COPYRIGHT © 2006 AXA ROSENBERG INVESTMENT MANAGEMENT

Linear algebra and rational investing

Mark HowardDirector of Software EngineeringBarr Rosenberg Research Center

April 1, 2009

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Background on the Firm

Founded in the US in 1985 to manage diversified equity portfolios Global equity specialist within AXA Investment Managers’ multi-expert

group of asset managers Offices located in major financial centers around the world Stable, committed team of 380 employees worldwide 341 clients including pension funds, governments, endowments and

foundations $118 billion assets under management as of March 2008

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AXA Rosenberg: History

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AXA Rosenberg: Sample Client List

* This client list is intended simply to indicate a sample of AXA Rosenberg's non-confidential clients; the selection of clients for the list is intended to demonstrate the range of our institutional client base both geographically and by type of institution (corporate, insurance, pension plans, universities, endowments, etc.). It is not known whether the listed clients approve or disapprove of the manager or the advisory services provided.

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Investment Philosophy and Process

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A Rational, Proven Foundation

Fundamental Analysis Through Expert Systems

“A time-tried investment principle…to discover and acquire undervalued individual securities as the result of comprehensive and expert statistical investigations.”

– Graham & Dodd, 1934

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Investment philosophy rooted in persistent economic principles

Fundamentally driven: Earnings Matter

Systematic security analysis and portfolio construction

Robust in different market environments Globally consistent with low regional correlations

Strategy Overview

Traditional Fundamental

Managers

Typical Quant ManagersTechnical Strategies

or Factor Models

CompanyFundamentals

Alpha Source

Subjective ObjectiveImplementation

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How Our Strategy Works: A Snapshot

More future earnings…

…result in superior performance

11.5%6.7%

12.8%8.6%

26.8%

18.4%

42.2%

29.3%

59.1%

41.6%

77.8%

55.3%

-25%

0%

25%

50%

75%

100%

AXA Rosenberg Market

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Earnings Forecast Model

Valuation Model

+Risk Model

=Forecast Company Return

Identify most attractively priced stocks in each industry

Identify companies with superior year-ahead earnings in each industry

Maximize return with minimum deviation from the benchmark

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Valuation Model

Identify most attractively priced stocks

Classic arbitrage analysis — identify stocks selling for less than the sum of a company’s parts

Analogous to real estate appraisal

Four-step comparison of prices to company fundamentals:

1. Identify and value 170 distinct business lines

2. Capture market value for each financial statement item

3. Identify value of unique features

4. Compare sum of company’s parts to current stock price

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Portfolios

For our purposes a portfolio is an allocation of our assets to securities. For example if our universe of securities is s1,…sn then a portfolio is a vector x1,…,xn of numbers in [0,1] which sum up to 1. xi is the proportion of our assets invested in si

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Returns

If you bought stock s for a dollars and sold it for b dollars then the return on s is (b-a)/a … in other words the amount gained (or lost) per dollar spent. If you have a portfolio x1,…,xn of stocks s1,…,sn with returns r1,…,rn then the portfolio return is

av i1 ... amv im

x irii1,n

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Expected returns

Assume that we have some way to predict returns … our prediction for stock is . Then the predicted or expected portfolio return is . We will refer to this as E.

i

si

x iii1,n

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Risk

Assume that for each pair of securities si,sj we have a number between 0 and 1 which measures the covariance between the predicted returns of si and sj. The variance or risk of the portfolio is

ij

ij

i1,n x ix j

j1,n ij

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Matrices

If we let the (column) vector denote our portfolio, denote the (column) vector of expected returns, the covariance matrix, the expected portfolio return, and the variance of the portfolio then

X

E

C

V

E 'X

V X 'CX

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Rational investing

The goal of rational investing is to maximize the expected return while minimizing the risk. A fundamental result of modern portfolio theory (for which Harry Markowitz won the nobel prize in economics) is that this tradeoff is meaningful.

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Optimizing portfolios

For a given return the problem of minimizing the risk is a quadratic optimization problem. One efficient method of solving this is to use convex piecewise linear programming.

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George Dantzig

In 1939 George Dantzig was a graduate student in statistics at Berkeley. One day he was late for class and wrote down the two problems on the board.

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George Dantzig

A few days later when he took them to his professor’s office he apologized for being late as they were somewhat harder than the other problems, and asked if he still wanted them. The professor (the great statistician Jerzy Neyman) told him to just toss them on the desk.

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George Dantzig

Several weeks later Neyman excitedly came to Dantzig’s house to tell him that those weren’t homework problems, but two of the most notorious open problems in statistics.

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Linear Programming

A few years later Dantzig invented the simplex method for solving (piecewise) linear programming problems. A linear programming problem is one of the formminimizesubject to

c'XAX bX 0

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Linear Programming

Here the linear equation is underspecified … in other words A usually has many more columns than rows, so many the equation has many solutions. The core of the simplex method is effectively Gaussian elimination.

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Financial models

Returns are very hard to predict. It is actually much easier to predict other variables, such as future earnings, which affect returns. Under the assumption that the market will eventually reward earnings this should be sufficient to produce a portfolio with superior returns.

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Earnings Forecast Model

Market Participant Indicators

What is the market telling usabout future earnings?

FundamentalIndicators

What can we tell about future earnings based on historic fundamentals?

• ProfitabilityMeasures

Trends in earnings, ROA, ROE, etc.

• OperatingRatiosMargins, debt coverage, etc.

• Analyst Forecasts• Earnings Revisions• Broker Recommendations• Prior Price Behavior

Forecast of next year’s earnings

Objective: Estimate of Forward Earnings

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Financial models

One common technique for predicting variables, such as future earnings, is to determine some collection of current variables such as book value, sales, etc., and to try to find a linear combination of these which is a reasonable approximation to the predicted value we are looking for.

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Financial models

A technique for doing this was invented by Gauss in 1795 (when he was 18), and is now known as the method of least squares, or linear regressions.

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Regression

Say that we think that future earnings is really a linear combination of m other current variables. Assume that

are the values for these variables for security i and that is the value of future earnings. Then there should be such that is close to for most i.

other words there should be m coefficients such that for any values of our variables

a1,...,am

a1v i1 ... amv im

v i1,...,v im

bi

bi

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Regression

One way to achieve this is take observations in the past and the actual values we are trying to predict and to find which minimize the sum of squares of differences between the predicted values and the actual values.

a1,...,am

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Regression

A little math shows that this sum of squares is minimized by

Where X is the matrix of variables.

(X 'X) 1X 'b

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Financial models

If we have found a reasonable set of explanatory variables then we can use this linear functional to predict future earnings…in other words we can try to maximize future earnings instead of the more elusive future returns, and then rely on the market to reward those earnings with returns.

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Factor Models

Usually when managing large portfolios the goal is not to minimize the volatility of the portfolio, but rather to minimize the difference between the portfolio return and the market return. One way to do this is to identify market factors and take into consideration every security’s exposure to those factors.

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Factor Models

A classic example of a market factor is which industry a company is in. We enumerate all of the industries and for each company give it a number between 0 and 1 indicating how much of the company is in that industry.

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Factor Models

Once we have our factors we can use the same techniques as before to minimize the difference between the average exposure to each factor and the market’s average … so if there are economic forces which cause a particular factor to behave differently then we will track the difference.

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Factor Models

Factors are constructed by hand by thinking about economic forces, but once you have created what you think are reasonable factors you can use eigenvalue decomposition of the factor covariance matrix to create a more precise set of factors.

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Solving equations

It is very common in portfolio optimization to be faced with the problem of solving sequences of systems of linear equations

A1x1 b1...Anxn bn

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Solving equations

Where each Ai+1 differs from Ai in only one column. The traditional way to solve Ax=b is to invert A, and use A-

1b, but there are many matrices for which solving the equation is much easier than inverting the matrix (e.g.triangular matrices).

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Solving equations

One very useful computational technique is to factor A into matrices which are easy to solve, for example represent A = LU where L and U are easy to solve… then Ax=b is the same as L(Ux) = b, so if Ly = b and Ux = y then Ax=b.

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Solving equations

One very useful computational technique is to factor A into matrices which are easy to solve, for example represent A = LU where L is lower triangular and U is upper triangular … then Ax=b is the same as L(Ux) = b, so if Ly = b and Ux = y then Ax=b.

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Solving equations

It turns out that one can slightly complicate this and get factorizations of Ai with the property that the i+1th factorization is easily computable from the ith factorization and the difference between Ai+1 and Ai, so solving the sequence of equations is fast.