HYGARCH Approach to Estimating Interest Rate and Exchange Rate Sensitivity of a Large Sample of U.S. Banking Institutions
Emina Cardamone, Daniel Folkinshteyn ∗,1 Temple University, Department of Finance, 1801 North Broad Street, Philadelphia, PA 19122
Abstract In this paper we will attempt to look at the effect of interest rate and foreign exchange changes on bank stock returns, using a HYGARCH model. It is natural to suppose that the stock return sensitivity will be higher for banks with larger interest rate exposures (duration gap) and foreign exchange rate exposures (foreign assets and interest income), which effects would be mitigated by the banks’ positions in interest rate and foreign exchange rate derivatives, if these off-balance sheet positions are used for hedging purposes. Alternatively, if interest rate and foreign exchange rate derivative positions are used speculatively at least in part, we would expect to see a smaller (if any) stabilizing effect of these OBSA exposures. This paper extends existing literature in several ways. First, we use a more sophisticated model that has fewer maintained assumptions with regard to constant variance and coefficients. Second, we use a wider and longer dataset, that enables us to look at broad historical trends in the banks’ usage of OBSA.
Key words: Banks, Performance, Stock return, Interest rate, Foreign exchange, Derivatives, Off-balance sheet activities
∗ Corresponding author. Email addresses:
[email protected] (Emina Cardamone),
[email protected] (Daniel Folkinshteyn). 1 We are grateful to Dr. Elyasiani and Dr. Buck for advice and consultation. We also thank Elsevier for providing this nice LATEX document template.
3 May 2007
1
Introduction
The issue of bank sensitivity to foreign exchange rate and interest rate risks, and the usage of derivatives to mitigate them, is of general interest to the banking and regulatory community. One question of particular importance is whether banks use off-balance sheet activities (OBSA) to hedge outstanding balance sheet risk, or to speculate in the hopes of increasing stock returns. In the former case, an increase in the amount of interest rate and foreign exchange rate derivatives would be expected to reduce bank sensitivity to interest rate and foreign exchange rate changes, while in the latter case, one would expect the opposite effect. The objective of this paper is to employ the hyperbolic GARCH (HYGARCH) methodology to investigate the bank stock return sensitivities to interest rate and foreign exchange rate changes and volatilities, controlling for balance sheet and off-balance sheet exposures. The results of this research may have a direct impact on bank regulation, wherein if we find that OBSA exposures are primarily used for speculation (or ineptly used for hedging), this would make a case for the imposition of regulatory guidelines on bank activities to prevent excessive risk exposure through OBSA. There has been some related research in this area, and the following section will summarize the most salient findings thereof. This paper is organized as follows. Section 2 gives an overview of existing literature, Section 3 develops a theoretical model, Section 4 sets the stage with an overview of the data and econometric technique, Section 5 introduces the empirical results, and Section 6 concludes.
2
Background
In the existing literature, researchers have employed varying empirical methods to measure the exposure of banking institutions to market, interest rate, and exchange rate risk. Several studies, with conflicting results, have used on-balance-sheet bank operations data to examine the sensitivity of banks to interest rate and exchange rate risks. Flannery (1983) argues that accounting variables, revenues and costs, adjust with a lag in response to changes in market conditions. Therefore, his study uses a lag structure specified as a partial adjustment process. His results fail to support the presumption that commercial banks are no more exposed to substantial interest rate risk than are large 2
banking organizations. Conflicting results are obtained by Flannery and James (1984). They use the duration approach to argue that the effect of interest rate changes on the equity value of a bank depends on the maturity of the bank’s assets and liabilities. They document a highly significant cross-sectional relationship between measured interest rate sensitivity and a bank’s balance sheet composition. Yourougou (1990) uses a different econometric technique to obtain results similar to those of Flannery and James (1984). More specifically, Yourougou (1990) uses the likelihood ratio test to present evidence and show that the interest rate risk is priced. Chamberlain et al. (1997) examine the foreign exchange exposure of a sample of U.S. and Japanese banking firms. They find that the stock returns of approximately one-third of thirty large U.S. bank holding companies are sensitive to exchange rate changes. They also find that banks with off-balance sheet activities in foreign exchange contracts exhibit less foreign exchange rate sensitivity. Stiroh (2004) attempts to empirically validate theoretical assumptions regarding potential diversification benefits from activities that generate fee income, trading revenue, and other types of noninterest income, in the US banking industry. His empirical results show evidence that the banking industry should not necessarily be counting on noninterest income to smooth revenue flows or reduce the risk exposure of banking institutions. More recently, there have been a number of studies that examine joint effects of banks’ interest rate and foreign exchange rate risk exposure using off-balancesheet bank operations data. Choi, Elyasiani, and Kopecky (1992) conduct a study that explicitly examines the joint interaction of exchange rates and interest rates on bank stock pricing. They estimate a multifactor model of bank stock returns and find that exchange rates exert an important influence on bank stock returns independent of the other market and interest rate factors. Grammatikos et al. (1986) conduct a similar study using on-balance-sheet data to look at the sensitivity of bank returns and profits to interest rate and exchange rate risks. Song (1994) uses an ARCH modeling technique to examine the effect of market and interest rate risks for a sample of depository institutions. He argues that since the market and interest rate risks depend on conditioning information, they are potentially time-varying, and therefore, it is appropriate to use an ARCH model for analysis of bank stock returns. His findings are in conflict with the results obtained from conventional event studies that assume constant betas in the market model. Our study builds upon the work of Choi and Elyasiani (1997). They estimate interest rate and exhange rate risk betas of 59 large U.S. commerical banks for the period of 1975-1992 using both on-balance-sheet and off-balance-sheet data. Their estimation procedure uses a modified seemingly unrelated simultaneous method. In our study, using a larger sample of banks over a longer time 3
period, we examine how derivative transactions of banks have affected their exchange rate and interest rate risk exposure. In addition, we use HYGARCH model, since it allows us to modify some important assumptions regarding linearity, independence, and constant conditional variance in modeling bank stock returns. Some researchers have already used several different types of GARCH models to estimate interest rate and exchange rate risk exposure of firms. One of the first studies to use a GARCH-M model to examine sensitivity of the U.S. bank stock returns to changes in interest rate was presented by Elyasiani and Mansur (1998). Their findings support the argument that interest rate volatility is an important determinant of the bank stock return volatility and bank stock risk premium. Ryan and Worthington (2004) obtain different results for a set of Australian banking institutions. Namely, they use the GARCH-M model to consider the time series sensitivity of Australian bank stock returns to market, interest rate and foreign exchange rate risks. Their results suggest that market risk is an important determinant of bank stock returns, along with short and medium term interest rate levels and their volatiliy. However, they find that long term interest rates and foreign exhcnage rates do not appear to be significant over their sample period from 1996-2001. Our contribution to the existing literature includes longer time series and a larger sample of banks. In addition, our study employs quarterly time series over a longer time period, while almost all of the studies cited used monthly and/or daily data. Lastly, we use a HYGARCH model to conduct our empirical estimation, and, to our knowledge, we are the first to employ this method to estimate interest rate and exchange rate risk exposure of the U.S. banking institutions.
3
Theoretical Model
The traditional theory of financial intermediation argues that the reason why banks and other intermediaries exist is because of their ability to efficiently reduce informational asymmetries and transaction costs in financial markets. Allen and Santomero (1997), on the other hand, argue that risk management, instead, has become the major activitiy of banking institutions. Banks have always been exposed to interest rate risk. When banks issue liquid deposits that are backed up by illiquid loans, they take a risk. Since the cost of funds, which depends on the short term interest rates, could increase or decrease above or below the interest income, which is determined by the fixed interest rate of the loans set by the bank, the bank that issues the loans faces interst rate risk. Consequently, banks have sought ways to mitigate their exposure to interest rate risk. According to Stiroh (2004), noninterest income has grown to account for 43 percent of net operating banking revenue in 2001, which is net interest income plus noninterest income, up from only 25 percent in 1984. 4
This statistical fact confirms that banks have been diversifying their portfolios and shifting toward noninterest income to increase their revenue. Noninterest income may help banks reduce the volatility of their profit and revenue, reduce risk, decrease leverage, and in some instances, escape regulation and taxes. Activities that generate most noninterest income arise from off-balance-sheet operations, such as loan commitments, credit lines, and guarantees. In addition, banks engage in derivatives transactions such as swaps, futures, forwards, which have become major sources of banks’ noninterest income. With increasing internationalization of business, banks’ exposure to foreign exchange fluctuations is also of some importance. Depending on the relevant foreign exchange rate changes and volatility, a bank’s return from its foreign operations and the value of its foreign assets may be quite volatile as well, and encourages derivatives positions to mitigate that risk. We are interested in examining whether this increased reliance on off-balancesheet transactions, specifically as related to derivatives positions, makes banks less or more sensitive to changes in interest rates and foreign exchange rate risks.
3.1
Basic Model Framework
We use a game theoretic approach to formalize a model that evaluates banks’ interest rate and foreign exchange rate risk exposure. 2 The basic framework is as follows. The bank manager may choose to either hedge, or speculate, and the market may move either favorably or unfavorably. In the case of interest rate risk, for example, most banks have long term assets (loans, mortgages) and shorter term liabilities (CDs, deposits), so the bank would be exposed to an upward movement of the interest rates, which would adversely affect their net balance sheet position, as well as their current net interest income. So the hegding or speculating will be done with respect to interest rates using IR derivatives. The bank’s foreign assets/liabilities differential would have a similar effect in exposing it to the foreign exchange risk, and it would be using FX derivatives for hedging or speculation. Figure 1 presents the payoffs that are facing a single bank in the market. We make a number of simplifying assumptions. When the banker hedges, we assume that he hedges the downside risk completely, while leaving open the possibility of upward risk. We further assume that the cost of investing in 2
We thank Martin J. Osborne for providing the LATEX style for formatting strategic games, available at http://www.economics.utoronto.ca/osborne/latex/.
5
Market
Bank
F avorable
U nf avorable
h
+B − C
−C
s
+B + S − C
−B − S − C
Fig. 1. Bank payoffs
derivatives for the purposes of hedging is the same as the cost of investing in same for speculation. Thus, when the bank hedges, and the market moves favorably, the bank’s position benefits from the balance sheet position (B), but loses the cost of the hedge (C). When the market moves unfavorably, the negative impact on the balance sheet positions is offset by the hedge position, but the bank is still out the cost of the hedge. If the bank chooses to speculate, and the market moves favorably, the bank benefits from its balance sheet position (B), and also gains from its speculative position (S), all net of the cost of the derivatives (C). When the market moves negatively, the bank loses on the balance sheet, the speculative position, and is still out the cost of the OBSA. Assuming efficient markets and rational expectations, the ex-ante probabilities facing the banker regarding the market move is a 50-50. Thus, the expected payoffs to the bank (note that these payoffs are distinct from those to the banker) from hedging and speculation, respectively, are: 1 1 1 Ph = (B − C) + (−C) = B − C 2 2 2 1 1 Ps = (B + S − C) + (−B − S − C) = −C 2 2 So far, this is not even a game, since there is only one player - it is merely a payoff table to help us set up the game structure. The game takes place between the managers of the multiple banks in the market. The expected personal incentives to each manager are modeled as: E U (a) = φPa + αW − βL a ⊂ {h, s} Where U(a) is the utility the manager gets from taking action a, which can be either hedge h, or speculate s. φ is a positive, finite, real-valued constant, Pa is the payoff to the bank of action a (payoffs are defined in Figure 1), W and L are relative measures of outperformance (Win) and underperformance 6
(Lose) of the bank (to be explicitly defined later), and α and β are also positive, finite, real-valued constants. The relative performance incentives for the manager need not be symmetric, which is why we separate this variable into two variables for under- and out-performance relative to the industry. Note that αW and βL need not be explicitly monetary - they are comprised of such things as banker reputation, promotions, employability, in addition to any monetary bonuses the banker may get. In words, then, the personal utility that the bank manager derives is some fraction of the payoff the bank received in the period, plus some extra incentives for performance relative to the rest of the industry. Note that we do not talk about the investors as a player in the game. We assume that investors will get the information on the payoff that the bank has achieved in the period, and reward the bankers correctly in accordance with the payoff. In a way, this is a situation similar to the ”uninformed moving first” framework described by Thakor (1991). So, let us look at the extensive form of the game as played between two managers (Figure 2).
Fig. 2. Complete game, extensive form
This is a simultaneous move game, so bankers 1 and 2 move at the same time. This is denoted by the dotted link between the two nodes of banker 2: the information partition of banker 2 is coarse; he cannot distinguish between his two nodes, as he is making his move at the same time as banker 1 is making 7
his. Nature (player N) moves last, but it is just a pseudoplayer, and does not care what moves the bankers make, so it also does not distinguish between the nodes. This is denoted by the linking of all four nodes, giving Nature an information partition of one coarse information set. For simplicity, we normalize the payoffs such that the incentives/disincentives for win/loss accrue only to one side, the side that chooses to speculate. To clarify the analysis, the partitions of this game, conditional on nature’s move, are presented in Figures 3 and 4, and the strategic form game, in expectation of Nature’s move, is shown in Figure 5. Bank 2
Bank 1
h
s
h
φ(B − C), φ(B − C)
φ(B − C), φ(B + S − C) + αW
s
φ(B + S − C) + αW, φ(B − C)
φ(B + S − C), φ(B + S − C)
Fig. 3. Bankers’ game when market moves favorably
Bank 2
Bank 1
h
s
h
φ(−C), φ(−C)
φ(−C), φ(−B − S − C) − βL
s
φ(−B − S − C) − βL, φ(−C)
φ(−B − S − C), φ(−B − S − C)
Fig. 4. Bankers’ game when market moves unfavorably
Bank 2
Bank 1
h
s
h
φ( 21 B − C), φ( 12 B − C)
φ( 12 B − C), φ(−C) + 21 (αW − βL)
s
φ(−C) + 12 (αW − βL), φ( 12 B − C)
φ(−C), φ(−C)
Fig. 5. Bankers’ game in expectation
What are the possible equilibria in this game (Figure 5)? They of course depend on the values of the variables. In the simplest case, if φB > αW − βL, there is a dominant strategy equilibrium of (h, h), and thus we would observe banks hedging all the time. If φB = αW − βL, then (h, h) would be a weak dominance equilibrium, and we would still observe the players hedging. If φB < αW −βL, then we have two asymmetric pure strategy Nash equlibria, (h, s) and (s, h), and will also have a mixed strategy equilibrium. The question of the relationship between φB, and αW −βL is an empirical one to determine: if we see banks speculate at least some of the time, we may conclude that the latter relationship holds, at least in some periods. 8
Now, we can extend this game to the setting with N identical banks, and mixed strategy equilibrium. In a mixed strategy, each banker will choose to speculate with probability γ, and to hedge with probability (1 − γ). In this setting, the measure of relative performance of a speculating bank (the W and L variables as defined above) will be the number of other banks that chose to hedge (and thus achieved different performance for the period). The distribution of the number of banks making the “hedge” decision is a binomial(N, (1 − γ)). In expectation, the number of banks that choose to hedge is thus N (1 − γ), and so we will define W and L in expectation as W = L = N (1 − γ) Using the payoff-equating method of solving for mixed equilibria (Rasmusen, 2006), we have: 1 π(h) = φ( B − C) 2 1 1 π(s) = −φC + (αW − βL) = −φC + (α − β)N (1 − γ) 2 2 π(h) = π(s) 1 1 φ( B − C) = −φC + (α − β)N (1 − γ) 2 2 φB (1 − γ) = = Pr(h) N (α − β) Where π(h) and π(s) indicate the expected payoffs from hedging and speculating, respectively. Thus, the probability of choosing to hedge (which is equal to (1 − γ) by definition), increases in the sensitivity of the balance sheet to market moves (B) and the sensitivity of the banker’s personal payoff to the bank’s payoff (φ), and decreases in the number of banks playing the game and the differential of reward between beating and losing to the market. It is encouraging that this result seems rather intuitive. Note, that as per the necessary condition of the existence of the mixed equilibrium, we need φB < [αW − βL = (α − β)N (1 − γ)] So the reward for outperformance (α) must be greater than the disincentive for underperformance (β). We have not said anything as yet about the form of the utility function of the bank manager. This is because it is largely irrelevant. One could either say that we assume risk-neutral agents, or say that our representations of the 9
payoffs to the manager already are in terms of utility values rather than raw monetary amounts. Of course, in all this, we have assumed for simplicity that the bank has a single binary decision to make: hedge or speculate. However, we can interpret the result of the mixed equilibrium also as saying that the bank will choose to hedge (1 − γ) fraction of its balance sheet exposure. This brings us to the idea that we could determine the value of γ empirically by looking at the degree of sensitivity of bank returns to the changes in FX and IR, through inferring the proportion of balance sheet risk that has been hedged by using OBSA. But first, let us look at some generalizations of the model.
3.2
Model Extension: Subjective Probabilities
In the previous section we have claimed that due to market efficiency, the probability of a favorable move by nature is half, as prices adjust “instantaneously” to take into account any information that may indicate a higher or lower probability. However, the assessment of that probability by bank managers may not be correct - managers may be either too optimistic or pessimistic in their subjective estimate of this probability. Extending this model, and denoting the probability of a favorable move by nature as ω, we find the following results. The bankers’ game in expectation becomes as shown in Figure 6 (we format the game as a column of payoffs, since the 2x2 table becomes too wide to fit on a page):
(h, h) (h, s) (s, h) (s, s)
φ(ωB − C), φ(ωB − C) φ(ωB − C), φ(2ω − 1)(B + S) + φ(−C) + ωαW − (1 − ω)βL φ(ωB − C), φ(2ω − 1)(B + S) + φ(−C) + ωαW − (1 − ω)βL, φ(ωB − C) φ(2ω − 1)(B + S) + φ(−C), φ(2ω − 1)(B + S) + φ(−C) Fig. 6. Bankers’ game in expectation: uneven probabilities
Again, using the payoff-equating method to solve the multiple player game, 10
we have the following. πh = φ(ωB − C) πs = φ(2ω − 1)(B + S) + φ(−C) + ωαW − (1 − ω)βL πh = πs φ(ωB − C) = φ(2ω − 1)(B + S) + φ(−C) + ωαW − (1 − ω)βL φ(ωB − C) = φ(2ω − 1)(B + S) + φ(−C) + [ωα − (1 − ω)β]N (1 − γ) φ(B + S − ωB − 2ωS) (1 − γ) = = Pr(h) N (ωα − (1 − ω)β)
In this case, it becomes more difficult to describe the interrelationship between the component variables. The probability hedging (or fraction of assets hedged) increases in φ, and decreases in N . As the difference between α and β increases, the probability of hedging decreases. With respect to ω, the effect is more complex. Both the numerator and denominator increase in ω, so the overall effect depends on the relative magnitudes of B, S, α, and β.
3.3
Model Extension: Variable Costs
Now we will extend the model to account for the price effects of banks competing for the same derivatives. As more banks choose to hedge their risk, they are competing for the same derivatives, thus driving the cost of hedging up. Thus, increasing cost with increasing overall hedging in the banking industry is also a factor that the bankers must take into account. We develop this extension of the model as follows. The payoffs to hedging or speculating, in the multiplayer framework, we can express as follows, directly extending the payoff functions from the previous section. πh = φ(ωB − Ch (Nh )) πs = φ(2ω − 1)(B + S) − φCs (Nh ) + ωαW − (1 − ω)βL where Ch (Nh ) defines the cost of hedging, as a function of the number of banks that choose to hedge, and Cs (Nh ) defines the cost of speculating, as a function of the number of banks that choose to hedge (since in the model structure, Ns = N − Nh ). We will assume a simple linear structure for the cost functions, and define 11
them as Ch (Nh ) = C + κh Nh Cs (Nh ) = C − κs Nh Nh = N (1 − γ) where κh and κs are positive, finite, real-valued constants. This specification shows that as the number of hedgers increases, the cost of hedging increases from the “baseline” C. At the same time, as the number of hedgers increases (and thus the number of speculators decreases), the cost of speculating decreases from the “baseline” C, but at a possibly different rate. The number of hedgers is, as before, simply the expected value of the binomial distribution, with Pr(h) = (1 − γ). Solving this game out using the same method as in the previous sections, we have πh = πs φ(ωB − Ch (Nh )) = φ(2ω − 1)(B + S) − φCs (Nh ) + ωαW − (1 − ω)βL φωB − φC − φκh N (1 − γ) = φ(2ω − 1)(B + S) − φC + φκs N (1 − γ) + [ωα − (1 − ω)β]N (1 − γ) φ(B + S − ωB − 2ωS) = N (1 − γ)(ωα − (1 − ω)β + φκh + φκs ) φ(B + S − ωB − 2ωS) (1 − γ) = = Pr(h) N (ωα − (1 − ω)β + φκh + φκs ) The probability of hedging decreases as the difference between α and β increases, and as N increases. The effect of φ and ω now depends on the complex interrelationship between B, S, κh , κs , α, and β.
4
4.1
Data and Econometric Specification
Methodology and features of GARCH models
The autoregressive conditional heteroscedastic model (ARCH) and the generalized autoregressive conditional heteroscedastic model (GARCH) have been used widely in the area of econometric analysis of financial markets. Elyasiani and Mansur (1998) argue that the reason these models are so appealing to the financial econometricians is their common feature of specifying the conditional variance as a function of the past shocks that allow volatility to evolve over time and permit volatility shocks to persist. In our current study, we choose to use a type of the GARCH model, since GARCH models are known to be 12
long memory models while ARCH models are short memory models. In other words, ARCH models allow for limited number of lags, while GARCH models permit us to use all lags including the past values of the conditional variance and the past values of the squares of the error term. In addition, we follow Ryan and Worthington (2004) and choose a univariate model over a multivariate model for simplicity reasons. If we were to use a multivariate model, we would have to model the interaction between bank portfolio volatility and the dependent variable (in one equation), in addition to modeling interaction between the dependent variable and other exogenous variables (in other equation(s)). According to these authors, it is unreasonable to assume the multivariate specification because, while the volatility of the bank portfolio may feed back into the market portfolio if the banks in the index are sufficiently represented in the market index, that might not be true for the relation between bank portfolio volatility and foreign exchange rate and interest rate volatility. Using a multivariate specification would assume exactly such an interaction between the bank portfolio volatility and the other independent variables. This would not be appropriate for our current purposes. To estimate our theoretical model empirically, we use the HYGARCH model, which was introduced by Davidson (2004) as a generalization of fractionally integrated GARCH model (FIGARCH) with hyperbolic convergence rates. A general specification of a FIGARCH model was presented by Baillie, Bollerslev, and Mikkelsen (1996). These models fall in the class of models where the conditional variance at time t is an infinite moving average of the squared realizations of the series up to time t − 1. In other words, let σt > 0, et ∼ iid(0, 1)
t = σt et ,
(4.1)
and
σt2 = ω +
∞ X
θi 2t−i
θi ≥ 0
(4.2)
i=1
where θi serve as the lag coefficients for the given set of parameters. Given this general framework, we define the conditional variance in a FIGARCH (p, d, q) model as follows: n
o
σt2 = ω[1 − β(L)]−1 + 1 − [1 − β(L)]−1 φ(L)(1 − L)d 2t
13
(4.3)
where (1 − L)d is the fractional differencing operator, defined as follows: ∞ X
Γ(d + 1)Lk k=0 Γ(k + 1)Γ(d − k + 1) 1 1 = 1 − dL − d(1 − d)L2 − d(1 − d)(2 − d)L3 − · · · 2 6
(1 − L)d =
=1−
∞ X
(4.4)
ck (d)Lk
k=1
with c1 (d) = d, c2 (d) = 12 d(1 − d), etc. Therefore,
σt2 = ω ∗ +
∞ X
λi Li 2t = ω ∗ + λ(L)2t
(4.5)
i=1
where 1 ≥ d ≥ 0. In the above equations, L serves as a lag operator such that β(L) = 1 − β1 − · · · − βq , and similarly is φ(L) a polynomial in L. A HYGARCH (p, d, q) model is obtained when λ(L) in the equation (4.5) is replaced with n
o
λ(L) = 1 − [1 − β(L)]−1 φ(L) 1 + k[(1 − L)d − 1]
(4.6)
Davidson (2004) shows that this proposed HYGARCH model generalizes the FIGARCH model, and also permits the existence of second moments at more extreme amplitudes than the simple IGARCH and FIGARCH models permit. More specifically, when k = 1 the HYGARCH model nests the FIGARCH, when 0 < k < 1 this process is stationary, and when k > 1 this process is non-stationary.
4.2
Model specification
Given our discussion in the previous subsection, we now specify the HYGARCH (1, d, 1) model that we use in our study to estimate the desired 14
parameters.
ERi,t = α0 +
n X
αi ERi,t−i + αm Rm,t + δ1 ∆IRt−1
(4.7)
i=1
+ δ2 ∆F Xt−1 +
n X
γi Xi +
∗
n
2 ψi Yi + λσi,t + i,t
i=0
i=0 2 σi,t
n X
n
−1
oo
= ω + 1 − [1 − β1 L] φ1 L 1 + k[(1 − L)d − 1]
i,t |Ωt−1 ∼ t(ν)
2i,t + d1 D1 (4.8) (4.9)
In this model, ERi,t is the excess return on the ith bank portfolio at time t, net of same-quarter 1-year treasury return (following Elyasiani and Mansur, 1998), Rm is the return on market 3 , ∆IR is the change in the ten-year Treasury composit yield, ∆F X is the rate of change in currency exchange rates, Xi is a collection of variables regarding a banks’s balance sheet positions, Yi is a collection of variables regarding a bank’s off-balance-sheet positions, and D1 is the dummy variable for the shift in the volatility equation due to the switch 2 of European countries to Euro in January 2002. σi,t measures the stock return risk of bank portfolio i at time t, and i,t is the error term. We assume the Student T distribution for the error term, which has fatter tails compared to the standard normal distribution, with zero mean and σt2 variance. According to Bao, Lee, and Saltoglu (2007), who evalate a number of GARCH-type model specifications using the Kullback-Leibler Information Criterion (KLIC), out of the four possible error distributions in the Ox G@rch package (Normal, Student T GED, Skewed T), Student T consistently shows the least KLIC value and is thus deemed closest to“the truth”. While a mixture of normal distributions is shown to perform even better, that is not available in the software, so we are not able to use it at this time. α0 , αi , αm , δ1 , δ2 , γi , ψi , λ, ω ∗ , β1 , φ1 , k, and d1 are parameters to be estimated, with d being the fractional integration parameter and L the lag or backshift operator. We extend the HYGARCH model to include the conditional variance in the 2 mean equation (the λσi,t term). Elyasiani and Mansur (1998) describe this as the “velocity feedback effect”, which captures the relationship between expected returns and the changing pattern of volatility.
3
equally weighted market return including dividends, CRSP data item EWRETD
15
4.3
Data sepcification
Our data comes from various sources. We collect data regarding bank balance sheet and off-balance sheet positions from the quarterly call reports issued by the Chicago Fed 4 , from 1986 Q2 to 2006 Q4, for a sample of 196 U.S. bank holding companies that we have identified as being present in the CRSP database as of Q4 2005. Due to lack of data availability on various BS and OBSA measures of bank exposure, we end up having to trim our dataset to drop everything prior to 1991 Q1. Data on market returns 5 and bank stock returns 6 is collected from the CRSP database. We measure changes in market interest rates through 10 year Treasury bonds (quarterly first difference in data item TCMNOM Y10 in the Federal Reserve Bank Reports data set), changes in FX rates through an index of trade-weighted USD vs. a basket currencies of major US trading partners (measured by a quarterly first difference in data item TWEXB, from the Federal Reserve Bank Reports data set). Since the CRSP stock return data is monthly, to produce quarterly stock returns we assumed reinvestment, and cumulated the returns as QR =
3 Y
(1 + Ri ) − 1
i=1
We use a similar procedure to cumulate 1 year treasury returns for our Ri t measure. Table 1 summarizes definitions and descriptive statistics for Rm , IR, and F X. Table 1. Descriptive statistics for market return, IR, and FX data
Name
Min
Mean
Max
S.Dev
-0.20403
0.047752
0.32559
0.10651
∆F X
-5.69
0.56
6.58
2.3615
∆IR
-0.0106
-0.00055
0.0094
0.0047331
Rm
Table 2 summarizes bank-specific exposure variables. Items have been normalized by bank total assets (data item BHCP2170). 4
Chicago Fed data source: http://www.chicagofed.org/economic research and data/bhc data.cfm. 5 equally weighted market return including dividends, CRSP data item EWRETD 6 including dividends, CRSP data item RET
16
Table 2. Definitions and descriptive statistics for bank-specific exposure variables
Name
Description
Min
Mean
Max
S.Dev
Fixed rate loans
3.4463
4.1818
4.9647
0.50949
a. BHCK3450
IR swaps
0.5356
1.2849
2.0503
0.42706
b. BHCK3809
OBSA IR 1 yr
0.47462
0.78467
1.2724
0.17474
Dom./tot. int. inc.
0.003446
0.001333
0.006810
0.001674
a. BHCK3826
FX swaps
0.010993
0.046581
0.085207
0.019169
b. BHCK3812
OBSA FX 1 yr
0.26958
0.67461
1.8703
0.38918
1. BS IR risk exposure BHCK1410 2. OBS IR risk exposure
3. BS FX risk exposure BHCK4105/BHCK4107 4. OBS FX risk exposure
While we would have liked to get better measures of balance sheet exposures, this data was very sparsely, if at all, available in the Chicago Fed dataset, so we had to make do with the best we could come up with.
5
Empirical Results
Upon creating an equally-weighted portfolio of all banks, we have attempted to run our HYGARCH model, as specified in Section 4. However, no matter how we tweaked the model, it refused to converge. As a result, here we present preliminary results in the form of Ox G@rch output 7 Ox version 4.1a (Linux) (C) J.A. Doornik, 1994-2007 Copyright for this package: S. Laurent and J.P. Peters, 2000-2006. G@RCH package version 4.2, object created on 3-05-2007 ---- Database information ---Sample: 1991(1) - 2006(4) (64 observations) Frequency: 4 Variables: 16 ******************** ** SPECIFICATIONS ** 7
Program code in appendix A.
17
******************** Dependent variable : ERit Mean Equation : ARMA (0, 0) model. 9 regressor(s) in the mean. Variance Equation : HYGARCH (1, d, 1) model of Davidson (Truncation order : 63). in-mean 1 regressor(s) in the variance. The distribution is a Student distribution, with 5.83451 degrees of freedom. No convergence (no improvement in line search) using numerical derivatives Log-likelihood = 112.304 No. Observations Mean (Y) Skewness (Y) Log Likelihood
: : : :
64 0.03944 -0.33334 112.304
No. Parameters Variance (Y) Kurtosis (Y)
: : :
18 0.00414 3.05294
Estimated Parameters Vector : 0.374183; 0.008782;-0.003803; 0.649945; -0.101655;-0.011782;-0.069163;-1.293508; -0.001359; 0.369600; 0.001769;-0.002534; 0.374223;-0.401566;-0.086650; 5.834514; -0.091147;-2.104382 Parameters Names Cst(M) ; OBSAIR1YR (M) ; TWEXB_D1 (M) ; d-Figarch ; Log Alpha (HY) ; The tests are not
ERitL1 (M) ; IRSWAPS (M) ; FXSWAPS (M) ; OBSAFX1YR (M) ; FIXEDIRLOANS (M) ; TCMNOM_Y10_D1 (M); MARKET_QR (M) ; Cst(V) ; D1 (V) ; ARCH(Phi1) ; GARCH(Beta1) ; Student(DF) ; ARCH-in-mean(var); reported since there is no convergence.
Table 3 presents the results in a more readable format. As the model has failed to converge (as have a number of other specifications), the meaningfulness of these results is dubious. However, we can make some notes about the signs of the coefficients. As expected, fixed rate loans are negatively associated with returns, as are changes in long term interest rates (TCMNOM Y10 D1), and changes in FX rates (TWEXB D1). FX swaps seem to have a large hedging component, as evidenced by their positive coefficient, while IR swaps do not appear to mitigate the risk of IR changes. Also as expected, the returns are positively associated with previous period’s returns, as well as with market returns. 18
Table 3. HYGARCH result Parameter
Value
Cst(M)
0.374183
ERitL1 (M)
0.008782
IRSWAPS (M)
-0.003803
FXSWAPS (M)
0.649945
OBSAIR1YR (M)
-0.101655
OBSAFX1YR (M)
-0.011782
FIXEDIRLOANS (M)
-0.069163
TCMNOM Y10 D1 (M)
-1.293508
TWEXB D1 (M)
-0.001359
MARKET QR (M)
0.3696
Cst(V)
0.001769
D1 (V)
-0.002534
d-Figarch
0.374223
ARCH(Phi1)
-0.401566
GARCH(Beta1)
-0.08665
Student(DF)
5.834514
Log Alpha (HY)
-0.091147
ARCH-in-mean(var)
-2.104382
6
Conclusions and Predictions
Due to the limited availability of results, which has forced us to use less-than ideal measures, and has rather significantly shortened out data series, as well as to the failure to converge of our empirical model (which certainly wasn’t helped by the shorter data series), we cannot draw any strong conclusions about our results yet. Some preliminary analysis of the model run shows that our hypotheses are at least on the right track. In future work, we plan on expanding our data set, and hope to get a converged model fit. 19
References Allen, F., Santomero, A. M., Dec. 1997. The theory of financial intermediation. Journal of Banking & Finance 21 (11/12), 1461–1485. 4 Baillie, R. T., Bollerslev, T., Mikkelsen, H. O., Sep. 1996. Fractionally integrated generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 74 (1), 3–30. 13 Bao, Y., Lee, T.-H., Saltoglu, B., Apr. 2007. Comparing density forecast models. Journal of Forecasting 26 (3), 203–225. 15 Chamberlain, S., Howe, J. S., Popper, H., Jun. 1997. The exchange rate exposure of u.s. and japanese banking institutions. Journal of Banking & Finance 21 (6), 871–892. 3 Choi, J. H., Elyasiani, E., Kopecky, K. J., Sep. 1992. The sensitivity of bank stock returns to market, interest and exchange rate risks. Journal of Banking & Finance 16 (5), 983–1004. 3 Choi, J. J., Elyasiani, E., Oct. 1997. Derivative exposure and the interest rate and exchange rate risks of u.s. banks. Journal of Financial Services Research 12 (2/3), 267–. 3 Davidson, J., Jan. 2004. Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business & Economic Statistics 22 (1), 16–29. 13, 14 Elyasiani, E., Mansur, I., May 1998. Sensitivity of the bank stock returns distribution to changes in the level and volatility of interest rate: A garchm model. Journal of Banking & Finance 22 (5), 535–563. 4, 12, 15 Flannery, M. J., Aug. 1983. Interest rates and bank profitability: Additional evidence. Journal of Money, Credit & Banking 15 (3), 355–362. 2 Flannery, M. J., James, C. M., Sep. 1984. The effect of interest rate changes on the common stock returns of financial institutions. Journal of Finance 39 (4), 1141–. 3 Grammatikos, T., Saunders, A., Swary, I., Jul. 1986. Returns and risks of u.s. bank foreign currency activities. Journal of Finance 41 (3), 671–. 3 Rasmusen, E., 2006. Games and Information: An Introduction to Game Theory. Blackwell Publishing Limited. 9 Ryan, S. K., Worthington, A. C., 2004. Market, interst rate and foreign exchange rate risk in australian banking: A garch-m approach. Inernational Journal of Applied Business and Economic Research 2, 81–103. 4, 13 Song, F. M., May 1994. A two-factor arch model for deposit-institution stock returns. Journal of Money, Credit & Banking 26 (2), 323–340. 3 Stiroh, K. J., Oct. 2004. Diversification in banking: Is noninterest income the answer? Journal of Money, Credit & Banking 36 (5), 853–882. 3, 4 Thakor, A. V., 1991. Game theory in finance. Financial Management 20 (1), 71–94. 7 Yourougou, P., Oct. 1990. Interest-rate risk and the pricing of depository financial intermediary common stock. Journal of Banking & Finance 14 (4), 803–820. 3 20
A
Ox G@rch Program Code
#import main() { decl garchobj; garchobj = new Garch(); //*** DATA ***// garchobj.Load("/data/allportfoliowitheverything_trimmed2.xls"); garchobj.Info(); garchobj.Select(Y_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(X_VAR, garchobj.Select(Z_VAR,
{"ERit",0,0}); {"ERitL1",0,0}); // REGRESSORS IN THE MEAN {"IRSWAPS",0,0}); {"FXSWAPS",0,0}); {"OBSAIR1YR",0,0}); {"OBSAFX1YR",0,0}); {"FIXEDIRLOANS",0,0}); {"TCMNOM_Y10_D1",0,0}); {"TWEXB_D1",0,0}); {"MARKET_QR",0,0}); {"D1",0,0}); // REGRESSOR IN THE VARIANCE
garchobj.SetSelSample(-1, 1, -1, 1); //*** SPECIFICATIONS ***// garchobj.CSTS(1,1); garchobj.DISTRI(1); garchobj.ARMA_ORDERS(0,0); garchobj.ARFIMA(0); garchobj.GARCH_ORDERS(1,1); // p order, q order garchobj.ARCH_IN_MEAN(1); garchobj.MODEL("HYGARCH"); garchobj.TRUNC(63);
//*** OUTPUT ***// garchobj.MLE(2); //*** PARAMETERS ***// garchobj.BOUNDS(0); 21
garchobj.FIXPARAM(0,); //*** ESTIMATION ***// garchobj.Initialization(); garchobj.PrintBounds(1); garchobj.DoEstimation(); garchobj.Output(); delete garchobj; }
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