A 1/n strategy and Markowitz'
problem in continuous time
Carl Lindberg
2006-10-06
Topics
Portfolio optimization and Markowitz’
problem’s problem
II. A new stock price model
III. The optimal strategy
I.
I. Portfolio optimization and
Markowitz’ problem’s problem
1.
2.
3.
4.
5.
Introduction to portfolio optimization
Markowitz’ one-period problem
The Black-Scholes model
Markowitz’ problem in continuous time
The problem with estimating rates of
return 
1. Introduction to portfolio
optimization
Natural idea: The investor wants to make
money on the stock market
•
•
•
•
Specify stochastic model for stock prices
Fit model to data
Choose optimality criterion
Calculate optimal trading strategy
2. Markowitz’ one-period problem
Stock return model
Correlated normal variables
One-period
Investor trades in n stocks at time 0, holds position until time T.
Trading strategies
i: Proportion of initial wealth W(0) held in stock i
1-1-...-n: Proportion of W(0) held in bond
Markowitz
Find strategy =(1,..., n) that minimizes variance of terminal wealth
W(T) while expected growth larger than predetermined constant
min R n VarW T
E
W TW0expT
3. The Black-Scholes model
Stock prices:
volatilities
independent
Brownian motions
S1
tS 1 
0exp

1  12  j1 21,j t  j1 1,j B j t
n
n

n
n
2
Sn
tS n 
0exp

n  12  j


t

 B t

1 n,j
j1 n,j j
Black-Scholes is the standard model in finance:
• Mathematically tractable: Stocks are geometric Brownian motions
• Theoretically convenient: Market arbitrage free and complete
• Fits data reasonably well
• Parameters can in principle be estimated using normal statistical
theory
4. Markowitz’ problem in
continuous time
Stock price model
Black-Scholes model
Continuous time
Investor can trade in all n stocks at all times
Trading strategies
i(t): Proportion of wealth W(t) held in stock i at time t
1-1(t)-...-n(t): Proportion of W(t) held in bond at time t
Progressively measurable and bounded
Markowitz
Find admissible strategy (t)=(1(t),..., n(t)) that minimizes variance of
terminal wealth W(T) while expected growth larger than predetermined
constant:
min A VarW T
E
W TW0expT
5. The problem with estimating
rates of return 
• Portfolio optimization has not received the same
attention from the financial industry as option
pricing theory
• The predominant explanation is that the rates of
return are very hard to estimate accurately
• This gives extremely unstable portfolio weights
when we apply the optimal strategies with
parameters estimated from data with standard
statistical methods
5. The problem with estimating
rates of return 
• Assume yearly returns for a stock are i.i.d.
N(,),  = 0.20
• A 95% confidence interval for  is
[Ŷ-1.96/√n, Ŷ+1.96/√n],
where n is the number of yearly observations
DATA FOR n=6147 YEARS IS NEEDED TO GET AN
INTERVAL OF WIDTH 0.01!!!
WE NEED MORE THAN NAÏVE STATISTICS
(A common approach in the industry is to use
linear models with some suitable predictors)
II. A new stock price model
1.
2.
3.
4.
The new model
Equal company risk premiums
Covariance and volatility
Examples
1. The new model
Black-Scholes parameterized as
continuous time n-index Arbitrage Pricing
Theory model with equal risk premiums:
S1 
tS 1 
0exp

r  12  j1 21,j 
t  j1 1,j 

 rt B j t

n
n

Same for all j
n
n
Sn 
tS n 
0exp

r  12  j1 2n,j 
t  j1 n,j 

 rt B j t

”Index”
2. Equal company risk premiums
• One interpretation: Stocks are governed by n
independent factors (indexes), each associated
with a specific underlying company
• Since we have no additional information on any
company, the risk premium  - r should be the
same for all of them
• Equal risk premiums gives coherence between
the rates of return. This is not obtained by
estimating the rates of return for each stock
individually
3. Covariance and volatility
• Classical Mathematical Finance: Volatility matrix  only
used to model dependence between stocks
• We can do more: A covariance matrix does not give a
unique volatility matrix
• Our model: Volatility matrix  decides rates of return, too
S1 
tS 1 
0exp

r  12  j1 21,j 
t  j1 1,j 

 rt B j t

n
n

n
n
Sn 
tS n 
0exp

r  12  j1 2n,j 
t  j1 n,j 

 rt B j t

• The investor can incorporate her views on specific
stocks in the market model: She can choose a
volatility matrix which implies relations between the
rates of returns that she believes to hold
4. Examples: Cholesky
decomposition
• Assume that the investor has specified
ranks for the stock risk premiums (SRP)

n
i,j
j1
n
2i,j
j1

 r 
for the n stocks: The stock with the lowest
SRP is assigned rank 1, the second lowest
rank 2, and so on.
4. Examples: Cholesky
decomposition
• Cholesky decomposition of the covariance
matrix sorted by ranks gives a volatility
matrix with clear tendency of assigning low
SRP to stocks with low ranks
• A stock with high rank is affected by some
factors that do not influence stocks with
lower rank. This makes the stock ”more
diversified”, hence it is likely to have
higher SRP
5. Examples: Matrix square root
• Condition: All n companies are (approximately)
equally important to each other
• Assumption: Stock i depends as much on what
happens to corporation j, as stock j depends on
what happens to corporation i
• Consequence: The volatility matrix should be
symmetric
• A symmetric volatility matrix is obtained by
applying the matrix square root to the
covariance matrix
5. Examples: Other alternatives
• Any volatility matrix with a specified
covariance matrix can be written as the
Cholesky decomposition multiplied by an
orthogonal matrix
• Sparse volatility matrix
– Assume conditional independence between
different stocks given some factors, verify with
statistical analysis
– Use to assign ranks to groups of stocks,
similarly to Cholesky decomposition
III. The optimal strategy
1. Main goals
2. The 1/n strategy
3. An explicit solution * to Markowitz’
problem in continuous time
4. The 1/n strategy in factors
5. Advantages
6. Increasing the number of stocks
7. Applying the strategy * to data
1. Main goals
• Goal I: Optimal strategies that are robust
to misspecification of model parameters
• Goal II: Investors can specify their unique
market views through the optimal strategy
without having to decide on exact numbers
for the individual rates of return
2. The 1/n strategy
• The investor puts 1/n of her wealth in each of n
available assets
• Advantage: No estimation of rates of return
• Disadvantages:
– Ignores dependence between stocks: Dependence
can be estimated well  We should use it
– Not very low risk: Few factors determine much of
dependence between stocks  1/n not really
diversified
3. An explicit solution to Markowitz’
problem in continuous time
We assume our new model
Markowitz:
Progressively measurable and bounded
min A VarW T
E
W TW0expT
Wealth:
Functions of 
t
t
0
0
W
tW
0exp
pj
s

 r 12 p 2j s
ds  p j 
s
dB j 
srt


 j1 
n
for the processes p j  
n
i1
i i,j .
3. An explicit solution to Markowitz’
problem in continuous time
But the optimal strategy * must be
constant, deterministic, and equal for
every stock j
This implies that the variance subject to the
growth constraint is minimized by
p j t, 
1
n
r
r
, 
3. An explicit solution to Markowitz’
problem in continuous time
Hence, for sufficiently wide bounds on the
admissible strategies the optimal strategy
* solves
r
1 1,1 
...
n n,1  1n 
 r

r ,
1 1,n 
...
n n,n  1n 
 r
for all t
3. An explicit solution to Markowitz’
problem in continuous time
Optimal wealth process:
n


W tW0expt
r 2 t  1  r B t

n  r j
 r 2n
j1
r 2 t  1  r Bt
L W0exp  1 
2n  r
n  r
Equality in law
4. The 1/n strategy in factors
The strategy *
• Prescribes that the investor should hold
1/n of her wealth in each factor/Brownian
motion. The optimal strategy * is in this
sense the most diversified strategy of all
• Gives the optimal expected wealth of all
portfolios with the same variance  * is
efficient
5. Advantages
• We can expect stability of the portfolio weights
when we use the optimal strategy * with
parameters estimated from data. The reason is
that the strategy * is based on estimates of the
volatility matrix
• The investor need not supply explicit estimates
of rates of return to apply the optimal strategies.
This is due to the natural coherence assumption
that all corporations have the same company
risk premiums
5. Advantages
• The optimal strategy * can be applied without
estimating values for the risk premium or
choosing the required expected return : The
investor’s views are obtained implicitly by the
fraction of her wealth that she chooses to invest
in the stock market. This choice corresponds
also to the investor's risk-aversion
• No utility function has to be selected. This is
often required in classical optimal portfolio
theory
• The investor may still play a part….
6. Increasing the number of stocks
•
•
•
The figures show that the higher expected return the investor requires, the
more she will have to risk. Nonetheless, the risk will still tend to zero as the
number of stocks n increases.
Left: Distributions of optimal wealth at time T=1 for different n with
parameters w=1, λ=0.1, μ=0.2, and r=0.03
Right: Distributions of optimal wealth at time T=1 for different n with
parameters w=1, λ=0.3, μ=0.2, and r=0.03
3
14
— n=200
— n=100
— n=50
— n=25
12
10
8
— n=200
— n=100
— n=50
— n=25
2.5
2
1.5
6
1
4
0.5
2
0
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0
0
0.5
1
1.5
2
2.5
3
7. Applying the strategy * to data
• 48 value weighted industry portfolios consisting
of each stock at NYSE, AMEX, and NASDAQ
• Data from 1963-07-01 to 2005-12-30. The
covariance matrix is estimated with five years of
data, and it is updated each month. The optimal
strategy * is applied out-of-sample
• We assume that the industry portfolios are
approximately equally important to each other:
We apply the matrix square root to the
covariance matrix to get the volatility matrix
7. Applying the strategy * to data
Normal distributions
estimated from
returns:
Blue: The wealth
process for the strategy
* with 1-1-...-n=0
Red: The wealth
process for the 1/N
strategy
Green: Industry portfolio
with smallest volatility
Magenta: Industry
portfolio with largest
volatility
55
50
45
40
35
30
25
20
15
10
5
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
7. Applying the strategy * to data
Portfolio processes:
Blue thick line: The wealth
process for the strategy *
with 1-1-...-n=0
Red thick line: The wealth
process for the 1/N strategy
Green thick line: The
industry portfolio with
smallest volatility
Magenta thick line: The
industry portfolio with largest
volatility
Colored thin lines: Individual
industry portfolios
7. Applying the strategy * to data
0.04
Sharpe ratios:
0.035
Blue horizontal line:
The wealth process
for the strategy *
with 1-1-...-n=0
Red horizontal line:
The wealth process
for the 1/N strategy
Blue stars: Individual
industry portfolios
0.03
0.025
0.02
0.015
0.01
0.005
0
5
10
15
20
25
30
35
40
45
Memmel’s corrected Jobson & Korkie test of the hypothesis of equal
Sharpe ratios between the wealth processes for the strategies * and  =
1/N gets a p-value of <<0.0001