% |
Max date | 2009m5 | 2008m11 | 2008m11 |
Min | −7.78% | −6.16% | −6.68% |
Min date | 1991m3 | 2008m12 | 2008m10 |
Results
This section presents the results I find when applying the GMM methodology to the USD and CHF exchange rate indices and to the EUR/CHF exchange rate.
Zero-stage regression results
Zero-stage regression: equity returns
S&P 500 | SPI | |||
---|---|---|---|---|
\(r_{t+1}^{\omega }\) | All | All | T<1999 | T≥1999 |
dp _{ t } | 0.012 | 0.005 | −0.006 | 0.002 |
[0.016] | [0.012] | [0.024] | [0.015] | |
\(r_{t}^{\omega }\) | 0.089 | 0.217*** | 0.134 | 0.267*** |
[0.089] | [0.044] | [0.084] | [0.054] | |
Cons | 0.053 | 0.024 | −0.010 | 0.007 |
[0.061] | [0.047] | [0.087] | [0.059] | |
R ^{2} | 0.011 | 0.046 | 0.020 | 0.070 |
Zero-stage regression: exchange rate returns
USD Index | CHF Index | EUR/CHF | |||||
---|---|---|---|---|---|---|---|
△e_{t+1} | All | All | T<1999 | T≥1999 | All | T<1999 | T≥1999 |
\(r^{*}_{f,t+1}-r_{f,t+1}\) | 0.147 | 5.447*** | 5.118*** | 7.750*** | 2.910* | 2.419* | 6.102* |
[0.820] | [1.326] | [1.848] | [2.818] | [1.507] | [1.365] | [3.503] | |
△e_{ t } | 0.137*** | −0.113 | 0.060 | −0.249** | −0.110* | 0.008 | −0.187*** |
[0.049] | [0.087] | [0.066] | [0.103] | [0.062] | [0.081] | [0.068] | |
Cons | 0.001 | −0.010*** | −0.010** | −0.012*** | −0.006*** | −0.006* | −0.010** |
[0.001] | [0.002] | [0.004] | [0.004] | [0.002] | [0.003] | [0.004] | |
R^{2} | 0.019 | 0.054 | 0.062 | 0.089 | 0.028 | 0.018 | 0.055 |
First-stage regression results
First-stage regression
USD Index | CHF Index | EUR/CHF | |||||
---|---|---|---|---|---|---|---|
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\) | All | All | T<1999 | T≥1999 | All | T<1999 | T≥1999 |
dp _{ t } | 0.001 | 0.000 | −0.000 | 0.001** | 0.000 | −0.000 | 0.001** |
[0.000] | [0.000] | [0.001] | [0.000] | [0.000] | [0.001] | [0.000] | |
\(r_{t}^{\omega }\) | −0.004* | −0.000 | 0.001 | −0.002 | −0.001 | 0.001 | −0.004* |
[0.002] | [0.001] | [0.001] | [0.001] | [0.001] | [0.001] | [0.002] | |
\({var}_{t}^{{\prime }e}\) | 0.005 | −0.046 | −0.097 | −0.177** | 0.026 | −0.092 | −0.077 |
[0.057] | [0.062] | [0.118] | [0.084] | [0.070] | [0.114] | [0.105] | |
\({var}_{t}^{{\prime }r}\) | 0.016 | −0.005 | 0.003 | −0.016 | −0.011 | −0.006 | −0.016 |
[0.031] | [0.009] | [0.011] | [0.015] | [0.009] | [0.013] | [0.018] | |
\({cov}_{t}^{\prime }\) | 0.061* | −0.071 | −0.166** | 0.034 | −0.054 | −0.133* | −0.025 |
[0.033] | [0.053] | [0.068] | [0.061] | [0.057] | [0.080] | [0.074] | |
Cons | 0.003 | 0.001 | −0.001 | 0.003** | 0.001 | −0.001 | 0.004** |
[0.002] | [0.001] | [0.002] | [0.001] | [0.001] | [0.002] | [0.002] | |
R ^{2} | 0.088 | 0.021 | 0.092 | 0.082 | 0.023 | 0.093 | 0.104 |
F-statistic | 12.08 | 2.231 | 4.285 | 2.447 | 1.660 | 7.478 | 2.947 |
χ^{2}-statistic | 60.400 | 11.155 | 21.425 | 12.235 | 8.300 | 37.390 | 14.735 |
p value (χ^{2}-stat.) | 0.000 | 0.048 | 0.001 | 0.032 | 0.140 | 0.000 | 0.012 |
Second-stage regression results
Second-stage regression—full sample
\(r^{*}_{f,t+1}+E_{t}\left [\triangle e_{t+1}\right ]-r_{f,t+1}\) | \(E_{t}\left [\triangle e_{t+1}\right ] = \triangle e_{t+1}\) | \(E_{t}\left [\triangle e_{t+1}\right ] = 0 \) | \(E_{t}\left [\triangle e_{t+1}\right ] = \widehat {E_{t}[\triangle e_{t+1}]} \) | ||||||
---|---|---|---|---|---|---|---|---|---|
\( + \frac {1}{2} \widetilde {Var_{t}} \left (\triangle e_{t+1} \right)\) | USD Index | CHF Index | EUR/CHF | USD Index | CHF Index | EUR/CHF | USD Index | CHF Index | EUR/CHF |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\) | 6.491** | −22.703*** | −15.975*** | 1.047*** | −1.582** | −0.963*** | 0.992 | −10.663*** | −2.053 |
[3.572] | [8.740] | [6.740] | [0.375] | [0.692] | [0.406] | [1.115] | [4.343] | [1.633] | |
Cons | 0.000 | 0.004*** | 0.003** | 0.000 | 0.002*** | 0.002*** | 0.001*** | 0.002** | 0.000 |
[0.001] | [0.002] | [0.001] | [0.000] | [0.000] | [0.000] | [0.000] | [0.001] | [0.000] | |
J-statistic | 4.097 | 0.825 | 1.390 | 4.369 | 0.894 | 1.880 | 5.439 | 1.141 | 5.269 |
p value | 0.393 | 0.935 | 0.846 | 0.358 | 0.925 | 0.758 | 0.245 | 0.888 | 0.261 |
Second-stage regression—subsamples
\(r^{*}_{f,t+1}+E_{t}\left [\triangle e_{t+1}\right ]-r_{f,t+1}\) | \(E_{t}\left [\triangle e_{t+1}\right ] = \triangle e_{t+1}\) | \(E_{t}\left [\triangle e_{t+1}\right ] = 0 \) | \(E_{t}\left [\triangle e_{t+1}\right ] = \widehat {E_{t}\left [\triangle e_{t+1}\right ]} \) | |||
---|---|---|---|---|---|---|
\( + \frac {1}{2} \widetilde {Var_{t}} \left (\triangle e_{t+1} \right)\) | CHF Index | EUR/CHF | CHF Index | EUR/CHF | CHF Index | EUR/CHF |
Subperiod 1 | ||||||
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\) | −5.972 | −7.086* | 0.637*** | 0.718** | 4.260*** | 2.482** |
[4.902] | [4.538] | [0.234] | [0.365] | [1.462] | [1.230] | |
Cons | 0.001 | 0.002* | 0.002*** | 0.002*** | 0.001* | 0.001*** |
[0.002] | [0.001] | [0.000] | [0.000] | [0.001] | [0.000] | |
J-statistic | 3.745 | 2.804 | 5.887 | 4.561 | 6.225 | 4.632 |
p value | 0.442 | 0.591 | 0.208 | 0.335 | 0.183 | 0.327 |
Subperiod 2 | ||||||
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\) | −0.101 | −1.530 | 0.050 | 0.328** | 2.610* | 3.211*** |
[3.789] | [4.463] | [0.198] | [0.150] | [1.881] | [0.926] | |
Cons | 0.000 | 0.000 | 0.001*** | 0.001*** | −0.001*** | −0.001*** |
[0.001] | [0.001] | [0.000] | [0.000] | [0.000] | [0.000] | |
J-statistic | 3.761 | 4.808 | 3.394 | 4.417 | 5.148 | 5.536 |
p value | 0.439 | 0.308 | 0.494 | 0.353 | 0.272 | 0.237 |
Remarkably, the price of risk estimates hardly change when the lagged exchange rate change is used as an instrument but not as a regressor in the Fama regression, or when the model is estimated with the “optimal” set of instruments (see the corresponding results in Additional file 1: Section C.2). They appear also to be quite robust across the two subsamples and the different CHF exchange rates. Thus, even tough there might be some concerns about the strength of the instruments and also the J-statistic (Hansen 1982), indicating in some, but not all cases that there might be some issues with the model specification, the CHF price of risk estimates seem to be credible to some extent.
Robustness
A range of five robustness tests is presented and discussed in detail in the Additional file 1: I performed the estimations using different sample periods (including one going back to 1975 and one excluding the Global Financial Crisis and the Great Recession), a different (global) benchmark return and a financially weighted instead of trade-weighted CHF exchange rate index. Overall, all robustness tests support the above finding that proxying the expected exchange rate change by the prediction of the zero stage regression yields at least as or even more realistic and reliable estimations of the price of risk as compared to measuring the expected exchange rate change by the actual ex-post exchange rate.
Time-varying price of risk
where κ is a 1 ×5 vector. Y_{ t } corresponds to a set of instruments including a constant, the market index dividend price ratio dp_{ t } and the change in the gap between long-term and short-term interest rates (yield spread) ys_{ t } measured by the yield of 10-year government bond in excess of the 1-month interbank rate. Furthermore, it includes the change in the home risk-free interest rate r_{ f,t } and the yield difference of Moody’s BAA-rated corporate bonds over Moody’s AAA-rated corporate bonds baa_aaa_{ t } (taken from FRED), which is used as a measure for default risk. This way of parameterizing the risk price allows to easily check for the time-variation of the coefficient by setting all κ’s except the first one (the one related to the constant) equal to zero.
Second-stage regression—time-varying price of risk
\(r^{*}_{f,t+1}+\widehat {E_{t}\left [\triangle e_{t+1}\right ]}-r_{f,t+1}\) | All | T<1999 | T≥1999 | ||||
---|---|---|---|---|---|---|---|
\( + \frac {1}{2} \widetilde {Var_{t}} \left (\triangle e_{t+1} \right)\) | USD Index | CHF Index | EUR/CHF | CHF Index | EUR/CHF | CHF Index | EUR/CHF |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)\) | 238.052* | −69.980 | −111.617 | −82.002 | −55.257 | −434.596* | −33.416 |
[155.262] | [103.549] | [110.291] | [112.052] | [108.330] | [290.049] | [88.693] | |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)*{dp}_{t}\) | 58.169* | −12.174 | −25.785 | −23.436 | −14.114 | −123.582* | −8.220 |
[37.989] | [23.571] | [24.536] | [26.110] | [23.680] | [79.216] | [23.181] | |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)*d.{ys}_{t}\) | −1245.245 | −1533.296 | −2908.650* | 1235.988 | 3613.389** | −6344.461* | −832.865 |
[1677.153] | [3471.034] | [2199.215] | [1388.052] | [1660.386] | [4052.069] | [905.723] | |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)*d.r_{f,t}\) | −1727.834* | −4939.896* | −2991.115 | −2342.645 | 5548.407*** | −4157.170 | 94.303 |
[1283.600] | [3510.700] | [3678.825] | [2073.427] | [1957.128] | [3612.278] | [967.194] | |
\(\widetilde {Cov_{t}} \left (r_{t+1}^{\omega }, \triangle e_{t+1} \right)* baa\mathrm {\_}{aaa}_{t}\) | −1776.649* | 866.314 | 698.838 | −2345.300 | 784.905 | −382.272 | 202.512 |
[1188.008] | [2014.862] | [1490.346] | [2138.844] | [2459.814] | [1902.112] | [556.533] | |
Cons | 0.001 | 0.002 | −0.000 | 0.002*** | 0.002*** | −0.008** | −0.001 |
[0.001] | [0.002] | [0.001] | [0.001] | [0.001] | [0.004] | [0.001] | |
J-statistic | 1.142 | 4.532 | 3.373 | 4.474 | 2.480 | 3.932 | 5.919 |
p value (J-stat.) | 0.767 | 0.209 | 0.338 | 0.215 | 0.479 | 0.269 | 0.116 |
χ^{2}-statistic | 5.194 | 4.624 | 7.656 | 9.964 | 15.077 | 2.881 | 5.603 |
p value (χ^{2}-stat.) | 0.268 | 0.328 | 0.105 | 0.041 | 0.005 | 0.578 | 0.231 |
where E_{ t }[△e_{t+1}] is proxied by the prediction of the augmented Fama regression \(\widehat {E_{t}[\triangle e_{t+1}]}\). Hardly any of the coefficients on the interaction terms is statistically significant and the Wald tests only provide evidence for them to be jointly significant in the first subperiod. Overall, this suggests that allowing the price of risk to change through time does not help to improve the model, in fact it even seems to be a deterioration compared to the constant price of risk model, as there all price of risk coefficients are significantly positive in the subperiods.
Comparison to a multivariate GARCH-in-mean specification
A straightforward drawback of the three-step GMM estimation procedure is the large number of orthogonality conditions that must hold for the estimation to be valid. While offering a high degree of flexibility, this makes this method also somewhat unreliable and is probably the reason why GMM has been given so little attention so far in the estimation of currency safety premiums. In finance, multivariate generalized autoregressive conditional heteroscedasticity (GARCH) models are by now a well-established method for calculating the covariance matrix of a conditional model. Maximum likelihood estimation under the assumption that the covariance matrix or the variances and covariances follow an autoregressive process allows to estimate the model in one single step and there is no need for instruments for calculating the conditional covariance.
This section presents the main results I obtain when the above safety premium model is for comparison estimated with a DCC^{20} multivariate GARCH-in-mean specification. A detailed exposition of the MGARCH set-up, the technical details of the estimation strategy and the complete results are provided in Additional file 1: Section D.
where \(xr^{c}_{t+1}\) corresponds to the excess return of investing abroad \(\left (=r_{f,t+1}^{*} + \triangle e_{t+1} - r_{f,t+1}\right)\), σ_{c,m,t+1} is the conditional covariance between currency returns and home market portfolio returns, and \(\gamma ^{c}_{1}\) corresponds to the price of currency risk. The model is estimated by quasi-maximum likelihood. During the estimation process for the first subsample, I encountered some convergence problems, which are a common issue of GARCH models when put into practice (see for example Silvennoinen and Teräsvirta (2009)), so I only present results for the full sample and the second subsample.
GARCH - constant price of risk
USD index | CHF index | EUR/CHF | |||
---|---|---|---|---|---|
\(xr^{c}_{t+1}\) | All | All | T≥1999 | All | T≥1999 |
σ _{c,m,t+1} | 6.150 | 3.758 | 10.168 | 2.208 | 3.482 |
[6.038] | [10.560] | [6.896] | [11.767] | [3.192] | |
Cons | 0.000 | −0.000 | −0.002 | 0.001 | 0.001 |
[0.001] | [0.002] | [0.002] | [0.002] | [0.001] |
Altogether, however, also this GARCH model finds only weak evidence that investors are rewarded for their exposure to currency risk, which is consistent with earlier GARCH literature. De Santis and Gérard (1998) estimate a BEKK GARCH model to find the magnitude of the premium for currency risk based on the international CAPM and only obtain insignificant results when estimating constant prices of risk^{22}. So, for completeness, I also let the price of risk in my GARCH specification change through time. The according results are presented and discussed in detail in the Additional file 1. Overall, the performance of the models is slightly better with a time-varying price of risk, which, however, goes with a higher complexity of the models, measured by the number of parameters. Various information criteria suggest going with the constant price of risk version. Thus, restricting the price of risk to be constant seems still to be justified.
Conclusions
In this paper, I show that the three-step GMM approach that Maggiori (2013) uses to calculate the USD safety premium does not work for the CHF. The price of risk estimates take unrealistic, negative values. One guess why this is the case is his choice of how to measure the expected exchange rate change. Taking the actual ex-post exchange rate to measure the expected exchange rate, the dependent variable incorporates the prediction error, which in the case of exchange rates is likely to be large as they are hard to forecast in the short term. This measurement error in the dependent variable is a potential source of imprecision. I try two alternative ways to proxy the expected exchange rate change. The results get more plausible, especially when using the predictions of an augmented Fama regression, but are still not fully convincing due to the poor performance of the instruments. A maximum likelihood-estimated GARCH model seems to be a better choice for estimating the conditional safety premium as it allows to estimate the model elegantly in one single step. Furthermore, there is no need for finding good instruments.
Once a potential structural break in the relationship between Swiss franc exchange rate returns and equity returns is taken into account, the above findings provide evidence that the conditional international CAPM can help to explain the dynamics of the CHF returns versus a basket of currencies and the Euro in particular. My results reveal that the conditional covariance between stock market shocks and the Swiss franc exchange rate varies significantly through time, is almost always positive and reaches its peaks during crisis times, confirming that investors expect it to appreciate after bad shocks. There is some evidence for the price of currency risk being time-varying, but at the same time, this evidence still justifies to go with a constant price of risk. Finally, my CHF safety premium estimates vary depending on which estimation strategy is used, being rather low when estimated with the three-step GMM approach, a finding that I mainly attribute to the weak instruments. The GARCH approaches finds slightly higher values, suggesting that the CHF safety premium was on average equal to around 2.5% (on an annual basis) between early 1999 and mid-2011, and around 4.5% with peaks of up to 12.5% during the recent financial crisis. Overall, these findings support the view of the CHF acting as a safe haven during crises.
Appendix
List of countries
USD index | CHF index | Selected euro area countries |
---|---|---|
Selected Euro Area Countries | Selected Euro Area Countries | Austria |
Australia | United States | Belgium-Luxembourg |
Canada | United Kingdom | Finland |
Denmark | China (including Hong Kong) | France |
Hong Kong | Japan | Germany |
Israel | Greece | |
Japan | Ireland | |
New Zealand | Italy | |
Norway | Netherlands | |
Singapore | Portugal | |
Sweden | Spain | |
Switzerland | ||
United Kingdom |
Episodes of crisis (stock market volatility shocks)
Event | Period |
---|---|
Gulf War I | August 1990–March 1991 |
Asian Crisis | June 1997–January 1998 |
Russian Crisis, LTCM Default | August–September 1998 |
Dotcom Bust* | April 2001 |
9/11 Terrorist Attacks | September 2001 |
Worldcom, Enron Bankruptcy | July–September 2002 |
Gulf War II | March–May 2003 |
Credit Crunch, Lehman Default | August 2007–March 2009, October 2008 |
Greek Government-Debt Crisis* | May 2010–end of sample |
Footnotes
- 1.
Note that the USD Index is a market-capitalization weighted index, while the CHF Index is a trade-weighted index. For justification (also for the choice of the stock market returns) and further details, see “Data” section.
- 2.
All subsequent findings also hold if D_{t+1} is defined as Open image in new window , instead.
- 3.
See section B in the Additional file 1 for equivalent figures with D_{t+1} defined as Open image in new window . The pattern is very similar.
- 4.
Although deeper analysis suggests that there could be other possible break dates, I decided to split the sample at January 1999 as this date stands for an exogenous change in international monetary conditions and as it gives me two roughly equally sized subsamples.
- 5.
These findings are robust to various measures of the pre-Euro EUR/CHF exchange rate. The one used in this paper is a trade-weighted index considering a sample of 11 future Euro area countries. The asymmetry pattern, however, is also present when the EUR/CHF exchange rate before 1999 only considers the Deutsche Mark.
- 6.
See section B in the Additional file 1 for an equivalent table with the correlation depending on whether the equity returns are below or above 0. The pattern is very similar.
- 7.
Duffee (2005) refers to the first step as the zero-stage regression as the two following steps are the typical GMM-IV-setup first- and second-stage regressions.
- 8.
For more details on the exact estimation procedure, see Maggiori (2013).
- 9.
This choice of the sample length is mainly driven by limited data availability.
- 10.
Even tough the years 2012 and 2013 are not part the sample I analyze, I include them when constructing my market capitalization weights as they might act as a counterbalance to the exceptional years of 2008 to 2010.
- 11.
Results obtained when using the same stock market index and the same time-span as Maggiori (2013) are provided in section C.8 in the Additional file 1.
- 12.
Typically, the interest differential is defined the other way round (home minus foreign). This is why, usually, they rather say that we should expect a coefficient of 1.
- 13.
For completeness, Additional file 1: Section C.1 provides first-stage and second-stage results obtained when the lagged exchange rate change is used as an instrument, but not as a regressor in the Fama regression.
- 14.
Maggiori uses an iterative GMM estimator, which Ferson and Foerster (1994) find to have the better properties in finite sample estimations. In my case, however, this estimator produces results that lie far from the ones produced by the simple or by the two-step GMM estimator, raising some doubts about its validity (see Cochrane (1996)).
- 15.
Results obtained when using the same stock market index, the same time-span and iterated GMM as Maggiori (2013) does are presented in section C.8 in the Additional file 1.
- 16.
A Monte Carlo exercise suggests that the estimates of the standard deviation based on the asymptotic GMM formula are reasonable.
- 17.
- 18.
See the bottom right figure in Figure A.3 and the last column in the lower part of Table A.7.
- 19.
Furthermore, there is evidence that risk aversion varies through time. See, for example, Bliss and Panigirtzoglou (2004), Bekaert and Hoerova (2016) and Kim et al. (2010).
- 20.
Dynamic Conditional Correlation model by Engle (2002).
- 21.
- 22.
While the GARCH specifications BEKK by Engle and Krone (1995) and DCC are shown to produce very similar results (see Caporin and McAleer (2008, 2012)), the DCC model is computationally more attractive.
Notes
Acknowledgements
I thank Philippe Bacchetta, Harris Dellas, Matteo Maggiori, Jean-Paul Renne, Pascal St-Amour, Cédric Tille, two anonymous referees and seminar participants at the University of Lausanne, SNB Research Seminar, SSES Annual Congress and RIEF Doctoral Meeting in International Trade and International Finance for valuable comments and suggestions. I would also like to thank Süleyman Ceran for his precious help concerning the data.
Competing interests
The author declares that she has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary material
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