Chain Merger Evaluation and Application to Banking Operations

Desheng Dash WuUniversity of Toronto, Reykjavik University

[with John R. Birge, Booth School of Business, University of Chicago]

Accepted and to appear at POMDSJ, 43(1) 2012

1

2

OutlineIntroduction

Problem, LiteratureBackground modelOur model

Conceptual model, Math model3 main contributions

▪ chain merger DEA model, leader-follower relations▪ efficiency at both chain and sub-chain levels, incentive

compatible▪ banking intra-firm division mergers

Case analysisConclusion & Further study

3

Outline

o Introduction

o Background

model

o Our model

o Case Study

o Conclusion 4

Industry facts

New York Times, March 11, 2013:

“the dollar value of U.S. mergers and acquisitions so far this year is $233 billion, more than double last year. But there were almost 10 percent fewer deals than last year.“

“Today's mergers and acquisitions are more about building up than cashing in.”

5

LiteratureIO: Salant (83), JOE; Deneckere and

Davidson (85) RandJ;Perry and Porter (85), AER; Farrell, (90), AER; Rothschild (00) Reg Sci&E; Benjamin et al. (09)

merger paradox

Finance: Sapienza (02), JOF; Guerard Jr. (89); Geppert and Kamerschen(08); Houston and Ryngaert (94) JBF; Duffie (07) JFE

stock

OR: Sherman and Rupert (06), EJOR; Cummins et al.(08) JBF; Ray (04); Bogetoft (05), JPA

efficiency6

Our study

A model for gauging merger efficiency of supply chainsdifferent structure

Supply chain view of banking operations Link operations to finance

Apply the model to banking operations with DEA, considering M&A with multiple metricsLink OR to IO/Finance

7

Question: banks or subdivisions merged, how business performance is affected considering such a banking chain? How to achieve potential gains?

8

Revisit Banking operations- supply chain

Borrower selection; Underwriting; Loan approval; Primary market

Securitization; Derivative Trading; Secondary market

Staff, asset

IT facility,staff

Information flow

Loans

Information flow Product flow

Sales, risks, availability, stock, customer/investor satisfaction level

Mortgage banking: a view of serial chain

Outline

o Introduction

o Background

model

o Our model

o Case Study

o Conclusion 9

Merger gains (Ray, 2004)2-step process: harmony effect, scale effect

10

Input 2

Input 1

B: (x1B, x2B)

A: (x1A, x2A)

D

G

yA

demonstration of the merger of two firms

yB

yD>1/2(yA+yB) ?

yG>(yA+yB) ?

Mathematically

The efficiency of merger can then be measured as denotes the harmony effect. represents the scale effect.

potential gains from the merger of the two firms positive if > 1.

11

Em( X

A, X

B)

yG

2y(

yD

y)(

yG

2yD

) H S (1)

H

yD

y

S

yG

2yD

Em

What1. a linear programming to measure the

efficiency of multiple decision-making units (DMU) when the DMUs present a structure of multiple inputs and outputs.

Different versions: Constant return to scale (CRS), Variable return to scale (VRS)

How2. Define DMU, input/output variables3. Define the efficiency frontier.4. A numerical weight coefficient is given to

each firm, computing its relative efficiency.

Data Envelopment Analysis (DEA)

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Outline

o Introduction

o Background

model

o Our model

o Case Study

o Conclusion 13

two-stage series-chain Conceptually

the ith DMU, i=1,2…N

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MathematicallyN : number of DMUs

: multiplier, to be solved, i=1,2…N; l=1,2 P, Q: price vector

In the lth stage, to evaluate the efficiency of the Ith DMU with 2- stage chain:

(2)

Here, are the decision variables 15

1 1 1 1 2 2 2 2

1, 1, 1, 2, 2, 2, 2, 1, 1,1 1 1

1 1, 1, 2 2, 2,1 1

1

( ) ( )max

. . ,

0 , 0 ,

I I I I

N N N

i i I i i I i i Ii i i

N NI I

i i i ii i

I

P f Q h P f Q h

y

y y

s t x x x x y

f f

h

1, 1, 2 2, 2,1 1

1, 2, 1, 2, 1 1 2 21 1

1 1 ,

0 , 0 ,

, , ; , , 0.

N NI

i i i ii i

N NI I I I

i i i ii i

z zh

f h f h

f1I ,h1

I , f2I ,h2

I

l ,i

Step 1: solve the DEA model for each chain and sub-chain, and construct the efficient input-output combination for each supply chain.

Step 2: Compute the average input bundle, intermediate output/input bundle and output bundle for each supply chain and members.

Step 3: Solve the series-chain DEA problem for the average input-output supply chain

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(x

l , I , y

l , I , z

l , I )

Solution approach: 6 steps

Step 4: Compute the total input and output bundle of the N Series-chain models.

Step 5: Solve the merger chain DEA problem for the whole chain with input and output bundle

Step 6: Compute the sub-chain efficiency, merger efficiency for the whole chain, the harmony and scale components.

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Solution approach: 6 steps

, ,( )Total Total Totall l lx y z

TheoremsTheorem 1. full two-stage chain is

efficient if and only if the sub-chain members are both efficient.

Theorem 2. Merger of the full two-stage chain is efficient if and only if the mergers of the sub-chain members are both efficient.

Similar theorems hold for the case with many sub-chain members

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Leader-follower relations

Direct input

Shared input

Intermediate output/input

Direct output

chain with constrained resources

Hierarchical structure

1DX Y

E1X 2X

2DX

1Z

Leader Follower

2Z

1DX 2DX

Y1Z

1X

The framework with limited resource E

2X

2Z19

o Constrained resource, leader-follower relation

Primary market business-leader

Secondary market business-followerOthers

IT Budget

Loans

Loan recovery

ProfitPersonal

A Canadian bank revist…

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Bilevel programming problem (BLP) : A hierarchical optimization problem consisting of two levels. The upper level/ the Leader’s level/ the dominant level

The lower level/ the Follower’s level/ the submissive level

A Bilevel Linear Programming given by Bard (88) is formulated as follows:

Bilevel programming

1 1

1 1 1

2 2

2 2 2

min ( , )

. .

min ( , )

. .

, 0

T T

x

T T

y

F x y p x q y

s t A x B y b

f x y p x q y

s t A x B y b

x y

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Proposition

The system efficiency is a convex combination of both the leader and follower efficiency.

The system is efficient iff the sub-systems are efficient.

Merging of the system is efficient iff merging of the sub-systems is efficient.

System-subsystem relation

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Dominant level (the Leader) gains much more potential improvement profit than what the lower level (the Follower) gains.

α -Strategy: To encourage the Follower to participate,

the Leader promises to share α percentage of his

profit to the Follower.

Incentive compatible ?

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α -Strategy:

Incentive compatible-example

00.

010.

020.

030.

040.

05

0.06

0000

0000

0000

010.

07

0.08

0000

0000

0000

010.

09 0.1

80%

90%

100%

110%

120%

130%

DMU1

DMU2

DMU3

DMU4

DMU5

DMU6

DMU7

DMU8

Efficiency ra-tio

α

00.

010.

020.

030.

040.

050.

060.

07

0.08

0000

0000

0000

010.

09 0.1

90%

100%

110%

120%

130%

140%

DMU1

DMU2

DMU3

DMU4

DMU5

DMU6

DMU7

DMU8

Efficiency ratio

α

adjusted optimized profitEfficiency ratio

observed profit

The efficiency ratio of the Leader under α strategy

The efficiency ratio of the Follower under α strategy

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Outline

o Introduction

o Background

model

o Our model

o Case Study

o Conclusion 25

Case study: a Canadian bank…

▪ Data from 36 branches (DMUs) for 6 variables

▪ Mortgage banking chain input-output framework

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a Canadian bank…36 branches

efficiency analysis of the mortgage banking operations

consider mergers of the branches as a form of intra-firm re-organization.

potential savings by merging two branches at a time

630 combinations using both the CRS and VRS DEA chain merger models 27

Efficiency distribution

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1 2 3 4 5 6 7 8 9 1011121314151617181920212223242526272829303132333435360

0.2

0.4

0.6

0.8

1

1.2

sub-chain CRSchain CRSsub-chain VRSchain VRS

Sub-chain and full chain comparison

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the 1st sub-chain (>100%) under CRS and

VRS.

full-chain (>100%) under CRS and VRS.

Sub-chain VRS

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merger efficiency distribution. Harmony efficiency

Scale efficiency

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VRS merger efficiencyFull-chain VRS

Harmony efficiency

Scale efficiency

Top 10 promising mergers The computation results recommend a merger of two strong DMUs, rather than two weak ones or a weak and strong one

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Code Merger ME Scale

D1 5 29 1.4943 0.95031

D2 5 28 1.4677 0.93631

D3 5 23 1.4453 0.93501

D4 5 30 1.3427 0.91358

D5 5 35 1.3309 0.87452

D6 5 31 1.3308 0.91817

D7 23 28 1.3155 0.95172

D8 4 5 1.3137 0.93138

D9 23 29 1.3095 0.94289

D10 5 17 1.3057 0.92251

Any incentive incompatible cases ? An example of 8 branch chains

Merger leader follower system4,5 1.125 0.99 1.0923,4 1.098 0.99 1.0725,8 1.09 1.001 1.0674,7 1.081 1.001 1.0593,8 1.06 1.001 1.0451,4 1.059 0.9 1.0421,8 1.057 0.99 1.0417,8 1.047 0.99 1.0355,6 1.042 0.99 1.0314,6 1.035 1.001 1.027

The top 10 promising mergers under CRS

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Coordinated effective merger

Merger efficiency scores of the Leader, the Follower and the whole system are all greater than 1.

Incentive compatible- coordinated mergers

Merger leader follower system5,8 1.09 1.001 1.0674,7 1.081 1.001 1.0593,8 1.06 1.001 1.0454,6 1.035 1.001 1.0276,8 1.023 1.001 1.0182,8 1.014 1.001 1.011

The promising coordinated mergers under CRS

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Outline

o Introduction

o Background

model

o Our model

o Case Study

o Conclusion 35

Conclusions & Further study3 things

Creation of chain merger DEA modelEffects captured and decomposed at

both chain and sub-chain levelsa case study in banking intra-firm

division merger operationsFuture work

Assumptions to be validatedBreakup of firmsComparison with other methods, e.g.,

game models.36

Thanks!

Questions?

37

Model

Bilevel programming efficiency analysis-optional

1 1 1 1

1 1 2 1 1 1 2 1

, , , ,

1 2 1 2

1 1

1 1

1

1 1

1

( 1) max ( ) ( )

. ,

,

,

T T T T

DJ J J J

DJ J J J

X X Y Z

n n

J J j j j jj j

nD DJ j j

j

n

J j jj

J

P Q Z Q Y P X P X

s t X X X X

X X

Z Z

Y

1

1

1 2

1 1

1 1 2 1 1 1 2 1 1 1 2 1 1 1 2 1

,

( .),

,

( ) ( ) ( ) ( )T T T T T T T T

n

j jj

J J

D DJ J

D DJ J J J J J J J

Y

X X E const

X X

Q Z Q Y P X P X Q Z Q Y P X P X

38

Bilevel programming efficiency analysis-optional

2 2 2 2

3 2 1 2 3 2 2 2

, , , ,

2 2

1

2

1

( 2) max ( )

. ,

,

T T T T

DJ J J J

DJ J J J

X X Y Z

nD DJ j j

j

n

J j jj

J

P Q Z P X P X Q Y

s t X X

Y Y

Z

2 2

1

2 2

2 1

3 2 1 2 3 2 2 2 3 2 1 2 3 2 2 2

1 2 1 2 1

,

,

,

( ) ( )

, , , , ,

T T T T T T T T

n

j jj

D DJ J

J J

D DJ J J J J J J J

D DJ J J J J

Z

X X

Y Y

Q Z P X P X Q Y Q Z P X P X Q Y

X X X X Y Y

2 1 2, , , , 0J J JZ Z

Model

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