post-modern portfolio theory - elizabethtown college
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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]
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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
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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
Post-Modern Portfolio Theory 2 / 38
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.
Post-Modern Portfolio Theory 3 / 38
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.
Post-Modern Portfolio Theory 4 / 38
MPT
Modified MPT
PMPT
Minimum Variance Portfolio
Post-Modern Portfolio Theory 5 / 38
MPT
Modified MPT
PMPT
E�cient Frontier
Post-Modern Portfolio Theory 6 / 38
MPT
Modified MPT
PMPT
E�cient Frontier
Post-Modern Portfolio Theory 7 / 38
MPT
Modified MPT
PMPT
E�cient Frontier
Post-Modern Portfolio Theory 8 / 38
MPT
Modified MPT
PMPT
E�cient Frontier
Post-Modern Portfolio Theory 9 / 38
MPT
Modified MPT
PMPT
E�cient Frontier
Post-Modern Portfolio Theory 10 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 11 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 12 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 13 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 14 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 15 / 38
MPT
Modified MPT
PMPT
E�cient Frontier with a Risk-Free Asset
Post-Modern Portfolio Theory 16 / 38
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
Post-Modern Portfolio Theory 17 / 38
MPT
Modified MPT
PMPT
MPT Can Produce Crazy Portfolios
Dow 30 returns for 2017
Post-Modern Portfolio Theory 18 / 38
MPT
Modified MPT
PMPT
Is Smaller Variance Really What We Want?
BA, CVX, and Portfolio Normal Distribution
Post-Modern Portfolio Theory 19 / 38
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
Post-Modern Portfolio Theory 20 / 38
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.
Post-Modern Portfolio Theory 21 / 38
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
Post-Modern Portfolio Theory 22 / 38
MPT
Modified MPT
PMPT
Example
Weights for 2019 Portfolio with and without the new constraint
Post-Modern Portfolio Theory 23 / 38
MPT
Modified MPT
PMPT
Compare with MPT (2 Assets)
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 24 / 38
MPT
Modified MPT
PMPT
Compare with MPT (2 Assets)
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 25 / 38
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.
Post-Modern Portfolio Theory 26 / 38
MPT
Modified MPT
PMPT
Probability Density Function for Returns
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 27 / 38
MPT
Modified MPT
PMPT
Probability Density Function for Returns
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 28 / 38
MPT
Modified MPT
PMPT
Value at Risk
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 29 / 38
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.
Post-Modern Portfolio Theory 30 / 38
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.
Post-Modern Portfolio Theory 31 / 38
MPT
Modified MPT
PMPT
Optimization Problem for PMPT
Based on Data for BA and CVX in 2018.
Post-Modern Portfolio Theory 32 / 38
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.
Post-Modern Portfolio Theory 33 / 38
MPT
Modified MPT
PMPT
MPT vs PMPT
MPT
PMPT
Post-Modern Portfolio Theory 34 / 38
MPT
Modified MPT
PMPT
Questions?
Post-Modern Portfolio Theory 35 / 38
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 ↵.
Post-Modern Portfolio Theory 36 / 38
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.
Post-Modern Portfolio Theory 37 / 38
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.
Post-Modern Portfolio Theory 38 / 38