Download Modeling Portfolios that Contain Risky Assets I: Risk and

Modeling Portfolios that Contain Risky Assets I:
Risk and Return
C. David Levermore
University of Maryland, College Park
ICERM Letcure, 14 November 2011
Extracted from Math 420: Mathematical Modeling
c 2011 Charles David Levermore
Main Street, not Wall Street
1. These lectures will be about investing, not trading.
2. They will be about stocks and bonds, not derivatives.
3. They will not provide a short cut to vast wealth.
4. They will not recommend specific stocks, bonds, or funds.
5. They might help you do better than most investors.
WARNING: I am not a stock broker, CFP, CFA, CPA, or CXX of any kind!
When investing, it is a good idea to seek professional investment advice.
These lectures might help you derive greater benefit from such advisors.
Acknowledgement: John Chadam introduced me to the foundations of
this material in a series of lectures at the IMA in early 2000. Thanks John!
A Practical Question
You have just taken a position as an Assistant Professor at an American
university. You are enrolling in their TIAA-CREF retirement plan. The first
question you will be asked that does not have an obvious answer is
“How do you want to split your withholding between TIAA and CREF?”
When faced with this question, many just say “50-50”, which may or may
not be a good answer depending on economic circumstances. (Usually it
is not a bad answer.) Some give this answer because they do not know the
difference between the two. (TIAA is bonds while CREF is stocks.) Most
know the difference, but do not know how to quantify their answer. These
lectures present tools that can help inform your answer to this question.
Mathmatical Perspective
These lectures are extracted from Math 420: Mathematical Modeling, an
undergraduate course at the University of Maryland, College Park. The
course teaches how use data-driven modeling to make the links between
data −→ information −→ knowledge.
They employ tools from vector calculus, linear algebra, interpolation, and
probability. (The probability is self-contained.) You do not need to know
this material to get something out of these lectures, but it would help.
They do not employ differential equations or tools from graduate courses
that are sometimes encountered in mathematical finance. For example,
they do not employ Ito calculus or Lévy processes. However, the modeling
ideas that are presented in these lectures do carry over to such settings.
Overview of the Lectures
1. Risk and Return: These basic economic ideas will be introduced and
quantified. We will cover model calibration and give a Madoff warning.
2. Efficient Frontiers for Various Models: This will present useful models
that are not included in many finance courses. We will discuss the
efficient market hypothesis.
3. Stochastic Models and Optimization: This will not be the usual utility
function approach of economics, nor will it be the usual model fitting
approach of statistics. There will be emphasis on model validation.
Lecture I: Risk and Return
Outline
1. Introduction
2. Markowitz Portfolios
3. Basic Markowitz Portfolio Theory
1. Introduction
Suppose you are considering how to invest in N risky assets that are
traded on a market that had D trading days last year. (Typically D = 255.)
Let si (d) be the share price of the ith asset at the close of the dth trading
day of the past year, where si(0) is understood to be the share price at the
close of the last trading day before the beginning of the past year. We will
assume that every si (d) is positive. You would like to use this price history
to gain insight into how to manage your portfolio over the coming year.
We will examine the following questions.
Can stochastic (random, probabilistic) models be built that quantitatively
mimic this price history? How can such models be used to help manage a
portfolio?
Risky Assets. The risk associated with an investment is the uncertainy
of its outcome. Every investment has risk associated with it. Hiding your
cash under a mattress puts it at greater risk of loss to theft or fire than
depositing it in a bank, and is a sure way to not make money. Depositing
your cash into an FDIC insured bank account is the safest investment that
you can make — the only risk of loss would be to an extreme national
calamity. However, a bank account generally will yield a lower return on
your investment than any asset that has more risk associated with it. Such
assets include real estate, stocks (equities), bonds, commodities (gold, oil,
corn, etc.), private equity (venture capital), and hedge funds. With the
exception of real estate, it is not uncommon for prices of these assets to
fluctuate one to five percent in a day. Any such asset is called a risky asset.
Remark. Market forces generally will insure that assets associated with
higher potential returns are also associated with greater risk and vice versa.
Investment offers that seem to violate this principle are always scams.
Here we will consider two basic types of risky assets: stocks and bonds.
We will also consider mutual funds, which are managed funds that hold a
combination of stocks and/or bonds, and possibly other risky assets.
Stocks. Stocks are part ownership of a company. Their value goes up
when the company does well, and goes down when it does poorly. Some
stocks pay a periodic (usually quarterly) dividend of either cash or more
stock. Stocks are traded on exchanges like the NYSE or NASDAQ.
The risk associated with a stock reflects the uncertainty about the future
performance of the company. This uncertainty has many facets. For example, there might be questions about the future market share of its products,
the availablity of the raw materials needed for its products, or the value of
its current assets. Stocks in larger companies are generally less risky than
stocks in smaller companies. Stocks are generally higher return/higher risk
investments compared to bonds.
Bonds. Bonds are essentially a loan to a government or company. The
borrower usually makes a periodic (often quarterly) interest payment, and
ultimately pays back the principle at a maturity date. Bonds are traded on
secondary markets where their value is based on current interest rates.
For example, if interest rates go up then bond values will go down on the
secondary market.
The risk associated with a bond reflects the uncertainty about the credit
worthiness of the borrower. Short term bonds are generally less risky than
long term ones. Bonds from large entities are generally less risky than
those from small entities. Bonds from governments are generally less risky
than those from companies. (This is even true in some cases where the
ratings given by some ratings agencies suggest otherwise.) Bonds are
generally lower return/lower risk investments compared to stocks.
Mutual Funds. These funds hold a combination of stocks and/or bonds,
and possibly other risky assets. You buy and sell shares in these funds
just as you would shares of a stock. Mutual funds are generally lower
return/lower risk investments compared to individual stocks and bonds.
Most mutual funds are managed in one of two ways: actively or passively.
An actively-managed fund usually has a strategy to perform better than
some market index like the S&P 500, Russell 1000, or Russell 2000. A
passively-managed fund usually builds a portfolio so that its performance
will match some market index, in which case it is called an index fund.
Index funds are often portrayed to be lower return/lower risk investments
compared to actively-managed funds. However, index funds will typically
outperform most actively-managed funds. Reasons for this include the
facts that they have lower management fees and that they require smaller
cash reserves.
Return Rates. The first thing you must understand that the share price of
an asset has very little economic significance. This is because the size of
your investment in an asset is the same if you own 100 shares worth 50
dollars each or 25 shares worth 200 dollars each. What is economically
significant is how much your investment rises or falls in value. Because
your investment in asset i would have changed by the ratio si(d)/si (d−1)
over the course of day d, this ratio is economically significant. Rather than
use this ratio as the basic variable, it is customary to use the so-called
return rate, which we define by
si(d) − si(d − 1)
.
ri(d) = D
si(d − 1)
The factor D arises because rates in banking, business, and finance are
usually given as annual rates expressed in units of either “per annum” or
1 years the factor of D makes r (d) a
“ % per annum.” Because a day is D
i
“per annum” rate. It would have to be multiplied by another factor of 100 to
make it a “% per annum” rate. We will always work with “per annum” rates.
One way to understand return rates is to set ri(d) equal to a constant µ.
Upon solving the resulting relation for si(d) you find that
µ
si(d) = 1 + D
si(d − 1)
for every d = 1, · · · , D .
By induction on d you can then derive the compound interest formula
µ d
si(d) = 1 + D si(0)
for every d = 1, · · · , D .
If you assume that |µ/D| << 1 then you can see that
whereby
D
1
µ µ
1+D
≈ lim (1 + h) h = e ,
h→0
D d
d
µ µ µD
µD
si(0) = eµt si(0) ,
si(d) = 1 + D
si(0) ≈ e
where t = d/D is the time (in units of years) at which day d occurs. You
thereby see µ is nearly the exponential growth rate of the share price.
We will consider a market of N risky assets indexed by i. For each i you
obtain the closing share price history {si(d)}D
d=0 of asset i over the past
year, and compute the return rate history {ri(d)}D
d=1 of asset i over the
past year by the formula
s (d) − si(d − 1)
ri(d) = D i
.
si(d − 1)
Because return rates are differences, you will need the closing share price
from the day before the first day for which you want the return rate history.
You can obtain share price histories from websites like Yahoo Finance or
Google Finance. For example, to compute the daily return rate history
for Apple in 2009, type “Apple” into where is says “get quotes”. You will
see that Apple has the identifier AAPL and is listed on the NASDAQ. Click
on “historical prices” and request share prices between “Dec 31, 2008”
and “Dec 31, 2009”. You will get a table that can be downloaded as a
spreadsheet. The return rates are computed using the closing prices.
Remark. It is not obvious that return rates are the right quantities upon
which to build a theory of markets. For example, another possibility is to
use the growth rates xi(d) defined by
xi(d) = D log
si(d)
si(d − 1)
!
.
These are also functions of the ratio si(d)/si (d − 1). Moreover, they seem
to be easier to understand than return rates. For example, if you set xi(d)
equal to a constant γ then by solving the resulting relation for si(d) you
find that
si(d) =
1γ
eD
si(d − 1)
for every d = 1, · · · , D .
By induction on d you can then show that
si(d) =
dγ
D
e si(0)
for every d = 1, · · · , D ,
whereby si(d) = eγt si(0) with t = d/D. However, return rates have
better properties with regard to porfolio statistics and so are preferred.
Statistical Approach. Return rates ri (d) for asset i can vary wildly from
day to day as the share price si (d) rises and falls. Sometimes the reasons
for such fluctuations are very clear because they directly relate to some
news about the company, agency, or government that issued the asset.
For example, news of the Deepwater Horizon explosion caused the share
price of British Petroleum stock to fall. At other times they relate to news
that benefit or hurt entire sectors of assets. For example, a rise in crude oil
prices might benefit oil and railroad companies but hurt airline and trucking
companies. And at yet other times they relate to general technological,
demographic, or social trends. For example, new internet technology might
benefit Google and Amazon (companies that exist because of the internet)
but hurt traditional “brick and mortar” retailers. Finally, there is often no
evident public reason for a particular stock price to rise or fall. The reason
might be a takeover attempt, a rumor, insider information, or the fact a large
investor needs cash for some other purpose.
Given the complexity of the dynamics underlying such market fluctuations,
we adopt a statistical approach to quantifying their trends and correlations.
More specifically, we will choose an appropriate set of statistics that will be
computed from selected return rate histories of the relevant assets. We will
then use these statistics to calibrate a model that will predict how a set of
ideal portfolios might behave in the future.
The implicit assumption of this approach is that in the future the market will
behave statistically as it did in the past. This means that the data should be
drawn from a long enough return rate history to sample most of the kinds
of market events that you expect to see in the future. However, the history
should not be too long because very old data will not be relevant to the
current market. To strike a balance we will use the return rate history from
the most recent twelve month period, which we will dub “the past year”.
For example, if we are planning our portfolio at the beginning of July 2011
then we will use the return rate histories for July 2010 through June 2011.
Then D would be the number of trading days in this period.
Suppose that you have computed the return rate history {ri(d)}D
d=1 for
each asset over the past year. At some point this data should be ported
from the speadsheet into MATLAB, R, or another higher level environment
that is well suited to the task ahead. The next step is to compute statistical
quantities; we will use means, variances, covariances, and correlations.
The return rate mean for asset i over the past year, denoted mi, is
D
1 X
ri(d) .
mi =
D d=1
This measures the trend of the share price. Unfortunately, it is commonly
called the expected return rate for asset i even though it is higher than the
return rate that most investors will see, especially in highly volatile markets.
We will not use this misleading terminology.
The return rate variance for asset i over the past year, denoted vi, is
D 2
X
1
vi =
ri(d) − mi .
D(D − 1) d=1
The reason for the D(D − 1) in the denominator will be made clear later.
The return rate standard deviation for asset i over the year, denoted σi , is
√
given by σi = vi. This is called the volatility of asset i. It measures the
uncertainty of the market regarding the share price trend.
The covariance of the return rates for assets i and j over the past year,
denoted vij , is
D X
1
ri (d) − mi rj (d) − mj .
vij =
D(D − 1) d=1
Notice that vii = vi. The N ×N matrix (vij ) is symmetric and nonnegative
definite. It will usually be positive definite — so we will assume it to be so.
Finally, the correlation of the return rates for assets i and j over the past
year, denoted cij , is
vij
cij =
.
σi σj
Notice that −1 ≤ cij ≤ 1. We say assets i and j are positively correlated when 0 < cij ≤ 1 and negatively correlated when −1 ≤ cij < 0.
Positively correlated assets will tend to move in the same direction, while
negatively correlated ones will often move in opposite directions.
We will consider the N -vector of means (mi) and the symmetric N ×N
matrix of covariances (vij ) to be our basic statistical quantities. We will
build our models to be consistent with these statistics. The variances (vi),
volatilities (σi ), and correlations (cij ) can then be easily obtained from
(mi) and (vij ) by formulas that are given above. The computation of
the statistics (mi) and (vij ) from the return rate histories is called the
calibration of our models.
Remark. Here the trading day is an arbitrary measure of time. From a
theoretical viewpoint we could equally well have used a shorter measure
like half-days, hours, quarter hours, or minutes. The shorter the measure,
the more data has to be collected and analyzed. This extra work is not
worth doing unless you profit sufficiently. Alternatively, we could have used
a longer measure like weeks, months, or quarters. The longer the measure, the less data you use, which means you have less understanding of
the market. For many investors daily or weekly data is a good balance. If
you use weekly data {si(w)}52
w=0, where si (w) is the share price of asset
i at the end of week w, then the rate of return of asset i for week w is
si(w) − si(w − 1)
.
ri(w) = 52
si(w − 1)
You have to make consistent changes when computing mi, vi, and vij by
replacing d with w and D with 52 in their defining formulas.
D
h
General Histories and Weights. We can consider a history {r(d)}d=1
over a period of Dh trading days and assign day d a weight w(d) > 0 such
Dh
that the weights {w(d)}d=1
satisfy
Dh
X
w(d) = 1 .
d=1
The return rate means and covariances are then given by
mi =
Dh
X
w(d) ri(d) ,
d=1
where
D
h
X
1
vij =
w(d) ri(d) − mi rj (d) − mj ,
D(1 − w) d=1
w=
Dh
X
d=1
w(d)2 .
In practice the history will extend over a period of one to five years. There
Dh
are many ways to choose the weights {w(d)}d=1
. The most common
choice is the so-called uniform weighting; this gives each day the same
weight by setting w(d) = 1/Dh. On the other hand, we might want to give
more weight to more recent data. For example, we can give each trading
day a positive weight that depends only on the quarter in which it lies, giving
greater weight to more recent quarters. We could also consider giving
different weights to different days of the week, but such a complication
should be avoided unless it yields a clear benefit.
You will have greater confidence in mi and vij when they are relatively
insensitive to different choices of Dh and the weights w(d). You can get
an idea of the magnitude of this sensitivity by checking the robustness of
mi and vij to a range of such choices.
Exercise. Compute mi, vi, vij , and cij for each of the following groups of
assets based on daily data, weekly data, and monthly data:
(a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009;
(b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007;
(c) S&P 500 and Russell 1000 and 2000 index funds in 2009;
(d) S&P 500 and Russell 1000 and 2000 index funds in 2007.
Give explanations for the values of cij you computed.
Exercise. Compute mi, vi, vij , and cij for the assets listed in the previous
exercise based on daily data and weekly data, but only from the last quarter of the year indicated. Based on a comparison of these answers with
those of the previous problem, in which numbers might you have the most
confidence, the mi, vi, vij , or cij ?
2. Markowitz Portfolios
A 1952 paper by Harry Markowitz had enormous influence on the theory
and practice of portfolio management and financial engineering ever since.
It presented his doctoral dissertation work at the Unversity of Chicago, for
which he was awarded the Nobel Prize in Economics in 1990. It was the
first work to quantify how diversifying a portfolio can reduce its risk without
changing its expected rate of return. It did this because it was the first work
to use the covariance between different assets in an essential way.
We will consider portfolios in which an investor can choose to hold one of
three positions with respect to any risky asset. The investor can:
- hold a long position by owning shares of the asset;
- hold a short position by selling borrowed shares of the asset;
- hold a neutral position by doing neither of the above.
In order to keep things simple, we will not consider derivative assets.
Remark. You hold a short position by borrowing shares of an asset from a
lender (usually your broker) and selling them immediately. If the share price
subsequently goes down then you can buy the same number of shares and
give them to the lender, thereby paying off your loan and profiting by the
price difference minus transaction costs. Of course, if the price goes up
then your lender can force you either to increase your collateral or to pay
off the loan by buying shares at this higher price, thereby taking a loss that
might be larger than the original value of the shares.
The value of any portfolio that holds ni(d) shares of asset i at the end of
trading day d is
Π(d) =
N
X
ni(d)si (d) .
i=1
If you hold a long position in asset i then ni(d) > 0. If you hold a short
position in asset i then ni(d) < 0. If you hold a neutral position in asset i
then ni(d) = 0. We will assume that Π(d) > 0 for every d.
Markowitz carried out his analysis on a class of idealized portfolios that are
each characterized by a set of real numbers {fi}N
i=1 such that
N
X
fi = 1 .
i=1
The portfolio picks ni(d) at the beginning at each trading day d so that
ni(d)si (d − 1)
= fi ,
Π(d − 1)
where ni(d) need not be an integer. We call these Markowitz portfolios.
The portfolio holds a long position in asset i if fi > 0 and holds a short
position if fi < 0. If every fi is nonnegative then fi is the fraction of the
portfolio’s value held in asset i at the beginning of each day. A Markowitz
portfolio will be self-financing if we neglect trading costs because
N
X
i=1
ni(d) si(d − 1) = Π(d − 1) .
Portfolio Return Rate. We see from the self-financing property and the
relationship between ni(d) and fi that the return rate r(d) of a Markowitz
portfolio for trading day d is
Π(d) − Π(d − 1)
r(d) = D
Π(d − 1)
=
=
N
X
n (d)si (d) − ni(d)si (d − 1)
D i
Π(d − 1)
i=1
N
X
ni(d)si (d − 1)
i=1
Π(d − 1)
N
X
si(d) − si(d − 1)
fi ri(d) .
D
=
si(d − 1)
i=1
The return rate r(d) for the Markowitz portfolio characterized by {fi}N
i=1
is therefore simply the linear combination with coefficients fi of the ri(d).
This relationship makes the class of Markowitz portfolios easy to analyze.
We will therefore use Markowitz portfolios to model real portfolios.
This relationship can be expressed in the compact form
r(d) = fTr(d) ,
where f and r(d) are the N -vectors defined by
f


f1
 .. 
= ,
fN


r1(d)
.. 
 .
rN (d)
r(d) = 

The N -vector of return rate means m and the N ×N -matrix of return rate
covariances V can be expressed in terms of r(d) as

V

m1
D
1 X
 . 
m= . =
r(d) ,
D d=1
mN


v11 · · · v1N
D T
X
1
 .

.
...
r(d) − m r(d) − m .
= .
. =
D(D − 1) d=1
vN 1 · · · vN N
Portfolio Statistics. Markowitz portfolios are easy to analyze because
r(d) = fTr(d) where f is independent of d. In particular, for the Markowitz
portfolio characterized by f the return rate mean µ and variance v can be
expressed simply in terms of m and V as
1
µ=D
D
X
d=1
1
r(d) = D
1
v = D(D−1)
D X
d=1
D X
1
= D(D−1)
d=1

= fT
1
D(D−1)
D
X
d=1
1
fTr(d) = fT D
r(d) − µ
T

2
1
= D(D−1)
T
D
X
r(d) = fTm ,
d=1
D X
d=1
T

T
d=1
T
T
r(d) − m r(d) − m
2
f r(d) − f m
f r(d) − f m r(d) f − m f
D X
T

 f = fT Vf .
Hence, µ = fTm and v = fTVf . Because V is positive definite, v > 0.
Remark. These simple formulas for µ and v are the reason that return
rates are preferred over growth rates when compiling statistics of markets.
The simplicity of these formulas arises because the return rates r(d) for
the Markowitz portfolio specified by the distribution f depends linearly upon
the vector r(d) of return rates for the individual assets as r(d) = fTr(d).
In contrast, the growth rates x(d) of a Markowitz portfolio are given by
Π(d)
x(d) = D log
Π(d − 1)
!
1
= D log 1 + D r(d)


N
X
T
1
1

fi ri (d)
= D log 1 + D f r(d) = D log 1 + D
i=1




N
N
X
X
1 x (d)
1 x (d)
i



D
D
fi e i  .
−1
= D log 1 +
= D log
fi e
i=1
i=1
Because the x(d) are not linear functions of the xi(d), averages of x(d)
over d are not simply expressed in terms of averages of xi(d) over d.
D
h to the trading days
Remark. Had we chosen to assign weights {w(d)}d=1
D
h then we would still obtain the formulas
of a return rate history {r(d)}d=1
µ = fTm ,
v = fTVf ,
but the N -vector of return rate means m and the N ×N -matrix of return
rate covariances V would be expressed in terms of r(d) as
m=
Dh
X
w(d) r(d) ,
d=1
D
h
T
X
1
V=
w(d) r(d) − m r(d) − m .
D(1 − w) d=1
D
D
h
h
The choices of return rate history {r(d)}d=1
and weights {w(d)}d=1
thereby determine m and V. They constitute a calibration of our models.
Ideally m and V will not be overly sensitive to these choices.
Remark. Aspects of Markowitz portfolios are unrealistic. These include:
- the fact portfolios can contain fractional shares of any asset;
- the fact portfolios are rebalanced every trading day;
- the fact transaction costs and taxes are neglected;
- the fact dividends are neglected.
By making these simplifications the subsequent analysis becomes easier.
The idea is to find the Markowitz portfolio that is best for a given investor.
The expectation is that any real portfolio with a distribution close to that for
the optimal Markowitz portfolio will perform nearly as well. Consequently,
most investors rebalance at most a few times per year, and not every asset
is involved each time. Transaction costs and taxes are thereby limited.
Similarly, borrowing costs are kept to a minimum by not borrowing often.
The model can be modified to account for dividends.
Remark. Portfolios of risky assets can be designed for trading or investing.
Traders often take positions that require constant attention. They might
buy and sell assets on short time scales in an attempt to profit from market
fluctuations. They might also take highly leveraged positions that expose
them to enormous gains or loses depending how the market moves. They
must be ready to handle margin calls. Trading is often a full time job.
Investors operate on longer time scales. They buy or sell an asset based
on their assessment of its fundamental value over time. Investing does
not have to be a full time job. Indeed, most people who hold risky assets
are investors who are saving for retirement. Lured by the promise of high
returns, sometimes investors will buy shares in funds designed for traders.
At that point they have become gamblers, whether they realize it or not.
The ideas presented in these lectures are designed to balance investment
portfolios, not trading portfolios.
Exercise. Compute µ and v based on daily data for the Markowitz portfolio
with value equally distributed among the assets in each of the following
groups:
(a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009;
(b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007;
(c) S&P 500 and Russell 1000 and 2000 index funds in 2009;
(d) S&P 500 and Russell 1000 and 2000 index funds in 2007.
√
Exercise. The volatility of a portfolio is σ = v. In the σµ-plane plot
(σ, µ) for the two 2007 portfolios and (σi , mi) for each of the 2007 assets
in the previous exercise.
Exercise. In the σµ-plane plot (σ, µ) for the two 2009 portfolios and
(σi, mi ) for each of the 2009 assets in the first exercise above.
3. Basic Markowitz Portfolio Theory
The 1952 Markowitz paper initiated what subsequently became known as
modern portfolio theory (MPT). Because 1952 was long ago, this name
has begun to look silly and some have taken to calling it Markowitz Portfolio
Theory (still MPT), to distinguish it from more modern theories. (Markowitz
simply called it portfolio theory, and often made fun of the name it aquired.)
Any portfolio theory strives to maximize the return of a portfolio for a given
risk, or equivalently, to mimimize risk for a given return. Portfolio theories
do this by quantifying the notions of return and risk, and by identifying a
class of idealized portfolios for which some kind of analysis is tractable.
Here we introduce MPT, the first such theory.
Markowitz chose to use the return rate mean µ as the proxy for the return
√
of a portfolio, and the volatility σ = v as the proxy for its risk. He also
chose to analyze the class that we have dubbed Markowitz portfolios. Then
for a portfolio of N risky assets characterized by m and V the problem of
minimizing risk for a given return becomes the problem of minimizing
σ=
q
fTVf
over f ∈ RN subject to the constraints
1Tf = 1 ,
mTf = µ ,
where µ is given. Here 1 is the N -vector that has every entry equal to 1.
Additional constraints can be imposed. For example, if only long positions
are to be considered then we must also impose the entrywise constraints
f ≥ 0, where 0 denotes the N -vector that has every entry equal to 0.
Constrained Minimization Problem. Because σ > 0, minimizing σ is
equivalent to minimizing σ 2. Because σ 2 is a quadratic function of f , it is
easier to minimize than σ. We therefore choose to solve the constrained
minimization problem
1 fT Vf :
min 2
f ∈RN
n
T
T
o
1 f = 1, m f = µ .
Because there are two equality constraints, we introduce the Lagrange
multipliers α and β, and define
1 fTVf − α
Φ(f , α, β) = 2
T
T
1 f −1 −β m f −µ .
By setting the partial derivatives of Φ(f , α, β) equal to zero we obtain
0 = ∇f Φ(f , α, β) = Vf − α1 − β m ,
0 = ∂αΦ(f , α, β) = −1Tf + 1 ,
0 = ∂β Φ(f , α, β) = −mTf + µ .
Because V is positive definite we may solve the first equation for f as
f = α V−11 + β V−1m .
By setting this into the second and third equations we obtain the system
α 1TV−11 + β 1TV−1m = 1 ,
α mTV−11 + β mTV−1m = µ .
If we introduce a, b, and c by
a = 1TV−11 ,
b = 1TV−1m ,
c = mTV−1m ,
then the above system can be expressed as
!
a b
b c
!
α
1
=
β
µ
!
.
Because V−1 is positive definite, the above 2×2 matrix is positive definite
if and only if the vectors 1 and m are not co-linear (m 6= µ1 for every µ).
We now assume that 1 and m are not co-linear, which is usually the case
in practice. We can then solve the system for α and β to obtain
!
!
1
α
c −b
=
β
ac − b2 −b a
!
!
1
1
c − bµ
=
µ
ac − b2 aµ − b
.
Hence, for each µ there is a unique minimizer given by
aµ − b −1
c − bµ −1
V 1+
V m.
f (µ) =
ac − b2
ac − b2
The associated minimum value of σ 2 is
2
T
σ = f (µ) Vf (µ) = α V
= α β
!
!
a b
α
b c
β
1
b
a
= +
µ−
2
a
ac − b
a
−1
1+βV
−1
T
m
V
−1
−1
αV 1 + β V m
!
!
c −b
1
=
1 µ
−b a
ac − b2
!2
.
1
µ
Remark. In the case where 1 and m are co-linear we have m = µ1 for
1 fT Vf subject to the constraint 1Tf = 1 then, by
some µ. If we minimize 2
Lagrange multipliers, we find that
Vf = α1 ,
for some α ∈ R .
Because V is invertible, for every µ the unique minimizer is given by
1
−1
V
1.
T
−1
1 V 1
The associated minimum value of σ 2 is
1
1
σ 2 = f (µ)TVf (µ) = T −1 = .
1 V 1
a
f (µ) =
Remark. The mathematics behind MPT is fairly elementary. Markowitz
had cleverly indentified a class of portfolios which is analytically tractable
by elemetary methods, and which provides a framework for useful models.
Frontier Portfolios. We have seen that for every µ there exists a unique
Markowitz portfolio with mean µ that minimizes σ 2. This minimum value is
1
2
σ = +
a
a
b
µ−
ac − b2
a
!2
.
This is the equation of a hyperbola in the σµ-plane. Because volatility is
nonnegative, we only consider the right half-plane σ ≥ 0. The volatility σ
and mean µ of any Markowitz portfolio will be a point (σ, µ) in this halfplane that lies either on or to the right of this hyperbola. Every point (σ, µ)
on this hyperbola in this half-plane represents a unique Markowitz portfolio.
These portfolios are called frontier portfolios.
We now replace a, b, and c with the more meaningful frontier parameters
1
σmv = √ ,
a
b
µmv = ,
a
νas =
s
ac − b2
.
a
The volatility σ for the frontier portfolio with mean µ is then given by
v
u
u
2
σ = σf (µ) ≡ tσmv
+
µ − µmv
νas
!2
.
It is clear that no portfolio has a volatility σ that is less than σmv . In
other words, σmv is the minimum volatility attainable by diversification.
2 to be the contribution to the volatility due to
Markowitz interpreted σmv
2 to be the
the systemic risk of the market, and interpreted (µ − µmv )2/νas
contribution to the volatility due to the specific risk of the portfolio.
The frontier portfolio corresponding to (σmv , µmv ) is called the minimum
volatility portfolio. Its associated distribution fmv is given by
b
1
2
V−11 .
= V−11 = σmv
a
a
This distribution depends only upon V, and is therefore known with greater
confidence than any distribution that also depends upon m.
fmv = f (µmv ) = f
The distribution of the frontier portfolio with mean µ can be expressed as
µ − µmv −1
V
ff (µ) ≡ fmv +
m − µmv 1 .
2
νas
Because 1TV−1
fT
mv V
m − µmv 1 = b − µmv a = 0, we see that
2 µ − µmv T −1
ff (µ) − fmv = σmv
1 V
m − µmv 1 = 0 .
2
νas
The vectors fmv and ff (µ)− fmv are thereby orthogonal with respect to the
V-inner product, which is given by (f1 | f2)V = fT
1 Vf2. In the associated
2 = (f | f ) , we find that
V-norm, which is given by kf kV
V
2
2
= σmv
,
kfmv kV
2
=
kff (µ) − fmv kV
µ − µmv
νas
!2
.
Hence, ff (µ) = fmv + (ff (µ) − fmv ) is the orthogonal decomposition
of ff (µ) into components that account for the contributions to the volatility
due to systemic risk and specific risk respectively.
Remark. Markowitz attributed σmv to systemic risk of the market because
he was considering the case when N was large enough that σmv would not
be significantly reduced by introducing additional assets into the portfolio.
More generally, one should attribute σmv to those risks that are common to
all of the N assets being considered for the portfolio.
Definition. When two portfolios have the same volatility but different means
then the one with the greater mean is said to be more efficient because it
promises greater return for the same risk.
Definition. Every frontier portfilio with mean µ > µmv is more efficient
than every other portfolio with the same volatility. This segment of the
frontier is called the efficient frontier.
Definition. Every frontier portfilio with mean µ < µmv is less efficient than
every other portfolio with the same volatility. This segment of the frontier is
called the inefficient frontier.
The efficient frontier is represented in the right-half σµ-plane by the upper
branch of the frontier hyperbola. It is given as a function of σ by
q
2
µ = µmv + νas σ 2 − σmv
,
for σ > σmv .
This curve is increasing and concave and emerges vertically upward from
the point (σmv , µmv ). As σ → ∞ it becomes asymptotic to the line
µ = µmv + νasσ .
The inefficient frontier is represented in the right-half σµ-plane by the lower
branch of the frontier hyperbola. It is given as a function of σ by
q
2
µ = µmv − νas σ 2 − σmv
,
for σ > σmv .
This curve is decreasing and convex and emerges vertically downward
from the point (σmv , µmv ). As σ → ∞ it becomes asymptotic to the line
µ = µmv − νasσ .
General Portfolio with Two Risky Assets. Consider a portfolio of two
risky assets with mean vector m and covarience matrix V given by
m=
m1
m2
!
,
V=
v11 v12
v12 v22
!
.
The constraints 1Tf = 1 and mTf = µ uniquely determine f to be
!
1
f1(µ)
m2 − µ
f (µ) =
=
f2(µ)
m2 − m1 µ − m1
!
.
Because there is exactly one portfolio for each µ, every portfolio must be
a frontier portfolio. Therefore there is no optimization problem to solve and
ff (µ) = f (µ). These portfolios trace out the hyperbola
σ 2 = f (µ)TVf (µ)
v11(m2 − µ)2 + 2v12(m2 − µ)(µ − m1) + v22(µ − m1)2
=
.
2
(m2 − m1)
The frontier parameters µmv , σmv , and νas are given by
(v22 − v12)m1 + (v11 − v12)m2
,
µmv =
v11 + v22 − 2v12
2
v11v22 − v12
(m2 − m1)2
2
2
σmv =
,
νas =
.
v11 + v22 − 2v12
v11 + v22 − 2v12
The minimum volatility portfolio is
2 V−11 =
fmv = σmv
=
1
2
σmv
2
v11v22 − v12
v22 −v12
−v12 v11
v22 − v12
v11 + v22 − 2v12 v11 − v12
!
!
1
1
!
.
Remark. Each (σi, mi ) lies on the frontier of every two-asset portfolio.
Typically each (σi , mi) lies strictly to the right of the frontier of a portfolio
that contains more than two risky assets.
Remark. The foregoing two-asset portfolio example also illustrates the
following general property of frontier portfolios that contain two or more
risky assets. Let µ1 and µ2 be any two return rate means with µ1 < µ2.
The distributions of the associated frontier portfolios are then f1 = ff (µ1)
and f2 = ff (µ2). Because ff (µ) depends linearly on µ we see that
µ2 − µ
µ − µ1
f1 +
f2 .
µ2 − µ1
µ2 − µ1
This formula can be interpreted as stating that every frontier portfolio can
be realized by holding positions in just two mutual funds with the portfolio
distributions f1 and f2. When µ ∈ (µ1, µ2) both funds are held long.
When µ > µ2 the first fund is held short while the second is held long.
When µ < µ1 the second fund is held short while the first is held long.
This observation is often called the Two Mutual Fund Theorem, which is
a label that elevates it to a higher status than it deserves. We will call it
simply the Two Mutual Fund Property.
ff (µ) =
Simple Portfolio with Three Risky Assets. Consider a portfolio of three
risky assets with mean vector m and covarience matrix V given by




m1
m−d



m=
m2 =  m  ,
m3
m+d


1 r r

V = s2 
r 1 r  .
r r 1
Here m ∈ R, d, s ∈ R+, and r ∈ (− 1
2 , 1). The last condition is equivalent
to the condition that V is positive definite given s > 0. This portfolio has
the unrealistic properties that (1) every asset has the same volatility s, (2)
every pair of distinct assets has the same correlation r, and (3) the return
rate means are uniformly spaced with difference d = m3 −m2 = m2 −m1.
These simplifications will make it easier to follow the ensuing calculations
than it would be for a more general three-asset portfolio. We will return to
this simple portfolio in subsequent examples.
It is helpful to express m and V in terms of 1 and n = −1 0 1
m = m1 + dn,
!
r
V = s (1 − r) I +
1 1T .
1−r
2
T
as
Notice that 1T1 = 3, 1Tn = 0, and nTn = 2. It can be checked that
!
r
1
1
T
−1
I
−
1
1
1,
,
V
1
=
2
2
s (1 − r)
1 + 2r
s (1 + 2r)
m
d
1
−1
−1
n,
V m= 2
1+ 2
n.
V n= 2
s (1 − r)
s (1 + 2r)
s (1 − r)
V−1 =
The parameters a, b, and c are therefore given by
3m
b = 1TV−1m = 2
,
s (1 + 2r)
2d2
3m2
T −1
+ 2
.
c=m V m= 2
s (1 + 2r)
s (1 − r)
3
,
a = 1TV−11 = 2
s (1 + 2r)
The frontier parameters are then
s
s
b
1
1 + 2r
=s
,
µmv = = m ,
σmv =
a
3
a
s
s
2
d
b
2
νas = c −
=
,
a
s 1−r
whereby the frontier is given by
σ = σf (µ) = s
v
u
u 1 + 2r
t
3
+
1−r µ−m
2
d
!2
for µ ∈ (−∞, ∞) .
Notice that each (σi , mi) lies strictly to the right of the frontier because
σmv = σf (m) = s
s
1 + 2r
< σf (m ± d) = s
3
s
5+r
< s.
6
2
Notice that as r decreases the frontier
moves
to
the
left
for
|µ−m|
<
√
3
2
and to the right for |µ − m| > 3 3d.
√
3d
The minimum volatility portfolio has distribution
2
fmv = σmv
V−11 =
1 1,
3
while the distribution of the frontier portfolio with return rate mean µ is

1 − µ−m
 3 1 2d 
µ − µmv −1
 .
ff (µ) = fmv +
V
m − µmv 1 = 


2
3
νas
µ−m
1+
3
2d

Notice that the frontier portfolio will hold long postitions in all three assets
2 d, m + 2 d). It will hold a short position in the first asset
when µ ∈ (m − 3
3
2
when µ > m+ 3 d, and a short position in the third asset when µ < m− 2
3 d.
In particular, in order to create a portfolio with return rate mean µ greater
than that of any asset contained within it you must short sell the asset with
the lowest return rate mean and invest the proceeds into the asset with
the highest return rate mean. The fact that ff (µ) is independent of V is a
consequence of the simple forms of both V and m. This is also why the
fraction of the investment in the second asset is a constant.
Remark. The frontier portfolios in the above examples are independent of
all the parameters in V. While this is not generally true, it is generally true
that they are independent of the overall market volatility. In particular, they
are independent of the factor D(D − 1) that was chosen to normalize V.
Said another way, the frontier portfolios depend only upon the correlations
cij , the volatility ratios σi /σj , and the means mi . Moreover, the minimum
volatility portfolio fmv depends only upon the correlations and the volatility
ratios. Because markets can exhibit periods of markedly different volatility,
it is natural to ask when correlations and volatility ratios might be relatively
stable across such periods.
Remark. The efficient frontier quantifies the relationship between risk and
return that we mentioned earlier. A portfolio management theory typically
assumes that investors prefer efficient frontier portfolios and will therefore
select an efficient frontier portfolio that is optimal given some measure of
the risk aversion of an investor. Our goal is to develop such theories.
Exercise. Consider the following groups of assets:
(a) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2009;
(b) Google, Microsoft, Exxon-Mobil, UPS, GE, and Ford stock in 2007;
(c) S&P 500 and Russell 1000 and 2000 index funds in 2009;
(d) S&P 500 and Russell 1000 and 2000 index funds in 2007.
On a single graph plot the points (σi, mi ) for every asset in groups (a) and
(c) along with the frontiers for group (a), for group (c), and for groups (a)
and (c) combined. Do the same thing for groups (b) and (d) on a second
graph. (Use daily data.) Comment on any relationships you see between
the objects plotted on each graph.
Exercise. Compute the minimum volatitly portfolio fmv for group (a), for
group (c), and for groups (a) and (c) combined. Do the same thing for
group (b), for group (d), and for groups (b) and (d) combined. Compare
these results for the analogous groupings in 2007 and 2009.