Desheng Dash Wu
University of Toronto, Reykjavik University
[with John R. Birge, Booth School of Business, University of Chicago]
Accepted and to appear at POM
DSJ, 43(1) 2012
0
1
Introduction
Problem, Literature
Background model
Our model
Conceptual model, Math model
3 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 analysis
Conclusion & Further study
2
o
Introduction
o
Background model
o
Our model
o
Case Study
o
Conclusion
3
 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.”
4
 IO: 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
efficiency
5
 A model for gauging merger efficiency of supply chains
different structure
 Supply chain view of banking operations
Link operations to finance
 Apply the model to banking operations with DEA,
considering M&A with multiple metrics
Link OR to IO/Finance
6

Question: banks or subdivisions merged, how
business performance is affected considering such
a banking chain? How to achieve potential gains?
IT facility,staff
Staff, asset
Borrower selection;
Underwriting;
Loan approval;
Primary market
Information flow
Loans
Securitization;
Derivative Trading;
Secondary market
Information flow
Product flow
Sales, risks, availability, stock, customer/investor satisfaction level
Mortgage banking: a view of serial chain
7
o
Introduction
o
Background model
o
Our model
o
Case Study
o
Conclusion
8
Input 2
G
yG>(yA+yB) ?
A: (x1A,
x2A)
D
yD>1/2(yA+yB) ?
B: (x1B, x2B)
yB
yA
Input 1
demonstration of the merger of two firms
9

The efficiency of merger can then be
measured as E ( X , X )  2yy  ( yy )  ( 2yy )  H  S
m
G
A


y
H D
y

S
yG
2 yD
D
G
B
(1)
D
denotes the harmony effect.
represents the scale effect.
potential gains from the merger of the two firms
positive if E m > 1.
10
What
1. 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)
How
1. Define DMU, input/output variables
2. Define the efficiency frontier.
3. A numerical weight coefficient is given to each
firm, computing its relative efficiency.
11
o
Introduction
o
Background model
o
Our model
o
Case Study
o
Conclusion
12
13


N : number of DMUs
l,i : 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)
max( P1 f1I  Q1h1I )  ( P2 f 2I  Q2 h2I )
N
s.t.
N
N
1,i x1,i  x1, I  2,i x2,i  x2, I  2,i y1,i  y1, I  ,

i 1
i 1
i 1
N
N
 f   1,i y1,i  0, f   2,i y2,i  0,
I
1
I
2
i 1
N
i 1
N
h   1,i z1,i  0,h   2,i z2,i  0,
I
1
N
N
i 1
i 1
i 1
I
2
i 1
 1,i  1, 2,i  1,1,i   2,i  ; f1I , h1I , f 2I , h2I  0.
Here, f1I ,h1I , f2I ,h2I are the decision variables
14

Step 1: solve the DEA model for each chain and subchain, and construct the efficient input-output
combination (x , y , z ) for each supply chain.

l,I


l,I

l,I
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
15

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
( xlTotal , ylTotal , zlTotal )

Step 6: Compute the sub-chain efficiency, merger
efficiency for the whole chain, the harmony and scale
components.
16

Theorem 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
17

Leader-follower relations
E
X1
X D1
Leader
X2
Y
Follower
Z1
X D2
Z2
The framework with limited resource E
Direct input
X D1
X D2
Shared input
X1
X2
Intermediate output/input
Direct output
Y
Z1
Z2
18
o
Constrained resource, leader-follower
relation
IT Budget
Personal
Others
Profit
Primary market
business-leader
Loans
Secondary market
business-follower Loan recovery
19

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:
min F ( x, y )  p1T x  q1T y
x
s.t. A1 x  B1 y  b1
min f ( x, y )  p2T x  q2T y
y
s.t. A2 x  B2 y  b2
x, y  0
20

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.


21


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.
22
Efficiency ratio 

α -Strategy:
Efficiency ratio
Efficiency ratio
130%
125%
120%
115%
110%
105%
100%
95%
90%
85%
80%
140%
135%
130%
125%
120%
115%
110%
105%
100%
95%
90%
DMU1
DMU2
DMU3
DMU4
DMU5
DMU6
DMU7
DMU8
α
0 0.010.020.030.040.050.060.070.080.09 0.1
The efficiency ratio of the
Leader under α strategy
adjusted optimized profit
observed profit
DMU1
DMU2
DMU3
DMU4
DMU5
DMU6
DMU7
DMU8
α
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
The efficiency ratio of the
Follower under α strategy
23
o
Introduction
o
Background model
o
Our model
o
Case Study
o
Conclusion
24
▪ Data from 36 branches (DMUs) for 6 variables
▪ Mortgage banking chain input-output framework
25

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
26
1.2
1
0.8
sub-chain CRS
chain CRS
0.6
sub-chain VRS
chain VRS
0.4
0.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
27
the 1st sub-chain (>100%)
under CRS and VRS.
full-chain (>100%) under CRS
and VRS.
28
merger efficiency distribution.
Harmony efficiency
Scale efficiency
29
VRS merger efficiency
Harmony efficiency
Scale efficiency
30
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
31
The top 10 promising mergers under CRS
Merger
4,5
3,4
5,8
4,7
3,8
1,4
1,8
7,8
5,6
4,6
leader
1.125
1.098
1.09
1.081
1.06
1.059
1.057
1.047
1.042
1.035
follower
0.99
0.99
1.001
1.001
1.001
0.9
0.99
0.99
0.99
1.001
system
1.092
1.072
1.067
1.059
1.045
1.042
1.041
1.035
1.031
1.027
32

Coordinated effective merger
Merger efficiency scores of the Leader, the Follower and the
whole system are all greater than 1.
The promising coordinated mergers under CRS
Merger
5,8
4,7
3,8
4,6
6,8
2,8
leader follower system
1.09
1.001
1.067
1.081 1.001
1.059
1.06
1.001
1.045
1.035 1.001
1.027
1.023 1.001
1.018
1.014 1.001
1.011
33
o
Introduction
o
Background model
o
Our model
o
Case Study
o
Conclusion
34
 3 things
 Creation of chain merger DEA model
 Effects captured and decomposed at both chain
and sub-chain levels
 a case study in banking intra-firm division merger
operations
 Future work
 Assumptions to be validated
 Breakup of firms
 Comparison with other methods, e.g., game
models.
35
Thanks!
Questions?
36

Model
( P1)
(Q1 Z 1J  Q 2 YJ1 )  ( P1 X 1J  P 2 X JD1 )
T
max
D1 1
X 1J , X J ,YJ , Z 1J , 
T
T
n
s.t
T
n
X  X   X  j   X 2j  j ,
1
J
2
J
j 1
1
j
j 1
n
X
D1
J
  X Dj 1 j ,
j 1
n
Z   Z 1j  j ,
1
J
j 1
n
Y   Yj j ,
1
J
j 1
X 1J  X J2  E (const.),
X JD1  X JD1 ,
(Q1 Z 1J  Q 2 YJ1 )  ( P1 X 1J  P 2 X JD1 )  (Q1 Z 1J  Q 2 YJ1 )  ( P1 X 1J  P 2 X JD1 )
T
T
T
T
T
T
T
T
37

Model
( P 2)
Q 3 Z J2  ( P1 X J2  P 3 X JD 2  Q 2 YJ2 )
T
max
D2
2
X J2 , X J ,YJ , Z J2 ,
T
T
T
n
s.t
X
D2
J
  X Dj 2 j ,
j 1
n
Y   Y j j ,
2
J
j 1
n
Z   Z 2j  j ,
2
J
j 1
X JD 2  X JD 2 ,
YJ2  YJ1 ,
Q3 Z J2  ( P1 X J2  P 3 X JD 2  Q 2 YJ2 )  Q3 Z J2  ( P1 X J2  P3 X JD 2  Q 2 YJ2 )
T
T
T
T
T
T
T
T
X 1J , X J2 , X JD1 , X JD 2 , YJ1 , YJ2 , Z 1J , Z J2 ,  ,   0
38