On mean–variance analysis of a bank’s behavior

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Highlights

  • Considering banks’ mean–variance problems for the portfolio return and accounting profit with the internalized balance sheet model.

  • The mean–variance problem for accounting profit provides good accuracy in the model fitting for cross banking companies.

  • The mean–variance problem for the portfolio return provides good accuracy in the model fitting for an individual banking company.

  • No significant difference in the accuracies of the model fitting for the portfolio return and accounting profit as long as the balance sheet is internalized.

Abstract

We consider the mean–variance utility maximization problem for banks. In particular, we consider the utility maximization problems of the portfolio return and accounting profit. Moreover, we consider balance sheet models for both conditions irrespective of whether the items on the liability side are internalized in terms of assets. The calibration result shows that there is no significant difference in the accuracies of the fit of the utility maximization models for the portfolio return and accounting profit as long as the balance sheet is internalized. Therefore, internalization of the balance sheet model is important to describe the bank’s behavior.

Introduction

We consider the classical optimal asset allocation problem for a bank. We suppose that the bank’s preference for risk is indicated by the mean–variance utility, and the bank seeks an optimal loan ratio to maximize the expected utility. The loan ratio in this study is measured by the proportion of the amount of lending to the total (risky) asset, which is the sum of the amounts of lending and investing in securities.

The mean–variance analysis by Markowitz (1952) has been traditionally used to analyze a bank’s behavior in asset allocation. Kane and Malkiel (1965), Kahane (1977), Koehn and Santomero (1980), Kim and Santomero (1988), Greenwald and Stiglitz (1989), and Keeley and Frederick (1990) consider the mean–variance utility maximization problem for the return of the bank’s asset portfolio. In contrast, Ishii (1971) and Halaj (2013) consider the mean–variance utility maximization problem for the bank’s accounting profit calculated by the difference of the portfolio return and the sum of funding costs. Thus, in the mean–variance framework, there are two types of models in which the agent maximizes its utility for the portfolio return and accounting profit. Besides the mean–variance utility maximization problem, Fischer (1983), Hartley and Walsh (1991), Jacques (2008), and Wang (2013) also consider the bank’s optimization problem for the accounting profit. Moreover, Ishii (1971)and Wang (2013) internalize the items on the liability side of balance sheets in terms of assets, while most previous studies treat all items on the liability side as exogenous variables (Kane and Malkiel, 1965, Fischer, 1983, Keeley and Frederick, 1990, Hartley and Walsh, 1991, Jacques, 2008, Halaj, 2013). Not only does this enable the model to contain more variables and parameters but also leads to optimal management of the bank’s funding.

Most previous studies focus on the effects of the central bank’s financial policy on the macroeconomy or of the bank regulations on bank behavior (i.e., lending and funding). However, few studies examine the fit of the theoretical model of the bank’s behavior with its actual behavior. This study addresses this gap in the literature. We verify whether the maximization problem for the portfolio return or accounting profit more closely describes the bank’s actual behavior.

We consider the bank’s mean–variance utility maximization problems for both the portfolio return and accounting profit. We also consider two balance sheet models for the bank regarding whether the items on the liability side are internalized in terms of assets. We construct the internalized balance sheet model following Ishii (1971). Thus, we treat three optimization problems for the bank: (i) its mean–variance utility maximization problem for the portfolio return without the internalized balance sheet model, (ii) its mean–variance utility maximization problem for the portfolio return with the internalized balance sheet model, and (iii) its mean–variance utility maximization problem for the accounting profit with the internalized balance sheet model. We then solve the optimal loan ratios for each optimization problem.

After obtaining the optimal loan ratios, we calibrate the model parameters to fit the model loan ratio to the actual one. This simultaneously verifies the accuracy of the model fit to the actual data. For the maximization problem for the portfolio return, we use only the balance sheet data. In contrast, for the maximization problem of the accounting profit, we use data from both the balance sheet and profit and loss statements. As examples, we use the financial statements of five large Japanese banking companies. Moreover, we perform two types of calibration. We implement the first under the assumption that all parameters to be estimated are common across the five banks (cross-bank data). In the other, we calibrate the models using the historical data of a bank’s financial statement (historical data).

The results are as follows: at first, the accuracy of fitting for the model maximizing the utility for the portfolio return without the internalized balance sheet model is the most inferior for both cross-bank data and historical data. We believe that this result is straightforward, as the model without the internalized balance sheet model has the least parameters and variables. Next, when using cross-bank data, the model optimizing the utility for the accounting profit with the internalized balance sheet model has the most accurate fit. However, the estimation errors for the model maximizing the utilities for the portfolio return and accounting profit with the internalized balance sheet model are quite close. Finally, when using historical data, the model optimizing the utility for the portfolio return with the internalized balance sheet model surprisingly has the most accurate fit. As the model optimizing the utility for the accounting profit takes into account more variables than that for the portfolio return, it is natural to expect that it has superior accuracy of fit than the latter. Therefore, our results show that reducing exogenous variables in modeling financial statements is more important than adding parameters and variables to the model in the context of the mean–variance framework, when modeling the actual behavior of banks.

Section snippets

Notations

The notations in our study follow those of Ishii (1971).

  • L: money amount of lending

  • B: money amount invested into securities

  • Dp: primary deposit

  • D: secondary deposit

  • Dg: total deposit defined by Dg=Dp+D

  • N: borrowed money at interbank markets

  • Cr: cash on the asset side of the balance sheet

  • r: proportion of cash to the total deposit, that is, r=CrDg

  • l: loan ratio defined as the proportion of the money amount of lending to the total (risky) asset, that is, l=LL+B

  • e: capital and liabilities with long

Optimal asset allocation under simple balance sheet model

In this section, we consider the bank’s utility maximization problem for the return of the asset portfolio, when the deposits and borrowed money are exogenously given, that is, the model given in Section 2.2.1.

From (2.3), (2.4), the gross return Rg for the bank’s asset portfolio is Rg=RLL+RBB=(RLRB)Xl+RBX,where X=(1r)Dg+N+e. Then, the expected gross profit Eg is EgE[Rg]=((μLμB)Xl+μBX),and the variance of Rg is σg2Var[Rg]=(σL2+σB22ρLBσLσB)X2l2+2(ρLBσLσBσB2)X2l+σB2X2.

The bank chooses the

Optimal asset allocation for portfolio return (Non-P/L)

In this section, we consider the bank’s utility maximization problem for the return of the asset portfolio when the primary deposit is exogenously given, that is, the model in Section 2.2.2. We do not incorporate the profit and loss statement into the model at this stage yet.

From (2.6), (2.5), the gross return Rg for the bank’s asset portfolio is Rg=RLL+RBB=RLM2lbM1l+RBM2(1l)bM1l=M2bM1l((RLRB)l+RB).Then, the expected gross return Eg is EgE[Rg]=M2bM1l((μLμB)l+μB),and the variance of Rg

Calibration result

We have now obtained the optimal lending ratio for each model. Next, we identify the model closest to the actual data through calibration. We use two different data sets1 of financial statements. In the first, we calibrate our formulae for the financial statements of five large Japanese banking companies in a fiscal year, assuming that the estimated parameters are common among those banks (cross-bank data). In the second, we

CRediT authorship contribution statement

Kazuhiro Takino: Conceptualization, Methodology, Software. Yoshikazu Ishinagi: Supervision, Preparation of balance sheets.

Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 20K02042.

References (13)

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This study was supported by a grant-in-aid from Zengin Foundation for Studies on Economics and Finance, Japan.

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