Markets and More Portfolio Optimization¶

BUSI 721: Data-Driven Finance I¶

Kerry Back, Rice University¶

Outline¶

  1. How markets work (and why your broker doesn't need to charge commissions)
  2. Stocks, bonds, and gold over a longer horizon
  3. Optimal portfolios without short sales

1. Markets¶

Limit orders versus market orders¶

  • Market order (usual order) is an order to trade at the market price. It will always execute.
  • A limit order is an order to trade at a specified (limit) price or better.
    • E.g., buy at $50 or less
    • E.g., sell at $50 or more
  • A marketable limit order is a limit order that can be executed immediately, because its limit price is available in the market.
  • In general, limit orders may or may not execute.

Limit order books¶

  • Each exchange keeps a book of limit orders
  • Orders to buy are called bids
  • Orders to sell are called offers
  • Incoming market orders are executed against the best available limit order
    • A market buy order executes against the lowest priced offer
    • A market sell order executes against the highest priced bid
  • The lowest price offer and highest price bid are called the best bid and offer or the inside quotes.

The best bid is $180.02 and the best offer is $180.03.

Bid-ask spread¶

  • Offer prices are also called ask prices.
  • The inside quotes can be called the best bid and ask.
  • The difference between the best ask and the best bid is called the bid-ask spread.
  • Some traders post bids and offers to earn the spread rather than to trade.
    • They try to make round trips to keep inventories low.
    • Called market makers or dealer.
    • Nowadays high frequency traders (fast machines and connections and trade via algorithms).

Nasdaq best bid and offer¶

CVX bid and offer

NMS (National Market System)¶

  • Reg NMS requires your broker to execute your order at the exchange (or non-exchange venue) that provides the best price.
  • The national best bid and offer are called the NBBO.
  • Reg NMS requires execution at the NBBO.

U.S. stock exchanges¶

  • NYSE
  • NYSE Mkt
  • NYSE Arca
  • Nasdaq
  • Nasdaq Boston
  • Nasdaq Philadelphia
  • BATS (owned by Chicago Board Options Exchange=CBOE)
  • IEX

Non-exchange trading venues¶

  • Trades can be executed outside of exchanges provided the execution is at the NBBO or better.
  • Institutions like Citadel fill orders to earn the spread.
  • To get orders, they kick back part of the spread to brokers (payment for order flow).

Make or take fees¶

  • Exchanges have make or take fees
    • Limit order = make liquidity
    • Market order = take liquidity
  • Some exchanges pay limit orders and charge market orders
  • Other exchanges pay market orders and charge limit orders
  • In general, a means of attracting order flow.
  • Fees are limited by the SEC to no more than 3/10 of a penny per share.

Broker order routing¶

  • Schwab{target="_blank"}
  • e-Trade{target="_blank"}
  • Interactive Brokers{target="_blank"}

Chairman of the SEC on order routing¶

Gary Gensler on YouTube

2. Stocks, bonds, and gold¶

  • IEF, SPY, and GLD history is not long enough, especially for estimating expected returns
    • IEF returns as illustration
  • Can use stock and bond indices over longer time period
    • Data from Aswath Damodoran (NYU)
  • Stock and bond correlation was > 0 in 20th century, < 0 in 21st century
    • What does the future hold?

IEF returns¶

In [41]:
import yfinance as yf
import seaborn as sns
import matplotlib.pyplot as plt
sns.set_style("whitegrid")

price = yf.download("IEF", start=1990)["Adj Close"]
price.plot()
plt.ylabel("IEF")
plt.show()

[*********************100%%**********************]  1 of 1 completed

20-year Treasury means from Damodoran's data¶

In [42]:
import pandas as pd
df = pd.read_csv(
    'https://www.dropbox.com/s/hgwte6swx57jqcv/nominal_sbb.csv?dl=1',
    index_col="Year" 
)

means = df.Treasuries.rolling(20).mean()
means.plot()   
plt.ylabel("Trailing 20-Year Mean")
plt.show()
In [43]:
df.head()
Out[43]:
S&P 500 TBills Treasuries Corporates
Year
1928 0.438112 0.0308 0.008355 0.032196
1929 -0.082979 0.0316 0.042038 0.030179
1930 -0.251236 0.0455 0.045409 0.005398
1931 -0.438375 0.0231 -0.025589 -0.156808
1932 -0.086424 0.0107 0.087903 0.235896

40-year means¶

In [44]:
means = df.Treasuries.rolling(40).mean()
means.plot()   
plt.ylabel("Trailing 40-Year Mean")
plt.show()

20-Year Stock and Bond Correlations¶

In [45]:
corrs = []
for i in range(20, len(df.index)):
    corr = df.iloc[(i-20):i]["S&P 500"].corr(df.Treasuries)
    corrs.append(corr)

plt.plot(df.index[20:], corrs)
Out[45]:
[<matplotlib.lines.Line2D at 0x1fdf64aa290>]

3. Optimal portfolios without short sales¶

Example¶

In [46]:
import numpy as np 

rf = 0.03
mu = [0.04, 0.10, 0.10]
stdevs = [0.2, 0.2, 0.2]
corrs = [
    [1., 0., 0.8],
    [0., 1., 0.3],
    [0.8, 0.3, 1.] 
]
Sigma = np.diag(stdevs) @ corrs @ np.diag(stdevs)

Define arrays¶

In [47]:
# example target expected return
r = 0.08

P = Sigma
q = np.zeros(3).reshape(3, 1)
A = (mu - rf*np.ones(3)).reshape(1, 3)
b = np.array([r-rf]).reshape(1, 1)

Compute the efficient portfolio (with short sales)¶

In [48]:
from cvxopt import matrix
from cvxopt.solvers import qp

sol = qp(
    P=matrix(P),
    q=matrix(q),
    A=matrix(A),
    b=matrix(b)
)
pd.Series(sol["x"], index=range(1, 4)).round(3)
Out[48]:
1   -0.497
2    0.109
3    0.676
dtype: float64

Compute the efficient portfolio (without short sales)¶

In [49]:
G = -np.identity(3)
h = np.zeros((3, 1))

sol = qp(
    P=matrix(P),
    q=matrix(q),
    G=matrix(G),
    h=matrix(h),
    A=matrix(A),
    b=matrix(b)
)
pd.Series(sol["x"], index=range(1, 4)).round(3)

     pcost       dcost       gap    pres   dres
 0:  7.0157e-03 -7.5505e-01  8e-01  0e+00  2e+00
 1:  7.0089e-03 -1.4634e-03  8e-03  8e-17  3e-02
 2:  6.7723e-03  6.5362e-03  2e-04  1e-17  3e-04
 3:  6.6347e-03  6.6316e-03  3e-06  8e-17  4e-06
 4:  6.6327e-03  6.6326e-03  3e-08  8e-17  4e-08
Optimal solution found.
Out[49]:
1    0.000
2    0.357
3    0.357
dtype: float64