Economic Scenario Generator

Economic Scenario

Generator

Ahmed Blanco, Caylee Chunga, Branden Diniz,

Brittany Mowe, Bowei Wei

Advisors:

Jon Abraham

Barry Posterro

Overview

• What is an Economic Scenario Generator?

• Basics

─ Regime Switching (Markov Chains)

─ Inverse Transform Methods

• Calibration

─ Maximum Likelihood Estimation

─ Covariance Matrix

─ Cholesky Decomposition

• Our ESG

─ Differences

─ Results

─ Recommendations

Worcester Polytechnic Institute

What is an Economic Scenario

Generator?

• Model that simulates correlated returns of multiple assets

• Life Insurance

Companies

─ Asset Liability

Management

• Property and

Casualty Insurance

Companies

─ Dynamic Financial

Analysis

• Banks

─ Balance Sheet

Management


Regime Switching (Markov Chains)

• A system of multiple states that switch based on fixed probabilities

• Growing regime and falling regime

• Movement between states is determined by random numbers and an application of inverse transform method

Ending in

Regime 1

Ending in

Regime 2

Transition Matrix

Starting in

Regime 1

.8

Starting in

Regime 2

.65

.2

.35

1

1

2

2

…

Sample Regime Switching

Starting in

Regime

Random

Number

Ending in

Regime

.76998

1

.82837

2

.21792

2

.57101

1

… …






Ending in

Regime 1

Ending in

Regime 2

Transition Matrix

Starting in

Regime 1

.8

Starting in

Regime 2

.65

.2

.35

1

1

2

2

…


Starting in

Regime

Random

Number

Ending in

Regime

.76998 1

.82837

2

.21792

2

.57101

1

… …






Ending in

Regime 1

Ending in

Regime 2

Transition Matrix

Starting in

Regime 1

.8

Starting in

Regime 2

.65

.2

.35

1

1

2

2

…


Starting in

Regime

Random

Number

Ending in

Regime

.76998

1

.82837 2

.21792

2

.57101

1

… …


Simulated Returns

• For our model we need to generate normal random numbers to use as stock returns

• We convert a uniform random number, x, to have a normal distribution.

• We use the Inverse Transform Method.

• 𝑌 = 𝑋 𝑓𝑟𝑜𝑚 𝐸𝑥𝑐𝑒𝑙

∗ 𝜎 + 𝜇


Inverse Transform Methods (Continuous)

Cumulative Distribution Function (CDF) of the

Normal Distribution


Inverse Transform Methods

(Continuous)


Normal Distribution

Begin with a uniform random number on

(0,1) and use the Inverse

Transform method to develop a random number that is normally distributed.



(Continuous)


Normal Distribution

0.80

µ=6 F -1 (0.8)=9

This is the random number in the Normal Distribution resulting from a uniform random number of 0.80



Ending in

Regime 1

Ending in

Regime 2

Starting in

Regime 1

.8

.2

Starting in

Regime

2

2

1

1

…

Starting in

Regime 2

.65

.35

Random

Number for

Regime

.76998

.82837

.91732

.57101

…

Ending in

Regime

2

1

1

2

… 𝜇 𝜎

Regime 1 Regime 2

.05

-.02

.01

.05

Random

Number for

Return

.65868

.27794

.35738

.66318

…

Transformed

Number

.05660

.02139

.02179

.04371

…



Ending in

Regime 1

Ending in

Regime 2

Starting in

Regime 1

.8

.2

Starting in

Regime

2

2

1

1

…

Starting in

Regime 2

.65

.35

Random

Number for

Regime

.76998

.82837

.91732

.57101

…

Ending in

Regime

2

1

1

2

… 𝜇 𝜎


.05

-.02

.01

.05

Random

Number for

Return

.65868

.27794

.35738

.66318

…

Transformed

Number

.05660

.02139

.02179

.04371

…



Ending in

Regime 1

Ending in

Regime 2

Starting in

Regime 1

.8

.2

Starting in

Regime

2

2

1

1

…

Starting in

Regime 2

.65

.35

Random

Number for

Regime

.76998

.82837

.91732

.57101

…

Ending in

Regime

2

1

1

2

… 𝜇 𝜎


.05

-.02

.01

.05

Random

Number for

Return

.65868

.27794

.35738

.66318

…

Transformed

Number

.05660

.02139

.02179

.04371

…


Calibration (Maximum Likelihood

Estimation)

• Definition: A method of estimating the parameters of a model given data.

─ In other words, finding the values of the parameter set with the highest probability of resulting in the observations.



Estimation)

• For regime switching with 2 regimes, there are 6 parameters:

─ 𝜇

1

& 𝜇

2

─ 𝜎

1

& 𝜎

2

─ 𝑝

1,2

- Means for Regime 1 & 2

- Standard deviations for Regime 1 & 2

- Probability of starting in Regime 1 and ending in 2

─ 𝑝

2,1

- Probability of starting in Regime 2 and Ending in 1



Estimation)

• The set up for our MLE:



Estimation)

• Invariant probabilities 𝜋

1

= 𝑝

2,1 𝑝

1,2

+𝑝

2,1 𝜋

2

= 𝑝

1,2 𝑝

1,2

+𝑝

2,1

• Normal pdfs 𝜙

𝑋−𝜇

1 𝜎

1 𝜙

𝑋−𝜇

2 𝜎

2



Estimation)

• We multiply the invariant by the normal… 𝜋

1

∗ 𝜙

𝑋−𝜇

1 𝜎

1

𝜋

2

∗ 𝜙

𝑋−𝜇

2 𝜎

2



Estimation)

• Then add these to get the first pdf value 𝜋

1

∗ 𝜙

𝑋−𝜇

1 𝜎

1

+ 𝜋

2

∗ 𝜙

𝑋−𝜇

2 𝜎

2



Estimation)

• Now we begin the recursion…

Currently in Regime 1 Currently in Regime 2

Previously in R1 Previously in R2 Previously in R1 Previously in R2



Estimation)

• Probability of being in the previous regime

─ Regime 1’s contribution to the pdf

𝑃. 𝑅. 1 𝑝

1,1

∗

𝑃𝐷𝐹

• Regime switching probability



Estimation)

• Multiply by the normal pdf… 𝑝

1,1

∗

𝑃. 𝑅. 1

∗ 𝜙

𝑃𝐷𝐹

𝑋 − 𝜇 𝜎

1

1

= 𝑅𝑒𝑔𝑖𝑚𝑒 1,1 𝑝𝑑𝑓



Estimation)

• Now in Regime 1, Previously in Regime 2 𝑝

2,1

∗

𝑃. 𝑅. 2

𝑃𝐷𝐹

∗ 𝜙

𝑋 − 𝜇 𝜎

1

1




Estimation)


1,2

∗

𝑃. 𝑅. 1

𝑃𝐷𝐹

∗ 𝜙

𝑋 − 𝜇 𝜎

2

2




Estimation)


2,2

∗

𝑃. 𝑅. 2

∗ 𝜙

𝑃𝐷𝐹

𝑋 − 𝜇 𝜎

2

2




Estimation)

• Estimated pdf values



Estimation)

• Natural log of estimated pdf values



Estimation)

Parameters which we solved for using Excel’s built-in “Solver”

Metric we maximized to by solving for the parameters


Where Next?


Covariance Matrix

• Definition: a matrix that depicts the covariance of an array of random variables relative to each other.

𝐶𝑜𝑣 𝑀𝑎𝑡𝑟𝑖𝑥 =

𝑉𝑎𝑟[𝑋

𝐶𝑜𝑣[𝑋

⋮

2

1

] 𝐶𝑜𝑣[𝑋

, 𝑋

1

⋮

1

, 𝑋

] 𝑉𝑎𝑟[𝑋

2

]

2

]

⋯

⋱

𝐶𝑜𝑣[𝑋

1

, 𝑋 𝑛

]

𝐶𝑜𝑣[𝑋

2

, 𝑋 𝑛

⋮

𝐶𝑜𝑣[𝑋 𝑛

, 𝑋

1

] 𝐶𝑜𝑣[𝑋 𝑛

, 𝑋

2

] ⋯ 𝑉𝑎𝑟[𝑋 𝑛

]

]


Cholesky Decomposition

• Theorem: every symmetric positive definite matrix can be decomposed into a product of a unique lower triangular matrix (the Cholesky factor) and its transpose.


Cholesky Example

0.015 0.009

A =

0.009 0.048

Decompose the matrix to get L and L T

L =

0.122

0 and L T =

0.122 0.073

0.073 0.206

0 0.206



0.015 0.009

A =

0.009 0.048


L =

0.122

0 and L T =

0.122 0.073

0.073 0.206

0 0.206

Stock 1 Stock 2

2.307

2.629

0.710

-0.796

1.614

-0.163

⋮ ⋮

-0.265

0.494

0.122 0.073

X

0 0.206

=



0.015 0.009

A =

0.009 0.048


L =

0.122

0 and L T =

0.122 0.073

0.073 0.206

0 0.206

Stock 1 Stock 2

2.307

2.629

0.710

-0.796

1.614

-0.163

⋮ ⋮

-0.265

0.494

0.122 0.073

X

0 0.206

=

Stock 1 Stock 2

0.283

0.712

0.087

-0.112

0.198

0.085

⋮ ⋮

-0.032

0.082



Covariance Matrix

0.015 0.009

0.009 0.048


The ESG

A

1

=

A

2

=

0.015 0.009

0.009 0.048

0.010 0.003

0.003 0.022

L

1

T =

L

2

T =

0.122

0.073

0 0.206

0.100

0.030

0 0.145

Regime Stock 1 Stock 2

⋮

2

1

1

2.307

0.710

1.614

⋮

2.629

-0.796

-0.163

⋮

2 -0.265

0.494

0.122 0.073

0 0.206

X OR =

0.100 0.030

0 0.145


1

1

0.283

0.712

0.087

-0.112

⋮

2 0.161

0.025

⋮ ⋮

2 -0.027

0.064


The ESG

A

1

=

A

2

=

0.015 0.009

0.009 0.048

0.010 0.003

0.003 0.022

L

1

T =

L

2

T =

0.122

0.073

0 0.206

0.100

0.030

0 0.145


⋮

2

1

1

2.307 2.629

0.710

-0.796

1.614

⋮

-0.163

⋮

2 -0.265

0.494

0.122 0.073

0 0.206

X OR =

0.100 0.030

0 0.145


1

1

0.283 0.712

0.087

-0.112

⋮

2 0.161

0.025

⋮ ⋮

2 -0.027

0.064


The ESG

A

1

=

A

2

=

0.015 0.009

0.009 0.048

0.010 0.003

0.003 0.022

L

1

T =

L

2

T =

0.122

0.073

0 0.206

0.100

0.030

0 0.145


⋮

2

1

1

2.307

2.629

0.710

-0.796

1.614 -0.163

⋮ ⋮

2 -0.265

0.494

0.122 0.073

0 0.206

X OR =

0.100 0.030

0 0.145


⋮

2

1

1

0.283

0.087

0.712

-0.112

0.161 0.025

⋮ ⋮

2 -0.027

0.064


Our ESG

• Uses 3 regimes instead of 2

─ 3 rd regime represents an economic crash and occurs rarely

─ µ

3

, σ

3

, third Covariance Matrix

• Uses 10 Exchange Traded Funds (ETFs)

─ ETFs track groups of stocks

• Outputs Daily Returns


Results: Mean & St. Dev. Difference

Regime 1 μ

1

Parameter μ

1

Simulation σ

1

Parameter σ

1

Simulation

SPY 0.00080

0.00079

0.00751

0.00726

VXX -0.00679

-0.00674

0.02517

0.03496

EFA 0.00064

0.00063

0.00899

0.00987

OIL 0.00004

0.00004

0.01466

0.01812

FEZ 0.00057

0.00055

0.01259

0.01379

EEM 0.00028

0.00027

0.01138

0.01149

HYG 0.00015

0.00015

0.00328

0.00409

TLT -0.00005

-0.00006

0.01210

0.00871

IWM -0.00043

-0.00045

0.02431

0.01007

GLD -0.00034

-0.00033

0.01474

0.01019


Results: Covariance Difference

Simulated Covariance Matrix

Regime

1

SPY VXX EFA OIL FEZ EEM HYG TLT IWM GLD

SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001222 -0.000223 -0.000145 -0.000297 -0.000232 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000223 0.000097 0.000071 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000145 0.000071 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000297 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000232 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000006

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000006 0.000010 0.000013 0.000104

Actual Covariance Matrix

Regime

1


SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001224 -0.000224 -0.000146 -0.000298 -0.000233 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000224 0.000097 0.000070 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000146 0.000070 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000298 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000233 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000005

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000005 0.000010 0.000013 0.000104




Regime

1


SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001222 -0.000223 -0.000145 -0.000297 -0.000232 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000223 0.000097 0.000071 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000145 0.000071 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000297 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000232 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000006

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000006 0.000010 0.000013 0.000104


Regime

1


SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001224 -0.000224 -0.000146 -0.000298 -0.000233 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000224 0.000097 0.000070 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000146 0.000070 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000298 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000233 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000005

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000005 0.000010 0.000013 0.000104




Regime

1


SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001222 -0.000223 -0.000145 -0.000297 -0.000232 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000223 0.000097 0.000071 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000145 0.000071 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000297 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000232 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000006

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000006 0.000010 0.000013 0.000104


Regime

1


SPY 0.000053 -0.000191 0.000060 0.000047 0.000080 0.000065 0.000017 -0.000025 0.000064 0.000008

VXX -0.000191 0.001224 -0.000224 -0.000146 -0.000298 -0.000233 -0.000068 0.000092 -0.000241 -0.000021

EFA 0.000060 -0.000224 0.000097 0.000070 0.000125 0.000093 0.000022 -0.000029 0.000073 0.000023

OIL 0.000047 -0.000146 0.000070 0.000328 0.000090 0.000084 0.000020 -0.000035 0.000060 0.000053

FEZ 0.000080 -0.000298 0.000125 0.000090 0.000190 0.000116 0.000028 -0.000042 0.000095 0.000024

EEM 0.000065 -0.000233 0.000093 0.000084 0.000116 0.000132 0.000024 -0.000028 0.000080 0.000030

HYG 0.000017 -0.000068 0.000022 0.000020 0.000028 0.000024 0.000017 -0.000004 0.000021 0.000005

TLT -0.000025 0.000092 -0.000029 -0.000035 -0.000042 -0.000028 -0.000004 0.000076 -0.000029 0.000010

IWM 0.000064 -0.000241 0.000073 0.000060 0.000095 0.000080 0.000021 -0.000029 0.000101 0.000013

GLD 0.000008 -0.000021 0.000023 0.000053 0.000024 0.000030 0.000005 0.000010 0.000013 0.000104


Recommendations

• Create a user-friendly interface

• Alternative platforms

─ Matlab would allow implementation on the WPI supercomputer

• Output results to a .txt file

─ Avoids Excel’s row limitations

• Introduce an automatic results checker

• More experimentation with regime 3

• Improve run time


Thank you for listening!

Questions?

Appendix I – Our 3 rd Regime

• Mean:

─ Twice the mean of Regime 2

• Standard Deviation:

─ .5 times the Std. Dev. of Regime 2

• Covariance:

─ .5 times the Covariances of Regime 2


Appendix II – Mean & St. Dev.

Differences

Results: Mean & St. Dev. Difference With Differences

Regime 1 μ

1

Parameter μ

1

Simulation Differences σ

1

Parameter σ

1

Simulation Differences

SPY 0.00080

0.00079

7.37E-06 0.00751

0.00726

-0.00024

VXX -0.00679

-0.00674

-4.2E-05 0.02517

0.03496

0.009786

EFA 0.00064

0.00063

1.11E-05 0.00899

0.00987

0.000874

OIL 0.00004

0.00004

-6E-06 0.01466

0.01812

0.003458

FEZ 0.00057

0.00055

1.7E-05 0.01259

0.01379

0.001194

EEM 0.00028

0.00027

1.07E-05 0.01138

0.01149

0.000107

HYG 0.00015

0.00015

4.33E-07 0.00328

0.00409

0.000821

TLT -0.00005

-0.00006

4.95E-06 0.01210

0.00871

-0.0034

IWM -0.00043

-0.00045

1.24E-05 0.02431

0.01007

-0.01424

GLD -0.00034

-0.00033

-7.5E-06 0.01474

0.01019

-0.00455


Appendix III – Covariance

Differences

Regime 1

SPY

VXX

EFA

OIL

FEZ

EEM

HYG

TLT

IWM

GLD


0.00000004 0.00000024 0.00000004 0.00000010 0.00000001 0.00000001 0.00000000 0.00000001 0.00000001 0.00000020

0.00000024 0.00000151 0.00000051 0.00000013 0.00000095 0.00000049 0.00000007 0.00000039 0.00000037 0.00000014

0.00000004 0.00000051 0.00000008 0.00000025 0.00000007 0.00000002 0.00000002 0.00000003 0.00000000 0.00000022

0.00000010 0.00000013 0.00000025 0.00000050 0.00000057 0.00000002 0.00000001 0.00000002 0.00000011 0.00000019

0.00000001 0.00000095 0.00000007 0.00000057 0.00000006 0.00000001 0.00000001 0.00000000 0.00000005 0.00000032

0.00000001 0.00000049 0.00000002 0.00000002 0.00000001 0.00000008 0.00000006 0.00000001 0.00000000 0.00000021

0.00000000 0.00000007 0.00000002 0.00000001 0.00000001 0.00000006 0.00000001 0.00000007 0.00000002 0.00000007

0.00000001 0.00000039 0.00000003 0.00000002 0.00000000 0.00000001 0.00000007 0.00000001 0.00000016 0.00000010

0.00000001 0.00000037 0.00000000 0.00000011 0.00000005 0.00000000 0.00000002 0.00000016 0.00000003 0.00000025

0.00000020 0.00000014 0.00000022 0.00000019 0.00000032 0.00000021 0.00000007 0.00000010 0.00000025 0.00000007

Minimum: 1.81E-09

Maximum: 1.5E-06


Appendix IV – ETFs Used

ETF/ETN

Underlying

Index/

Commodity

SPY

S&P 500

Features of the Index

Largest 500

U.S. companies

IWM

Russel 2000

Smallest 2000 companies in the Russel

3000 index of small-cap equities

TLT HYG

Barclays U.S.

20+ Year

Treasury

Bonds

U.S. Treasury

Bonds that will not reach maturity for twenty or more years

Markit iBoxx

USD Liquid

High Yield

High yield corporate bonds for sale in the U.S.

GLD

Gold bullions spot price

Bars of gold with a purity of

99.5% or higher

ETF/ETN EFA VXX OIL FEZ EEM

Underlying

Index/

Commodity

MSCI EAFE

Features of the Index

Large-cap and medium-cap equities

S&P 500 VIX

Short-Term

Futures

S&P GSCI

Crude Oil Total

Return

CBOE Volatility

Index which measures the volatility of

S&P 500 futures

Returns of oil futures contracts with

West Texas

Intermediate

EURO STOXX

50

50 of the largest and most liquid

Eurozone stocks

MSCI Emerging

Markets

Medium-cap and large-cap equities from emerging markets


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• The set up for our MLE:

𝜋

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Results: Mean & St. Dev. Difference

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