MSBA7021 Prescriptive Analytics
Background
In order to mitigate risk while ensuring a reasonable level of return, investors purchase
a variety of securities and combine these into an investment portfolio
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Securities often change together with some classes of securities, and in the opposite
direction of other classes of securities
A portfolio that contains many positively correlated securities is more risky (and
potentially more rewarding) than one that contains a mix of positively and negatively
correlated securities
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Goals
Most investors have two goals in forming portfolios:
to obtain a large expected return, and
to obtain a small variance (to minimize risk)
At a given expected return, there is one portfolio which has the lowest risk
The risk-return characteristics of a portfolio change in a nonlinear fashion
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Index vs. active funds
Index Fund Active Fund
Goal
Match the performance of a specic
market benchmark or "index" (e.g.
S&P500) as closely as possible
Outperform its benchmark
Strategy
Buys all (or a representative sample) of
the stocks or bonds in the index it's
tracking
Uses the portfolio manager's deep
research and expertise to hand-select
stocks or bonds for the fund
Risk
Aligns directly to the risks involved with
the specic stock or bond market the
fund tracks
Adds the risk that the portfolio
manager may underperform its
benchmark
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Asset
Asset: an investment instrument that can be bought and sold
Suppose we invest into an asset now and sell it some time later, receiving X
0
X
t
total return = R =
rate of return = r =
It is sometimes possible to sell an asset that you do not own through the process of
short selling the asset
Borrow the asset from someone (the lender) who owns it
Sell the borrowed asset to someone else, receiving an amount
At a later date, purchase the asset for and return it to the lender
Short selling is protable if the asset price declines
If , you'll make a prot of
Note: Short selling is quite risky
The potential loss is unlimited: can increase arbitrarily so can the loss
Short selling is forbidden or supplemented by restrictions within certain nancial
institutions
Suppose we invest to form a portfolio of different assets
The amount invested in the th asset is
Dene as the fraction of asset in the portfolio:
If short selling is allowed, then some of the 's can be negative
Date AAPL AXP BA CAT CSCO CVX DIS GE
5087 22/3/2018 165.6652 89.7283 311.6044 142.5569 41.5126 108.4317 99.0218 12.5799 171
5088 23/3/2018 161.8289 88.7860 312.9596 140.0241 40.8861 107.7451 96.9941 12.3160 167
5089 26/3/2018 169.5112 90.9160 320.7300 144.7792 42.4668 110.0053 99.0710 12.1464 172
5090 27/3/2018 165.1648 89.7381 313.0766 142.6442 41.1367 109.3472 97.8012 12.6647 170
5091 28/3/2018 163.3399 90.5136 312.0042 140.8684 40.1536 106.9059 96.9941 12.8908 170
5 rows × 21 columns
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import plotly.express as px
import plotly.io as pio
pio.templates.default = 'plotly_white'
S_melt = S.melt(id_vars = 'Date', var_name = 'Stock', value_name = 'Price')
fig_price = px.line(S_melt, x = 'Date', y = 'Price', color = 'Stock', height = 600)
fig_price.update_layout(xaxis_title = '')
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Computing stock returns
Let denote the price of stock at time
Let denote the rate of return of asset from time to time
stocks = S.columns[1:]
tmp = S[stocks]
ret = (tmp.diff()[1:] /tmp.shift(1)[1:])
ret = pd.concat([S['Date'][1:], ret], axis=1)