Quantitative Analysis

Quantitative Analysis

Finding Alpha Fundamental Analysis

Forecast Dividends and find PV Look for stocks for which price PV

Ratio Analysis Compare accounting ratios Buy neglected stocks, short glamorous stocks

Quantitative Analysis Estimate regressions to identify equity styles that earn alpha

Quantitative Analysis Find anomalies to the CAPM Use regressions to forecast when the

anomalies will be particularly strong.

Tilt towards styles with positive alpha Tilt away from styles with negative alpha

Misvaluation View Misvaluation View:

Some stocks are just neglected by analysts Over-reaction by investors

Foundation of most hedge fund equity strategies

“Once an anomaly is discovered in academics, it takes about 5 to 10 years for traders to actually make it go away” (Bob Turley, Global Alpha Fund)

Value StrategiesAverage returns of Book-to-Market Portfolios

1927-2009

Growth Value

Value Strategies

Alphas of Book-to-Market Portfolios1927-2009

Growth Value

Momentum Portfolios Portfolio Formation at the end of month t:

Calculate the return for every stock over the period from month t-12 to month t-2. Rank all stocks by their past return Divide stocks into 10 value-weighted portfolios

Worst 10% in one portfolio Next 10% in next portfolio (etc.)

Momentum PortfoliosAverage Momentum Portfolio Returns

1927-2009

Momentum PortfoliosMomentum Portfolio Alphas

1927-2009

Quantitative Analysis Pros:

Straight forward to implement We know how to maximize Sharpe Ratios We don’t have to forecast dividends

Cons What has happened historically may not continue

in the future

Value and Momentum: Why do value and momentum strategies earn

positive alpha to begin with?

Misvaluation view Neglect and Over-reaction

Broken CAPM View: The CAPM doesn’t correctly model investor behavior. The CAPM was not built to tell us how people should

behave, but to model how they actually behave. Perhaps the model isn’t a realistic reflection of reality.

CAPM Assumptions1) Everyone has same forecast of expected

returns, standard deviations, and correlations. Probably not true When this assumption is relaxed, we still get the a

CAPM-like model, but in equilibrium, the VWP is no longer MSRP. Further, it is not obvious what portfolio is MSRP.

Roll’s Critique: Non-zero alpha can be the result of1) Prices out of equilibrium (markets inefficient)2) You are using the wrong portfolio in your test (Value-weighted

portfolio is not MSRP in equlibrium). It’s impossible to tell which is the true cause

CAPM Assumptions2) The R-Investor only cares about maximizing Sharpe

ratio Perhaps the R-investor has other objectives.

Arbitrage Pricing Theory The CAPM should hold

APT: Ross (1976)

Makes minimal assumptions: In equilibrium, no arbitrage opportunities Conclusion: alphas should be zero

Arbitrage Pricing Theory Notation:

Alpha is measured as the intercept in the following regression:

fme

m

fie

i

rrr

rrr

ie

me

i ebrr

Arbitrage Pricing Theory Suppose we measure the alpha of a “well

diversified” portfolio. Most of the variation is systematic Variance of ei is small Examples:

value portfolio top 30% (HW #17): 82% of the total variance is systematic

For Ford Stock (HW #16) 28% of the total variance is systematic

To an approximation, we can ignore ei in our regression for well diversified portfolios.

ExamplesY variables: returns on different stocksX variable: return on market

Asset 1:Lot’s of variation in e11% of variation is systematicR-squared=0.1189% of variation is unsystematic

Asset 2:Not a lot of variation in e82% of variation is systematicR-squared=0.8218% of variation is unsystematic

Extreme Example

Asset 3:All error terms are 0100% of variation is systematicR-squared=10 of variation is unsystematic

Examples For the “extreme example” there is no error

term:

For example 2 (well diversified portfolio): Error terms are small Most variation is systematic

em

e rar 33

emi

ei rr

Arbitrage APT:

Assume: In equilibrium there are no arbitrage opportunities (chance to make free money)

Claim: for well diversified portfolios, alpha must be approximately zero, or in other words, CAPM view of equilibrium is approximately correct.

Arbitrage Assume there is a well diversified portfolio

(A) whose alpha is 1% Assume beta=1.5 (arbitrary)

Assume there is some other well diversified portfolio (B) with alpha of -0.5% Assume beta=3 (arbitrary)

eM

eB

eM

eA

rr

rr

3005.

5.101.

Arbitrage The beta of a portfolio with weight w in A and

(1-w) in B has a portfolio beta of:

and a portfolio alpha of

BAp ww )1(

BAp ww )1(

Arbitrage Consider a portfolio (z) with a weight of 2 in

asset A and a weight of -1 in asset B

But since z is well diversified:

%5.2)005.()1(01.203)1(5.12

z

z

%5.2 zmzzez rr

Arbitrage But z by definition, is E[rz]-rf But portfolio z is risk-less (approximately)

Almost all systematic risk is diversified away, because it is composed of two well-diversified portfolios.

Systematic risk has been perfectly hedged Hence, E[rz]=rz

Because portfolio is approximately riskless, what you expect is what you get.

But since alpha=0.01 rz =rf +0.01 Borrow at risk-free rate, invest in this portfolio. Free money.

Arbitrage As many investors do this, what happens to

the cost of portfolio z? Goes up.

What happens to the realized return? Goes down.

In equilibrium rz=rf

Example 2 Suppose there are two well diversified portfolios:

How do you attack the arbitrage opportunity?

alpha betaA -0.06% 1.5B 0.50% 0.75

B)in (weight 2)1(A)in (weight 1

75.75.075).1(5.1

ww

www

Example 2 Portfolio z:

Weight in A = -1 Weight in B = 2 Approximately riskless Excess Return=2*.50+(-1*-.06)=1.06(%) Borrow all you can at risk-free rate Invest in portfolio z

Long-Short Hedge Funds Common hedge fund strategies:

Long value, short growth Long long/short momentum strategies

AQR Capital Management Barclay’s Global Investors Global Ascent Goldman Sachs Global Alpha Fund

Anomaly Even though many funds are attacking these

apparent arbitrage opportunities, opportunities apparently have not gone away.

Why? Industry view: pervasive mispricing

Anomaly Academic view:

These portfolios are not really close to risk-free

Why doesn’t the risk go away?

If the “e” in portfolio A is not independent to the “e” in portfolio B, then you can actually magnify the risk of the “arbitrage” portfolio by combining A and B. Related to stat rule number 5 - the variance of a sum of

random variables depends on the covariance.

the value-weighted portfolio is not the right portfolio to measure alpha against.

Arbitrage Pricing Theory Pros of APT:

Only relies on assumption of no-arbitrage Tells us the general form of any model of expected returns.

Expected returns must be a linear function of slope coefficients (betas).

Cons of APT What is the right portfolio or portfolios to use in

measuring alpha? APT tells us nothing about this.