A 1/n strategy and Markowitz' problem in continuous time

Carl Lindberg

2006-10-06

Topics

I. Portfolio optimization and Markowitz’ problem’s problem

II. A new stock price model

III. The optimal strategy

I. Portfolio optimization and Markowitz’ problem’s problem

1. Introduction to portfolio optimization

2. Markowitz’ one-period problem

3. The Black-Scholes model

4. Markowitz’ problem in continuous time

5. 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 modelCorrelated normal variables

One-periodInvestor trades in n stocks at time 0, holds position until time T.

Trading strategiesTrading strategiesii: Proportion of initial wealth W(0) held in stock i: Proportion of initial wealth W(0) held in stock i1-1-11-...--...-nn: Proportion of W(0) held in bond: Proportion of W(0) held in bond

MarkowitzFind strategy =(=(11,..., ,..., nn)) that minimizes variance of terminal wealth W(T) while expected growth larger than predetermined constant

Stock prices:Stock prices: volatilitiesvolatilities Brownian motionsBrownian motionsindependentindependent

3. The Black-Scholes model

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 modelBlack-Scholes model

Continuous time Investor can trade in all n stocks at all times

Trading strategiesTrading strategiesii(t): Proportion of wealth W(t) held in stock i at time t(t): Proportion of wealth W(t) held in stock i at time t1-1-11(t)-...-(t)-...-nn(t): Proportion of W(t) held in bond at time t(t): Proportion of W(t) held in bond at time t

MarkowitzFind admissible strategy (t)=((t)=(11(t),..., (t),..., nn(t))(t)) that minimizes variance of terminal wealth W(T) while expected growth larger than predetermined constant:

Progressively measurable and bounded

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 observationsDATA 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. The new model

2. Equal company risk premiums

3. Covariance and volatility

4. Examples

1. The new model

Black-Scholes parameterized as continuous time n-index Arbitrage Pricing Theory model with equal risk premiums:

Same for all j

”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

• 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

• 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

3. Covariance and volatility

4. Examples: Cholesky decomposition

• Assume that the investor has specified ranks for the stock risk premiums (SRP)

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 goals2. The 1/n strategy3. An explicit solution * to Markowitz’

problem in continuous time4. The 1/n strategy in factors5. Advantages6. Increasing the number of stocks7. 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 modelMarkowitz:

Wealth:

for the processes

Progressively measurable and bounded

Functions of

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

3. An explicit solution to Markowitz’ problem in continuous time

Hence, for sufficiently wide bounds on the admissible strategies the optimal strategy * solves

for all t

3. An explicit solution to Markowitz’ problem in continuous time

Optimal wealth process:

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….

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

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

0.8 0.9 1 1.1 1.2 1.3 1.4 1.50

2

4

6

8

10

12

14

— n=200— n=100— n=50— n=25

— n=200— n=100— n=50— n=25

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

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

5

10

15

20

25

30

35

40

45

50

55

7. Applying the strategy * to data

Normal distributions estimated from returns:Blue: The wealth process for the strategy * with 1-1-11-...--...-nn=0Red: The wealth process for the 1/N strategy Green: Industry portfolio with smallest volatilityMagenta: Industry portfolio with largest volatility

7. Applying the strategy * to data

Portfolio processes:Blue thick line: The wealth process for the strategy * with 1-1-11-...--...-nn=0Red thick line: The wealth process for the 1/N strategy Green thick line: The industry portfolio with smallest volatilityMagenta thick line: The industry portfolio with largest volatilityColored thin lines: Individual industry portfolios

7. Applying the strategy * to data

5 10 15 20 25 30 35 40 450

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Sharpe ratios:Blue horizontal line: The wealth process for the strategy * with 1-1-11-...--...-nn=0Red horizontal line: The wealth process for the 1/N strategy Blue stars: Individual industry portfolios

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