#!/usr/bin/env python
# coding: utf-8

# ** The following example is based on Karl Borch (1969, J. Economic Studies), where mean variance model fails to select efficient portfolios.**

# - Consider the investment universe with 3 assets,
# - with mean return vector:
# 
# $$ \mu = \left[1,4,9\right]^T $$
# 
# - and the covariance matrix:
# 
# $$\Sigma=\left[\begin{array}{ccc}
# 1 & 0 & 0\\
# 0 & 4 & 0\\
# 0 & 0 & 9
# \end{array}\right]$$

# - Let's consider long position only
# - Efficient frontier can be obtained by solving the following quadratic programming problem:
# 
# \begin{align*}
# \underset{w}{\mathrm{argmin}}\quad & w^{T}\Sigma w\\
# \mathrm{subject\ to}\quad & \left[\begin{array}{ccc}
# 1 & 1 & 1\\
# 1 & 4 & 9
# \end{array}\right]w=\left[\begin{array}{c}
# 1\\
# a
# \end{array}\right]\\
#  & w_{i}\ge0,\quad i=1,2,3
# \end{align*}
# with $a$ varying from $1$ to $9$

# **Let's use python package cvxopt to obtain the efficient frontier**

# In[1]:


from cvxopt import solvers, matrix, spdiag
from numpy import linspace, random
solvers.options['show_progress'] = False
solvers.options['abstol'] = 1e-10
solvers.options['reltol'] = 1e-9
solvers.options['feastol'] = 1e-10
P = spdiag([1.0, 4.0, 9.0])
q = matrix([0.0, 0.0, 0.0])
G = -1.0 * spdiag([1.0, 1.0, 1.0])
h = matrix([0.0, 0.0, 0.0])
A = matrix([1.0, 1.0, 1.0, 4.0, 1.0, 9.0], (2, 3))
Rs_ef, Vs_ef, Rs, Vs = linspace(1.0, 9.0, 100), [], [], []
for R in Rs_ef:
    b = matrix([1.0, R])
    sol = solvers.qp(2.0 * P, q, G=G, h=h, A=A, b=b)
    Vs_ef.append(sol['primal objective'])
    ws = random.uniform(0.0, 1.0, 3)
    ws = ws / sum(ws)
    Rs.append(ws.dot([1.0, 4.0, 9.0]))
    Vs.append((ws*ws).dot([1.0, 4.0, 9.0]))


# **Plot the efficient frontier**

# In[2]:


get_ipython().run_line_magic('matplotlib', 'inline')
get_ipython().run_line_magic('config', "InlineBackend.figure_format = 'svg'")

import matplotlib.pyplot as plt
plt.plot(Vs_ef, Rs_ef)
plt.xlabel('Variance')
plt.ylabel('Mean')
plt.scatter(Vs, Rs, s=0.2, c='k')
#fig = plt.scatter(9.0, 9.0, s=10, c='r', zorder = 3)
plt.scatter(9.0, 9.0, s=10, c='r', zorder = 3)


# **However, let's take a look at one possible return distribution with the required moments**
# 
# | $P$ | $0.5$ | $0.5$ |
# | :------: | :------: | :------: |
# | $X_1$ | $0$ | $2$ |
# | $X_2$ | $2$ | $6$ |
# | $X_3$ | $6$ | $12$ |
# 
# - You don't need any math knowledge to get the obvious best portfolio, all in with $X_3$
# - How come we got a continuum of other choices equally attractive as this one?

# ** What happened here? **
# 
# Modern Portfolio Theory was introduced by Harry Markowitz in 1952, lead to Nobel price in economics. It is the first mean variance model. 
# 
# In this model, portfolio selection is only based the first two moments of the random return. This is also true for some other more sophisticated models (Black–Litterman, etc).
# 
# Here the variance is chosen as risk measure, which choose non-efficient portfolios when the random return is not Gaussian. 
# 
# Is it the return normally distributed in the real world?
# 
# We know it is not (skewed, fat tail, you name it). 
# 

# ** Can PyMCEF handle this situation? **
# - PyMCEF use risk measures that will produce efficient portfolios according to stochastic dominance theory. (Hadar and Russell 1969, Hanoch and Levy 1969 and Rothschild and Stiglitz 1970) 
# 
# ** Two risk measures are used in PyMCEF **
# - Absolute semi-deviation (Ogryczak and Ruszczyński 1999)
# - Fixed-target under-performance (Fishburn 1977)

# ** Let's use PyMCEF to solve this problem **

# In[3]:


from pymcef import SimpleEF, RiskMeasure
import numpy as np
# notice that each row corresponds of all realizations of one asset return

returns = np.array([0,2,6,2,6,12]).reshape((3, 2))
labels = ['Asset 1', 'Asset 2', 'Asset 3']
sol = SimpleEF(training_set = returns, \
               risk_measure = RiskMeasure.FixTargetUnderPerformance, 
               target_return = 0,\
               asset_names = labels)


# The efficient frontier only contains one portfolio.

# In[4]:


for prt in sol.frontier_in_sample:
    print(prt['weight'])


# ## Next example:
# 
# [2. Black Scholes revisit, delta hedging as an efficient portfolio](http://nbviewer.jupyter.org/github/hzzyyy/pymcef/blob/master/examples/2.%20Black%20Scholes%20revisit%2C%20delta%20hedging%20as%20an%20efficient%20portfolio.ipynb)