ORF 307: Lecture 3
Linear Programming: Chapter 13, Section 1
Portfolio Optimization
Robert Vanderbei
February 12, 2019
Slides last edited on February 12, 2019
https://vanderbei.princeton.edu
Portfolio Optimization: Markowitz Shares the 1990
Nobel Prize Press Release - The Sveriges Riksbank (Bank of Sweden) Prize in Economic Sciences
in Memory of Alfred Nobel
KUNGL. VETENSKAPSAKADEMIEN
THE ROYAL SWEDISH ACADEMY OF SCIENCES
16 October 1990
THIS YEAR’S LAUREATES ARE PIONEERS IN THE THEORY OF FINANCIAL ECONOMICS
AND CORPORATE FINANCE
The Royal Swedish Academy of Sciences has decided to award the 1990 Alfred Nobel Memorial Prize
in Economic Sciences with one third each, to
Professor Harry Markowitz, City University of New York, USA,
Professor Merton Miller, University of Chicago, USA,
Professor William Sharpe, Stanford University, USA,
for their pioneering work in the theory of financial economics.
Harry Markowitz is awarded the Prize for having developed the theory of portfolio choice;
William Sharpe, for his contributions to the theory of price formation for financial assets, the so-called,
Capital Asset Pricing Model (CAPM); and
Merton Miller, for his fundamental contributions to the theory of corporate finance.
Summary
Financial markets serve a key purpose in a modern market economy by allocating productive resources
among various areas of production. It is to a large extent through financial markets that saving in
different sectors of the economy is transferred to firms for investments in buildings and machines.
Financial markets also reflect firms’ expected prospects and risks, which implies that risks can be spread
and that savers and investors can acquire valuable information for their investment decisions.
The first pioneering contribution in the field of financial economics was made in the 1950s by Harry
Markowitz who developed a theory for households’ and firms’ allocation of financial assets under
uncertainty, the so-called theory of portfolio choice. This theory analyzes how wealth can be optimally
invested in assets which differ in regard to their expected return and risk, and thereby also how risks can
be reduced.
Copyright© 1998 The Nobel Foundation 1
Historical Data—Some ETF Prices
Notation: Sj (t) = share price for investment j at time t.
2
Return Data: Rj (t) = Sj (t)/Sj (t − 1)
Important observation: volatility is easy to see, mean return is lost in the noise. 3
Risk vs. Reward
Reward: Estimated using historical means:
T
1X
rewardj = Rj (t).
T t=1
Risk: Markowitz defined risk as the variability of the returns as measured by the historical
variances:
T
1X 2
riskj = Rj (t) − rewardj .
T t=1
However, to get a linear programming problem (and for other reasons) we use the
sum of the absolute values instead of the sum of the squares:
T
1X
riskj = Rj (t) − rewardj .
T t=1
4
Why Make a Portfolio? ... Hedging
Investment A: Up 20%, down 10%, equally likely—a risky asset.
Investment B: Up 20%, down 10%, equally likely—another risky asset.
Correlation: Up-years for A are down-years for B and vice versa.
Portfolio: Half in A, half in B: up 5% every year! No risk!
5
Explain
Explain the 5% every year claim.
6
Return Data: 50 days around 01/01/2014
1.03
1.02
1.01
1
Returns
0.99
XLU
0.98 XLB
XLI
XLV
XLF
XLE
MDY
0.97 XLK
XLY
XLP
QQQ
S&P500
0.96
2013.96 2013.98 2014 2014.02 2014.04 2014.06 2014.08 2014.1 2014.12 2014.14
Date
Note: Not much negative correlation in price fluctuations. An up-day is an up-day and a
down-day is a down-day. 7
Portfolios
Fractions: xj = fraction of portfolio to invest in j
X
Portfolio’s Historical Returns: Rx(t) = xj Rj (t)
j
T T X
1X 1X
Portfolio’s Reward: reward(x) = Rx(t) = xj Rj (t)
T t=1 T t=1 j
T
X 1X X
= xj Rj (t) = xj rewardj
j
T t=1 j
8
What’s a Good Formula for the Portfolio’s Risk?
9
Portfolio’s Risk:
T 2
1X
risk(x) = Rx(t) − reward(x)
T t=1
2
T T
1 X X 1 XX
= xj Rj (t) − xj Rj (s)
T t=1 j
T s=1 j
2
T T
1X X 1X
= xj Rj (t) − Rj (s)
T T
t=1 j s=1
2
T
1 X X
= xj (Rj (t) − rewardj )
T t=1 j
10
A Markowitz-Type Model
Decision Variables: the fractions xj .
Objective: maximize return, minimize risk.
Fundamental Lesson: can’t simultaneously optimize two objectives.
Compromise: set a lower bound µ for reward and minimize risk subject to this bound con-
straint:
• Parameter µ is called reward happiness parameter.
• Small value for µ puts emphasis on risk minimization.
• Large value for µ puts emphasis on reward maximization.
Constraints:
T X
1X
xj Rj (t) ≥ µ
T t=1 j
X
xj = 1
j
xj ≥ 0 for all j
11
Optimization Problem
2
T
1 X X
minimize xj (Rj (t) − rewardj )
T t=1 j
T X
1 X
subject to xj Rj (t) ≥ µ
T t=1 j X
xj = 1
j
xj ≥ 0 for all j
12
AMPL: Model
reset;
set Assets; # asset categories
set Dates; # dates
param T := card(Dates);
param mu;
param R {Dates,Assets};
param mean {j in Assets} := ( sum{t in Dates} R[t,j] )/T;
param Rdev {t in Dates, j in Assets} := R[t,j] - mean[j];
param variance {j in Assets} := ( sum{t in Dates} Rdev[t,j]^2 )/T;
var x{Assets} >= 0;
minimize risk: sum{t in Dates} (sum{j in Assets} Rdev[t,j]*x[j])^2 / T;
s.t. reward_bound: sum{j in Assets} mean[j]*x[j] >= mu;
s.t. tot_mass: sum{j in Assets} x[j] = 1;
13
AMPL: Data
data;
set Assets := mdy xlb xli xlu spy qqq xle xlk xlv xlf xlp xly;
set Dates := include 'dates';
param R: mdy xlb xli xlu spy qqq xle xlk xlv xlf xlp xly :=
include 'returns.data' ;
printf {j in Assets}: "%10.7f %10.5f \n",
mean[j]^(12), sum{t in Dates} (Rdev[t,j])^2/T > "assets";
14
AMPL: Solve, and Print
set assets_min_var ordered := {j in Assets: variance[j] == min {jj in Assets} variance[jj]};
param maxmean := max {j in Assets} mean[j];
param minmean := mean[first(assets_min_var)];
display mean, variance;
display minmean, maxmean;
printf {j in Assets}: " %5s ", j > "portfolio_minrisk";
printf " | reward risk \n" > "portfolio_minrisk";
for {k in 0..20} {
display k;
let mu := (k/20)*minmean + (1-k/20)*maxmean;
solve;
printf {j in Assets}: "%7.4f ", x[j] > "portfolio_minrisk";
printf " | %7.4f %7.4f \n",
(sum{j in Assets} mean[j]*x[j])^(12),
sum{t in Dates} (sum{j in Assets} Rdev[t,j]*x[j])^2 / T
> "portfolio_minrisk";
}
15
Efficient Frontier
Varying risk bound µ produces the so-called efficient frontier.
Portfolios on the efficient frontier are reasonable.
Portfolios not on the efficient frontier can be strictly improved.
XLU XLB XLI XLV XLF XLE MDY XLK XLY XLP QQQ SPY Risk Reward
1.00000 0.00715 1.00063
0.91073 0.08927 0.00705 1.00063
0.80327 0.19673 0.00696 1.00063
0.64003 0.35997 0.00686 1.00063
0.52089 0.03862 0.44049 0.00676 1.00062
0.50041 0.01272 0.06919 0.41768 0.00667 1.00062
0.48484 0.04132 0.07129 0.40254 0.00657 1.00061
0.46483 0.06857 0.07658 0.39002 0.00647 1.00060
0.44030 0.09633 0.08232 0.38105 0.00638 1.00059
0.42825 0.12917 0.08171 0.36086 0.00628 1.00059
0.39737 0.16114 0.08506 0.35643 0.00619 1.00058
0.36890 0.19318 0.09133 0.34659 0.00609 1.00057
0.33802 0.22223 0.00451 0.09494 0.34030 0.00599 1.00056
0.29959 0.23687 0.01707 0.10664 0.33984 0.00590 1.00055
0.27975 0.26587 0.02543 0.10951 0.31943 0.00580 1.00054
0.25688 0.28212 0.03974 0.12461 0.29666 0.00570 1.00053
0.24677 0.30348 0.05438 0.13634 0.25903 0.00561 1.00052
0.23570 0.32960 0.07273 0.13670 0.22527 0.00551 1.00051
0.21978 0.36630 0.09093 0.12719 0.19580 0.00541 1.00049
0.21069 0.40713 0.10881 0.12695 0.14641 0.00532 1.00048
0.18010 0.46128 0.12077 0.13760 0.10025 0.00522 1.00046
16
Efficient Frontier
17
Downloading the AMPL model and data
AMPL Model:
https://vanderbei.princeton.edu/307/ampl/markL2 minrisk.txt
List of dates:
https://vanderbei.princeton.edu/307/ampl/dates.txt
Monthly return data:
https://vanderbei.princeton.edu/307/ampl/returns.txt
Data from
Yahoo Groups Finance
18
Alternative Formulation
Maximize reward subject to a bound on risk and use least absolute deviations as the risk
measure:
T X
1X
maximize xj Rj (t)
T t=1 j
T
1X X
subject to xj (Rj (t) − rewardj ) ≤ µ
T t=1 j
X
xj = 1
j
xj ≥ 0 for all j
Because of absolute values not a linear programming problem.
Easy to convert...
19
Main Idea For The Conversion
Using the “greedy substitution”, we introduce new variables to represent the troublesome
part of the problem
X
yt = xj (Rj (t) − rewardj )
j
to get
T X
1X
maximize xj Rj (t)
T t=1 j
X
subject to xj (Rj (t) − rewardj ) = yt for all t
j
T
1X
yt ≤ µ
T t=1
X
xj = 1
j
xj ≥ 0 for all j.
We then note that the constraint defining yt can be relaxed to a pair of inequalities:
X
−yt ≤ xj (Rj (t) − rewardj ) ≤ yt.
j 20
A Linear Programming Formulation
T X
1X
maximize xj Rj (t)
T t=1 j
X
subject to −yt ≤ xj (Rj (t) − rewardj ) ≤ yt for all t
j
T
1X
yt ≤ µ
T t=1
X
xj = 1
j
xj ≥ 0 for all j
yt ≥ 0 for all t
21
