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Forward Premium And Structure Of The Interest Rates Finance Essay


Since the choice of bond yields to use in the estimation is arbitrary, it is better if the estimates are consistent with the yields of other bonds. Therefore, in the fitted bond yield data there can be misspecifications and measurement errors (Brennan and Xia, 2006). In order to minimise these, Chen and Scott method is used to estimate the time series of the unobservable state variables and their dynamics from bond yield data.

“Ang and Piazzesi (2003) Dai and Philippon (2004) use factor models with both observed and latent factors to study how yields respond to macroeconomic variables”


Enormous literature tests the validity of Uncovered Interest Rate (UIP) and the unbiasedness in the foreign exchange market. Surveys of Hodrick (1987) and Engel (1996) provide strong empirical finding that there is a large conditional bias if forward rate is used to forecast the future spot rate. Forward premium is defined as the difference between forward and spot exchange rates and in stark contrast with UIP, forward premium negatively forecast subsequent exchange rate changes which is also known as forward premium puzzle (Fama, 1984).

One of the most extensive explanations for this anomaly is about how it reflects time-varying rational premium for systematic risk which initially put forth by Fama (1984). In the following years Hodrick (1987) states that he could not have a model of expected return which fits the data and empirical tests routinely reject the null hypothesis that forward rate is an unbiased estimator of future spot rates and risk premium models is unsuccessful at explaining the magnitude of the failure of unbiasedness. Engel (1996) also records negative relation between forward premium and future spot rates regardless of the degree of risk aversion of agents.

Short horizon- Long Horizon difference

Failure of unbiasedness hypothesis to hold at short horizon can be explained in 3 different categories: risk premium, Forecast Errors and non-linearities.

Risk Premium:

Static CAPM Model

Consumption based risk premium approach

Forecast Error:

Vast majority of the studies reject unbiasedness hypothesis consistent with Hodrick (1987) and Engel (1996) such as Bilson (1981) and Gregory and McCurdy (1984), Hsieh (1984), Froot and Frankel (1989), Lin (1999) and Lin et. al (2002) and Chernenko et. Al (2002) providing mixed result. Cornell (1989) argues that estimates of slope coefficient in Uncovered Interest Parity Equation is biased due to the measurement error that empirical studies use an average of bid and ask rates, however, bid-ask spread reflects transaction costs and he also claimed that previous studies do not find exact value of the future spot rate fits to forward rate in their data. Therefore, he suggests lagged forward premium in forex-bias equation. Same as Cornell (1977), Kohlagen (1978) and Mayfield and Murphy (1992) could not reject the UH.


Several studies including Mark et. al (1993) and Baillie and Bollerslev (1994) report forward premium is stationary and and are co-integrated with CI(-1, 1). However, Crowder (1994) proves that forward premium is non-stationary. Liu and Maddala (1992) and Barnhart, McNown and Wallace (1999) conclude Fama’s regression suffers from endogeneity problem. According to Barnhart et. al. (1999) co-integrating relation between and should be tested and then if this relation holds, residual correlation should also be tested. Consistent with Liu and Maddala (1992), compared to Fama’s regression, unbiasedness test on co-integrating framework (to allow for endogeneity problem) provides evidence of forward rate is an unbiased predictor of the future spot rate.

CHAPTER 3: Theoretical Framework

3. 1. Forward Premium Puzzle

Expectation Hypothesis (EH) or Uncovered Interest Parity (UIP) is one of the most commonly tested theories in the literature (Hansen and Hodrick, 1980). Under UIP investors are risk-neutral, they only care about expected return not risk and they have rational expectations. Expected return from investing the foreign money market and bearing the risk equal to the return on home money market investment. Therefore, if there is an increase/decrease in home interest rates compared to foreign country interest rates, it will lead to depreciation/appreciation of home currency versus foreign currency in the future. This UIP equation which is also known as Forex-bias equation is defined as:


Following Fama and Farber (1979) and Stulz (1981), and denotes logarithms of spot exchange rate (by using direct quote) observed and time t+1 and t, is interest differential between domestic and foreign country and is the residual term.

Under UIP, in the presence of unexploited economic profit opportunities (assuming that Covered Interest Parity holds) forward rates become unbiased estimates of future spot rates which imply that expected forecast error is zero. This can also be called as Unbiased Hypothesis, in order to prove this hypothesis both UIP and CIP should hold and interest differential between two countries should be equal to the exchange rate difference for same periods. Unbiased Expectation Hypothesis (UEH) equation is represented as follows (assuming no-risk premium):


is defined as logarithm of forward rate observed at time t. This regression should give an intercept of zero and coefficient of unity in order to prove UEH. However, null hypothesis of = 0 and b = 1 and E() = 0 (consistent with rational expectation) is rejected in most cases (Chaboud and Wright, 2003). There is a profitable trade in which investors borrow currency with low interest rate, converts the fund into high interest rate currency and invest in foreign money market. In many cases, b found as significant but negative which means that the currency with higher interest rate tends to appreciate. This empirical failure of UIP to the existence of time-varying currency premium is termed by Fama (1984) as forward premium puzzle.

3. 2. Fama Conditions:

Slope coefficient can be defined as:

b = (0)

Since is the future spot rate at time t+1, it is uncertain. In order to split this certainty, this value can be equalized to the sum of expected future spot rate and a premium which implies as follows:

b = (0)

Fama (1984) illustrates that forward premium can also be expressed as:


is stated as risk premium () and is expected depreciation (). Therefore, in the context of linear projection, slope coefficient b is:

b = = (0)

Moreover, negative coefficient of b has two conditions, which are

covariance between and is negative

var () > var ( which means risk premium must be more volatile than expected depreciation.

At this stage it is useful to consider currency pricing and pricing kernel in more general context and translate Fama’s conditions into restriction on pricing kernel and examine that pricing kernel satisfy these restrictions.

3. 3. Asset-Pricing in Multi-currency Setting

To examine the implications of the dynamic asset pricing models, several methods are introduced in asset pricing literature. Consumption-based model is one of the predominant approaches to approximate “asset pricing kernel”. Assumptions about the form of unobservable utility function and on the observability of aggregate consumption such as markets are complete, the economy is in equilibrium enables pricing kernel to be tested by examining the conditional or unconditional co-variation of estimated intertemporal marginal rates of substitution with or without habit factor. In other words, within consumption-based models pricing kernel is equal to the intertemporal marginal rate of substitution that (Breeden, 1979). However, this approach uses strong assumptions and it is difficult to measure consumption and marginal utility that in most cases the model predictions are not consistent with the data on asset returns and consumption (Chapman, 1997).

As an alternate approach, traditional factor models can be introduced such as CAPM of Sharpe (1965) which defines the pricing kernel as linear function of the market portfolio, APT of Ross (1976) which determines the pricing kernel as linear function of state variables and ICAPM of Merton (1973) which defines the pricing kernel with dynamic state variables for interest rate options. However, these models based on erroneous assumptions such as continuous trading and portfolio rebalancing, no transaction costs and taxes, no inflation or deflation… Furthermore, these models could not explain forward premium puzzle. Therefore, vast majority of the studies use direct specification of pricing kernel which is discussed in the following section.

In the absence of arbitrage and market frictions, there exist minimum variance pricing kernel (, stochastic discount factor) that gives a complete description of asset prices, expected returns and risk premia which is proved by Harrison and Kreps (1979).


is the domestic currency stochastic discount factor (SDF), is the nominal future payoff of the asset at time t+1 and is the price of domestic bond at time t. Moreover, gross return on the financial instrument can be defined as . Therefore, the usual equilibrium asset pricing condition is determined by:


Following Backus, Foresi and Telmer (2001), equilibrium asset pricing condition for financial instruments denominated in foreign currency (CHF) can be stated as:


, the price payoff in the foreign currency, can be formed by scaling domestic currency SDF by the gross growth in nominal exchange rate where denotes spot rate (USD per CHF)


Therefore, if the market is complete or M is the minimum variance stochastic discount factor, the relationship between and and exchange rate growth can be expressed as:


Taking natural logarithms of both sides of equation (11) yields the depreciation rate


where s is the natural logarithm of S, m is the natural logarithm of M and is the first difference. Equation (12) holds for minimum variance pricing kernels and indicates that exchange rate differential can be obtained by the two pricing kernels, the domestic and the foreign which means decrease in domestic pricing kernel leads to depreciation in domestic currency.

3.4. Affine Models for Foreign Exchange Returns

Duffie and Kan (1996) affine term-structure model is translated into discrete time by several studies. This paper follows the Ang and Piazzesi (2003) and Dong (2006). Therefore, discrete factor representation for the stochastic discount factor in order to model the exchange rate and term structure can be stated as follows [1] :


where is the short term interest rate of the country (domestic and foreign), is k×1 vector of the time-varying the market prices of risk associated with the sources of uncertainty and is the shock to the unobservable state variables, . In line with Ang, Dong and Piazzesi (2005), the dynamics of , in other words unobservable or latent factors, can be defined as a first order Gaussian Vector Autoregressive Process (VAR) (Hamilton, 1994):


is an k×1 vector, is block diagonal k×k matrix with elements for i: 0,1,2 and the zeros in this matrix are coming from the assumption of foreign country variables do not affect the domestic country variables and vice versa (Dong, 2006). is also square matrix which is the result of shocks to the unobservable state variables and ~ IID N(0, I).

Time-varying market prices of risks and short rate which is modelled in nominal terms are assumed to be affine functions of k unobserved state variables :



for a scalar and a k×1 vector and for a k dimensional vector and k×k matrix .

Using Equation (5) and (12), components of forward premium, conditional mean of the expected depreciation ( and risk premium (), can be expressed in terms of domestic and foreign pricing kernels or function of market price of risk. Hence, expected depreciation rate is:


Excess return from investing foreign exchange markets or the foreign exchange risk premium can be expressed as:


Combining equation (13), (17) and (18) Uncovered Interest Parity (UIP) condition augmented by risk premia can be derived:


when investors are risk-neutral or if there is no uncertainty in the market therefore the expected rate of depreciation is equal to interest rate differential and Uncovered Interest Parity (UIP) holds. Furthermore, if , and risk premia are constant UIP holds. However, risk premium is time-varying term and this implies deviations from UIP.

3.5. Bond Prices

In an arbitrage-free environment, discrete factor representation for the nominal pricing kernel (equation (13)) prices all nominal assets (And and Piazzesi, 2003). However, compared to other assets, bond returns do not span foreign exchange market returns. Therefore, exchange rate factor is introduced which is orthogonal to bond market in this study (Gravaline, 2006). Let be the price of a zero coupon bond at time t. After one period, investors sell this bond and get , since the remaining maturity is n. Thus, holding period gross return becomes , so that


Ang and Piazzesi (2003) define that bond prices are exponential affine functions of the state variables, . By using equation (13) – (16), it is possible to define bond prices as follows [2] 


where and are a scalar and k dimensional vector that [3] 



Continuously compounded yield to maturity of zero coupon bonds , can be computed from bond prices:


Where denotes logarithms of the price of zero coupon bond, . Consequently, by using equation (21) - (24) yield curve can be defined as:


Market price of risk cannot be identified only with short rates and the exchange rate but long term yields help identify market price of risk. Hence, long term yield is important in order to estimate the system (Dong, 2006). Moreover, Ang and Piazzesi (2003) demonstrate that state variables for long term yields explain significant amount of bond yield volatility, bond prices reveal important state variables and there exists small measurement errors. For this reason, affine term structure model (discrete time exponential Gaussian model) depend on bond pricing is used in this study however lags or macroeconomic aggregates are not incorporated.


4. 1. Data Description

The empirical analysis is based on monthly observations from January 1995 to June 2010 (186 months). Monthly interest rates data were collected from Bloomberg, spot and forward rates for two currencies, Swiss Franc (CHF) and United States Dollar (USD), were collected from Global Financial Data and Datastream database. Every second observation of the month for interest rates and forward rates are used. Forward rate time horizon considered are 1, 3, 6, 12 month, 2, 3 and 5 year and interest rate time horizon considered are 1, 3, 6 month, 1, 2, 3, 4, 5, 10 year. For short-term interest rates 1 month interbank offered rates (LIBOR) for both Switzerland and USA are used. For long-term interest rates 3 month to 10 year zero coupon bond yields are used.

Standard errors are robust to heteroscedasticity and autocorrelation as they were corrected using the Newey and West (1987) procedure.

4. 2. Descriptive Statistics

In order to examine the US-Swiss term structures Table 1 reports summary statistics for estimated annual percentage zero-coupon bond yields for USD and CHF. Yields curves are upward sloping on average. For instance, average zero-coupon bond yield for USD (CHF) increases from 3.86 % (1.44%) for the three-month bond to 5.66% (3.52%) for ten-year bond. Standard deviation decreases with maturity for both of the currencies. Moreover, domestic-correlation and values of US and Swiss yields are presented in Table 2. An important stylized fact is reported due to the high correlation between yields of near maturities [4] . Lowest value for Swiss yield is 0.71 and 0.79 for US yield suggest that there is potential advantage of international diversification using both US and Swiss foreign exchange markets. Since correlations are not equal to 1, there is a not parallel shift in the yield curve which means one-factor model would be insufficient to explain US and Swiss term structure of interests (Cassala and Luis, 2001). Meanwhile, Panel C in Table 2 presents cross-correlations between the US and Swiss yields. Yield correlations are upward sloping with maturity. Highest cross-correlation value is 0.80 and lowest is 0.59. Compared to Ahn (2004) findings for US and Germany yields correlations, cross-correlations are still lower for US-Swiss yields.

Table 3 reports descriptive statistics of spot rate (denominated in U.S. Dollar per Swiss Franc), interest rate differential defined as US interest rate minus Swiss interest rate, forward premium and depreciation rate. Monthly spot and forward rates and continuously compounded interest rates are used in annualized percent terms.

Hsieh (1991) and Cont (2001) provide evidence that spot returns generally have fat tailed distributions, negative skewness and autocorrelation of spot returns have significantly different from zero. In line with their findings, spot rate has excess kurtosis (leptokutic) (5.119) and negative skewness (-0.023) which is the evidence of exchange rate is not normally distributed (Jarque Bera test also reject the null hypothesis of normality at 1% level) and it also has high autocorrelation (0.980). Moreover, short term interest rate differential between US and Switzerland exhibits strong serial relationship, while depreciation rate is roughly uncorrelated.

Summary statistics for forward premium and depreciation rate are very similar that is important for resolving the forward premium puzzle, implies exchange rates follow near-random-walk (Thornton, 2007). Both have means close to zero (Backus, Foresi and Telmer, 2001). More importantly, depreciation rate is highly volatile compared to forward premium that standard deviation is equal to 3.9%. Figure 1 is a plot of forward premium and depreciation rate which is also consistent with these findings. Different values of variance do not suggest forward rate on average is rational predictor of the future spot rate (Fama, 1984). For this reason, in order to capture unexplained exchange rate volatility time-varying risk premium included to the model.

4. 3. Unit Root Tests

Mean reversion of forward premium also provides evidence of cointegration between spot rate and forward rate which suggest implementation of unit root test. As it is stated by Philips (1986) regressing one non-stationary series against another such series can lead to spurious results in which significance test may indicate relationship between variables even if there is no relationship. Therefore, presence of unit root in and was tested by using the Augmented Dickey-Fuller (ADF) test with constant which allows for serial correlation in error term (Dickey and Fuller,1979) and KPSS test (Kwaitkiwski, Philips, Schmidt and Shin, 1992). These tests are estimated based on equation (1) and the results of these tests are reported in Table 4. Null hypothesis for ADF test is that time series is a unit root process. Akaike information criteria was used to select p values for this test under the null hypothesis which suggested the optimal p level as zero. The 5% and 1% critical values are -2.87 and -3.46 respectively. Therefore, unit root hypothesis is rejected at 1% level for depreciation rate whereas it cannot be rejected at the 1% level for interest rate differential. Even if these results suggest interest rate differential is non-stationary, both of those rates are stationary in first differences. Since, the ADF test cannot distinguish between unit root and near unit root processes, inability to reject null hypothesis cannot be considered as evidence against the stationary of depreciation rate and interest rate differential (Engel, 1996).

On the other hand, for KPSS, null hypothesis is no unit root which can be used in order to distinguish short memory processes from long memory processes (Lee and Schmidt, 1996). The 5% and 1% critical values are 0.463 and 0.739 respectively. The KPSS test does not support presence of unit root at 1% level but it reject null hypothesis at 5% for interest rate differential. As it is stated by Lee and Schmidth (1996) larger sample is needed for more reliable results. Therefore, findings of unit root in depreciation rate and interest rate differential can be the result of small sample problem. This paper considers those as stationary processes that correction of the cointegration regression estimator is not included.

4. 4. Estimation Procedure

Empirical exercise consists of three steps which are estimation the forward premium regression, modelling of the foreign exchange risk premium and parameter by using bond yields information and estimation of extended forward premium regression.

Estimation of Forward Premium Regression

In order to estimate UIP, exchange rate data for two countries (Switzerland and USA) is used and equation (1) is estimated with Ordinary Least Square (OLS) by assuming that individuals are risk neutral and there is perfect capital market. Results are reported in section 5.1.

Modelling of the Foreign Exchange Risk Premium

Following Baker et. al. (2006), principle component analysis is used to determine the number of latent factors. Due to the imperfect correlation between interest rates, one factor model is not adequate most of the times. As can be seen in Table 6 first two components explain 99% of the USD yields and 98.5% of the CHF yields. Since principle component analysis also suggest two-factor model, same as Brennan and Schwatz (1979), Heath, Jarrow and Morton (1990), Chen and Scott (1992) two-factor model is developed to allow consistent and simultaneous pricing of the fundamental options observed in the market (Rebonoto and Cooper, 1995).

Given that the state variables determining the dynamics of the yield curve are non-observable, Kalman filtering model and Chen and Scott approach can be chosen for the estimation of the model. Kalman filter does not generate linear model and it also has a bias even though it minimizes the mean squared errors. Therefore, Chen and Scott (1993) approach is followed in assuming yields as unobservable factors are measured exactly with bond yield data and remaining yields are measured with error.



Given USD and CHF yields, equation (26) and (27) is inverted. In particular, zero coupon bonds with maturity 3 and 60 months are measured accurately and are obtained as initial values and whereas 1, 6, 12, 24, 36, 60 and 120 month yields are measured with normally distributed measurement errors which are uncorrelated through time.

In the two-factor model, factors are usually defined as the level and the slope because of the effects of these factors on the yield curve (Knez, Litterman and Scheinkman, 1994). Therefore, model is estimated by replacing the two factors by 3 and 60 month zero coupon bond yields that are observed without error. As can be seen in Figure 5 and 6, first components denoted by red line clearly accounts for a parallel shift of both USD and CHF yield curves while the second component denoted by blue line seems to account for the slope of the yield curve.

Using equation (14), latent factor can be rewritten as [5] :


Based on the initial parameter estimates, to find the vector of parameters for USA and Switzerland, Ψ= {, , , , μ, Φ, Σ, Ω}, that maximizes the likelihood function, MATLAB, fminsearch function, is used. However, initial values of parameters are important for numerical optimization especially in the case of likelihood functions with more than one local maximum. Starting values for market price of risk, and , are set to zero which means investors are risk-neutral that there is no uncertainty in the market therefore the UIP holds. Starting values for scalar terms in the affine short rate equation (16), , are restricted to be the mean of one-month continuously compounded yields which is equal to 0.0032 and 0.0011 for USD and CHF respectively. It is established that are nonnegative. Consequently, first order Gaussian Vector Autoregressive Process (VAR) is estimated in order to compute starting values of and Σ.

Estimating Extended Forward Premium Regression

Following Brennan and Xia (2006), it is considered that forward premium puzzle originated from the omission of the risk premium term in the UIP regression. Two factors are assumed to be conditionally heteroscedastic and thus they are modelled with GARCH specifications. GARCH (1,1) model which is estimated by maximum likelihood controls the time variation in risk premia by estimating pricing kernel volatilities. In other words, extended forward premium regression model estimates whether foreign exchange risk premium depends on market prices of risk of the relevant currencies in integrated markets. In order to test this hypothesis mean equation is generated as follows:


Explanatory power of risk premium is tested by Wald test with null hypothesis of 0.

In addition, variance of error term in equation (29) is a quadratic function of market prices of risks. Therefore, variance equation is:


in which and are expected to be equal to zero which suggest volatilities of pricing kernels cannot explain exchange rate volatility whereas exchange rate volatility can only be driven by forward exchange risk premium (Brennan and Xia, 2006).


5. 1. Testing of Unbiased Estimation Hypothesis

Forward premium equation can be used to test for UIP.

Table 5 presents the OLS estimates of , and the values for different maturities, short maturity, middle maturity and long maturity by using the entire sample period. It is observed that the constant term is close to zero and often statistically insignificant, whereas estimates of slope coefficient are negative for most of the maturities. For instance, the slope coefficient for one-month forex regression is -2.31. Consistent with the evidence that spot rates are near-random walk processes, negative slope coefficient suggests that exchange rates are not pure random walk processes. Compared to Fama’s (1984) and Inci’s (2007) estimate of b which are -1.14 and -1.42, estimation in short and middle-horizon regression is more negative on USD-CHF parity. The results on middle-maturity group lie between those on short and long maturities. For longer horizons negative slope coefficients up to 5-year also suggest forward premium bias but still short-horizon bias is reduced and brings long-horizon regression more positive and closer to unity. In addition, Wald test is performed in order to test hypothesis that b equals to unity, up to ten-year horizon it is rejected that confirms the findings of Bansal (1997), ten-year horizon UIP is not rejected and standard errors of parameters are quite large.

Small values of indicates weak relationship between and conflicting with relative variance of depreciation rate and interest rate differential reported in Table 3. Overall, the results in Table 5 suggest that estimation of the Fama regression using OLS rejects unbiasedness which is consistent with the stylized facts leading to the forward bias puzzle. However, due to a simultaneity problem estimators can be inconsistent and biased. Therefore, this might be problematic in the presence of an omitted risk premium in the Fama regression which drives negative slope coefficient that is different from unity (Fama, 1984; Liu and Maddala, 1992).

5.3. Modelling of Foreign Exchange Risk Premium

To estimate the parameters of the system (..........) by Maximum Likelihood using equation (..........) and bond yields of different maturities.

Constuct the likelihood function given data setas a function of model parameters (

Risk premium is generated by the estimated pricing kernel volatilities.

Market Price of Risk and Forward Premium Regression:

Term Structure Analysis:

5.2. Testing Fama Conditions

In order to test whether the Fama (1984) conditions are satisfied, the forward premium is decomposed into . Recall that the two Fama’s (1984) necessary conditions for <0 are Var()>Var() and Cov()<0. Using the expressions of implied risk premium of a currency () and its expected rate of depreciation () in section 2, Table 7 illustrates that and are negatively correlated and have greater variance which means Fama`s conditions can easily be satisfied for USD-CHF currency pair.

5. 4. Estimating Extended Version of UIP

Theoretical value of risk premium, -, is derived following Fama(1984). In order to test whether forward premium puzzle remains, after considering time variation in risk premium extended GARCH(1,1) model for the exchange rate risk premium by using equation (29) and (30) is estimated by maximum likelihood. Panel A in Table 8 reports mean equation estimates. Different than basic forward premium regression risk premia which is generated by the estimates of pricing kernel volatilities is added to the right hand side of the equation. The slope coefficient of the forward premium is still negative and lower than one in absolute terms (-4.73), whilst its theoretical value is unity. However, it is less negative compared to the slope coefficient of basic forward premium regression (Table 5). Still, after accounting for time variation in risk premia captured by the estimated pricing kernel volatilities, forward premium puzzle remains.

The Wald test for the mean equation tests whether a log likelihood ratio rejects the null hypothesis that . At 1% significance level null hypothesis is rejected. This provides strong evidence that the pricing kernel volatility terms have significant explanatory power in mean equation

Panel B reports the results of variance equation. The Wald test for the significance of and terms rejects the null of no significance which means there is theoretical relation between exchange rate volatility and pricing kernel. However, statistical significance of and implies that pricing kernels are not sufficient in explaining variation in exchange rate risk premium.

In summary, results reported in Table 8 are consistent with the theoretical equations (29) and (30) that US and Swiss pricing kernel volatilities carries information about exchange rate volatility and foreign exchange risk premium. However, adding additional variable into the forward premium regression does not solve forward premium puzzle which might be because of estimation errors in market price of risk calculation or model misspecification.


Table 1: Descriptive Statistics for USD and CHF Yield Curves

This table report descriptive statistics for zero coupon bond yields for different maturities (1, 3,6 month, 1, 2, 3, 4, 5, 10 year). The bond yields are taken from Bloomberg and expressed as in annualized and percent terms. The sample is from January 1995 to June 2010.

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