Reexamining risk-return relationship in banks using quantile regressi

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By Ming-Yuan Leon Li

Service Industries Journal (a SSCI journal), accepted and forthcoming in 2010

Motivations

• banks make money in one of two ways: providing services to customers and taking risks.

• if a bank takes more risk it can expect to make more money

• empirical finding regarding the risk-return relationship is controversial and it has posed a longstanding problem in research field.

Motivations

• a positive risk-return relationship – Pettway, 1976; Kim, 1978; Jahankhani and Lynge, 19

80; Schneller, 1980; Bradley, et al., 1984; Brewer and Lee, 1986; Karels et al., 1989; Hassan and Bashir, 2003) show a positive risk-return relationship

• a negative relationship between risk and bank performance.– Bourke (1989), Molyneux and Thornton (1992), Canto

r and Johnson (1992), Berger (1995), Golin (2001) and Goddard et al. (2004)

Motivations

• This investigation 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 questions regarding the relationship dynamics between risk and bank performance.

Motivations

• this study examines whether the risk-return relationship in bank industry is consistent with different levels of bank profitability quantile.

• the quantile is a statistical term describing a division of observations into certain defined intervals based upon the values of the data

• the profitability quartile of a specific bank could show the relative magnitude of its profitability in comparison with the entire set of bank observations .

Motivations

• Our idea – Adizes (1988) proposes the firm life cycle the

ory: business strategies and organizational structures of firms vary according to the problems faced at different life cycle stages of the organization.

• The concept of life-cycle stage has generated considerable applied interest

Motivations

• To analyze the effect of life-cycle stage on profitability, these aforementioned studies invariably apply criteria such as earnings and/or age to segment sample companies into various subsets before performing traditional optimization techniques such as ordinary least squares (OLS) and least absolute deviation (LAD) to fit their subsets.

Motivations

• The analytical framework in these studies is based on unconditional distribution of firm samples.

• The findings of the current study suggest that this form of “truncation of samples” may yield invalid results.

• As demonstrated by Heckman (1979), such methods often exhibit sample selection bias.

Motivations

• A valid alternative is the quantile regression (QR hereafter), which segments the sample into subsets defined by conditioning covariates.

• Moreover, in comparison with the least square method, the QR approach offers a relatively rich description of the conditional mean for extreme cases in the samples

Motivations

• In addition, we posit that the behavior of banks with higher profitability differs from banks with lower profitability.

• First, banks at the growth (decline) stage tend to exhibit higher (lower) profitability.

• Further, according to lifecycle theory, profitable banks should differ from profitless banks in their strategies for enhancing profitability.

Motivations

• this investigation departs from previous related studies in proposing the QR framework to questions regarding the risk-return relationships in banks.

• the empirical results of this study could satisfactorily account for the existing risk-return relationship puzzle among numerous prior studies.

Empirical methods: Non-quantile models: OLS and LAD

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Empirical methods: Non-quantile models: QR (quantile regression)

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• a key feature of the quantile regression technique: the estimator vector of β,θ varies with θ.

• Moreover, by comparing the behaviors with different θ, one could thus characterize the dynamic estimator vector, namely β,θ, in various output-quantile regimes.

• the LAD estimator is a special case of the quantile-varying estimator with a quantile of 0.5.

Data and empirical results

• Samples for publicly-traded U.S. bank holding companies (BHCs hereafter) from March 2001 to September 2007 are analyzed.

• The final sample includes 18,108 quarterly BHS observations.

• Return on equity (RoE) is selected as the proxy variable for bank profitability performance.

• employ loan loss reserves to gross loans ratio (LLRGL) as a proxy variable for the risk taken by a bank – Mansur, et al. (1993), Hassan (1993), Stiroh (2006), H

irtle (2007)

• All data are obtained from the U.S. Federal Reserve Y-9C reports

Table 1 Definition of dependent/independent variables

Variables Definitions

Dependent variable

RoE Return on equity= Net income/ shareholders' Equity

Independent variables

LLRGL The ratio of loan-loss-reserves-to-gross-loans

Tables

Table 2 Descriptive statistics of dependent/independent variables

Variables Mean Standard error Median Minimum Maximum

RoE 0.0808 0.0592 0.0725 -1.3029 0.7027

LLRGL 0.0136 0.0086 0.0127 0.0000 0.2572

Table 3 Impact of LLRGL on bank profitability (RoE) across various quantile levels

Estimation results of the QR and OLS methods Statistic tests of the equality of slope

estimates across various quantiles

Quantile Estimate (p-value) Quantile Estimate (p-value)

Intercept Slope Intercept Slope Quantile F-statistics (p-value)

0.05 0.0290 (0.0000)*** -0.7218 (0.0000)*** 0.95 0.1543 (0.0000)*** 1.5344 (0.0000)*** 0.05 vs. 0.95 64.63 (0.0000)***

0.10 0.0309 (0.0000)*** -0.3406 (0.0000)*** 0.90 0.1315 (0.0000)*** 1.1856 (0.0000)*** 0.10 vs. 0.90 73.46 (0.0000)***

0.15 0.0354 (0.0000)*** -0.3121 (0.0010)*** 0.85 0.1210 (0.0000)*** 0.8426 (0.0000)*** 0.15 vs. 0.85 62.23 (0.0000)***

0.20 0.0396 (0.0000)*** -0.2823 (0.0110)** 0.80 0.1114 (0.0000)*** 0.6648 (0.0000)*** 0.20 vs. 0.80 37.9 (0.0000)***

0.25 0.0444 (0.0000)*** -0.2519 (0.0450)** 0.75 0.1025 (0.0000)*** 0.5721 (0.0010)*** 0.25 vs. 0.75 31.73 (0.0000)***

0.30 0.0494 (0.0000)*** -0.2049 (0.1680) 0.70 0.0967 (0.0000)*** 0.3613 (0.0230)** 0.30 vs. 0.70 16.62 (0.0000)***

0.35 0.0548 (0.0000)*** -0.0848 (0.6080) 0.65 0.0893 (0.0000)*** 0.3202 (0.0830)* 0.35 vs. 0.65 7.86 (0.0050)***

0.40 0.0611 (0.0000)*** -0.0707 (0.6770) 0.60 0.0826 (0.0000)*** 0.2832 (0.0710)* 0.40 vs. 0.60 11.51 (0.0007)***

0.45 0.0657 (0.0000)*** 0.0459 (0.7440) 0.55 0.0762 (0.0000)*** 0.2303 (0.1720) 0.45 vs. 0.55 4.36 (0.0368)**

0.50 0.0716 (0.0000)*** 0.0669 (0.6950) OLS 0.0750 (0.0000)*** 0.4237 (0.0000)***

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Some explanations

• In theoretical, a bank taking a relatively high risk is supposed to earn high profits, but is also exposed to certain costs; therefore its profitability might be reduced

• In particular, bankruptcy costs may be relatively high for a bank maintaining higher risk exposure.

• A subsequent increase in risk taking should lead to a decrease in profitability by heightening insurance expenses on uninsured debt.

• Our empirical findings show that highly profitable banks can increase their profitability by taking more risks

• by contrast, the superior policy in less profitable banks is to decrease rather than increase their risk exposures

• According to the life-cycle theory, a bank in the growth (declining) stage is denoted by a higher (lower) profitability.

• Moreover, this analysis further reveals that, in banks in a growth stage, namely, banks at higher quantile levels: 0.60 to 0.95, a significantly positive risk-return relationship is defined.

• Therefore, risk exposures positively affect profitability.

• In contrast, in banks at lower quantile levels, from 0.25 to 0.05, or banks in a declining stage

• increased risk exposure has a minimal effect on marginal profit enhancement; thus, the potential bankruptcy cost effect of risk taking would be dominant and would negatively affect profitability.

• Last, in banks at moderate quantiles, from 0.30 to 0.55, or banks in a mature stage,

• the profit enhancement effect derived from risk taking might be fully offset by its cost.

• Consequently, the effect of risk exposures on profitability would be insignificant.

• the existing risk-return puzzle among earlier studies could be satisfactorily accounted for our “V” shape relationship between risk and profitability, as shown in Fig. 2.

• We further indicate that pooling data together without considering the impact of bank life cycle is one of the reasons for the inconsistent bank risk-return relationship findings presented by prior empirical works.

Other applications of QR models

• Li, Ming-Yuan Leon* (2008) Reexamining asymmetric effects of monetary and government spending policies on economic growth using quantile regression, Journal of Developing Areas, accepted and forthcoming 【 SSCI】

• Value Strategy or Volume Strategy? A Dynamic Perspective Using Quantile Regression, submitted to Journal of Empirical Finance 【 SSCI】

• Reexamining the dynamic relationships between capital, size and earnings in banking using the quantile regression model, submitted to Journal of Money, Credit and Banking 【 SSCI】

• Establishing A Hybrid Bankruptcy Prediction Model with Dynamic Loadings Based on Accounting-ratio-based and Market-based Information Using Binary Quantile Regression, submitted to Journal of Financial Services Research 【 SSCI】