An Intellectual Capital Model of Bank Lending
Dr. Chao-Hui Yeh, I-Shou University, Taiwan
ABSTRACT
Purpose: Intellectual Capital can be said to constitute valuable factors not shown in accounting financial
statements, but which are of critical importance to a company's long-term profitability. This study models the
relationship between intellectual capital(IC hereafter) and various loans in bank lending. Method: Pioneering
and exhaustive research by Yeh (2001) discusses the residual income valuation model (RIVM hereafter). This
paper extends the RIVM to derive mathematically the accounting items of IC for lending banks. The RIVM
analyses the relationship between market value of equity and accounting activities (investments) in the
manufacturing firms. This study adds to the RIVM literatures by applying it to lending banks where operating
assets are various loans, which are of critical difference to a manufacturing firm. Result: We find that IC (the
difference between market and book value of equity) is a function of fee income, non-performing loans, gross
loans and new loan investment. This implies that the most important assets of banking industry are reputation,
honesty, and commitment. Conclusion: We conclude that the sales of fee-based services, non-performing loans,
gross loans and new loan investment are valuable factors in bank lending. With our mathematical model, the
empirical analysis is more disciplined than that of many prior ad-hoc valuation studies. Our model could be
extended to encompass other valuable factors in banks that are also likely to drive market and book values
differently. Directions for Future Research: We suggest a number of value-driving activities of banks,
including deposit taking, credit card services, mortgage servicing rights and trust activities. The combination of
fee income and new deposit investments in our model will partially describe these activities. Nevertheless, it is
potentially fruitful to use our model to incorporate these and other factors of value more explicitly. Within our
modeling framework this would require identification of the investing activities that give generate these hidden
values. These activities could take the form of new investments in social capital, that is, it is social capital that
transforms human capital into producing positive career outcomes and increasing investors' perception of
potential... Such an extension would likely result in a valuation model that contains more financial statements
reports items, especially the profit/loss account and balance-sheet of the banks.
Key words: Intellectual Capital, Residual Income Valuation Model, Loans.
1. INTRODUCTION
Most intellectual capital approaches have problems with meaningful measurement.
Big differences often exist between a company's market value and its book value. Many of these are explained
by intellectual capital assets not shown in the balance -sheet. Assets like employee knowledge, expertise and
creativity, customer confidence in the company and its products, brands and franchises, Information and
knowledge management systems, administrative procedures, copyrights, patents and trademarks, the efficiency
of company business processes and the effectiveness of company planning, forecaster and strategy-making. This
has presented companies with a new challenge - how to measure, account for, manage and develop intellectual
capital. This paper of Intellectual Capital will help you do exactly that.
The rise of the "new economy", one principally driven by information and knowledge, has led to an
increased interest in intellectual capital (IC). IC is an area of interest to numerous parties, such as
shareholders, institutional investors, scholars, policymakers and managers.
However, there have been very few papers that have studied IC of banks. The implications of IC are
more prominent in banks as banks have abundant human capital at their disposal. Therefore, it becomes
necessary to understand what the factors of creating value in banks are. Banks happen to be one service
sector that uses a huge amount of human capital and customer capital for its survival. Thus, this paper
evaluates the intellectual capital model of bank lending using the residual income valuation model
(RIVM).
This paper provides a strong case for reporting of the value creation through intellectual capital in
the financial statements. The paper would be a useful tool for benchmarking the performance of the
banks across various countries. When in 1988 the Swiss food-products company Nestle bought the
British confectionery group Rowntree, the price paid included well over $1 billion for something that
had never appeared on Rowntree's balance sheet, this hidden value now recognized as intellectual
capital.
With the advent of knowledge economy era, tangible assets are no longer the factors that companies
rely on to create high value, while intangible assets and intellectual capital play more significant roles
that even surpass tangible assets in determining companies’ future competitiveness. Intellectual capital
includes intellectual properties, intellectual assets and other information assets or intangible assets, such
as human capital, capability of management team, relationships with customers and suppliers, employee
devotedness and innovation ability. Most of these items are absent from or cannot be evaluated by
traditional accounting, which is also the major cause of differences between corporate market value and
book value. Therefore, assessment of intellectual capital is rather important for companies.
Value is decided by the quantitative analysts who work in the financial markets developing
mathematical models to assist the activities of traders and risk managers within banks and other large
corporate institutions, while price be affected by market conditions, then the quantity of the two are
usually unequal. There are seven intellectual capital valuation models in the literature, including Tobin’s
Q, Market-to-book ratio (M/B ratio), Economic Value Added (EVA™), Calculated Intangible value
(CIV), Knowledge Capital Earnings (KCE), Value Added Intellectual Coefficients (VAIC™) and
Financial Method of Intangible Assets Measuring (FiMIAM). This paper will discuss a different model.
The next of our paper is the model. The paper concludes with Section 3.
2. MODEL(The Residual Income Valuation Model)
Given that accounting profit is computed on a "comprehensive income" or "clean surplus" basis. This means that
all revenues, expenses, gains, and losses recognized during the period are included in income, regardless of
whether they are considered to be the result of operations for the period. Therefore, under clean surplus
accounting, earning after tax for the period ( I t ) contain all changes in book value during the period. That is,
BVEt − BVEt −1 = I t − Dt
(1)
Where Dt is cash paid to (net of contributions by) the firm’s equity capital providers for period t;
and ( BVE t − BVEt −1 ) equals the change in accounting book value of net assets between the end of
period t-1 and t.
The movement of revenues and expenses following GAAP fails to account for the opportunity cost of
equity capitals, so residual income was developed to account for the time value of money. Residual income for
R
R
the period, I t , is defined as follows: Residual income ( I t ) is earning after tax ( I t ) less equity book value at
the beginning of the period ( BVEt −1 ) multiplied by cost of equity ( rE ).Residual income is defined as
follows: I t ≡ (I t ) − (rE × BVEt −1 )
R
(2)
To see how a firm’s future residual income can be tied back to its financially rewarding value on
date t we begin by defining market value of equity ( MVEt ) as the present value of all future cash flows
~
⎡ D
⎤
t+s
to equity claimants, i.e., MVEt = ∑ E t ⎢
s ⎥
s =1
⎣ (1 + rE ) ⎦
∞
(3)
Substituting from equations (1) and (2) into equation (3) produces the RIVM. That is,
~
⎡ I t +Rs ⎤
MVEt = BVEt + ∑ Et ⎢
s ⎥
s =1
⎣ (1 + rE ) ⎦
∞
(4)
Thus equation (4) stated that market value of equity ( MVE t ) is book value of equity ( BVEt ) plus
the discounted value of the firm's future residual income over all future periods.
For many authors, the difference between the market value of companies' shares and their book
value is the consequence of intellectual capital (IC). That is, MVEt = BVEt + IC
(5)
Substituting
from
~
I t +Rs
equation
(5)
into
equation
(4)
produces
the
equation
(6).
⎡
⎤
ICt ≡ ∑ Et ⎢
s ⎥
s =1
⎣ (1 + rE ) ⎦
∞
(6)
Given (6)and R E = 1 + rE , we get that the following： ICt
[
= RE−1Et I tR+1 + ICt +1
]
(7)
Given financial debt is negative financial asset, this paper assume that net financial assets are
financial assets minus financial debts and use financial assets as net financial assets. Given the financial
assets are marked to market value, the market value of financial assets ( MVFA t ) is the book value of
financial assets ( BVFA t ). If the assumption is correct, then we get that the following ：
MVFA t = BVFA t
(8)
Given that financial assets is net financial assets and does not use financial debts. Firm’s equity is Firm’s
financial assets plus operating assets of the firm, so book value of equity is book value of operating assets
(9)
( BVOAt ) plus book value of financial assets： BVE t = BVOA t + BVFA t
Market value of equity is market value of operating assets plus market value of financial assets：
(10)
MVE t = MVOA t + MVFA t
Substituting from equations (8) and (9) into equation (10) produces the equation (11).
MVEt − BVEt = MVOAt − BVOAt
(11)
The equation (11) implies that cost of equity equals cost of operating assets. That is,
rE = rOA
(12)
Substituting from equation (12) into equation (7) produces the equation (13). That is,
[
−1
ICt = ROA
Et I tR+1 + ICt +1
]
(13)
Substituting from equations (5) into equation (11) produces the equation (14).
ICt = MVOAt − BVOAt
(14)
The following persistent value driver dynamic, PVDD, lays out the evolution of free cash flows:
I tL+1 = rLθ1PLt + FEEt +1
(15)
PLt +1 = NLI t +1 + θ1PLt
(16)
NLI t +1 = CLL NLI t
(17)
NPLt +1 = (1 − θ1 )PLt + θ 2 NPLt
(18)
FEEt +1 = CFF FEEt + CFL NLI t
(19)
L
In equation (15) the mark, I t +1 takes the abbreviation of income from loans at t+1, superscript L
refers loans. Interest income from loans at time t+1 is a fraction ( θ1 ) of current performing loans ( PLt )
at time t multiplied by the stated interest rate on loans ( rL ) plus fee income at time t+1( FEEt +1 ).
This implies the default rate on loans is (1 − θ1 ) .
In equation (16) the mark, PLt +1 takes the abbreviation of performing loans at t+1.Performing
loans at t+1 is a fraction ( θ1 ) of current performing loans ( PLt ) at time t plus new loan investments at
t+1 ( NLI t +1 ).At time t+1, new loans investment, NLI t +1 , are invested (lent) that add to PLt +1 . They
are subject to default and start paying interest at t + 1, one can think of lending as investment for banks,
and θ 1 as the persistence of performing loans. Note that new loans are net of repayments. We do not
take account of the maturities of existing loans because they are assumed to rollover. Any net
repayments of loans would be reflected in NLI t +1 .
In equation (17) the mark, NLI t +1 takes the abbreviation of new loan investments at t+1. New loan
investments at t+1 is a fraction ( C LL ) of new loan investments at t ( NLI t ).Growth in lending is
described through the parameter C LL , which is
NLI t +1
.
NLI t
In equation (18) the mark, NPLt +1 takes the abbreviation of non-performing loans at t+1.
Non-performing loans at t+1 is a fraction ( θ 1 ) of current non-performing loans ( NPLt ) at time t plus
default rate on loans (1 − θ 1 ) multiplied by current performing loans ( PLt ) at time t. The stock of
non-performing loans includes a proportion ( θ 2 ) of the prior period non-performing loan balance.
Implicitly, (1− θ 2 ) of the prior period non-performing loans are charged off the books completely. For
the moment, non-loan net assets such as deposits or investment securities of the lending bank are
assumed to be financial in nature, marked-to-market and are zero net present value. ( θ 1 ∈ 0,1 )
[ ]
In equation (19) the mark, FEEt +1 takes the abbreviation of fee income from loaners at t+1. Fee
income at time t+1 is a fraction ( CFF ) of current fee income ( FEEt ) at time t plus a fraction ( CFL ) of
new loan investments at t ( NLI t ).
Substituting from equation (15) into equation (19) produces the equation (20).
I tL+1 = rLθ1PLt + CFF FEEt + CFL NLI t
(20)
L
For a lending bank, if the equation (20), I t +1 is the total income, the market value of operating assets is a
linear combination of the bank’s performing loans, new loan investments and fee-based services. This is,
MVOAt = CPL PLt + C NLI NLI t + CFEE FEEt
(21)
The advantage of the RIVM is that we can describe our model directly in terms of balance sheets.
The balance sheets provide gross loans ( GLt ), non-performing loans ( NPLt ) and the loan loss
allowance ( LLRt ). We focus on the loan loss allowance because it is the major bank accrual.
Practicably banks set LLRt according to GLt and NPLt , we assume that bank managers set the
LLRt each period through the use of two given policy parameters. The first, 1 − CGL , describes the
ratio of LLRt in GLt , and the second, C NPL , describes the ratio of LLRt in NPLt , the fraction
of non-performing loans set aside:
LLRt ≡ (1 − C GL )GLt + C NPL NPLt
Substituting from equation NLt
(22)
= GLt − LLRt into equation (22) produces the equation (23).
NLt ≡ (C GL )GLt − C NPL NPLt
(23)
The mark, NLt , takes the abbreviation of net loans at t.
If loans is the only operating assets, then book value of operating assets equal net loans.
(24)
NLt ≡ BVOAt
This modeling of the accrual process describes the banking practice of setting a LLPt +1 based on the
level of net loan at time t+1, net loan at time t and new loan investments at time t+1. Equation (25)
shows that net loans at time t+1 can be expressed as follows:
LLPt +1 = NLt + NLI t +1 − NLt +1
(25)
When the sole activity generating value for a bank is its lending activities, the residual income for time
t+1 is expressed as follows:
I tR+1 = I tL+1 − LLPt +1 − rOA × NLt
(26)
Substituting from equation (20) into equation (26) produces the equation (27).
I tR+1 = rLθ1PLt + CFF FEEt + CFL NLI t − LLPt +1 − rOA × NLt
144444244444
3
(27)
( 20 )
Substituting
from
equation
(25)
into
equation
(27)
produces
the
equation
(28).
I tR+1 = rLθ1PLt + CFF FEEt + CFL NLI t + NLt +1 − NLI t +1 − ROA × NLt
144444244444
3
(28)
( 20 )
Substituting from equations (23) and (24) into equation (14) produces the equation (29).
ICt ≡ MVOAt − BVOAt = MVOAt − C GL GLt + C NPL NPLt
(29)
Substituting from equation (21) into equation (29) produces the equation (30).
ICt ≡ MVOAt − BVOAt = C PL PLt + C NLI NLI t + C FEE FEEt − CGLGLt + C NPL NPLt
1444442444443
(30)
( 21)
Substituting from equation PL = GL − NPL into equation (30) produces the following equation.
IC t ≡ MVOAt − BVOAt = C PL (GLt − NPLt ) + C NLI NLI t + C FEE FEEt − C GL GLt + C NPL NPLt
Collecting the coefficients of GL and NPL produces the following equation.
ICt = C NLI NLI t + C FEE FEEt + (C PL − C GL ) × GLt + (C NPL − C PL ) × NPLt
(31)
The equation (31) state that IC of banks is described by NLI t , FEEt , NPLt and GLt .
ICt ≡ c NPL NPLt + cGL GLt + c NLI NLI t + c FEE FEEt
(32)
Where
c FEE = (ROA − C FF ) C FF = C FEE
−1
(33)
c NLI = C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
{ [
]
[
]}
cNLI = (ROA − CLL ) CLL (ROA − θ1 ) rLθ1 − 1 + C FL (ROA − CFF ) C FF + 1
(34)
cGL = C PL − CGL = (ROA − θ1 ) rLθ1 − CGL
(35)
cNPL = C NPL − CPL = C NPL − (ROA − θ1 ) rLθ1
(36)
−1
−1
−1
−1
−1
Where C NPL and CGL are given values in equation (22).Proofs of this are in the Appendix.
3 Conclusion and Directions for Future Research
Successfully implementing a method for the valuation or measurement of intellectual capital is not an easy
task. Practitioners yet receive little support from the intellectual capital research community. Little research has
been done into the factors that influence the success of a method. This paper is a first attempt at mathematically
applying the residual income valuation model to bank’s IC. For the purpose of developing our bank’s IC model,
we assume the major value creating activities are fee-based services, non-performing loans, gross loans and new
loans. In addition, we assume the major accounting bias is loan loss allowance. When we combine these
assumptions we arrive at an IC valuation model that depends on loan, new loans, non-performing loans and fee
income. Our model could surely be extended to include other value generating activities and accounting biases
of banks. We have shed light on including other bank activities that are also likely to drive market and book
values apart.
Appendix
[
]
−1
ICt = ROA
Et ICt +1 + I tR+1 (13)
ICt = C NLI NLI t + CFEE FEEt + (C NPL − CPL ) × NPLt + (CPL − CGL ) × GLt (31)
I tR+1 = I tL+1 + NLt +1 − NLI t +1 − ROA × NLt
(28)
Substituting from equations (28) and (31) into equation (13) produces the equation (A1).
C NLI NLI t + C FEE FEEt + (C NPL − C PL ) NPLt + (C PL − CGL )GLt
−1
= ROA
[ C NLI NLI t +1 + CFEE (CFF FEEt + CFL NLI t ) + (CNPL − CPL ) × ( NPLt +1 )
+ (CPL − CGL ) × (GLt +1 ) + rLθ1PLt + (C FF FEEt + CFL NLI t )
+ NLt +1 − NLI t +1 − ROA × NLt
(A1)
]
We collect variable FEEt in (A1).
LHS of variable FEEt is C FEE
RHS of variable FEEt is ROA (C FEE C FF + C FF )
−1
−1
(CFEE CFF + CFF )
⇒ CFEE = ROA
⇒ CFEE = (ROA − CFF ) C FF
−1
Deleting variable FEEt in (A1) produces the equation (A2).
C NLI NLI t + (C NPL − C PL ) × ( NPLt ) + (C PL − CGL ) × (GLt )
−1
[ C NLI NLI t +1 + CFEE (CFL NLI t ) + (C NPL − CPL ) × ( NPLt +1 )
= ROA
(A2)
+ (C PL − CGL ) × (GLt +1 ) + rLθ1 PLt + (C FL NLI t )
]
+ NLt +1 − NLI t +1 − ROA × NLt
Substituting from equations NLt
≡ C GL GLt − C NPL NPLt and NLt +1 ≡ C GL GLt +1 − C NPL NPLt +1 into
equation (A2) produces the equation (A3).
C NLI NLI t + (C NPL − C PL ) × ( NPLt ) + (C PL − C GL ) × (GLt )
−1
[ C NLI NLI t +1 + C FEE (C FL NLI t ) + (C NPL − C PL ) × ( NPLt +1 )
= ROA
+ (C PL − C GL ) × (GLt +1 ) + rLθ 1 PLt + (C FL NLI t )
+ C GL GLt +1 − C NPL NPLt +1 − NLI t +1 − ROA × (C GL GLt − C NPL NPLt )
Substituting from equations GLt
(A3)
]
= PLt + NPLt and GLt +1 = PLt +1 + NPLt +1 into equation (A3)
produces the equation (A4).
C NLI NLI t + (C NPL − C PL ) × ( NPLt ) + (C PL − C GL ) × ( PLt + NPLt )
−1
[ C NLI NLI t +1 + C FEE (C FL NLI t ) + (C NPL − C PL ) × ( NPLt +1 )
= ROA
+ (C PL ) × ( PLt +1 + NPLt +1 ) + rLθ1 PLt + (C FL NLI t )
− C NPL NPLt +1 − NLI t +1 − ROA × (C GL PLt + C GL NPLt − C NPL NPLt )
(A4)
]
Collecting variable NPLt in (A4) produces the equation (A5).
C NLI NLI t + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt )
−1
[ C NLI NLI t +1 + C FEE (C FL NLI t ) +
= ROA
+ (C PL ) × ( PLt +1 ) + rLθ1 PLt + (C FL NLI t )
− NLI t +1 − ROA × (C GL PLt + C GL NPLt − C NPL NPLt )
(A5)
]
= NLI t +1 + θ1 PLt into equation (A3) produces the equation (A6).
− C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt )
Substituting from equation PLt +1
C NLI NLI t + (C NPL
−1
[ C NLI NLI t +1 + C FEE (C FL NLI t ) +
= ROA
+ (C PL ) × ( NLI t +1 + θ 1 PLt ) + rLθ1 PLt + (C FL NLI t )
− NLI t +1 − ROA × (C GL PLt + C GL NPLt − C NPL NPLt )
(A6)
]
We collect variable PLt .
− C GL .
LHS of variable PLt is C PL
(rLθ1 + C PLθ1 ) − CGL .
(rLθ1 + C PLθ1 ) ⇒ C PL = (ROA − θ1 )−1 rLθ1 .
−1
OA
RHS of variable PLt is R
⇒ C PL = R
−1
OA
Deleting variable PLt in (A6) produces the equation (A7).
C NLI NLI t + (C NPL − C GL ) × ( NPLt )
−1
[ C NLI NLI t +1 + C FEE (C FL NLI t ) +
= ROA
(A7)
+ (C PL ) × ( NLI t +1 ) + (C FL NLI t )
]
− NLI t +1 − ROA × (C GL NPLt − C NPL NPLt )
= C LL NLI t into equation (A7) produces the equation (A8).
− C GL ) × ( NPLt )
Substituting from equation NLI t +1
C NLI NLI t + (C NPL
−1
[ C NLI C LL NLI t + C FEE (C FL NLI t ) +
= ROA
(A8)
+ (C PL ) × (C LL NLI t ) + (C FL NLI t )
− C LL NLI t − ROA × (C GL NPLt − C NPL NPLt )
]
We collecting variable NLI t .
LHS of variable NLI t is C NLI
−1
[
RHS of variable NLI t is ROA C NLI C LL
+ C LL (C PL − 1) + C FL (C FEE + 1)]
⇒ C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
{ [
]
[
]}
C NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ 1 − 1 + C FL (ROA − C FF ) C FF + 1 (4.2.13)
−1
−1
−1
Deleting variable NLI t in (A8) produces the equation (A9).
(C NPL − C GL ) × ( NPLt )
−1
[ − ROA × (CGL NPLt − C NPL NPLt )
= ROA
(A9).
]
We collecting variable NPLt .
LHS of variable NPLt is
(C NPL − C GL )
−1
[
RHS of variable NPLt is ROA ( ROA C NPL
−1
⇒ C NPL − C GL = ROA
[ROA (C NPL − CGL )]
− ROA C GL )]
⇒ C NPL − C GL = C NPL − C GL
REFERENCES
Yeh, Chao-Hui (2000). The Non-Linear residual income valuation model. Unpublished paper. Taiwan.
The National Sun Yat-Sen University.
An Intellectual Capital Model of Bank Lending
Dr. Chao-Hui Yeh, I-Shou University, Taiwan
ABSTRACT
Purpose: Intellectual Capital can be said to constitute valuable factors not shown in accounting financial
statements, but which are of critical importance to a company's long-term profitability. This study models
the relationship between intellectual capital(IC hereafter) and various loans in bank lending. Method:
Pioneering and exhaustive research by Yeh (2001) discusses operating free cash flows valuation model
(FCFVM hereafter). This paper extends the FCFVM to derive mathematically the accounting items of IC
for lending banks. The FCFVM analyses the relationship between market value of equity and accounting
activities (investments) in the manufacturing firms. This study adds to the FCFVM literatures by applying
it to lending banks where operating assets are various loans, which are of critical difference to a
manufacturing firm. Result: We find that IC (the difference between market and book value of equity) is a
function of fee income, performing loans and new loan investment. This implies that the most important
assets of banking industry are reputation, honesty, and commitment. Conclusion: We conclude that the
sales of fee-based services, performing loans and new loan investment are valuable factors in bank
lending. With our mathematical model, the empirical analysis is more disciplined than that of many prior
ad-hoc valuation studies. Our model could be extended to encompass other valuable factors in banks that
are also likely to drive market and book values differently. Directions for Future Research: We suggest
a number of value-driving activities of banks, including deposit taking, credit card services, mortgage
servicing rights and trust activities. The combination of fee income and new deposit investments in our
model will partially describe these activities. Nevertheless, it is potentially fruitful to use our model to
incorporate these and other factors of value more explicitly. Within our modeling framework this would
require identification of the investing activities that give generate these hidden values. These activities
could take the form of new investments in social capital, that is, it is social capital that transforms human
capital into producing positive career outcomes and increasing investors' perception of potential... Such an
extension would likely result in a valuation model that contains more financial statements reports items,
especially the profit/loss account and balance-sheet of the banks.
Key words: Intellectual Capital, Free Cash Flows Valuation Model, Loans.
1. INTRODUCTION
Most intellectual capital approaches have problems with meaningful measurement. Big differences
often exist between a company's market value and its book value. Many of these are explained by
intellectual capital assets not shown in the balance sheet. Assets like employee knowledge, expertise and
creativity, customer confidence in the company and its products, brands and franchises, Information and
knowledge management systems, administrative procedures, copyrights, patents and trademarks, the
efficiency of company business processes and the effectiveness of company planning, forecaster and
strategy-making. This has presented companies with a new challenge how to measure, account for,
manage and develop intellectual capital. This paper of Intellectual Capital will help you do exactly that.
The rise of the "new economy", one principally driven by information and knowledge, has led to
an increased interest in intellectual capital (IC). IC is an area of interest to numerous parties, such as
shareholders, institutional investors, scholars, policymakers and managers.
However, there have been very few papers that have studied IC of banks. The implications of IC
are more prominent in banks as banks have abundant human capital at their disposal. Therefore, it
becomes necessary to understand what the factors of creating value in banks are. Banks happen to be
one service sector that uses a huge amount of human capital and customer capital for its survival.
Thus, this paper evaluates the intellectual capital model of bank lending using the residual income
valuation model (FCFVM).
This paper provides a strong case for reporting of the value creation through intellectual capital
in the financial statements. The paper would be a useful tool for benchmarking the performance of the
banks across various countries. When in 1988 the Swiss food-products company Nestle bought the
British confectionery group Rowntree, the price paid included well over $1 billion for something that
had never appeared on Rowntree's balance sheet, this hidden value now recognized as intellectual
capital.
With the advent of knowledge economy era, tangible assets are no longer the factors that
companies rely on to create high value, while intangible assets and intellectual capital play more
significant roles that even surpass tangible assets in determining companies’ future competitiveness.
Intellectual capital includes intellectual properties, intellectual assets and other information assets or
intangible assets, such as human capital, capability of management team, relationships with
customers and suppliers, employee devotedness and innovation ability. Most of these items are absent
from or cannot be evaluated by traditional accounting, which is also the major cause of differences
between corporate market value and book value. Therefore, assessment of intellectual capital is rather
important for companies.
Value is decided by the quantitative analysts who work in the financial markets developing
mathematical models to assist the activities of traders and risk managers within banks and other large
corporate institutions, while price be affected by market conditions, then the quantity of the two are
usually unequal. There are seven intellectual capital valuation models in the literature, including
Tobin’s Q, Market-to-book ratio (M/B ratio), Economic Value Added (EVA™), Calculated Intangible
value (CIV), Knowledge Capital Earnings (KCE), Value Added Intellectual Coefficients (VAIC™)
and Financial Method of Intangible Assets Measuring (FiMIAM). This paper will discuss a different
model.
The next of our paper is the model. The paper concludes with Section 3.
2. THE MODEL
In finance, the discounted cash flow valuation describes a method to value a project, company, or
asset using the concepts of the time value of money. All future cash flows are estimated and discounted to
give them a present value. The discount rate used is generally the appropriate cost of capital, and
incorporates judgments of the uncertainty (risk) of the future cash flows.
In this paper, we use the discounted cash flow valuation to describe the operating asset of a bank.
To see how a bank’s future operating free cash flows can be tied back to its intrinsic value on date t we
begin by defining market value of operating assets as the present value of all future operating free cash
flows to operating assets. This is,
∞
⎡ FCF ⎤
MVOAt = ∑ Et ⎢ s t + s ⎥
s =1
⎣ ROA ⎦
(1)
Formula (1) indicates that the market value of operating assets (
MVOAt ) is the net present value of
the expected operating free cash flows (
FCFt + s
the required return on operating assets (
rOA = ROA − 1 ). In formula (1), operating free cash flows are
) available for that operating assets (
OAt
) discounted at
operating cash flows minus capital expenditures.
Given (1), we get that the following：
[
−1
MVOAt = ROA
Et FCFt +1 + MVOAt +1
]
(2)
Given financial debt is negative financial asset, this paper assume that net financial assets are
financial assets minus financial debts and use financial assets as net financial assets. Given the financial
assets are marked to market value, the market value of financial assets (
MVFA t ) is the book value of
financial assets (
BVFA t ). If the assumption is correct, then we get that the following：
MVFA t = BVFA t
(3)
Given that financial assets is net financial assets and does not use financial debts. Firm’s equity is
Firm’s financial assets plus operating assets of the firm, so book value of equity is book value of operating
assets (
BVOAt ) plus book value of financial assets：
BVE t = BVOA t + BVFA t
(4)
Market value of equity is market value of operating assets plus market value of financial assets：
MVE t = MVOA t + MVFA t
(5)
Substituting from equations (3) and (4) into equation (5) produces the equation (6).
MVEt − BVEt = MVOAt − BVOAt
(6)
For many authors, the difference between the market value of companies' shares and their book value
is the consequence of intellectual capital (
ICt ).
Substituting from equation (5) into equation (6) produces the equation (7).
ICt = MVOAt − BVOAt
(7)
The following persistent value driver dynamic, PVDD, lays out the evolution of free cash flows:
I tL+1 = rLθ1PLt + FEEt +1
(8)
PLt +1 = NLI t +1 + θ1PLt
(9)
NLI t +1 = CLL NLI t
(10)
NPLt +1 = (1 − θ1 )PLt + θ 2 NPLt
(11)
FEEt +1 = CFF FEEt + CFL NLI t
In equation (8) the mark,
(12)
I tL+1 takes the abbreviation of income from loans at t+1, superscript L
refers loans. Interest income from loans at time t+1 is a fraction ( θ 1 ) of current performing loans (
at time t multiplied by the stated interest rate on loans ( rL ) plus fee income at time t+1(
PLt )
FEEt +1 ). This
(
)
implies the default rate on loans is 1 − θ1 .
In equation (9) the mark,
PLt +1 takes the abbreviation of performing loans at t+1.Performing loans
at t+1 is a fraction ( θ1 ) of current performing loans (
(
PLt ) at time t plus new loan investments at t+1
NLI t +1 ).At time t+1, new loans investment, NLI t +1 , are invested (lent) that add to PLt +1 . They are
subject to default and start paying interest at t + 1, one can think of lending as investment for banks,
and θ 1 as the persistence of performing loans. Note that new loans are net of repayments. We do not take
account of the maturities of existing loans because they are assumed to rollover. Any net repayments of
loans would be reflected in
NLI t +1 .
In equation (10) the mark,
NLI t +1 takes the abbreviation of new loan investments at t+1. New loan
investments at t+1 is a fraction ( C LL ) of new loan investments at t (
NLI t ).Growth in lending is
NLI t +1
NLI t . New loan investments in a lending bank are just
described through the parameter C LL , which is
like capital expenditures in a non-bank firm.
In equation (11) the mark,
NPLt +1 takes the abbreviation of non-performing loans at t+1.
Non-performing loans at t+1 is a fraction ( θ1 ) of current non-performing loans (
NPLt ) at time t plus
PLt ) at time t. The stock of
(
)
default rate on loans 1 − θ1 multiplied by current performing loans (
non-performing loans includes a proportion ( θ 2 ) of the prior period non-performing loan balance.
Implicitly, (1− θ 2 ) of the prior period non-performing loans are charged off the books completely. For the
moment, non-loan net assets such as deposits or investment securities of the lending bank are assumed to
[ ]
be financial in nature, marked-to-market and are zero net present value. ( θ 1 ∈ 0,1 )
In equation (12) the mark,
FEEt +1 takes the abbreviation of fee income from loaners at t+1. Fee
income at time t+1 is a fraction ( CFF ) of current fee income (
new loan investments at t (
NLI t
).
For the lending bank, each period’s operating free cash flows (
(
FEEt ) at time t plus a fraction ( CFL ) of
FCFt + s ) are equal to the interest received
rLθ1 PLt ) on loans plus fee income ( FEEt +1 ) and minus new investments in loans ( NLI t +1 ). If the
PVDD is correct, then we get that
FCFt +1 = I tL+1 − NLI t +1 = rLθ1 PLt + C FF FEEt + C FL NLI t − NLI t +1
(13)
Substituting from equation (13) into equation (10) produces the equation (14).
FCFt +1 = I tL+1 − NLI t +1 = rLθ1PLt + CFF FEEt + (CFL − CLL ) NLI t
In equation (1), we have showed that
(14)
MVOAt is a function of FCFt +1
Substituting from equation (14) into equation (1) produces the equation (15).
MVOAt = C PL PLt + C FEE FEEt + C NLI NLI t
(15)
Substituting from equation (14) and (15) into equation (2) produces the equation (16), (17) and (18).
C PL = (ROA − θ1 ) rLθ1
(16)
C FEE = (ROA − C FF ) C FF
(17)
−1
−1
C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
(18)
Proofs of this are in the Appendix.
In words, the market value of the lending bank’s operating assets is a linear combination of the
bank’s performing loans, fee-based services and new loan investments.
Some implications of the model are described as follows.
The coefficient on performing loans, C PL reflects the discounted future interests earned on
where discounting considers the persistence of current loans
default (i.e.,
,
If performing loans never
θ1 = 1) then the income the bank receives should be discounted as a perpetuity at rate rOA .
C PL PLt =
(i.e.,
(ROA − θ1 )−1
PLt
rL PLt
rOA ). Accordingly the numerator, rL PLt , is a fixed income and C PL > 0 .
The coefficient on fees, C FEE reflects the discounted future fees earned on
discounting considers the persistence of current loans
(ROA − CFF )−1 . If C FF =1, then the fees income
the bank receives should be discounted as a perpetuity at rate
Accordingly the numerator,
FEEt , where
rOA (i.e.,
C FEE FEE t =
FEEt
rOA ).
rL PLt is a fixed income and C FEE > 0 .
The coefficient on net new loans
C NLI is positive when new loan investments are positive net
present value investments, and when there is a potential for growth in these activities. To see this, we
rewrite the equation (18) as follows:
C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
(19)
C NLI = (ROA − C LL ) C LL (C PL − 1) + (ROA − C LL ) C FL (C FEE + 1)
(20)
−1
−1
−1
Substituting from equations (16) and (17) into equation (20) produces the equation (21).
(
)
(
)
CNLI = (ROA − CLL ) CLL (ROA − θ1 ) rLθ1 −1 + (ROA − CLL ) CFL (ROA − CFF ) CFF + 1
−1
−1
−1
−1
(21)
Continuing to reorganize, we get that the following：
C NLI = (ROA − C LL )
−1
(ROA − θ1 )−1 C LL [θ1 (1 + rL ) − (1 + rOA )]＋
(ROA − C LL )−1 (ROA − C FF )−1 C FL ROA
(22)
Inspection of the term in (22) reveals that because of
C LL (ROA − θ1 )
−1
(ROA − C LL )−1 and (ROA − C LL )−1 (ROA − C FF )−1 Z FL ROA
sign of equation (22) and the term (
[θ1 (1 + rL ) − (1 + rOA )]) in brackets are the same.：Inspection of the
term in (22) reveals that positive net present value occurs when
Accordingly
if
are always positive, the
rL > rOA .
[(1 + rL ) > (1 + rOA )], we adjust [(1 + rL )] to [θ1 (1 + rL )] , which means
[θ1 (1 + rL ) > (1 + rOA )] , then C NLI
> 0 . Zero net present value occurs if C NLI = 0 .
The
valuation effect of a one dollar investment in positive net present value loans is increasing in the growth
NLI t +1
NLI t . When loans are zero net present value investments, it is easy to
parameter C LL , which is
show that the coefficient on current performing loans, C PL equal one and C FL = 0 .
3 Conclusion and Directions for Future Research
Successfully implementing a method for the valuation or measurement of intellectual capital is not
an easy task. Practitioners yet receive little support from the intellectual capital research community. Little
research has been done into the factors that influence the success of a method. This paper is a first attempt
at mathematically applying the residual income valuation model to bank’s IC. For the purpose of
developing our bank’s IC model, we assume the major value creating activities are fee-based services,
performing loans and new loans. In addition, we assume the major accounting bias is loan loss allowance.
When we combine these assumptions we arrive at an IC valuation model that depends on loan, new loans,
non-performing loans and fee income. Our model could surely be extended to include other value
generating activities and accounting biases of banks. We have shed light on including other bank activities
that are also likely to drive market and book values apart.
Appendix
−1
[
Given (2), MVOAt = ROA E FCFt +1 + MVOAt +1
]
Substituting from equation (14) and (15) into equation (2) produces the following:
−1
MVOAt = ROA
E [FCFt +1 + MVOAt +1 ]
−1
⇒ C PL PLt + C FEE FEEt + C NLI NLI t = ROA
[FCFt +1 + C PL PLt +1 + C FEE FEEt +1 + C NLI NLI t +1 ]
r θ PLt + C FF FEEt + C FL NLI t − C LL NLI t + C PL (C LL NLI t + θ1 PLt )⎤
−1 ⎡ L 1
= ROA
⎢+ C (C FEE + C NLI ) + C C NLI
⎥
t
FL
t
NLI LL
t
⎣ FEE FF
⎦
Collecting variable PLt
LHS of variable PLt is C PL
RHS of variable PLt is ROA (rLθ1 + C PLθ1 )
−1
−1
(rLθ1 + C PLθ1 )
⇒ C PL = ROA
⇒ C PL = (ROA − θ 1 ) rLθ1
−1
Collecting variable FEE t
LHS of variable FEE t is C FEE
RHS of variable FEE t is ROA (C FEE C FF + C FF )
−1
−1
(CFEE CFF + CFF )
⇒ CFEE = ROA
⇒ CFEE = (ROA − CFF ) CFF
−1
Collecting variable NLI t
LHS of variable NLI t is C NLI
[
]
RHS of variable NLI t is ROA C NLI C LL + C LL (C PL − 1) + C FL (C FEE + 1)
−1
⇒ C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
{ [
]
[
]}
C NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ 1 − 1 + C FL (ROA − C FF ) C FF + 1
−1
−1
−1
REFERENCES
Yeh, Chao-Hui (2000). The Non-Linear residual income valuation model. Unpublished paper. Taiwan.
The National Sun Yat-Sen University.
2007/9/27
1
放款銀行之商譽模型
葉兆輝
義守大學
摘要
本 文 研 究 銀 行 商 譽 與 放 款 之 關 係 並 建 立 模 型 ， 本 研 究 援 引 Feltham and
Ohlson[1996] (此後簡稱為FO model)並擴大以數學式建立放款銀行之商譽模型。FO
model這個領域先前研究的假設都是針對製造業，目的在檢視股東權益評價與會計科
目之間的關係。本文的主要貢獻在於，是最先找出如何將FO model由製造業推展至
放款銀行的研究，而放款銀行的營運資產是各類的放款。我們的模型發現銀行商譽(定
義為股東權益市值與面值之差額)為手續費收入、不良債權、放款總額及新增加放款
的函數。
關鍵詞：商譽模型、放款銀行、FO mode、放款
A Goodwill Model of Bank Lending
Chao-Hui Yeh
I-Shou University
Abstract
This study models the relationship between goodwill and various loans in bank
lending. It extends the Feltham and Ohlson [1996] model (hereafter the FO model) to
derive mathematically the basic principles of goodwill for lending banks The FO model
analyses the relationship between market value of equity and accounting activity under the
manufacturing conditions. This study contributes to the FO mode research by applying it to
lending banks where operating assets are various loans. We find that goodwill (the
difference between market and book value of equity) is a function of fee income,
non-performing loans, gross loans and new loan investment.
Keywords: Goodwill model, Bank lending, FO model, Loans.
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2
壹、 前言
定期花錢做公益、員工上班時間應邀到無利害關係人，無償無私針對邀約人需
求，提供商品與勞務並實質創造價值、效用，滿足其需求，上述活動均能增加企業
的附加價值，對當期是錢出口袋，有入帳(recorded)，但效益只是好的形象在心頭，
依據一般公認應計基礎會計原則(GAAP)，無法入帳(unrecorded)；而好的形象在心
頭轉成錢入口袋，可能是數十期後，因為好的形象=好的社會觀感=好的口碑=值的
信賴=好的商譽，所以商譽是企業最重要的資產之一，企業的價值，除來自有形資
產外，更包含了商譽的經營效益，雖然商譽是無形的，但是在知識經濟的風潮下，
其價值卻遠超過有形的動產或不動產。
自1895年英國首次將商譽(goodwill)運用於司法實踐以來，國際社會對商譽的性
質、影響因素、價值確認以及商譽的會計處理等方面的研究取得了一定的成果。在商
譽研究的成果中，國內外的共識為：商譽是公司總價值與其淨資產公允價值之間的差
額，表示企業獲取超額利潤能力的價值。雖然概念上商譽的價值計量問題解決了，但
更深層次的問題則仍然未解決，如企業獲取超額利潤的能力從何而來？能否將企業所
有產生超額利潤的因素均作為商譽的組成要素？為解決上述問題，本文認為，將產生
商譽的各種未入賬的隱含因素凸現出來，使用會計科目建構商譽價值的數學模型。
商譽是企業擁有的各種未入賬的經濟資源或優勢所帶來的超額獲利能力的價值
體現。這些經濟資源或優勢：包括優秀的管理隊伍、特有的營運網路或組織、良好的
勞資關係；信用好、高瞻遠矚的員工培訓計畫、通過向慈善機構捐款或派員工參與公
益活動而建立起崇高的社會威望、與政府的良好關係等。商譽是由企業過去所擁有的
各種“優勢”綜合作用的結果，這些“優勢”既有有形資產，又有無形資產，還可能有偶
然的因素。儘管這些“優勢”在會計的帳面上並未出現，在企業的資產負債表上也無反
映，但它們對企業超額利潤的貢獻是不爭事實。
商譽是有多種因素共同作用的超額利潤能力價值表現，是人們早期對企業產生超
額利潤的諸多因素無法分割為無形資產具體存在形式時的一種認識，以商譽命名之，
並且一直影響至今。然而，隨著科學技術的發展和人們對無形資產認識的逐步深入，
構成商譽的諸多因素完全可以分割為相應的會計科目，並進行單個的確認與計量，以
體現企業各種資產對企業收益的貢獻能力，而不必再以較為模糊的商譽來反映。
雖然商譽評價已有初步研究成果，但皆以製造業為主，尚未見到以銀行業為研究
企業，因此本文希望能提出一個以銀行業為研究企業的商譽評價模型。為了能客觀且
準確的衡量銀行之商譽，本文發展了一個應計會計基礎的評價模型，這是一個為了銀
行所量身訂作的評價模型。這個模型依循的是FO model以製造業為基礎所提出的方
法，而FO model依據的是剩餘損益評價模型(Residual Income Valuation Model，此後
簡稱為RIVM)及應計基礎會計保守原則，本文模型之所以能表現出銀行的保守原則，
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是由於備抵呆帳的偏誤(bias)及放款投資活動產生正淨現值的可能性，放款是銀行的
營運資產(operating assets)，此外，本文是第一篇將FO model 應用至非製造業(銀行)。
具體而言，本文主要目的在於：純理論上發展應計基礎會計的評價模型，可以客觀且
準確地衡量銀行的價值與商譽。
除第一節為緒論外，本文其餘之架構如下：第二節為文獻回顧，第三節為模型概
述，第四節為單獨考慮放款之評價模型，第五節則提出本文結論。
貳、文獻回顧
應用銀行內在財務報表至銀行評價的文獻，最早且常為後來引用者，本文認為是
Beaver et al.(1989)一文。在Beaver et al.(1989)一文中提供了一個重要的銀行評價實證
模型，而該模型所依據的是應計基礎會計資料，他們的模型為一迴歸式，其中的依變
數(Y)為商譽，而自變數則分別為逾期放款(X 1 )、備抵呆帳(X 2 )及資產與負債到期日
不配合缺口(X 3 )，他們發現這三個自變數在解釋銀行的市場價值上全都是顯著的，在
他們的研究中一個有重大影響的發現：備抵呆帳(X 2 )這個自變數具有遠大零的迴歸係
數，表示這個變數對於銀行可能扮演一個重要的角色；逾期放款(X 1 )這個變數具有一
個顯著的負迴歸係數；而資產與負債到期日缺口(X 3 )之迴歸係數符號則是每年都會不
同。銀行業財報與價值攸關性之文獻中尚有下列的研究：Barth (1994)，Venkatachalam
(1996)，Nelson (1996)，Beaver and Engle (1996)，Eccher, Ramesh, and Thiagarajan
(1996)，Beaver (1999)，Liu and Ohlson. (2000)及Begley and Feltham. (2002)。唯一使用
RIVM的研究只有Kohlbeck and Warfield (2003)：其係以Ohlson(1995)模型為架構將商
譽表達為本期剩餘損益的乘數且確認出銀行商譽的四個來源：包括忠誠户之存款、房
貸抵押放款、信託業務及信用卡，並藉由估計每個來源的現金流程折現值來衡量這商
譽。然而Ohlson(1995)是不偏的(unbiased)應計基礎會計模型，即假設市場價值等於帳
面價值，因此與Kohlbeck and Warfield (2003)主張：商譽等於市場價值減帳面價值，
產生衝突。
本文不以不偏會計模型Ohlson(1995) 1 為架構；而以FO model為架構，因FO model
已將導致應計基礎會計偏誤的因子納入模型之中，並且定義商譽為公司價值與股東權
益帳面價值之差額，等於未來超常盈餘的折現值。
1
Ohlson [1995] demonstrated that with unbiased accounting, firm value can be expressed as
a linear combination of net income, dividends and book value. Feltham and Ohlson [1995, 1996]
and others (e.g. Zhang [2000], Feltham and Pae [2000]) incorporate accounting realism such
as conservatism.
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商譽在應計基礎會計上所產生的問題，主要是商譽不能公平且正確的評估，例如
向外購買取得之商譽可以入帳(Recorded assets)；然而內部形成之商譽卻無法入帳
(Unrecorded assets)，以致在財務報表上未能看出公司的實際價值，往往造成公司市場
價值與帳面價值的重大差距。向外購買者，通常發生於企業間購併交易或資產交換
時，在GAAP下，可依購買或交換價格認列為商譽，並在財務報表上入帳為商譽；然
而內部形成者，則不能入帳為商譽。內部形成之商譽會使財務報表失去可靠性與攸關
性，本文以內部形成之商譽 2 為研究對象。
內部形成之商譽究竟是如何發生？FO model舉出產生商譽之三項因素：超常盈餘
持續程度( X 1 )、保守應計基礎會計原則下營業資產價值低估程度( X 2 )、其它與價值
攸關且尚未影響到權益帳面價值和淨利的資訊( X 3 )。其中 X 1、 X 2 二項因素與保守應
計基礎會計有關； X 3 則與保守應計基礎會計無關。透過持續性發展的價值攸關動態
(Persistent Value Driver Dynamic，此後簡稱為PVDD)：即也就是說將 X 1 、 X 2 、 X 3 間
相互關係式加以線性限制，Feltham and Ohlson(1996)推導出公司的權益價值區分成營
業資產帳面價值、 X 1、與 X 3 等三個部份。何以 X 1 是決定商譽因素之一？因為公司市
場價值是現有資產價值與未來成長機會價值所構成，成長機會大小會影響應計基礎會
計資訊與股價之關聯程度，但因現行一般公認應計基礎會計原則過於保守，財務報表
資訊無法及時反應成長機會對於企業價值之影響，採用保守應計基礎會計原則通常會
使公司資產低估，將使公司之營業資產帳面價值會小於市場價值；而營業資產面值市
值之差額等於商譽。
而葉兆輝（2001）質疑PVDD模型沒有載明投資的影響，且Ohlson (1995)假設
∂B
∂I t +1
= − r ，因為每增加$1的現金股息支出會減少權益資本$1( t = −1 )，及權益資金
∂Dt
∂Dt
成本$r，所以被放棄的投資報酬率也必定是r，這隱含了所有的投資淨現值(NPV)平均
起來為零的假設，但是假設投資淨現值平均起來為零是不合理的（至少在短期內）。
因此葉兆輝(民90)，於第三章中擴充PVDD模型，納入投資NPV不等於零的前提假設。
更仔細地說，經理人的投資決策是伺機而動的，而該機會與本期經濟租（剩餘損益）
有關，若本期經濟租大好則擴充規模；反之則縮小規模。而擴充、縮小或等候是經理
人的實質選擇權(Real Option)。
綜合這些文獻，我們可以獲得下列結論：
2
In general, the accounting for intangible assets is based on the historical cost principle
(AICPA, 1970 and FASB, 2001), and as a result, the vast majority of internally generated
intangible assets are not recognized in the financial statements.
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一、在上述文獻中有一共同的特點，即商譽皆以公司市場價值與帳面價值的差異表示
之，是故本文也將以銀行市場價值與帳面價值的差異作為商譽來進行研究。
二、另外，從上述文獻中可以發現在商譽的研究中，並沒有出現以銀行業作為研究
對象的文獻。
參、模型概述
一、剩餘損益評價模型
RIVM指出權益的市場價值(Market Value of Equity, MVE)等於權益的帳面價值
(Book Value of Equity, BVE）加上該帳面資本創造未來剩餘損益(Residual Income, I tR )
的能力，而 I tR 即為公司創造財富的能力。因此，在給定資金成本率r的第t期剩餘損益
之定義如下：
I tR ≡ (I t ) − (rE × BVEt −1 )
(3.1.1)
其中
I tR 為第t期的剩餘損益(Residual Income)；
I t 為第t期的稅後淨利(Net Income)；
rE 為股東使用權益資金的成本率(Stockholders Required Rate of Return)，且
rE +1 = RE ；
BVEt −1 為上一期的公司權益的會計帳面價值(Book Value of Equity)；
(3.1.1)為第三章第一節第一個公式，之後依此類推；
在(3.1.1)公式中，正的 I tR 表示公司創造價值。
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為了要更清楚地研究剩餘損益與公司市場價值之關係，本文從Miller, M. and F.
Modigliani (1961)權益評價的折現股息模型（Discounted Dividend Valuation Model，此
後簡稱為DDVM）開始說明，茲將(DDVM)模型介紹如下：
~ ⎤
⎡D
MVEt = ∑ E t ⎢ t +ss ⎥
s =1
⎣ RE ⎦
∞
(3.1.2)
其中
MVEt 為第t期的股東權益市值；
~
Dt + s 為第t+s期流向權益股東的現金流程(現金流程有源自營運及金融性資產)
本文所謂金融性資產(Financial Assets)，是指調整金融負債後淨金融資產，
因 此 可 以 是 負 數 ， 例 如 表 3.1.1 中 的 BVFAt −1 = −70 ， 3.1.2 中 的
BVFAt = −70 ，如果第 t－1 期資產負債表給定如表 3.1.1：
表 3.1.1
現金=20
存款户存款=90
放款=80( BVOAt −1 = 80 )
權益=10( BVEt −1 = 10 )
則第 t－1 期金融性資產為現金=20；金融性負債為現金=90；因此調整金融
負債後淨金融資產=-70； BVFAt −1 = −70 。放款營運性資產之帳面價值=80
； BVOAt −1 = 80 。 股 東 權 益 面 值 =10 ； BVEt −1 = 10 。 表 3.1.1 符 合
BVE t −1 = BVFAt −1 + BVOAt −1 的要求。
如果第t期新增加備抵壞帳=10時，則第t－1期的表3.1.1將進化到第t期的表
3.1.2：
表3.1.2
現金=20
存款户存款=90
放款=80；備抵壞帳=10；( BVOAt = 70 )
權益=0( BVEt = 0 )
則第 t 期金融性資產為現金=20；金融性負債為現金=90；因此調整金融負債後淨
金融資產=-70； BVFAt = −70 。營運性資產之帳面價值=放款－壞帳=80－10=70；
BVOAt = 70 。股東權益面值=10； BVEt = 0 。表 3.1.2 符合 BVEt = BVFAt + BVOAt 的
要求 。
rE 為股東使用權益資金的成本率；
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E t [ ] 為是期望值；
DDVM所表示的是，股東權益的市值會等於一連串未來流向權益股東之現金流程
的淨現值總和。
Ohlson(1995)合併DDVM與淨盈餘關係（Clean Surplus Relation，此後簡稱為
CSR），發展出剩餘損益評價模型(RIVM)。而CSR是指前後期股東權益總額的差額
( BVEt − BVE t −1 )可被表示成淨利與股息之差額( I t − Dt )。
(CSR)： BVEt − BVEt −1 = I t − Dt
(3.1.3)
(3.1.3)式之目的是用權益的淨值( BVE t ，之後簡寫成 Bt )和稅後淨利( I t )來表示股
息( Dt )。( Bt − Bt −1 )是存量的變化；而( I t − Dt )是流量的變化。合併(3.1.1)、(3.1.2)、(3.1.3)
可得(3.1.4)如下：
~
~
~
~
∞
⎡ I t +Rs + (1 + r )Bt + s − Bt + s +1 ⎤
⎡ I t +Rs ⎤
MVE t = ∑ E t ⎢
⎥ = Bt + ∑ Et ⎢ s ⎥ = Bt + U .I . A.
R Es
s =1
s =1
⎣
⎦
⎣ RE ⎦
∞
(3.1.4)
其中 U .I . A. 代表尚末入帳之無形資產(Unrecorded Intangible Assets)。(3.1.4)式的涵
意是將公司的價值區分成二部份，第一部份是公司的帳面價值( Bt )部份，在虧損公司
中帳面價值較淨利重要；第二部份是公司創造財富的能力，即淨利超過所使用資金成
本的程度，名為剩餘損益( I R )，剩餘損益在有獲利的公司中較帳面價值變數重要。
(3.1.4)式有六個特色值得注意：
~
1. 在解釋公司價值方面， I t +Rs 指標可能優於 I t 指標，因為 I t 未扣除所使用的股東權益
~
~
成本，但 I t +Rs 則扣除了使用的股東權益成本，因此 I t +Rs 才是投資人所關心的經濟利
潤。
~
2. 在t期時，只有 Bt 是已知值；r為常數，所有的未來 I t +Rs 皆為預測值。
3. 與傳統的模型不同點在於唯一的限制是CSR。
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4. 在公司的永續經營期間，CSR可以確保所有非業主交易的股東權益價值之變化，
依序全部呈現在公司的報表上。
5. 只要CSR成立，則RIVM可免於應計基礎會計操弄：若應計基礎會計過於誇張而使
I t 額外增加了一單位，將造成 Bt 也額外增加一單位，因此將造成 Bt 較高的基礎而
~
導致較低的未來剩餘損益 I t +Rs 。同理若應計基礎會計過於保守而使 I t 的認列較它
的實際發生較慢，Bt 將會被較低估，因此將造成 Bt 較低的基礎而導致較高的未來
剩餘損益。
6. RIVM的預測與文獻上的預測一致：例如享有較高本益比的公司是那些未來會有較
高ROE的公司，即未來有較高剩餘損益的公司(Fama and French, 1995；Fairfield,
1994)。
在 DDVM(3.1.2) 中 ， 權 益 的 市 值 被 表 示 成 未 來 股 息 ( Dt ) 的 淨 現 值 ； 然 而 在
~
RIVM(3.1.4)中，權益的市值被表示成帳面價值( Bt )和未來剩餘損益( I t +Rs )，因為在
(3.1.4)中既沒有載明期望值的函數型態，也沒有提供所需的資訊來估計未來的剩餘損
~
益，而且 I t +Rs 皆為預測值，為了使剩餘損益的未來值與當期已知值有所關連，
Ohlson(1995)首先提出：持續性發展的價值攸關動態(Persistent Value Driver Dynamic,
PVDD)
~
I t +R1 = C I R I tR + Ot + ε~I R ,t +1
~
Ot +1 =
C O Ot + ε~O ,t +1
(3.1.5)
(3.1.6)
(3.1.5)和(3.1.6)聯合表示價值攸關因子持續性發展的動態，PVDD 3 是Ohlson對
RIVM的貢獻，PVDD是多變量自我迴歸過程【AR(1)】，其中的變數 Ot 是其它與價
~
值攸關且尚未影響到權益帳面價值( Bt )和淨利( I t )的資訊，而且又可用來估計 I t +R1 ，
C I R 及 C O 是自我迴歸係數(Coefficient)，在本文係數皆以符號 C 表示，ε~I R ,t +1 及 ε~O ,t +1 是
零期望值隨機誤差項，假設 C I R 及 C O 是介於零與壹間固定已知的常數，即令0＜ C I R 、
C O ＜1；C I R 及 C O 小於1可使第(3.1.5)與(3.1.6)有恒定性，C I R 及 C O 大於零是經濟的推
理及實証的觀察， C I R 及 C O 被定義為個別公司的基本面及應計基礎會計原則。因此
Ohlson假設 I tR 及 Ot 遵循線性一階自我迴歸過程，Dechow et al.(1999)求得 C I R =0.62，
3
Ohlson(1995)首先提出(3.1.5)式和(3.1.6)式，並命名為(ID1)和(ID2)，(ID1)和(ID2)聯合表示
為(Linear Information Dynamic, LID)；然而本文認為(3.1.5)式和(3.1.6)式的精神在於持續性發
展的價值攸關動態(Persistent Value Driver Dynamic, PVDD)，因此在本文中不稱 LID 而稱 PVDD。
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而且Myers (1999)求得 C I R =0.30，因此這二篇文獻皆告訴我們(3.1.5)符合共變異平穩
過程(Covariance-stationary)，然而這二篇文獻皆將(3.1.6)視為零（其實不只這二篇視
為零，絕大都數這一類的實証做法皆視為零），且建議(3.1.5)未來研究應加入創造價
值的變數（即找出 Ot 是什麼）。
Ohlson (1995)年首先將(3.1.4)、(3.1.5)及(3.1.6)合併後，可表示如下：
Vt = Bt + C1 I tR + C2Ot
其中 C1 =
C2 =
(R
CI R
E
− CI R )
RE
(RE − C I R )(RE − CO )
(3.1.7)
(3.1.8)
(3.1.9)
(3.1.7)式即文獻上著名的Ohlson Model，(3.1.8)及(3.1.9)是係數(Coefficient)。
由上述Ohlson所發展的剩餘損益模型，可以瞭解Ohlson模型引人入勝之處在於，
Ohlson模型提供了一個精簡的線性式子連結了市值、權益總額、盈餘及其它資訊變
數。總而言之，剩餘損益之優點如下：
1. 在精神上剩餘損益與NPV息息相關，最接近公司理財理論的主張：即若公司採用
NPV為正數之計畫則可增加公司價值。
2. 許多公司喜愛採用剩餘損益法，因為該方法可促使經理人追求絕對數額（剩餘損
益金額）最大化，而非儘量提高某一百分比（投資報酬率）。但若公司以投資報
酬率來做為經理人的績效，則可能促使某些只顧私利的經理人，拒絕採用對公司
整體有利之方案（代理問題），例如不執行正NPV計畫。
3. 經理人無法控制股價的高低，也無須負責；但是剩餘損益是經理人可以努力且控
制的，例如降低資金成本與提高投資報酬。
4. 剩餘損益受經理人在公司內所做的所有決策影響，例如投資決策及股息決策（透
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過現金存量）影響投資報酬率；而融資決策影響資金成本。
二、銀行商譽評價雛型
自從Ohlson(1995)首度提出以應計基礎會計為基礎的評價模型後，公司評價的許
多研究都認為公司的市場價值是由公司的帳面價值與淨利所組成的線性方程式。然
而，這種評價方法不但太過於簡單化，而且這種評價方法的假設也是不切實際的，因
為Ohlson(1995)假設應計基礎會計制度對於現金流程之認列與衡量是不偏的 4，可以公
正客觀地衡量公司的價值。
FO model的研究結果顯示，如果應計基礎會計方法是有偏誤的(biased)，則被用
來描述公司價值的應計基礎會計變數將不只是包括帳面價值及淨利而已，本文擴充了
FO model的模型，假設應計基礎會計是會產生偏誤的，而不偏應計基礎會計(unbiased
accounting)只是一種特例，FO model的研究強調，應計基礎會計偏誤產生的因素有二
個。第一個因素為公司投資能產生淨現值的投資案，因為應計基礎會計都是記錄過去
已經發生的事情，對於「公司現行投資案」所能產生之未來現金流程，因尚未實現無
法入帳於是才會造成應計基礎會計的偏誤 5 ，對於「公司現行投資案」就銀行而言是
指放款，放款是銀行的營運性資產，放款是銀行的投資案，有可能會產生正淨現值
(NPV)。第二個造成偏誤的因素則為應計制政策中的穩健保守原則(conservatism in
accruals)，例如應計基礎制才有的主觀判斷備抵呆帳(Loan Loss Allowance)。
為了要更清楚地研究銀行商譽，本文DDVM開始說明：
~ ⎤
⎡ D
給定DDVM模型如(3.1.2) ： MVE t = ∑ Et ⎢ t + s s ⎥
s =1
⎣ (R E ) ⎦
∞
~
如果在(3.1.2)中的現金流程( Dt + s )只有營運現金流程，則營運資產市場價值可表
示如下(3.2.1)所示：
4
本研究假定公司第t期時的商譽會等於公司市場價值與帳面價值的差異。本研究對於不偏會計的定義
(
)
與Ohlson(1995)相同。詳細地說，如果 lim E UIAt + s Ot = 0 ，其中ψ t 表示的是第t期時可獲得的資
s →∞
訊集合，則代表會計是不偏的。
如果能產生正淨現值的投資案在入帳時是依據投資時的成本，而不是市場預期這些投資案在未來所
能收到的收益，則商譽會不等於零(即公司的市場價值不等於帳面價值)。
5
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∞
⎡ FCF ⎤
MVOAt = ∑ Et ⎢ s t + s ⎥
s =1
⎣ ROA ⎦
(3.2.1)
其中
MVOA = 銀行營運性資產之市場價值；
FCF =調整投資後的自由營運現金流程 (operating free cash flows net of investments)；
ROA = 1 + rOA ：營運性資產之資金成本。
−1
給定(3.2.1)，可得 MVOAt = ROA
E t [FCFt +1 + MVOAt +1 ].
(3.2.2)
⎡ I~t +Rs ⎤
給定(2.1.4)： MVEt = BVEt + ∑ E t ⎢
s ⎥ ，可得 BVE t 等於金融性資產之帳面價
(
)
1
r
+
s =1
E
⎣
⎦
∞
值( BVFAt )，加上營運性資產之帳面價值( BVOAt )。其中權益資本( E )可分為金融性
資產(Financial Asset)與營運性資產(Operation Asset)，
由於金融性資產是當日結算的，因此金融性資產帳面價值( BVFA )等於市場上金融性
資產的價值( MVFA )，以圖 3.1 第一欄代表 (BVFA ≡ MVFA) ；
由於營運性資產不是當日結算的，所以營運性資產帳面價值( BVOA)不等於營運性資
產市場上的價值( MVOA )，以圖 3.1 第二欄代表 (BVOA ≠ MVOA) ；
其中營運性資產市場上的價值( MVOA )減營運性資產帳面價值( BVOA)等於商譽，以
圖 3.1 第三欄代表 (GW = MVOA − BVOA) 。
其關係可如圖 3.1 表示
MVE
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12
(BVFA ≡ MVFA)
(BVOA ≠ MVOA)
(GW = MVOA − BVOA)
BVE
圖3.1：商譽等於營運性資產市場上的價值( MVOA )減營運性資產帳面價值( BVOA )關係圖，
其中 MVE 代表普通股市值； BVE 代表普通股帳面價值，
MVE = MVFA + MVOA BVE = BVFA + BVOA MVFA = BVFA 。
給定(2.1.4)，可得下列關係式：
MVOA ≡ MVE − MVFA
(3.2.3)
⎡ I~t +Rs ⎤
GWt ≡ MVOAt − BVOAt = ∑ Et ⎢
s ⎥
s =1
⎣⎢ (1 + rOA ) ⎦⎥
∞
(3.2.4)
其中
rOA ：營運性資產之資金成本， 1 + rOA = ROA 。
由(3.2.4)式可得商譽評價的唯一因子為 I tR+ s ，而 I tR+ s 是一個可測量的應計基礎會計
科目。
[
]
−1
給定(3.2.4)，可得 GWt = ROA
E t I tR+1 + GWt +1 .
(3.2.5)
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肆、單獨考慮放款之評價模型
本章依據放款投資案所能產生之未來現金流程，詳述於第4章第一節：單獨考慮
放款之現金流程模型；將現金流程轉變為應計基礎並納入備抵呆帳，詳述於第4章第
二節單獨考慮放款應計基礎評價模型。本文假設金融性資產無法產生商譽，因為金融
性資產是當日結算(Mark to Market)，所以淨值會等於市值。對銀行來說，放款是營業
性資產，本文假設放款是創造價值的活動。
一、單獨考慮放款之現金流程模型
1. 現金流程模型假設
放款是唯一的營運資產。
2. 現金流程模型結果
營運性資產市場上的價值( MVOA )可由自變數 X 1 、 X 2 與 X 3 表示如下：
MVOAt ≡ f ( X 1 , X 2 , X 3 ) ≡ C PL PLt + C NLI NLI t + C FEE FEEt
其中 X 1 為銀行未違約之放款積累金額，即利息與本金如期支付之銀行放款金額(符號
為 PLt )、 X 2 為新增加放款金額(符號為 NLI t )與 X 3 為第t期的手續費收入(符號為
FEE t )，符號 PL 是取Performing Loans的縮寫，符號 NLI 是取是New Loan Investment
的縮寫， PLt 是存量； NLI t 是當期流量， C PL 、 C NLI 、 C FEE 為係數項(Coefficient)，稍
後於公式(4.1.7)詳述。底下以表4.1.1來說明 NLI 對 BVOA的影響，如果第t+1期新增加
放款=17時，則第t期的表3.1.2 BVOAt = 70 將進化到第t＋1期的表4.1.1 BVOAt +1 = 87 ：
表 4.1.1
現金=3
存款户存款=90
放款=97；備抵壞帳=10；新增加放款=17；( BVOAt +1 = 87 ) 權益=0( BVE t +1 = 0 )
因此第 t+1 期金融性資產為現金=3；金融性負債為現金=90；因此調整金融負債
後淨金融資產=-87； BVFAt +1 = −87 。營運性資產之帳面價值=放款－壞帳=97－
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10=87 ； BVOAt +1 = 87 。 股 東 權 益 面 值 =0 ； BVEt +1 = 0 。 表 4.1.1 符 合
BVE t +1 = BVFAt +1 + BVOAt +1 的要求 。
3. 現金流程模型詳細說明
銀行與存款人、貸款人的現金流程關係如下：(1)貸款人付貸款利息給銀行、(2)
貸款人付服務費(services fees)給銀行、(3)銀行付利息給存款人、(4)存款人付服務費
(services fees)給銀行。銀行提供存(貸)款人的服務例如提供支票、自動櫃員機機器、
基金、信用卡、房貸、房屋保險、人身保險等。如果放款是唯一的營運資產，那麼可
以先單獨考慮上述(1)、(2)而不考慮 (3)、(4)，則放款部對銀行現金流程的影響，可
以4.1.1、4.1.2、4.1.3、4.1.4、4.1.5聯合表示如下：
I tL+1 = rLθ1 PLt + FEEt +1 + ε CR , t +1
(4.1.1)
PLt +1 = NLI t +1 + θ1 PLt + ε PL, t +1
(4.1.2)
NLI t +1 = C LL NLI t + ε NLI , t +1
(4.1.3)
NPLt +1 = (1 − θ1 )PLt + θ 2 NPLt + ε NPL, t +1
(4.1.4)
FEEt +1 = C FF FEEt + C FL NLI t + ε FEE , t +1
(4.1.5)
符號說明如下：
I tL+1 = 第t+1期時銀行根據沒有違約之放款金額所收到的現金利息(interest income from
loans )，符號 I tL+1 是取interest income的縮寫，上標L指Loans， θ1 為 PLt 持續性發展的
程度(persistence of Performing Loans)， rL = 放款的報酬利率，符號 rL 取rate on Loans
的縮寫， ε CR , t +1 是零期望值隨機誤差項，之後 ε 依此類推皆為零期望值隨機誤差項。
PLt +1 = 第t+1期時未違約的銀行放款積累金額，符號 PL 因為是取Performing Loans的
縮寫，所以又稱為良債權。
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NLI t = 新放款投資金額， NLI 是New Loan Investment的縮寫，投資案=放款， C LL 為
NLI t +1
放款的成長率(
)。
NLI t
NPLt = 第t期時因借款人拖欠利息或本金而形成的逾期放款金額，符號 NPL 因為是取
Non-Performing Loans的縮寫，所以又稱為不良債權。
上述(4.1.1)、(4.1.2)、(4.1.3)、(4.1.4)、(4.1.5)之後合併簡寫成PVDD1，PVDD1所
聯合表示的是：單獨考慮放款營運資產對銀行現金流程之影響模型，PVDD1聯合公
式之設定完全沒有用到損益表科目，這與銀行業的利息收入是來自於資產負債表中放
款 資 產 科 目 之 觀 念 相 符 ； 然 而 FO model 模 型 PVDD 中 有 (3.1.5) ：
~
I t +R1 = C I R I tR + Ot + ε~I R ,t +1 ，可知下一期的收入與本期的收入有連結，而本期的收入是
來自於損益表不是資產負債表，會產生這種結果是因為FO model模型是針對傳統製造
業。
(4.1.1)是說明：第t+1期的現金流入來源(利息收入， I tL+1 )等於第t期的積累良債權
放款金額，乘上放款的報酬利率，然後再乘上放款未違約率。而雖然銀行付利息給存
款人是銀行唯一的現金流出源頭，但是在此先簡化暫不考慮，放款是唯一的營運資
產，放款以外的資產，如存款或投資有價證券，皆被假設是金融性的資產，即結算到
當日之市價，且會產生零淨現值。
(4.1.2)是說明：第t+1期良債權積累金額( PLt +1 )等於第t+1期的新增加放款金額
( NLI t +1 )，加上良債權積累金額( PLt )乘上不違約率( θ1 )，即( θ1 × PLt )， θ1 是 PLt 存量
的不違約率，這表示放款存量的違約率為1- θ1 。本文將 θ1 稱為放款不違約率，即現有
沒有違約之放款金額的持續程度或存留比率( θ1 ∈ [0,1] )， θ1 為 PLt 持續性發展的程度
(persistence of Performing Loans)，θ1 為 PL 上下二期之慣性，θ1 為 PL 不違約率，1 − θ1
為 PL 違約率。
(4.1.3)是說明：第t+1期的新放款( NLI t +1 )等於第t期的新放款( NLI t )，乘上 C LL，C LL
為 NLI 上下二期之慣性，第t+1期 NLI t +1 依第t期 NLI t 比率 C LL 持續下去，用 C LL 代表
放款的成長率，而 C LL 是介於0到 ROA . (也就是 1 + rOA )之間， NLI 、 C LL 皆恆為正數，
此外，本文的 C 皆為係數項。
(4.1.4)是說明：第t+1期不良債權積累金額( NPLt +1 )等於第t期的良債權積累金額
( PLt )乘上違約率( 1 − θ1 )，即 [(1 − θ1 ) × PLt ] ，然後再加上第t期的不良債權積累金額
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( NPLt )乘上 θ 2 ， θ 2 為 NPLt 持續性發展的程度，隱含表示，前一期逾期放款餘額的
( 1 − θ 2 )會在應計基礎會計帳中完全地被沖銷(write-off, charge-off)。
(4.1.5)是說明：第t+1期的手續費收入( FEEt +1 )等於第t期的手續費收入( FEE t )，乘
上 C FF ，然後再加上第t期的新放款( NLI t )乘上 C FL 。 C FF 為 FEE 為上下二期之慣性，
持續性發展的程度，第t+1期 FEEt +1 依第t期 FEE t 比率 C FF 持續下去，而 C FL 是每一元
NLI t 所帶來的第t+1期的手續費收入( FEEt +1 )，而 C FF 是介於0到 ROA (也就是 1 + rOA )之
間， C FL ≥ 0 ，C得下標有兩個F；即指上下兩期之FEE。
4. 現金流程模型推導過程
如果PVDD1是對的，則可得下列公式：
FCFt +1 = I tL+1 − NLI t +1 = rLθ 1 PLt + C FF FEE t + C FL NLI t − NLI t +1
(4.1.6)
(4.1.6)說明：第t+1期單獨考慮放款營運自由現金流程等於由放款所收到的利息減
掉第t期時新放款，因此 FCFt +1 可由 PLt 、 FEE∞t 、 NLI t 來描述；因為 FCFt +1 是描繪
⎡ FCFt + s ⎤
MVOAt 的唯一因子，詳公式(3.2.1)： MVOAt = ∑ Et ⎢
⎥
s
s =1
⎣ ROA ⎦
所以給定(4.1.6)、(3.2.1)：可得(4.1.7)關係式：
MVOAt = C PL PLt + C NLI NLI t + C FEE FEEt
(4.1.7)
(4.1.7)說明：銀行的放款營業資產市場價值( MVOAt )可由(4.1.2)所描繪之 PLt 、
(4.1.3)所描繪之 NLI t 與(4.1.5) FEE t 來表示。其中係數項(Coefficient)
C PL = (ROA − θ1 ) rLθ1
(4.1.8)
C FEE = (ROA − C FF ) C FF
(4.1.9)
−1
−1
C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
(4.1.10)
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(4.1.8)、(4.1.9)與(4.1.10)之推導詳細證明放在附錄4.1中(附錄4.1為第四章第一個
附錄，之後依此類推)。觀察(4.1.8)、(4.1.9)與(4.1.10)，得知 C PL > 0、C FEE > 0、C FF > 0、
C LL > 0 ，但 C NLI 不一定大於零， C NLI 可以等於零，而 C NLI 等於零的充分條件為
C PL = 1 且 C FL = 0 ，而 C FL = 0 說明了(4.1.5)中 NLI t 對 FEEt +1 沒影響，C NLI 為零表示放
款是淨現值為零的投資案。
5. 模型之經濟意涵
(1) C PL 之經濟意涵：未違約放款金額( PLt )的係數( C PL )反映的是折現值，是 PLt 資產
所賺得未來利息之今日現值，該折現值考慮了 PLt 續借比率： θ1 ，如果 PLt 一直沒有
被違約(即 θ1 =1)，則 PLt 資產所賺得未來利息之今日現值等於以 rOA 為折現率之永續年
r PL
金(perpetuity)，即 C PL PLt = L t ，分子是定額收入。
rOA
(2) C FEE 之經濟意涵：第t期的手續費收入( FEE t )的係數( C FEE )反映的是折現值，是
FEE t 所賺得未來手續費收入之今日現值，該折現值考慮了 FEE t 持續比率： C FF ，如
果 C FF =1，則 FEE t 所賺得未來手續費收入之今日現值等於以 rOA 為折現率之永續年金
FEEt
(perpetuity)，即 C FEE FEEt =
。
rOA
(3) C NLI 之經濟意涵：當新的放款投資案(New Loan Investment)是會產生正淨現值的
投資案，則 C NLI 大於零，詳細證明如下：
給定(4.1.10) ： C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]，本文將 C NLI 重寫
−1
−1
為： C NLI = (ROA − C LL ) C LL (C PL − 1) + (ROA − C LL ) C FL (C FEE + 1)
−1
將(4.1.8) 、(4.1.9)帶入上式，可得下列關係式：
(
)
(
)
C NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ1 − 1 + (ROA − C LL ) C FL (ROA − C FF ) C FF + 1
繼續整理，可得下列關係式：
−1
−1
−1
−1
C NLI = (ROA − C LL ) (ROA − θ1 ) C LL [θ1 (1 + rL ) − (1 + rOA )] ＋
(ROA − C LL )−1 (ROA − C FF )−1 C FL ROA
−1
−1
(4.1.11)
因 C LL (ROA − θ1 ) (ROA − C LL ) 與 (ROA − C LL ) (ROA − C FF ) C FL ROA 恆為正數永遠
大於零，所以(4.1.11)式與方括號中的項目： [θ1 (1 + rL ) − (1 + rOA )] 同向：
−1
−1
−1
−1
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(i)
18
顯示當 rL > rOA 時，則 [(1 + rL ) > (1 + rOA )] ；調整放款的違約率( θ1 )後，放款
的報酬率 [θ1 (1 + rL )] 如果大於放款的資金成本 [(1 + rOA )] ，則 C NLI 大於零，
然而 rL > rOA 表示新的放款投資案淨現值大於零。
當新的放款投資案淨現值為零時，可以容易地證明 C PL = 1 且 C FL = 0 。換
句話說，當放款的淨現值為零時，則放款可以被視為是金融資產，因此表
3.1.1有新的銓釋如表4.1.2：
表 4.1.2
現金=20
存款户存款=90
放款=80；( BVOAt −1 = 0 )
權益=10( BVEt −1 = 10 )
則第 t－1 期金融性資產為現金＋放款=20＋80=100；金融性負債為存款户存款
=90；因此調整金融負債後淨金融資產=100－90=10； BVFAt −1 = 10 。放款營運性資
產之帳面價值=0； BVOAt −1 = 0 。股東權益面值=10； BVEt −1 = 10 。表 4.1.2 符合
BVEt −1 = BVFAt −1 + BVOAt −1 ⇒ 10 = 10 + 0 的要求。注意表 4.1.2 之 BVOAt −1 = 0 ；但
是表 3.1.1 之 BVOAt −1 = 80 。
(ii)
在本文的模型中，逾期放款不良債權( NPL )被假設都不能變回未逾期放款優良債
權( PL )且假設以(1- θ 2 )的比率將之沖銷掉，因此(4.1.7)現金流程評價法是不會受沖銷
比例 θ 2 所影響的，但這並不是指 NPL 在評價過程中是無關緊要的，而是指 NPL 的影
響可以完全由 θ1 來代表。
如果 NPL 被假設有可能變回 PL ，且假設以 θ12 的比率將 NPL 回沖成 PL ，則
PVDD1中之第二式須改寫如下： PLt +1 = NLI t +1 + θ1 PLt + θ12 NPLt + ε PL, t +1 ，由於僅增
加複雜度而無新發現，故本文就此打住。
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二、單獨考慮放款應計基礎評價模型
1. 應計基礎模型假設
放款是唯一的營運資產。
2. 應計基礎模型結果
放款銀行之商譽模型可由自變數 X 1 、 X 2 、 X 3 與 X 4 表示如下：
GWt ≡ f ( X 1 , X 2 , X 3 , X 4 ) ≡ c NPL NPLt + cGL GLt + c NLI NLI t + c FEE FEEt
其中 X 1 為逾期放款，即違約之放款積累金額，即利息與本金未如期支付之銀行
放款金額(符號為 NPLt )、 X 2 為放款總額(符號為 GLt )、 X 3 為新增加放款金額(符號為
NLI t )， X 4 為第t期的手續費收入(符號為 FEE t )，符號 NPL 是取是Non-Performing
Loans的縮寫，符號 GL 是取Gross Loans的縮寫， NPLt 及 GLt 是存量； NLI t 是當期流
量， c NPL 、 cGL 、 c NLI 與 c FEE 為係數項(Coefficient)，稍後於公式(4.2.10)詳述。
3. 應計基礎模型詳細說明
由於上一段章節是自由營業現金流程，因此無法使用財報上之會計科目來評價，
所以本段章節使用RIVM，RIVM是應計會計基礎評價模型之一例。
檢視公式(4.1.7)： MVOAt = C PL PLt + C NLI NLI t + C FEE FEEt ，其中 PLt 是潛在變數
(latent variable)，財報中沒有 PLt 這一種會計科目；本段章節使用財報中的 GLt 、
LLRt (Loan Loss Reserve)及 NPLt 這三種會計科目，其中 GLt － LLRt ＝ NLt 。而符號
NLt 是取Net Loan的縮寫。使用RIVM需要將現金流程模型中之現金流程(operating
cash flows)轉變為權責制應計所得(accrual income)例如逾期放款( NPL )、放款總額
( GL )及備抵呆帳( LLR )，所以本段章節會把焦點集中在備抵呆帳( LLR )上，因為 LLR
是銀行主要的應計金額(accrual)。實務上銀行設定 LLRt 的根據是 GLt 及 NPLt ，所以本
段章節假定銀行經理每期都會透過二個備抵壞帳參數的使用來設定 LLR：第一個參數
是1－ C GL ，代表的是放款總額中提列為備抵呆帳的比例；而第二個參數則是 C NPL ，
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20
代表的是逾期放款中提列為備抵呆帳的比例，即備抵呆帳覆蓋率 6 ：
LLRt ≡ (1 − C GL )GLt + C NPL NPLt
(4.2.1)
因為 GL = NL + LLR ，所以淨放款(放款總額與備抵呆帳的差額)可被表示為下列
的式子：
NLt ≡ (C GL )GLt − C NPL NPLt
(4.2.2)
如果放款是唯一的營運資產，則 NLt ≡ BVOAt 。
(4.2.3)
備抵應計基礎會計的結構呈現在圖4.2.1，並附有實際的數值分析如表4.2.1。放款
總額( GL )，即為不良債權違約逾期放款金額( NPL )和優良債權未違約逾期放款金額
( PL )的加總( GL = PL + NPL )。放款總額科目會因為當期新增加的放款而增加，會因
為本金的償還及逾期放款的註銷而減少，以圖4.2.1第1欄之T字帳來說明GL之前後期
關係。備抵呆帳是資產的抵銷科目，它會因為備抵呆帳準備金 LLP (Loan Loss
Provision)的提列而增加(記入貸方credited)，會因為備抵呆帳的註銷而減少(記入借方
debited)。備抵呆帳準備金提列的方式與製造業呆帳費用的提列類似，以圖4.2.1第2欄
之T字帳來說明LLR之前後期關係，而以圖4.2.1第3欄之T字帳來說明NL之前後期關
係。
Gross Loans
GLt −1
writeoffst
NLI t
Loan Loss Reserve
LLRt −1
writeoffst
LLPt
GLt
LLRt
Net Loans
NLt −1
NLI t
LLPt
NLt
圖 4.2.1：備抵應計基礎會計 T 字帳
其中
GL : 放款總額(Gross Loans)
NLI : 新放款投資(New Loan Investment)
LLR : 備抵壞帳累積庫存總額(Loan Loss Reserve= Loan Loss Allowance)
NPL : 不良債權違約逾期放款金額(Non-Performing Loan=Overdue Loan)
LLP ：本期壞帳準備金，即銀行中的壞帳費用(Loan Loss Provision)
NL ：放款淨額(Net Loans)，如果放款是唯一的營運資產，則 NLt ≡ BVOAt
由圖 4.2.1 可得公式(4.2.4)： NLt = NLt −1 + NLI t − LLPt 稍後用得到。
6
備抵呆帳覆蓋率=備抵呆帳/逾期放款。 C NPL =
LLR
NPL
第 20 頁
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21
表 4.2.1 備抵應計基礎會計的實際數值分析
2004 第三季(t)
2004 第二季(t-1)
Gross Loans (GL)
$5,981,395
$5,783,141
Less: Loan Loss Reserve(LLR)
83,834
84,139
Net Loans (NL)
5,897,561
5,699,002
Loans 30-89 days overdue (NPL)
49,632
46,740
Net charge-offs(write-offs)
23,169
15,776
資料來源：http://www.fdic.gov 美國聯邦存款保險公司對 9,025 家總體銀行所作的統
計資料
由圖 4.2.1 中的「Gross Loans」T 字帳，可計算出 NLI t
NLI t = GLt − GLt −1 + writeoffs t
＝5,981,395－5,783,141＋23,169＝221,423
由圖 4.2.1 中的「Loan Loss Reserve」T 字帳，可計算出 LLPt
LLPt = LLRt − LLRt −1 + writeoffs
＝83,834－84,139＋23,169＝22,864
由圖 4.2.1 中的「Net Loans」T 字帳，可計算出 NLt
NLt = NLt −1 + NLI t − LLPt
＝5,699,002+221,423－22,864＝5,897,561
與 FDIC 所公布的數據一致，故證明了圖 4.2.1 的三種關係式在實務上也是能夠成立
的。
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22
4. 應計基礎剩餘損益會計評價模型推導過程
本段章節以 LLPt 來代表當期提列之備抵呆帳，而 rOA × NLt −1 代表的是使用放款營
運資產的成本。當銀行唯一能產生價值的活動就是它的放款活動時，則剩餘損益( I tR )
可表示如下：
I tR = I tL − LLPt − rOA × NLt −1
(4.2.5)
合併(4.2.4)： NLt = NLt −1 + NLI t − LLPt 及(4.2.5)可得(4.2.6)如下：
I tR = I tL + NLt − ROA × NLt −1 − NLI t
(4.2.6)
而 I tR+ s 是一個可測量的應計基礎會計科目。
合 併 (3.2.4) ： GWt ≡ MVOAt − BVOAt 、 (4.2.2) ： NLt ≡ (C GL )GLt − C NPL NPLt 及
(4.2.3)： NLt ≡ BVOAt ，可得(4.2.7)商譽評價模型表示如下：
GWt ≡ MVOAt − BVOAt = MVOAt − C GL GLt + C NPL NPLt
(4.2.7)
如 果 放 款 是 唯 一 的 營 運 資 產 ， 那 麼 將 (4.1.7) ：
MVOAt = C PL PLt + C NLI NLI t + C FEE FEEt 代入(4.2.7) 則可得放款部對銀行應計基礎
剩 餘 損 益 會 計 的 商 譽 評 價 模 型 (4.2.8) 如 下 ：
GWt ≡ MVOAt − BVOAt = C PL PLt + C NLI NLI t + C FEE FEE t − C GL GLt + C NPL NPLt
(4.2.8)
將 PL = GL − NPL 帶 入 (4.2.8) 上 式 ， 可 得 下 列 關 係 式 ：
GWt ≡ MVOAt − BVOAt = C PL (GLt − NPLt ) + C NLI NLI t + C FEE FEE t − C GL GLt + C NPL NPLt
整理 GL 、 NPL 的係數，可得下列關係式：
GWt = C NLI NLI t + C FEE FEEt + (C PL − CGL ) × GLt + (C NPL − C PL ) × NPLt
(4.2.9)
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23
(4.2.9) 說明：放款銀行之商譽模型( GWt )可由自變數 NLI t 、 FEE t 、 NPLt 與 GLt 表示
如下：
GWt ≡ c NPL NPLt + cGL GLt + c NLI NLI t + c FEE FEEt
其中係數項 c NPL 、 cGL 、 c NLI 與 c FEE
(4.2.10)
cGL = C PL − C GL
(4.2.11)
c NPL = C NPL − C PL
(4.2.12)
c NLI = C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
−1
−1
c NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ1 − 1 + C FL (ROA − C FF ) C FF + 1
(4.2.13)
c FEE = (ROA − C FF ) C FF = C FEE
(4.2.14)
−1
{ [
]
[
]}
−1
(4.2.11)、(4.2.12)、(4.2.13)與(4.2.14)之推導詳細證明放在附錄4.2中。
(4.2.13)式與(4.1.10)式相同，(4.2.14)式與(4.1.9)式相同，(4.2.11)與(4.2.12)中的
C PL 與(4.1.8)相同，(4.2.11)與(4.2.12)中的 C GL 、 C NPL 與(4.2.2)中的 C GL 、 C NPL 相同。
比較(4.2.6)與(4.2.10)，可得 NLI t 是唯一共同的因子，而 GWt 與 I tR 之關係，可由
⎡ I~t +Rs ⎤
(3.2.4)： GWt ≡ MVOAt − BVOAt = ∑ Et ⎢
來連結。
s ⎥
s =1
⎢⎣ (1 + rOA ) ⎥⎦
∞
5. 模型之經濟意涵
本段章節4.2的重點是(4.2.10)式；而上一段章節4.1的重點是(4.1.7)式，比較(4.2.10)
式與(4.1.7)式的係數有下例發現：(1)、兩式第t期時新增加放款金額( NLI )的係數是相
等的(即 c NLI = C NLI )。(2)、兩式第t期時新增加放款金額( FEE )的係數是相等的(即
c FEE = C FEE )。(3)、(4.2.10)式中放款總額的係數(即 cGL )、逾期放款的係數(即 c NPL )不
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24
等於(4.1.7)式中 PLt 的係數（即 C PL ），之所以不同(不等於)是反映(4.2.10)式有備抵呆
帳的應計基礎會計政策；而(4.1.7)式沒有備抵呆帳的應計基礎會計政策。(4)、(4.2.10)
式與(4.1.7)式都僅用到資產負債表科目；而沒有用到損益表科目，會產生這種結果是
因為本文PVDD1之設定完全沒有用到損益表科目，這與銀行業的利息收入是來自於
資產負債表中放款資產科目之觀念相符。因此本文銀行商譽評價根據的不是損益表而
是資產負債表科目，會產生這種結果是因為本文假設放款銀行所收到的現金收入與放
款業務量有著緊密的關係，這與銀行需要放款來獲得利息收入的觀念相符；然而FO
model評價根據的不是資產負債表而是科目損益表。
如果應計基礎會計是不偏的，則本文銀行評價模型中的商譽價值隱含是零，即
(4.2.10)式中的係數全部都是零( c NPL = cGL = c NLI = c FEE =0)。 c NLI 為零表示放款是淨現值
為零的投資案，當放款的淨現值為零時，可以容易地證明 C PL = 1 且 C FL = 0 。換句話
說，當放款的淨現值為零時，則放款可以被視為是金融資產。因此，正如Feltham和
Ohlson(1996)指出，就製造業而言，任何投資的淨現值為零是不偏應計基礎會計的必
要條件。此外觀察(4.2.11)、(4.2.12)、(4.2.13)得知，如果 c NPL = cGL = c NLI = c FEE =0，則
C GL = C NPL = C PL =1 ， 將 C GL = C NPL =1代入(4.2.1)： LLRt ≡ (1 − C GL )GLt + C NPL NPLt 與
(4.2.2) ： NLt ≡ (C GL )GLt − C NPL NPLt ， 得 知 如 果 C GL = C NPL =1 ， 則
NLt = GLt − NPLt ⇒ NPL = GL − NL = LLR ，因此每一期的 LLR 皆相等於每一期的
NPL ，這隱含備抵呆帳百分之百正確的(不偏的)。
伍、結論
本文證明出將 FO model 之剩餘損益評價模型(RIVM)運用到特別產業(在本研究
的例子中這個特別的產業就是銀行)的解釋力。就本研究所知，本文是第一篇著手作
此嘗試的論文。FO model 所提出之模型針對的是製造業。當本研究應用 FO model
到銀行時，發現解釋銀行價值的變數與解釋製造業公司價值的變數有很大的不同。
什麼樣的變數解釋了銀行價值，所根據的是什麼樣的活動會產生大於零的淨現值，
以及會計原則從現金制轉變為應計制改變了什麼樣的變數。為了發展本研究的銀行
評價模型，本研究假設主要的價值創造活動是放款。除此之外，本研究也假設主要
的會計誤差是由備抵呆帳會計所導致。當本研究將這些假設合併起來時，本研究產
生了一個商譽的評價模型，該模型根據的是放款總額、當期增加的放款與逾期放款。
正如同仍有人能擴張價值的驅動因子，並運用在FO model所提出之會計評價誤
差，本研究也是將其他學者所提出的評價模型加以延伸，本研究的模型同樣也可以被
擴張到包含銀行其他產生價值的活動和會計評價的誤差，本研究一直有個想法，就是
第 24 頁
2007/9/27
25
有關於將模型合併其他的銀行活動，這些活動也可能分別地產生銀行的市場價值與帳
面價值。Kohlbeck and Warfield (2003)的研究結果建議了一些銀行中能產生價值的活
動，包括存款、信用卡服務、不動產抵押證券，以及信託活動。將本研究的模型與上
述活動及價值創造的其他來源加以合併，會使模型更加完整，在本研究的模型架構裡
面，需要確認究竟是那些投資活動會產生這些無形資產。這些活動可能是其他放款的
新投資形式（例如，房貸），或是人力資本的投資形式（也許可藉由員工薪資費用來
測量），這個擴張或許能將損益表的會計項目也納入。
第 25 頁
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26
附錄
附錄 4.1 放款之現金流程模型
−1
合 併 (3.2.2) ： MVOAt = ROA
E [FCFt +1 + MVOAt +1 ]. 、 (4.1.6) 與 (4.1.7) 可 得
−1
MVOAt = ROA
E [FCFt +1 + MVOAt +1 ]
−1
[FCFt +1 + C PL PLt +1 + C FEE FEEt +1 + C NLI NLI t +1 ]
⇒ C PL PLt + C FEE FEEt + C NLI NLI t = ROA
r θ PLt + C FF FEEt + C FL NLI t − C LL NLI t + C PL (C LL NLI t + θ1 PLt )⎤
−1 ⎡ L 1
= ROA
⎢+ C (C FEE + C NLI ) + C C NLI
⎥
t
FL
t
NLI LL
t
⎣ FEE FF
⎦
集合變數 PLt
左手邊(之後皆以LHS代表之)的 PLt 變數係數為 C PL
−1
右手邊(之後皆以RHS代表之)的 PLt 變數係數為 ROA
(rLθ1 + C PLθ1 )
−1
(rLθ1 + C PLθ1 )
⇒ C PL = ROA
⇒ C PL = (ROA − θ1 ) rLθ1
−1
集合變數 FEE t
LHS 的 FEE t 變數係數為 C FEE
RHS 的 FEE t 變數係數為 (C FEE C FF + C FF )
−1
(C FEE C FF + C FF )
⇒ C FEE = ROA
⇒ C FEE = (ROA − C FF ) C FF
−1
集合變數 NLI t
LHS的 NLI t 變數係數為 C NLI
−1
RHS的 NLI t 變數係數為 ROA
[C NLI C LL + C LL (C PL − 1) + C FL (C FEE + 1)]
⇒ C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
{ [
]
[
]}
C NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ1 − 1 + C FL (ROA − C FF ) C FF + 1
−1
−1
−1
第 26 頁
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27
附錄 4.2 放款之應計基礎評價模型
給定
−1
GWt = ROA
E t GWt +1 + I tR+1
GWt = C NLI NLI t + C FEE FEEt + (C NPL − C PL ) × NPLt + (C PL − C GL ) × GLt
[
]
I tR+1 = I tL+1 + NLt +1 − NLI t +1 − ROA × NLt
合併(3.2.5)、(4.2.6)與(4.2.9)可得
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C PL ) × ( NPLt ) + (C PL − C GL ) × (GLt ) = ROA
(3.2.5)
(4.2.9)
(4.2.6)
C NLI NLI t +1 + C FEE FEEt +1 + (C NPL − C PL ) × ( NPLt +1 ) + (C PL − C GL ) × (GLt +1 )
+ I tL+1 + NLt +1 − NLI t +1 − ROA × NLt
⇒
]
將(4.2.2)： NLt ≡ C GL GLt − C NPL NPLt 帶入上式，可得下列關係式：
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C PL ) × ( NPLt ) + (C PL − C GL ) × (GLt ) = ROA
C NLI NLI t +1 + C FEE FEEt +1 + (C NPL − C PL ) × ( NPLt +1 ) + (C PL − C GL ) × (GLt +1 )
+ I tL+1 + (C GL GLt +1 − C NPL NPLt +1 ) − NLI t +1 − ROA × (C GL GLt − C NPL NPLt )
]
⇒
將 GLt = PLt + NPLt 、 GLt +1 = PLt +1 + NPLt +1 帶 入 上 式 ， 可 得 下 列 關 係 式 ：
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C PL ) × ( NPLt ) + (C PL − C GL ) × ( PLt + NPLt ) = ROA
C NLI NLI t +1 + C FEE FEEt +1 + (C NPL − C PL ) × ( NPLt +1 ) + (C PL − C GL ) × ( PLt +1 + NPLt +1 )
+ I tL+1 + C GL (PLt +1 + NPLt +1 ) − C NPL NPLt +1 − NLI t +1
− ROA × (C GL PLt + C GL NPLt − C NPL NPLt )
]
⇒
將屬於 NPL 的係數聯合，可得下列關係式：
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
C NLI NLI t +1 + C FEE FEEt +1 + (C NPL − C GL ) × ( NPLt +1 ) + (C PL − C GL ) × ( PLt +1 )
+ I tL+1 + C GL PLt +1 + C GL NPLt +1 − C NPL NPLt +1 − NLI t +1
− ROA C GL PLt − ROA C GL NPLt + ROA C NPL NPLt
]
⇒
繼續整理 NPL 的係數，可得下列關係式：
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
(
C NLI NLI t +1 + C FEE FEEt +1 + C
)
− C NPL
+ (C PL − C GL ) × ( PLt +1 )
NPL
NPL
t +1
GL
t +1
+ I tL+1 + C GL PLt +1 + C NPL
−C
− NLI t +1
NPL
GL
t +1
NPL
t +1
− ROAC GL PLt + ( ROA C NPL − ROA C GL ) NPLt ]
(
)
⇒
繼續整理 NPL 的係數，可得下列關係式：
第 27 頁
2007/9/27
28
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
C NLI NLI t +1 + C FEE FEEt +1 + (C PL − CGL ) × ( PLt +1 )
+ I tL+1 + CGL PLt +1 − NLI t +1
− ROA CGL PLt + ( ROA C NPL − ROA CGL ) NPLt
]
⇒
將屬於 PL 的係數聯合，可得下列關係式：
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
C NLI NLI t +1 + C FEE FEEt +1 + C PL PLt +1
+ I tL+1 − NLI t +1
− ROA C GL PLt + ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將(4.1.2)： PLt +1 = NLI t +1 + θ1 PLt 帶入上式，可得下列關係式：
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
C NLI NLI t +1 + C FEE FEEt +1 + C PL NLI t +1 + C PLθ1 PLt
+ I tL+1 − NLI t +1
− ROA C GL PLt + ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將屬於 PLt 的係數聯合，可得下列關係式：
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
C NLI NLI t +1 + C FEE FEEt +1 + C PL NLI t +1 + (C PLθ1 − ROA C GL )PLt
+ I tL+1 − NLI t +1
+ ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將屬於 NLI t +1 的係數聯合，可得下列關係式：
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
C FEE FEEt +1 + (C NLI + C PL − 1)NLI t +1 + (C PLθ1 − ROA C GL )PLt
+ I tL+1
+ ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將(4.1.3)： NLI t +1 = C LL NLI t
將(4.1.5)： FEEt +1 = C FF FEEt + C FL NLI t
將 I tL+1 = (rLθ 1 PLt + C FF FEE t + C FL NLI t ) 帶入上式，可得下列關係式：
第 28 頁
2007/9/27
29
−1
[
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
C FEE C FF FEEt + C FEE C FL NLI t + (C NLI + C PL − 1)C LL NLI t + (C PLθ1 − ROA C GL )PLt
+ (rLθ1 PLt + C FF FEEt + C FL NLI t )
+ ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將屬於 NLI t 的係數聯合，可得下列關係式：
−1
[
C NLI NLI t + C FEE FEE t + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
C FEE C FF FEE t + (C FEE C FL + C FL + C NLI C LL + C PL C LL − C LL )NLI t
+ (C PLθ1 − ROA C GL )PLt + (rLθ 1 PLt + C FF FEEt ) + ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將屬於 PLt 的係數聯合，可得下列關係式：
−1
[
C NLI NLI t + C FEE FEE t + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
C FEE C FF FEE t + (C FEE C FL + C FL + C NLI C LL + C PL C LL − C LL )NLI t
+ (C PLθ1 − ROA C GL + rLθ1 )PLt + C FF FEE t + ( ROA C NPL − ROA C GL ) NPLt
]
⇒
將屬於 FEEt 的係數聯合，可得下列關係式：
−1
C NLI NLI t + C FEE FEEt + (C NPL − C GL ) × ( NPLt ) + (C PL − C GL ) × ( PLt ) = ROA
[
(C FEE C FF + C FF )FEEt + (C FEE C FL + C FL + C NLI C LL + C PL C LL − C LL )NLI t
+ (C PLθ1 − ROA C GL + rLθ1 )PLt + ( ROA C NPL − ROA C GL ) NPLt ]
選擇相關的項目形成下列的 4 個等式：
(1)集合變數 PLt
左手邊(之後皆以LHS代表之)的 PLt 變數係數為 (C PL − CGL )
−1
(rLθ1 + C PLθ1 − ROA CGL )
右手邊(之後皆以RHS代表之)的 PLt 變數係數為 ROA
−1
(rLθ1 + C PLθ1 ) − CGL ⇒ C PL = (ROA − θ1 ) rLθ1
⇒ C PL − CGL = ROA
(2)集合變數 FEE t
LHS 的 FEE t 變數係數為 C FEE
RHS 的 FEE t 變數係數為 (C FEE C FF + C FF )
−1
−1
(C FEE C FF + C FF )
⇒ C FEE = ROA
(3)集合變數 NLI t
LHS 的 NLI t 變數係數為 C NLI
⇒ C FEE = (ROA − C FF ) C FF
−1
−1
RHS 的 NLI t 變數係數為 ROA
[C NLI C LL + C LL (C PL − 1) + C FL (C FEE + 1)]
⇒ C NLI = (ROA − C LL ) [C LL (C PL − 1) + C FL (C FEE + 1)]
−1
{ [
]
[
]}
⇒ C NLI = (ROA − C LL ) C LL (ROA − θ1 ) rLθ1 − 1 + C FL (ROA − C FF ) C FF + 1
(4)集合變數 NPLt
LHS 的 NPLt 變數係數為 (C NPL − C GL )
−1
−1
−1
−1
RHS 的 NPLt 變數係數為 ROA
[( ROA C NPL − ROA C GL )]
−1
[ROA (C NPL − CGL )]
⇒ C NPL − CGL = ROA
⇒ C NPL − CGL = C NPL − CGL
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30
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