Kapitalmarktgleichgewichtsmodell¶
CAPM - Capital Asset Pricing Model¶
Entwickelt in den 1960er Jahren von William F. Sharpe, John Lintner und Jan Mossin.
Auf der Portefeuilletheorie (Portfolio Selection) basierendes Modell des Kapitalmarktes
- zur Erklärung von Wertpapierrenditen und
- zur Ableitung von Handlungsempfehlungen;
- theoretische Alternative u.a. zur Arbitrage Pricing Theory (APT)
CAPM - introduction¶
Based on Markowitz' modern portfolio theory
aka. mean-variance analysis
Harry Markowitz, 1927
Nobel Memorial Prize in Economic Sciences, 1990
Problemstellung Portfoliooptimierung¶
$$\begin{align} \text{maximiere}_{x\in\mathbb R^J}\ & \mathbb E(x^\top \xi) \\ \text{wobei }\ & \mathsf{var}(x^\top\xi)\le q & \text{risk constraint}\\ & x^\top \mathbb 1 = 1\,€ & \text{budget constraint} \\ & (x \ge 0) & \text{ no shortselling constraint} \end{align}$$Dabei ist:
- $\xi_j=e_j^\top \xi $ return des assets $j$ ($j=1,\dots,J$),
- $x=(x_1,\dots,x_J)$ der Allokationsvektor,
- $q>0$ eine obere Schranke für das Risiko, ein Modellparameter.
- $\mathbb 1=(1,\dots,1)$ der Einsvektor.
Problemstellung von Markowitz¶
$$\begin{align} \text{minimiere}_{x\in\mathbb R^J}\ & \mathsf{var}(x^\top\xi) & \text{ minimiere Risiko}\\ \text{wobei }\ & \mathbb E(x^\top \xi)\ge \mu & \text{ untere Schranke für den return}\\ & x^\top \mathbb 1 = 1\,€ & \text{budget constraint}\\ &(x \ge 0) & \text{ no shortselling constraint} \end{align}$$Dabei ist:
- $\xi$ Vekter aller returns; $\xi_j:=e_j^\top \xi $ return des assets $j$ ($j=1,\dots,J$),
- $x=(x_1,\dots,x_J)$ der Allokationsvektor,
- $\mu\in\mathbb R$ ein Parameter für den Return des Modelles.
- $\mathbb 1=(1,\dots,1)$ der Einsvektor.
In [1]:
import quandl
import pandas as pd # time series
import datetime
import numpy as np
import matplotlib.pyplot as plt # Import matplotlib
#quandl.ApiConfig.api_key = "..." - this is the personal Quandl API key
quandlList= [["Microsoft", "Close", "EOD/MSFT"],
["SAP", "Close", "FSE/SAP_X"],
["gold", "Value", "WGC/GOLD_DAILY_EUR"],
["oil", "Value", "EIA/PET_RWTC_D"],
["E.ON", "Close", "FSE/EON_X"],
["deutsche Bank", "Close", "FSE/DBK_X"],
["Allianz", "Close", "FSE/ALV_X"],
["Munich Re", "Close", "FSE/MUV2_X"],
["€/$", "Value", "ECB/EURUSD"]
]
# We look at stock prices over the past year
start = datetime.datetime(2018,1,2)
#start = datetime.datetime(2018,1,2)
end = datetime.date.today()
stocks = pd.DataFrame()
for s in quandlList: # this call includes the personal Quandl API key
tmp= quandl.get(s[2], start_date=start, end_date=end, authtoken="...")
stocks= pd.concat([stocks, pd.DataFrame({s[0]: tmp[s[1]]})], axis=1, sort=True)
stocks['Microsoft']= stocks['Microsoft']/ stocks['€/$']
stocks['€/$']=1/stocks['€/$']
stocks # stocks.head(); stocks
Out[1]:
Visualize stock data¶
In [2]:
import plotly #import plotly.plotly as py
import plotly.graph_objs as go
# Must enable in order to use plotly off-line (vs. in the cloud... hate cloud)
plotly.offline.init_notebook_mode()
stock_return = stocks.apply(lambda x: (x / x[0] ))
data = [go.Scatter( x=stock_return.index, y=stock_return[x[0]], name= x[0]) for x in quandlList]
fig = go.Figure(data=data, layout = go.Layout(title=go.layout.Title(text='assets, S')))
plotly.offline.iplot(fig, filename='stock data')