Banks’ equity performance and the term structure of interest rates

Using an extensive global sample, this paper investigates the impact of the term structure of interest rates on bank equity returns. Decomposing the yield curve to its three constituents (level, slope and curvature), the paper evaluates the time-varying sensitivity of the bank’s equity returns to these constituents by using a diagonal dynamic conditional correlation multivariate GARCH framework. Evidence reveals that the empirical proxies for the three factors explain the variations in equity returns above and beyond the market-wide effect. More specifically, shocks to the long-term (level) and short-term (slope) factors have a statistically significant impact on equity returns, while those on the medium-term (curvature) factor are less clear-cut. Bank size plays an important role in the sense that exposures are higher for SIFIs and large banks compared to medium and small banks. Moreover, banks exhibit greater sensitivities to all risk factors during the crisis and postcrisis periods compared to the pre-crisis period; though these sensitivities do not differ for market-oriented and bank-oriented financial systems.


Introduction
The specialness of financial intermediaries and banking in particular is well discussed in the finance literature with emphasis on the unique structure of the bank's balance sheet (Beston, 2004;Saunders and Cornett, 2017). The "new" originate-to-distribute model, adopted by banks, has enabled them to tap into new funding channels (e.g. asset-backed securities, derivatives, etc.), which in turn has broadened their investment activities via the creation of new asset classes such as collateralised asset obligations and other structured products (Shin, 2009;Purnanandam, 2011). Recent findings on the causes of the 2007 financial crisis point to the balance sheet structure of the banking firm (Farhi and Tirole, 2012;Brunnermeier and Oehmke, 2013), while the inherent leverage-adjusted duration/ convexity gap of the bank's assets and liabilities underlines its exposure to interest rate fluctuations (Flannery and James, 1984a;Anderson and Cakici, 1999;Entrop et al., 2008, Alessandri andNelson, 2015;English et al., 2018).
The significance of interest rate changes was documented much earlier by Merton (1973) and Long (1974) where, under the assumption of a stochastic risk-free rate, investors are exposed to another kind of risk, namely, the risk of unfavorable shifts in the investment opportunity set. In reality, the bank's portfolio (assets/liabilities) contains a wide range of instruments with different maturities and, thus, broader yield curve features highlight the evolution of market expectation in response to changing economic conditions and the bank's risk exposure. This point is reinforced by the recent Basel Committee on Banking Supervision (2016) where banks are required to measure their 12-month net interest income while balancing the multiple maturities in their portfolio.
Yield curve properties have a distinctive influence on the investors' perception about risk-return relationships as they are linked to business cycle conditions (Dewachter and Lyrio, 2006;Aguiar-Conraria et. al., 2012) and consequently to the bank's equity performance. Thus, using a single point of the yields' distribution (e.g. threemonth T-bill) overlooks the impact of the whole spectrum of yield changes on the market value of the bank's overall portfolio. This issue becomes nontrivial when investment portfolios with expected and contingent cash flows of different maturities are considered.
Therefore, the limitations of analysing bank equity's yield sensitivity on the basis of yield point changes, as opposed to yield curve changes, become economically relevant (see surveys by Staikouras (2003Staikouras ( , 2006 on financial intermediaries' interest rate risk exposure).
The current paper investigates the potential exposure of banks' stock returns to interest rate risk by explicitly taking into account the level, slope and curvature of the entire term structure of interest rates (yield curve). More specifically, the present study contributes to the literature in four fronts. First, it deploys the level, slope and curvature of the term structure of interest rates, derived from a three-factor interest rate model, to examine the exposure of bank's equity to yield curve fluctuations across all maturities. These threefactors are used as independent risk factors in the banks' equity return generating process.
The decomposition of the yield curve into its three components provides a research design that aims to overcome the caveats of earlier work focusing on fixed maturity yield changes and ignoring the effect of changes in the shape of the term structure or that of a "twist" in the yield curve 1 . Previous empirical studies have tried to resolve the issue by considering multiple yield measures with different maturities and/or term spreads with the exception of Dzaja et al. (2009). Yet, an important consideration is that yield changes across different maturities are not perfectly correlated and, thus, using different maturities in isolation can lead to misleading results. Second, the paper sheds light on the interface between the dynamics of the wider economy and the yield curve exposure of the banking firm by incorporating a period long enough to embrace different phases of the business cycle, as well as both the crisis and non-crisis periods. One stylized fact of the yield curve is that its shape is intimately connected to the cyclical dynamics of the economy . The yield curve tends to be steeper near the trough of the business cycle, while relatively flat near its peak. This feature directly influences banks' risk profile, since their leverage and credit generating capacity (balance-sheet size) are determined by the interest rate environment where they operate 2 . Thus, when the yield curve is upward sloping during an economic boom, banks expand their balance sheet through leverage, subject to regulatory capital requirement, to take advantage of the carry spread (Adrian and Shin, 2008). On the other hand, during an economic downturn, banks may experience difficulties to rollover these debts as a result of shortage in funding liquidity (Acharya and Viswanathan, 2011).
Using a dataset covering both the pre-, during and post-financial crisis periods allows this research to present fresh evidence of the banks' yield sensitivities during different business cycles. Third, the research setup allows for potential time-variation in yield betas by employing the diagonal dynamic conditional correlation multivariate GARCH (diagonal DCC-MGARCH) model. Unlike the conventional regression models, this econometric 2 Maddaloni and Peydro (2011) argue that the main driver behind the recent financial crisis (2007-09) was the prolonged and low short-term interest rates as a result of the monetary policy observed in the U.S. and the Euro-area. The low short-term rates soften the lending standard for household and corporate loans and encourage banks to rely heavily on short-term borrowing that leverages up their balance sheets. framework allows for the dynamic evolution of the institutions' interest rate risk exposure and facilitates pair-specific correlation dynamics and asymmetries in the conditional variances. Moreover, it accommodates the heteroscedastic nature of equity returns and overcomes the issue of multicollinearity among exogenous variables 3 . Fourth, the analysis is based on a global sample of banking firms across major market-oriented (U.S./U.K.) and bank-oriented (Japan/Europe) financial systems 4 . To this end, equally weighted country banking portfolios are constructed, which are further divided into size portfolios, based on total asset value, in order to differentiate between systemically important financial institutions (SIFI), large, medium and small size banks.
The remainder of the paper is organized as follows. Section 2 presents the methodological framework employed. Section 3 describes the data. Section 4 discusses the empirical findings, while Section 5 concludes the paper.

Yield Curve Term Structure Model
Interest rate risk exposure has been traditionally measured by the coefficients from a two-factor multiple regression model between equity returns and changes in the market factor and interest rate factor with a fixed maturity (Flannery and James, 1984b;Elyasiani and Mansur 1998;Oertmann et al., 2000;Elyasiani et al., 2007;Bessler and Kurmann, 2014 among others). Banking institutions, however, hold assets and liabilities across a wide 3 Previous studies try to sidestep the issue of interdependence between the market and interest rate risk factors by orthogonalizing the risk factors (Flannery and James, 1984b;Oertamnn et al., 2000), but the orthogonalization approach can introduce estimation bias to the regression model (Giliberto, 1985). 4 Under a bank-oriented financial structure the main contributor of capital allocation, provision of risk management platforms and savings' mobilization is the bank. On the other hand, under a market-oriented system security markets alongside banks function to move savings to corporations and to exercise corporate control (Thakor, 1996;Allen, 1999). spectrum of maturities. Therefore, measuring changes in interest rates of a specific maturity fails to recognize the full impact of the yield curve changes on the bank's equity value.
Thus, to capture the changes in the entire yield curve the paper employs the Nelson-Siegel (1987) three-factor model. The paper uses Diebold and Li's (2006) parsimonious representation, which imposes only a small number of parameters and provides flexibility to reflect a range of monotonic, humped and S-type shapes typically observed in yield data.
Let three latent factors 1,t, 2,t and 3,t be the long-term (level), short-term (slope) and medium-term (curvature) factors of the yield curve at time t, with corresponding factor loadings [1, (1 − e − )/ , (1 − e − )/ − e − ] and parameter t is an optimal-fit parameter (decay factor) governing the shape of the second and third factor loadings 5 at time t. In this setting, the spot zero-yield curve 6 yt () with maturity  at time t is formulated as follows: Eq.
(1) can be estimated via the OLS with fixed . To obtain the time series of the interest rate risk factors, first 1,t, 2,t and 3,t are estimated by fitting Eq.(1) to the yield curve. Then, the first order differences of 1,t (level), 2,t (slope) and 3,t (curvature) are computed to capture yield curve changes. Nelson-Siegel components have a clear interpretation as proxies of long, short and medium-term yields. In particular, a shock in 5 The parameter λ determines the maximum loading of the curvature factor and the exponential decay rate of the slope. Large (small) values of λ generate fast (slow) decay and can better fit the curve at short (long) maturities. We follow Diebold and Li (2006) who fix λ at 0.0609 so that the loading on the curvature component is maximized at the medium term; that is, 30 months. 6 A zero-coupon yield curve is the yield curve that maps zero-coupon Treasury bond yields to different maturities. Zero-coupon bonds have a single payment at maturity, so these curves enable us to price fixedincome instruments. To obtain a continuous yield curve, and since zero coupon bonds are available for a limited number of maturities, bootstrapping and interpolation techniques are employed.
1,t affects uniformly all maturity yields, thereby causing a parallel shift in the location of the yield curve; as such, it is viewed as a long-term yield factor (this is called level). Loading 2,t is viewed as a short-term yield factor because it has a maximal impact on short maturities and a minimal effect on the distant yields, thereby causing a flattening/steepening of the curve (this is called slope). Finally, 3,t achieves its maximum at medium maturities thereby affecting medium term yields more than the short-and long-term rates (this is called curvature).

A Dynamic Conditional Correlation Model for Time-Varying Betas
Multiple regression models typically employed in the banking literature do not explicitly address the time-varying nature of the bank's market and interest rate risk exposure. Betas obtained within these models are constant over the entire estimation period or defined over lengthy sub-samples by either using binary dummy variables (Faff et al., 2005) or by splitting the sample period (Oertmann et al., 2000). Alternative approaches, such as rolling window estimation, although they do allow for time-variation in the coefficients, they restrict the betas to be constant over the embedded sub-samples. Song (1994) is the first to apply the ARCH estimation framework arguing that betas should change as new information arrives in the market. Subsequent research such as Flannery et al. (1997), Faff et al. (2005) and Carson et al. (2008) use different GARCH models to investigate banks' interest rate exposure. For example, Elyasiani and Mansur (1998) deploy a GARCH in mean approach to study the effect of yield changes and their associated volatilities on bank stock returns distributions.
In the current study, we derive the yield betas from the conditional covariances between the interest rate (IR) risk factors (level, slope and curvature) and the banks' equity (BK) returns: , = cov ( , , , )/var ( , ); where βIR,t is the time-varying IR beta of the bank equity return (rBK,t) upon changes in the IR risk factor (rIR,t) at time t; covt(rBK,t,rIR,t) is the conditional covariance between equity return (rBK,t) and the IR risk factor (rIR,t) at time t, while the vart (rIR,t) is the conditional variance of the IR risk factor (rIR,t) at time t. Conditional covariances and variances are obtained from a diagonal dynamic conditional correlation multivariate GARCH model, henceforth, DCC-MGARCH model. The model has the following functional form (for more technical details, see Engle, 2002): where rt is a [5  1] vector containing bank portfolio returns, market returns, and level, slope and curvature factors of the term structure in week t. Ht is the [5  5] conditional covariance matrix among the five series. Dt is a [5  5] diagonal matrix with its main diagonal equal to the standard deviation (ℎ , 1/2 ) of the five variables in rt generated by an EGARCH (1,1) model; to accommodate asymmetries in the conditional variance dynamics. Rt is a [5  5] conditional correlation matrix, which is derived as: where Qt* is a

Data
The dataset includes bank equity prices from the United States (US) and the United Japan is represented by the zero-yield curves with 11 maturities from 3-month to 10-year (3, 6, 12 months, 2, 3, 4, 5, 7, 8, 9, 10 years), while for the European market, the yield curve is based on European AAA-rating treasuries. The zero-yield curve is derived from government treasury strips with all data provided by Bloomberg.
Equally weighted banking portfolios are constructed within each market 10 . Four size portfolios are formed: the systemically important financial institutions (SIFIs), the large, 9 The paper employs weekly data because daily returns departures from the normal distribution are more pronounced (Fama, 1976;Trzcinka, 1986). In particular, daily returns are subject to a high level of skewness and results of the APT tests improve when every other observation is used (Roll and Ross, 1980). Moreover, the use of lower sampling frequency (monthly compared to weekly data) reduces not only the noisiness of the data but also the number of observations, which might reduce the significance of the interest rate beta estimates and consequently the reliability of the coefficient estimates and tests. 10 The choice of the equal weighting is based on the presence of size-homogeneity since the equity portfolios are grouped according to bank's size. One may also argue that stocks within a portfolio are not under-/overweighted due to mispricing or mirroring emotions over the short-term and thus pricing errors remain random. medium and small bank portfolios for each country where possible. The SIFIs are identified by the Financial Stability Board (FSB) across the four markets examined 11 . The large, medium and small banks are grouped according to the average size of their asset value over the sample period. Following Elyasiani and Mansur (1998), banks with average asset value exceeding 50 billion US dollars are categorized as either SIFIs or large banks; to avoid overlapping, SIFIs are excluded from the large bank portfolio. Banks with average asset value in excess of 10 billion US dollars, but no more than 50 billion US dollars, are categorized as medium. The remaining banks, with asset value less than 10 billion US dollars, form the small portfolio. Banks with an average asset value less than 1 billion US dollars are excluded from our sample. These are community banks and their yield exposure may be smaller or larger because they do not have access to derivatives-based hedging (they could restrict exposure, though, through balance sheet choices and/or asset-liability management). The UK banks are all in the same portfolio given that all of them are categorized as SIFIs.

Term Structure Model Estimates
The yield curves and the estimated level, slope and curvature factors over the sample period are presented in Figure  policies and the bailout decisions during and after the crisis were all aimed to restore stability and ease the liquidity pressure in financial markets. In contrast, the interest rate environment in Japan appears relatively stable with the short-term rate almost zero over the whole sample period apart from a small jump before the crisis. Since the mid-90s, the low interest rate environment observed in Japan is mainly due to the central bank's policy to stimulate economic growth. This policy, however, has been unable to pull the country out of deflation and fuel economic boom over the last decade.

Figure 1
Looking at the changing economic conditions (business cycles), one can notice that the slope tends to reach its peak just before the recession, while the yield curve tends to flatten near the top of the business cycle. There is prior literature on the interaction between the term structure factors and macro-variables/business cycles where different relationships, including directional influences, are identified (Estrella and Mishkin, 1998;Ang and Piazzesi, 2003;Evans and Marshall, 2007). The literature documents that slope relates to business cycle conditions, while level relates to inflation expectations. It also identifies monetary policy as a factor explaining most of the changes in the slope and thus relating economic expansion/contraction to interest rate increases/decreases (Dewachter and Lyrio, 2006;Aguiar-Conraria et. al., 2012). Actually, the interbank borrowing cost increased dramatically following the liquidity squeeze in August 2007, which led to the financial panic in the Fall of 2008 (Acharya and Merrouche, 2013).
To investigate the credibility of the estimated factors to represent the level, slope and curvature, the correlation between the estimated factors and standard empirical proxies is examined in Table 2. We use y t (120), y t (120) − y t (3) and 2y t (24) − y t (3) − y t (120) as empirical proxies for the level, slope and curvature, respectively; where y t (3), y t (24) and y t (120) respresent the short-(3-month), medium-(2-year) and long-term (10-year) yields (Diebold and Li, 2006). The estimated factors are highly correlated with the corresponding empirical proxies (in line with  and as such they sufficiently represent the shape of the yield curve. It is worth noting, however, that since the long-end of the yield curve is relatively stable over time and as term to maturity approaches infinity, the three-factor term structure model collapses to a single factor model represented by the level factor loading (i.e. yt() = β1,t see Eq.1). Thus, the slope factor is mainly driven by the short-term rates and is a proxy for either the yield spread or just the short-end of the yield curve .

Table 2
Moreover, Table 2 reports the descriptive statistics of the yield factor data. Nonparametric unit root tests (Phillips and Perron, 1988) indicate that all series contain unit roots, while their first differences are stationary. First differences exhibit signs of serial correlation and heteroscedasticity supporting the use of time-varying conditional variance. Table 2 reports the Root Mean Squared Errors (RMSE) in cumulative terms i.e. start with the level factor, then add the slope and finally add the curvature factor. The results imply that all three factors are essential to describe the yield curve. That is, the RMSE reduction is in the range of 55% to 68% (83% to 91%) when comparing the two (three) factor model to the one factor benchmark.

The Dynamics of Conditional Correlation
Before analyzing the bank's time-varying yield exposure, the estimation results from the diagonal DCC-MGARCH model, across all markets, are presented. A two-stage procedure is employed (Engle, 2002). The first step involves the estimation of univariate EGARCH (1,1) models for the dynamics in conditional variances of the bank portfolios as well as market, level, slope and curvature factors (results available upon request). The second step involves the estimation of conditional correlations dynamics. Table 3 reports the estimation results from the second step of the MGARCH estimation. bank portfoliorisk factor, that correspond to sensitivities of correlations to market shocks is given by the product (see Eq. 6), which ranges between 0.0048 (EU small; ) to 0.0468 (UK SIFI; ) and tend to be higher for the portfolio-level and the portfolio-slope pairs. The coefficients that measure the sensitivity of current correlation to past correlation values ( ), range from 0.6078 (JP small; ) to 0.9902 (JP SIFI; ).
As shown in Table 3

Banks' Time-Varying Yield Curve Exposure
The conditional beta estimates for the market factor and the three yield curve components (level, slope and curvature), derived from the DCC-MGARCH model, are plotted in Figure 2. Since same factor betas exhibit similar patterns across size-portfolios, and due to space limitations, only the graphs for the SIFIs are presented (full results available upon request). Figure 2 illustrates the evolutions of the time-varying conditional betas for SIFIs.

Figure 2
The impact of the level factor (long-term rates) on banks' equity returns is largely positive, implying that an increase in long-term yields increases the value of the bank's equity, while changes in the slope factor (short-term rates) have the opposite effect 12 . The time-variation in the estimates is in line with previous studies (Song, 1994;Oertmann et al., 2000). In particular, the model-implied time series of betas (market, level, slope and curvature) is closely linked to the global economic cycles. That is, banks' market and yield exposures (betas' absolute values) are lower during the pre-crisis period compared to the period from the onset of the crisis and beyond.
Following from the above, the average weekly betas over the whole sample period are presented in Table 4. Market betas are highly significant and positive, while their magnitude increases with the bank's size. For example, the market beta of the US SIFIs is 1.4589 whereas the ones of the large, medium and small banking portfolios are 1.1379, 0.9843 and 0.7687, respectively. This is in line with banks (especially large ones) being highly leveraged with an increased appetite for risky investments and engagement in off- 12 The positive relationship between long-term rates and equity returns can be attributed to the negative maturity gaps, that banks usually run, which in turn increases the bank's net interest income.
balance-sheet activities (Demsetz and Strahan, 1997;De Nicoló et al., 2004). It is also worth noting that most of the largest banks are included in the market index and, thus, their correlation with the market return is naturally higher.
Turning to the banks' overall yield exposure, the equity returns of all portfolios are significantly affected by all three components of the yield curve. Bank size is an important determinant of the extent of yield curve exposure as the factor loadings are higher for SIFIs and large banks compared to medium and small banks. For the curvature, the effect is overall positive, with the exception of the Japanese portfolios where the impact is negative.
The effect of curvature is less clear-cut, however, and there is no macroeconomic variable associated with this latent factor . Shocks to the yield curve level (slope) factor have a significant positive (negative) impact on the banking portfolio's equity returns over the whole sample period.

Table 4
Looking closer at the slope and level factors of the yield curve, a couple of arguments can be put forward. The significant negative beta associated with the yield curve's slope (short-term) factor may be due to the fact that changes in the term structure of interest rates are closely related to the business cycle (Estrella and Mishkin, 1998;Diebold et al., , 2008 and inflation (Bernanke and Gertler, 2001). At the same time, future real economic activity is driven by the current monetary policy (Kormendi and Meguire, 1985;Fischer, 1993). An expansionary monetary policy will steepen the yield curve and enhance the short-term real economic growth (Estrella, 2005). Steepening of the yield curve, for banks experiencing a negative maturity gap, will enhance their profitability. That is, shortterm financing costs will remain below the long-term investment returns, which in turn this widening yield spread (steepening yield curve) will serve as a conduit of increasing the bank's net interest income. During the 2002-07 economic boom, banks chose to expose themselves to yield curve changes as they took advantage of the low short-term rate environment by rapidly expanding their balance sheets through short-term borrowing (Adrian and Shin, 2008;Maddaloni and Peydro, 2011). Banks are, therefore, more likely to ride the steepening yield curve (negative slope factor beta) 13 as a result of frequently refinancing their short-term liabilities .
Turning to the level factor, its positive and significant beta can be attributed to the fact that an increase in long-term interest rates mirrors higher long-term inflation expectation . Moreover, an increase in the long-end of the yield curve (higher level factor) can be associated with loosening of monetary policy, which implies a reduction in the short-end of the yield curve. That means banks can benefit from positive shocks in the level factor indirectly through the increase in short-term credit supply. To this end, an impulse response function is employed to endorse the inverse relationship between the level and slope of the yield curve. The results (available upon request) indicate that there is a lead-lag effect between changes in the level and slope factors. Specifically, shocks in the level factor have a negative and long-lasting impact on the slopein line with .

Finally, yield curve fluctuations can influence banks' equity returns via macro
factors. Corporate default rates are highly related to the business cycle, which in turn are influenced by the real economic activity (Pesaran et al., 2006). Since relative changes in the level and the slope of the yield curve are leading indicators of the real economic activity 13 Under these circumstances, banks enjoy an increased net interest income from the carry trade, where long-term high yield investments are financed by short-term low-cost liabilities. and business cycle (Estrella and Mishkin, 1998), they can also be deployed to assess the business loans' default risk (Carling et al., 2007). A prolonged flattening yield curve can have a negative impact on real economic activity, which may point out towards an economic recession. In particular, positive shocks in the slope factor can mirror an increase in corporate default rates, which in turn have a negative impact on the bank equity capital through the rise in debt write-offs (Drehamann et al., 2010).

The Impact of Crisis on Yield Curve Risk Exposure
August 9, 2007 and June 30, 2009 mark the start and the end of the recent financial crisis 14 . The financial crisis began when BNP Paribas stopped the redemption of its investment funds, followed by the liquidity squeeze in global financial markets; while the end of the crisis is based on the S&P 500 having noticeably bypassed its lowest point (March 2009) and following an upward trend well above the last trough. Table 5 summarizes the average weekly beta by splitting the sample into pre-crisis, crisis and post-crisis periods to further investigate the bank's equity yield exposure over different market conditions.

Table 5
Starting with the wide market effect, the US SIFIs' exposure increased from 1.334 during the pre-crisis period to 1.921 during the crisis and then fell to 1.508 after 2009. In 14 For a broader academic discussion see Acharya and Merrouche (2013). For market reactions, regarding the start of the crisis, see the financial press: The New York Times, August 9, 2007 "BNP Paribas suspends funds because of subprime problems"; Financial Times, August 12, 2007 "Scramble for cash reflects fears for system"; The Guardian, August 5, 2008 "Credit crisis -how it all began". After the BNP announcement, The European Central Bank pumped €95bn into the Eurozone banking market to allay fears about a sub-prime credit crunch (see BBC News, August 9, 2007 "ECB moves to help banking sector"). During June 2009, a number of events took place that contributed to signify the end of the financial turmoil. Such events (in addition to the S&P500 upward trend) include: ten large banks allowed to exit the Troubled Asset Relief Program (TARP) with the Treasury receiving $68.3 billion (09/06)more than a quarter of the bailout funds that banks have received since October 2008; the Fed extends and modifies a number of its liquidity programs (09/24); and AIG and the Federal Reserve Bank of New York entered into an agreement to reduce AIG's debt (09/25); and finally the TED spread fell at 35 basis pointa decrease of more than 400 basis points from October 2008.
other words, banks were almost twice as risky as the general market. Although a few were included in the market index, some were too big or too systemically relevant to fail and were expected to be bailed out. Their riskiness was due to the fact that they were highly engaged in the asset-backed securities market. Turning to the bank's yield exposure, over these distinctive economic phases, there is a worth noting swift in the sign of the slope risk factor, while its significance remains high. Interestingly, the slope factor beta turns from negative to positive for the US SIFIs and large banks during the financial crisis. In tranquil periods, banks' short-term funding cost (LIBOR rates) is tied to the short-end of the Treasury yield curve. With New Century Financial and Lehman Brothers filing for bankruptcy, Wachovia and Washington Mutual's fall, the near collapse of AIG and every other major US financial institution reflecting on the consequences of this turmoil, investor's confidence weakened as the crisis deepened in mid-September 2008. As a result, the difference between the US dollar LIBOR rates and the Treasury yields widened in late 2008 15 . Thus, during the peak of the crisis, the shortend of the Treasury yield curve has an inverse relationship with the short-term interbank 15 The LIBOR rates and short-term Treasury yields commonly move in the same direction. The TED spread (LIBOR minus T-bill rates) is regarded as a measure of liquidity and credit risk. In other word, an increasing TED mirrors the lack of interbank trust and a corresponding credit tightening for all other counterparties. funding costs. The slope factor proxies changes in short-term yields, hence, its relationship with the US SIFIs and large banks' equity returns became positive during the crisis period.
Looking at the size of the yield beta estimates, the level beta has increased significantly from 0.037 (until 2007)  Merrouche; 2013), which have forced banks to rely more on short-term liabilities (demand deposits, commercial papers etc.) for liquidity, compared to the pre-crisis period. Therefore, banks' equity returns experience an inverse relationship with changes in short-term interest rates. Banks also issued a large amount of loan commitment that did not, in general, expect to be exercised but serious liquidity shortages (i.e. market freeze) led loan commitment holders to draw down on the commitments, forcing banks to seek funds more vigorously.
This may have made bank stocks more sensitive to changes in yields.
Finally, changes in investors' behavior during the crisis period may have contributed to the increased level effect. The flight-to-quality phenomenon is commonly observed during economic downturns as investors switch from risky investments to safe securities with lower credit risk exposure (Chari et al., 2008). Since the demand for Treasury bonds is closely related to the liquidity condition in the stock market 16 , investors' flight-to-quality 16 When there is high selling pressure in the stock market, equity liquidity drops; but liquidity in the Treasury bond market increases as buying pressure is high (Li et al., 2009). behavior tends to depress the banks' equity prices, while pushing the Treasury bond prices up (long-term yields drop). As a result, the relationship between banks' equity returns and long-term yields would be strengthened during the crisis, which is evident from the increase in the level factor betas.
One interesting finding is that although all banking institutions become more vulnerable to short-term rate changes (i.e. negative slope factor betas), during and after the crisis period, the effect seems to be more notable for the SIFIs and large banks. One could argue that the bank's yield sensitivity is the direct result of its nominal contracting (Flannery and James, 1984a;Gomez et al., 2016). That is, some of the banks will unavoidably have wider/narrower maturity gaps than others resulting in a more sensitive balance sheet structure when short-term yield changes hit the market. To take the matter further, such yield sensitivity also depends on the banks' liquidity positions as well as on their loan commitment obligations. For instance, the slope factor effect for the Japanese SIFIs banking portfolio has increased in magnitude from -0.158 to -0.309 during the crisis period, while the one for the small banking portfolios had a marginal change of 0.009 (-0.056 to -0.065).
Moreover, the observed size effect (i.e. the increase in the absolute value of slope factor beta is higher for SIFIs and large banks during the crisis) may stem from the precautionary hoarding 17 of liquidity during the recent financial crisis. It is indeed true that banks were reluctant to lend money to each other during the 2007 financial crisis due to liquidity constraints and needs for self-insurance against payment uncertainties, especially for 17 Precautionary hoarding occurs when banks hold more reserve and liquidity than the level needed to selfinsure against shocks. This hoarding reduces the amount of available funding for the interbank loan market. As a result, liquidity shortages in the interbank market had a greater impact on the US banks than those in Japan. The roots of hoarding liquidity, in general, can be either speculative or precautionary (Ashcraft et al., 2011;Acharya and Merrouche, 2013).
In general, during the crisis market participants become more responsive to any event and require a higher premium for a given unit of risk, which translates into higher sensitivity.
Large banks are usually net borrowers and rely heavily on short-term interbank funding, so the funding liquidity shortage during and after the crisis increased the associated fear of being left with no liquidity, therefore, heightening those institutions' short-term yield sensitivity relative to the smaller banks.

Concluding Remarks
Over the last three decades, global financial markets have witnessed a variety of events making the banking sector an interesting landscape to observe and analyze. To this end, the paper provides new information about a) the bank's equity behavior when accounting for the yield curve's short-, long-and medium-term components (slope, level, curvature), which they enter the equity return generating process as exogenous risk factors; b) the bank's equity function using the comprehensive econometric framework of DCC-MGARCH allowing for time variation in factor loadings, c) the bank's yield sensitivity when banks are separated according to size and their systemic risk, d) banks belonging to market-oriented and bank-oriented economic systems, and e) the importance of economic cycles on banks' stock market performance in relation to yield curve changes.
The findings suggest that the empirical proxies for level, slope and curvature of the yield curve are statistically able to explain variation in equity prices above and beyond the wide market effect. There is evidence of time variation in the interest rate risk factors whose magnitude is linked to the global economic cycles. That is, absolute values of pre-crisis betas are lower than those in the period from the onset of the crisis and beyond. Shocks to the slope factor have overall a negative impact on bank equity returns with the findings being pretty much consistent across markets and institutions. For the US SIFIs and large banks, the changing sign as well as the size of the slope factor coefficient is something worth mentioning, as it turns (from negative) positive and becomes larger at the peak of the economic crisis. Unexpected changes to the level factor unveil a positive correlation with expected returns, a consistent finding across our international banking sample. Shocks in the level factor have a negative impact on the slope and banks are indirectly benefited from such shocks through an increase in the short-term credit supply. The impact of the curvature is less clear-cut since there is no precise pattern either among banks with different sizes or between financial systems (bank-oriented versus market-oriented). Market risk exposure is, as expected, positively correlated with bank size irrespective of the financial system observed. The systematic market exposure is more pronounced with larger banks, across all markets, and in many cases over twice as strong when we look at small banks versus SIFIs. Finally, when distinct economic cycles are taken into account, the bank equity exposure to all systematic risks is noticeably different across these cycles. In all cases considered, more than 80% of the time, the sensitivities during the crisis and the post-crisis periods are, on average, higher in magnitude than the corresponding pre-crisis estimates.
The present work has important implications for various aspects of modern financial markets and opens avenues for future research. First, by recognizing that the yield curve can be independently treated as a systematic risk factor, inevitably one recognizes its interface with areas such as central bank policy, investment theory, regulation and bank management. Given that monetary policy affects the short-end of the yield curve (Fed fund rate), its impact via the yield curve spread on real economic activity is not questionable (Friedman, 1996). Second, the yield curve as a systematic risk factor provides insight into the investment/corporate arena by a) simultaneously assessing the importance of yield and market risk to guide fund managers towards their portfolio mix (debt/equity) or to embrace income-oriented equities, b) evaluating performance measurement within an asset pricing framework and/or when funds mix bonds and low-beta securities, c) analyzing the extent to which the risk premia are priced by the market, d) looking at the market portfolio as a "risk surrogate", and e) emphasizing the wider contribution of corporate risk management to shareholder value (Bartram, 2000). Third, and within an asset-liability management framework, bankers and supervisors can a) use the yield curve to establish short-and longterm margin targets, as well as to evaluate the maturity mix of their assets and liabilities along with their respective repricing intervals and b) regularly assess the impact of the yield curve changes on the banking book and subsequently on value of the bank's net-worth.
Fourth, and as a consequence of the above discussion, regulators can consider bank performance as well as capital requirements, small business finance and economic growth by embracing a more comprehensive risk-return structure assuming an impartial macroprudential framework.

Table 1. Summary Statistics of Bank Portfolio Returns
The dataset contains bank equity prices from the United States (US) and the United Kingdom (UK) during the period from December 10, 1997 to June 15, 2016, and from Europe (EU) and Japan (JP) during the period February 5, 2003 to June 15, 2016. "Mean" and "Std. Dev." stand for the annualized average return and standard deviation for each banking portfolio, respectively. "JB" refers to the Jarque-Bera (1980) normality test statistics. Q (n) refers to the Ljung-Box (1978) test for the n th order serial correlation in the return series. Q 2 (n) is the Engle's (1982) test for ARCH effects with n lags. "N" is the number of banks that form each of the portfolios across markets. *** , ** and * indicate significance at the 1%, 5% and 10% level of significance, respectively.   Phillips and Perron (1988) unit root test, which tests the null hypothesis that the variable is non stationary, I(1), against the alternative that the variable is stationary, I(0); PP 1 and PP 2 refers to the series tested i.e. levels or changes, respectively.  is the unconditional correlation of the yield factor with the empirical proxies proposed by ; the row "Level" refers to the unconditional correlation between the estimated level factors and the 10-year yields, y t (120); row "Slope" refers to the unconditional correlation between the estimated slope factors and the difference between the 10-year and 3-month yields, y t (120)y t (3); row "Curve" refers to the unconditional correlation between the estimated curvature factors and twice the two-

Mean
year yield minus the sum of the ten-year and three month yields, 2 y t (24)y t (3)y t (120). The significance of  is based on the Student's t-test with t-statistic =  / [(1- 2 ) / (N-2)], where N is the number of weekly observations. RMSE is the Root Mean Squared Error of fitting the Nelson and Siegel (1987) model of Eq.
(1) to the term structure of zero coupon yields; this is reported cumulatively i.e. when only the level factor is considered -row "Level", when both level and slope factors are considered -"Slope" and when all three factors are taken into account -"Curve". Ljung-Box (1978) and Engle's (1982) test for ARCH effects are performed on the factor changes. See also notes in Table 1  Pane A refers to the estimated coefficient from the diagonal DCC-MGARCH model (Eqs. 3 -6) over the sample period; "ɑ i " and "b i " refer to the elements within the parameter matrix A and B in the dynamic of conditional correlation (Eq. 6); "Portfolio" represents the corresponding banking portfolio; "Market" refers to the market risk represented by the equity market return; "Level", "Slope" and "Curve" are the first difference of the level, slope and curvature factors which represents the changes in the yield curve. In Panel B, the persistence in the correlation is measured by the sum of cross-products of parameter ai and bi between the portfolio and corresponding risk factors, i.e. the level of persistence in correlation with portfolio for the slope factor is equal to a Portfolio x a Slope + b Portfolio x b Slope . All the estimated coefficients are significant at conventional significant levels except for the ones with " †" sign, which are insignificant.

Table 4. Average Weekly Beta for Banking Portfolios
The table presents the average weekly beta over the whole sample period derived from conditional variance-covariance matrices estimated from the diagonal DCC-MGARCH model; weekly beta equals to the covariance between the risk factor (F) and portfolio return (R p ) divided by the variance of the risk factor (i.e. Cov(R p ,F)/Var(F)); "Market" refers to the market risk represented by the equity market return; "Level", "Slope" and "Curve" are the first difference of the level, slope and curvature factors which represent the changes in the yield curve. The statistical significance of the average betas is computed by the t-stat. = ̅ / SE( ̅ ), where ̅ is the average weekly beta estimates over the estimation period and SE( ̅ )=stdev( ̅ )/ √N; N is the number of weekly observations. All the estimated betas are significant at the 1% significance level.

Table 5. Average Weekly Betas for Banking Portfolios pre-and post-crisis
The table presents to the average weekly beta over the pre-crisis, crisis and post-crisis period. The differences between the pre-crisis betas against the betas during and after the crisis for each risk factor are investigated based on Welch's Student's t-test; all highly significant at the 5% confidence level of significance except for the ones with " †" sign, which are insignificant. Additional t-tests are also reported in the form a " ‡" sign under the pre-crisis and post-crisis columns implying that the null hypothesis of equality of the average weekly betas over the pre-and during crisis, and over the pre-and post-crisis periods, respectively, cannot be rejected at 5% significance level. See also notes in Table 4.

Figure 1. Yield Curve and Factor Loading Estimates
The diagrams illustrate the weekly zero-yield curves as well as the level, slope and curvature factor loading estimates across the US, UK, EU and JP.

. Dynamics of Conditional Betas
The diagrams illustrate the weekly conditional beta estimates derived from the diagonal DCC model for the US, UK, JP and EU SIFIs. The magnitude for market beta is on the right-hand side axis, while the magnitude for level, slope and curvature factor betas is on the left-hand side axis.