Examining the risk-return relationship in Taiwan banks using quantile
regression
Chin-Yuan Lin
Ph.D. Candidate, Department of Finance, National Sun Yet-Sen University
Wa-Lan Chang
Ph.D. Candidate, College of Business, Feng Chia University
Yang-Che Wu*
Assistant Professor, Department of Finance, Feng Chia University
Abstract
The bank risk-return relationship puzzle has posed a long-standing problem in this
banking research field. Using the data of the banks in Taiwan for the years 2005- 2010,
this study employs the LO and the RoE as the proxy variables of risk taken by a bank
and bank profitability, respectively. This paper is one of the first to employ a
conditional QR to explore changing distribution in bank RoE, across banks and over
time. Our empirical findings show that the risk-return relationship is negative across
the quantiles of bank profitability before the subprime crisis, and positive across the
low and middle quantiles of bank profitable after the subprime crisis. The finding
shows that highly profitable banks can increase their profitability by taking less risks;
by contrast, the superior policy in less-profitable banks is to increase rather than
increase their risk exposures. In addition, this study reveals that traditional OLS
optimization techniques capture central behaviors only, and misidentify the
relationship between the bank risk and profitability, such as size, significance and
even sign, particularly for extremely profitable/profitless banks.
Keywords: quantile regression, risk-return relationship
___________________________________________________________
*Correspondence author, Assistant Professor, Department of Finance,
Feng Chia University, 100 Wenhwa Rd., Seatwen, Taichung, Taiwan
40724, R.O.C. Email: wuyangche@fcu.edu.tw.
Introduction
Banks make money in two ways: providing services to customers and taking
risks. Theoretically, if a bank takes more risk, it will be expected to make more money.
However, empirical findings regarding the risk-return relationship are contradictory
and it has posed a long-standing problem in this research field. In particular, while
certain empirical studies (Hassan and Bashir, 2003) show a positive risk-return
relationship, some studies, such as Goddard, Molyneux and Wilson (2004) indicate a
negative relationship between risk and bank performance. Li (2010) use quantile
regression (QR hereafter) shows that there exists a positive risk-return relationship for
the profitable banks, negative for the profitless banks. The above empirical data is
limited in U.S.
As far as Taiwan banks are concerned, this study departs from the more
conventional research in the way that the parameters of the risk-return regression are
modeled, and proposes a new approach to survey the relationship dynamics between
the risk and bank performance in Taiwan banks. In particular, this study examines
whether the risk-return relationship in the banking industry is consistent with different
levels of the bank profitability quantile. It must be noted that the quantile is a
statistical term describing a division of observations into certain defined intervals
based upon the values of the data and the profitability quartile of a specific bank could
show the relative magnitude of its profitability in comparison with the entire set of
bank observations. In comparison with the ordinary least square (OLS), the QR
approach offers a relatively rich description of the conditional mean for extreme cases
in the samples.
This research departs from previous related studies in proposing the QR
framework to questions regarding the risk-return relationships in Taiwan banks. The
empirical results of this study can account for the existing risk-return relationship
puzzle in Taiwan banks. The rest of this paper is organized as follows: The second
section reviews the underlying model. The third section presents the empirical results,
and the fourth section concludes the paper.
QR model
Constant-coefficient regression models have been applied extensively in
statistics, while various random-coefficient models have also emerged as viable
competitors in particular fields of application. One variant of the latter class of models,
although perhaps not immediately recognizable as such, is the QR model. This study
employs a QR model in which the parameter of explanatory variables can be
expressed as a monotone function of a single, scalar random variable. The model
captures systematic influences of conditioning variables on location, scale and shape
of the condition distribution of the response. The model is thus significantly extended
with a constant coefficient in which the effects of conditioning are confined to a
location shift.
A QR technique developed by Koenker and Bassett (1978) is used in this study
to examine the dynamic relationship between the risk and bank profitability
performance. Assuming that the  th quantile of the conditional distribution of the
explained variable is linear in xi , the conditional QR model can be expressed as
follows:
yi  xi    u i
Quant ( yi |xi )  inf{ y : Fi ( y | x) }  xi  
Quant (u i |xi )  0
(1)
Where Quant ( yi |xi ) denotes the  th conditional quantile of yi on the regressor
vector xi ;  is the unknown vector of parameters to be estimated for different values
of  in (0,1); u i is the error term assumed to be continuously differentiable c.d.f.
(cumulative density function) of Fu (.| x) and a density function Fu (.| x) . The value
Fi (. | x) denotes the conditional distribution of y conditional on x . Varying the
value of  from 0 to 1 reveals the entire distribution of y conditional on x .
The estimator for  is obtained from
min
   u   (1   )  u
i
i:u i0


i: yi  xi   0
i
i:ubi0
  yi  xi   

i: yi  xi    0
(1   )  yi  xi  
(2)
Notably, the estimators do not have an explicit form, but the resulting
minimization problem can be solved by liner programming techniques. The design
matrix bootstrap method is employed to estimate the standard errors for the
coefficients in QR. In a Monte Carlo study, Buchinsky (1994) first recommended
bootstrap methods for relatively small samples, because the methods are robust to
changes in bootstrap sample size relative to the data sample size. Further, the
percentile method proposed by Koenker and Hallock (2001) was used in this study to
construct confidence intervals for each parameter in  , where the intervals are
computed from the empirical distribution of the sample of the bootstrapped estimates.
Notably, in comparison with standard asymptotic confidence intervals, the bootstrap
percentile intervals are not generally symmetric around the underlying parameter
estimate. These bootstrap procedures can be extended to deal with joint distribution of
various QR estimators, which allows the equality of slope parameters to be tested
across various quantiles.
Data and empirical results
Samples for publicly-traded bank in Taiwan from March 2005 to December 2011
are analysed. Certain banks are excluded from the analysis if they are not available for
the whole testing period. The final sample includes 720 quarterly observations. Return
on equity (RoE) is selected as the proxy variable for bank profitability performance.
Furthermore, this study follows Bikker & Haaf (2002) to employ the ratio of loans to
total assets (LO) as a proxy variable for the risk taken by a bank and the relationship
between RoE and LO is examined. All data are obtained from the Taiwan Economic
Journal. Table 1 summarizes the definitions of dependent/independent variables
selected for this study and certain descriptive statistic of those variables are presented
in Table 2.
Table 3 lists the estimation results of the QR model for the impact of the LO on
the RoE. For comparison, the OLS estimates are also presented. First, the OLS slope
estimate of the LO is significantly negative at the 1% level. This result indicates that
as more risks are taken by a bank, less money could be made. However, the OLS
estimator, by focusing only on the central tendency of the distributions, does not allow
the impact of the LO on bank profitability to differ for more/less profitable banks.
By contrast, the quantile-varying estimates of the LO variable derived by the
QR model, as shown in Table 3, reveal considerable variation in size, significance
and even in sign. In particular, by using the 10% level of significance as a criterion,
The OLS slope estimate of the LO is significantly negative at the 1% level. By using
the 10% level of significance as a criterion, while the LO variable is associated with
negative coefficients at all quantiles, which are most significantly. This finding
reveals some abnormal phenomena. In theory, a bank taking a relatively high risk is
supposed to earn high profits, but the empirical evidence shows an opposite
conclusion. It is necessary to further research how the risk-return relationship in
banks changes.
Because the subprime crisis occurs in 2007, we divided the sample period into
two sub-periods: before the crisis (2005Q1~2007Q3) and after the crisis
(2009Q2~2011Q4). Table 4 lists the estimation results of the QR model for the
impact of the LO on the RoE before the subprime crisis. The OLS slope estimate of
the LO is significantly negative at the 1% level. By using the 10% level of
significance as a criterion, while the LO variable is associated with negative
coefficients at all quantiles, which are most significantly. However, after the
subprime crisis, as shown in Table 5, the OLS slope estimate of the LO is
insignificantly positive at the 1% level. By using the 10% level of significance as a
criterion, while the LO variable is associated with positive coefficients at the
quantiles, from 0.05 to 0.55, but they are insignificant. The coefficients become
significantly negative at the quantiles, from 0.8 to 0.95.
The finding requires some explanations. A bank taking a relatively high risk
may not earn high profits. The cause can be the exposure to bad loans; therefore, its
profitability might be reduced. In particular, before the subprime crisis, banks
loosened the loan credit. The costs caused by bad debt may be relatively high for a
bank maintaining higher risk exposure. A subsequent increase in risk taking should
lead to a decrease in profitability due to more costs. However, after the subprime
crisis, banks heighten the credit quality. The risk-return relationship gets better. But,
Our empirical findings show that highly profitable banks can increase their
profitability by taking less risks; by contrast, the superior policy in less-profitable
banks is to increase rather than increase their risk exposures.
Conclusions and future research
The bank risk-return relationship puzzle has posed a long-standing problem in
this banking research field. Using the data of the Taiwan banks for the years 20052010, this study is one of the first to employ a conditional QR to explore changing
distribution in bank RoE, across banks and over time. Our empirical findings show
that the risk-return relationship varies differently across the quantiles of bank
profitability before and after the subprime crisis. In particular, for less and middle
profitable banks, a positive risk-return relationship is presented after the subprime
crisis. Importantly, this study reveals that traditional OLS optimization techniques
capture central behaviors only, and misidentify the relationship between the bank risk
and profitability, including size, significance and even sign, particularly for extremely
profitable/profitless banks.
There are four important caveats in interpreting the analytical results of this
investigation. First, the present empirical results are limited to the banks in Taiwan.
Future works could examine a range of countries. Second, although two alternatives –
the OLS and the QR techniques – are adopted and compared, comparisons should also
be made with models with other dynamic parameter designs. Third, this study
employs the LO and the RoE as the proxy variables of risk taken by a bank and bank
profitability, respectively. Future areas of work might use different proxy variables.
Finally, QR techniques are intended for cross-sectional data. The panel data in this
study are adapted to this requirement by stacking the quarterly cross-sectional
observations from 2005 to 2010 and treating the pooled data set as a large
cross-section. Capturing bank-specific fixed effects is encouraged for future research.
Reference
Bikker J. A. and Haal K., 2002.“Competition, concentration and their relationship: An
empirical analysis of the banking industry”, Journal of Banking and Finance Vol.
26, Issue 11 pp.2191~2214.
Buchinsky, M., 1994. “Changes in U.S. wage structure 1963~1987: An application of
quantile regression”, Econometrica, 62(2), pp.405~458.
Goddard, J., Molyneux, P., and Wilson, J. O. S., 2004. “The profitability of European
banks: A cross-sectional and dynamic panel analysis”, Manchester School, 72(3),
pp. 363~381.
Hassan, M. K. and Bashir, A.-H. M., 2003. “Determinants of Islamic banking
profitability”, The Economic Research forum (ERF) 10th Annual Conference (pp.
16~18). Marrakesh, Morocco.
Koenker, R., and Hallock, K. F. 2001. “Quantile regression”, Journal of Economic
Perspectives, 15(4), pp.143~156.
Li. M. Y., 2010. “Re-examining the risk-return relationship in banks using quantile
regression”, The Service Industries Journal Vol. 30. No. 11, pp. 1871~1881.
Table 1. Definition of dependent/independent variables.
Variables
Definitions
Dependent variable: RoE
Independent variables: LO
The ratio of net income to shareholders’ equity
The ratio of loans to total assets
Table 2. Descriptive statistics of dependent/independent variables.
Variables
Mean
Standard error
Median
Minimum
Maximum
RoE
LO
-0.3161
0.6077
7.6374
0.0967
1.36
0.6119
-101.43
0.0967
14.46
0.7794
Note: Data source is consistent with Table 1.
Table 3. Impact of LO on bank profitability (ROE) across various quantile levels.
Estimation results of the QR and OLS methods
Estimate(p-value)
Quantile
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Intercept
2.0995
(0.8382)
1.7219
(0.8036)
0.8706
(0.7109)
0.1615
(0.9388)
1.0559
(0.2270)
0.8674
(0.1604)
1.1254
(0.0306)
1.5378
(0.0037)
0.0657
(0.0001)***
1.8799
(0.0003)***
Slope
-21.6666
(0.1949)
-5.9084
(0.5996)
-5.1189
(0.1802)
-1.2023
(0.7251)
-1.6294
(0.2513)
-0.6954
(0.4884)
-0.7175
(0.3955)
-0.9981
(0.2445)
0.0459
(0.1247)
1.2267
(0.2069)
Estimate(p-value)
Quantile
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
OLS
Intercept
9.3828
(0.0000)***
5.5616
(0.0000)***
5.7092
(0.0000)***
4.3706
(0.0000)***
4.0435
(0.0000)***
3.8987
(0.0000)***
3.2893
(0.0000)***
2.5239
(0.0000)***
2.2636
(0.0000)***
2.281
(0.208)
Slope
-5.3554
(0.0133) **
-2.3533
(0.0043)***
-3.4374
(0.0002)***
-5.3517
(0.0030)***
-2.6262
(0.0012)***
-2.7076
(0.0021) ***
-2.1019
(0.0073) ***
-1.1733
(0.1131)
-1.1161
(0.1997)
-4.273
(0.147)
Notes: The value in the parenthesis denotes the p-value. The OLS slope estimate of the LO is insignificantly positive at the 1%
level. By using the 10% level of significance as a criterion, while the LO variable is associated with an insignificant negative
coefficient at the quantiles, from 0.05 to 0.60, it becomes a significantly negative coefficient at higher quantile levels from 0.65
to 0.95. The data source is consistent with Table 1.
*Significance at the level of 10%.
**Significance at the level of 5%.
***Significance at the level of 1%
Table 4. Impact of LO on bank profitability (ROE) across various quantile levels before subprime crisis
Estimation results of the QR and OLS methods
Estimate(p-value)
Quantile
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Intercept
15.5482
(0.0302)**
8.2260
(0.0523)*
6.9526
(0.2269)
5.1009
(0.0596)*
3.9821
(0.2277)
3.1690
(0.1559)
3.0021
(0.0192)**
2.6507
(0.0187)***
3.0047
(0.0015)***
3.2786
(0.0003)***
Estimate(p-value)
Slope
Quantile
-53.7129
(0.0000)***
8.2260
(0.0523)*
-20.2524
(0.0312)**
-13.0440
(0.0033)***
-8.8712
(0.0994)*
-5.5021
(0.1306)
-4.3486
(0.0371)**
-3.1284
(0.0877)*
1.2267
(0.2069)
-3.2706
(0.0245)***
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
OLS
Intercept
5.6670
(0.0000)***
4.9294
(0.0000)***
4.9601
(0.0000)***
5.1624
(0.0000)***
4.3945
(0.0000)***
4.3969
(0.0008)***
4.4268
(0.0001)***
4.6593
(0.0000)***
4.2714
(0.0000)***
15.1762
(0.0000)***
Slope
-2.3087
(0.1756)
-1.8958
(0.1340)
-2.6299
(0.1238)
-3.4485
(0.0184)**
-2.6929
(0.0792)*
-3.1767
(0.1336)
-3.8456
(0.0395) **
-4.6830
(0.0029)***
-4.4918
(0.0018)***
-15.3227
(0.0000)***
Notes: The value in the parenthesis denotes the p-value. The OLS slope estimate of the LO is significantly negative at the 1%
level. By using the 10% level of significance as a criterion, while the LO variable is associated with negative coefficients at all
quantiles, which are most significantly. The data source is consistent with Table 1.
*Significance at the level of 10%.
**Significance at the level of 5%.
***Significance at the level of 1%
Table 5. Impact of LO on bank profitability (ROE) across various quantile levels after subprime crisis.
Estimation results of the QR and OLS methods
Estimate(p-value)
Quantile
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
Intercept
Slope
-7.9450
(0.4478)**
-6.8027
(0.1712)
-4.0312
(0.0791)*
-0.9397
(0.4113)
0.1793
(0.8480)
0.5504
(0.2393)
0.5334
(0.2382)
0.7594
(0.2129)
0.6509
(0.3033)
0.7239
(0.1714)
1.1134
(0.9477)
6.5759
(0.4145)
4.6409
(0.2122)
1.3587
(0.4639)
0.2225
(0.8834)
0.1319
(0.8619)
0.3988
(0.5865)
0.3338
(0.7355)
0.8796
(0.3910)
1.0636
(0.2153)
Estimate(p-value)
Quantile
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0.55
OLS
Intercept
6.3873
(0.0000)***
5.9065
(0.0000)***
5.5935
(0.0000)***
4.0970
(0.0000)***
3.3484
(0.0000)***
2.9043
(0.0000)***
2.3136
(0.0000)***
1.9703
(0.0001)***
0.9208
(0.0791)*
0.0873
(0.965)
Slope
-4.0254
(0.0020)***
-4.1916
(0.0467)**
-4.3111
(0.0004)***
-2.4573
(0.0303)**
-1.6361
(0.1103)
-1.2810
(0.1748)
-0.6378
(0.4097)
-0.3391
(0.6814)
1.0025
(0.2381)
0.3583
(0.912)
Notes: The value in the parenthesis denotes the p-value. The OLS slope estimate of the LO is insignificantly positive at the 1%
level. By using the 10% level of significance as a criterion, while the LO variable is associated with positive coefficients at the
quantiles, from 0.05 to 0.55, but they are insignificant. The coefficients become significantly negative at the quantiles, from 0.8
to 0.95. The data source is consistent with Table 1.
*Significance at the level of 10%.
**Significance at the level of 5%.
***Significance at the level of 1%