Post-Modern Portfolio Theory - Elizabethtown College

Elizabethtown College Elizabethtown College

JayScholar JayScholar

Summer Scholarship, Creative Arts and Research Projects (SCARP) Programs and Events

Summer 2020

Post-Modern Portfolio Theory Post-Modern Portfolio Theory

Cassandra DeBacco Elizabethtown College, [email protected]

Follow this and additional works at: https://jayscholar.etown.edu/scarp

Part of the Mathematics Commons

Recommended Citation Recommended Citation DeBacco, Cassandra, "Post-Modern Portfolio Theory" (2020). Summer Scholarship, Creative Arts and Research Projects (SCARP). 8. https://jayscholar.etown.edu/scarp/8

This Presentation is brought to you for free and open access by the Programs and Events at JayScholar. It has been accepted for inclusion in Summer Scholarship, Creative Arts and Research Projects (SCARP) by an authorized administrator of JayScholar. For more information, please contact [email protected].

https://jayscholar.etown.edu/

https://jayscholar.etown.edu/scarp

https://jayscholar.etown.edu/scarp

https://jayscholar.etown.edu/programs-events

https://jayscholar.etown.edu/scarp?utm_source=jayscholar.etown.edu%2Fscarp%2F8&utm_medium=PDF&utm_campaign=PDFCoverPages

http://network.bepress.com/hgg/discipline/174?utm_source=jayscholar.etown.edu%2Fscarp%2F8&utm_medium=PDF&utm_campaign=PDFCoverPages

https://jayscholar.etown.edu/scarp/8?utm_source=jayscholar.etown.edu%2Fscarp%2F8&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

MPT

Modified MPT

PMPTPost-Modern Portfolio Theory

Cassy DeBacco

July 27, 2020

Post-Modern Portfolio Theory 1 / 38

MPT

Modified MPT

PMPT

How Should I Invest My Money?

AAPL

BA

DISGE

KO

MSFT

0.1 0.2 0.3 0.4volatility

0.10.20.30.40.50.60.7

mean return


MPT

Modified MPT

PMPT

Modern Portfolio Theory

The mean and variance for portfolio

P = w1X1 + . . .+ wnXn

is µP = µTw and �2P = wT⌃w.

Modern Portfolio Theory (MPT) solves the quadratic program

Minimize �2P

such that µP � ↵

and 1Tw = 1.


MPT

Modified MPT

PMPT

Solution with Shorting Allowed

If short sales are allowed, then the optimal weights for a

minimum variance portfolio are

w =1

1T⌃�11⌃�11.

The corresponding mean and variance are

µP =µT⌃�11

1T⌃�11and �

2P =

1

1T⌃�11.


MPT

Modified MPT

PMPT

Minimum Variance Portfolio


MPT

Modified MPT

PMPT

E�cient Frontier


MPT

Modified MPT

PMPT

E�cient Frontier


MPT

Modified MPT

PMPT

E�cient Frontier


MPT

Modified MPT

PMPT

E�cient Frontier


MPT

Modified MPT

PMPT

E�cient Frontier


MPT

Modified MPT

PMPT

E�cient Frontier with a Risk-Free Asset


MPT

Modified MPT

PMPT



MPT

Modified MPT

PMPT



MPT

Modified MPT

PMPT



MPT

Modified MPT

PMPT



MPT

Modified MPT

PMPT



MPT

Modified MPT

PMPT

Some Shortcomings of MPT

MPT can produce crazy portfolios

Variance as a measure of risk is debatable

Proceeds from short sales cannot be reinvested in practice


MPT

Modified MPT

PMPT

MPT Can Produce Crazy Portfolios

Dow 30 returns for 2017


MPT

Modified MPT

PMPT

Is Smaller Variance Really What We Want?

BA, CVX, and Portfolio Normal Distribution


MPT

Modified MPT

PMPT

Proceeds from the Short Sale are Reinvested

Based on Data for BA and CVX in 2018

119% BA and -19% CVX


MPT

Modified MPT

PMPT

Modified MPT

For n assets, modify MPT to minimize �2P such that

µTw + 1.5rnX

i=1

R(�wi)

| {z }µP

� ↵

nX

i=1

[wi] = 1,

where

[x] =

8<

:x x � 0

�0.5x x < 0and R(x) =

8<

:x x � 0

0 x < 0.


MPT

Modified MPT

PMPT

Example Illustrating Constraints

Principal: $1000

Spot Prices: Asset 1: $100 Asset 2: $100

Long $500 in asset 1 (w1 = 0.5) and short $1000 of asset 2

(w2 = �1).

w1 + 0.5(�w2) = 0.5 + 0.5(+1) = 1.

Margin: $0|{z}deposit

for asset 1

+ $500|{z}deposit

for asset 2

+ $1000| {z }proceeds from

short sale

= $1500

1.5 [R(�w1) +R(�w2)] = 1.5 [R(�0.5) +R(1)] = 1.5


MPT

Modified MPT

PMPT

Example

Weights for 2019 Portfolio with and without the new constraint


MPT

Modified MPT

PMPT

Compare with MPT (2 Assets)

Based on Data for BA and CVX in 2018.


MPT

Modified MPT

PMPT

Compare with MPT (2 Assets)



MPT

Modified MPT

PMPT

Solution

Solutions must be found numerically, but the existence of many

similar local minima can make the global minimum di�cult to

find.


MPT

Modified MPT

PMPT

Probability Density Function for Returns



MPT

Modified MPT

PMPT

Probability Density Function for Returns



MPT

Modified MPT

PMPT

Value at Risk



MPT

Modified MPT

PMPT

Value at Risk

If the asset returns are normally distributed, then

Value at Risk = P (portfolio return < 0)

= P

✓return� µp

�p<

�µp

�p

◆

= P

✓Z < �µP

�P

◆

= �

✓�µP�P

◆,

where � is the CDF for the standard normal distribution.

Note that minimizing the value at risk is equivalent to

maximizingµP�P

or minimizing�PµP

, the coe�cient of variation

for the portfolio.


MPT

Modified MPT

PMPT

Post-Modern Portfolio Theory (PMPT)

Minimize�PµP

=

pwT⌃w

µTw + 1.5rPn

i=1R(�wi)such that

µP � ↵ and

nX

i=1

[wi] = 1, where

[x] =

8<

:x x � 0

�0.5x x < 0and R(x) =

8<

:x x � 0

0 x < 0.


MPT

Modified MPT

PMPT

Optimization Problem for PMPT



MPT

Modified MPT

PMPT

Almost Closed-Form Solution

w = ⌃�1 [µ� 1.5rH(�w)] t, t > 0,

where

H(x) =

8<

:1 x � 0

0 x < 0

is the Heaviside step function and t is chosen so thatnX

i=1

[wi] = 1.

If r > 0, then we can use fixed point iteration to quickly find

the solution for any r using

wk = ⌃�1 [µ� 1.5rH(�wk�1)] t, w0 = ⌃�1µt0.


MPT

Modified MPT

PMPT

MPT vs PMPT

MPT

PMPT


MPT

Modified MPT

PMPT

Questions?


MPT

Modified MPT

PMPT

PMPT Weights

Note that the weights are piecewise linear in ↵ and, except for

GE, they achieve exact values of zero for larger values of ↵.


MPT

Modified MPT

PMPT

r = 0.02

It looks like the contours are still lines even with r > 0.

Perhaps we can get a closed form solution here too?

Setting r = 0.02 doesn’t seem to do much to the portfolio in

this case.


MPT

Modified MPT

PMPT

Back Testing

There may not be room for this in your presentation, but it’s

the kind of thing that a person from the audience may ask, so

it might be good to prepare a slide on it.

Pictures comparing the results for all three models would be

good to have.


Documents

Post-Modern Portfolio Theory - Elizabethtown College